Log24

Friday, August 16, 2019

Stanza Romanza

Filed under: General — Tags: — m759 @ 7:49 PM

Wallace Stevens's 'a million diamonds' quote in Woodman's 'Stanza My Stone'

For those who prefer greater clarity than is offered by Stevens . . .

The A section —

The B section —

"A paper from Helsinki in 2005 says there are more than a million
3-(16,4,1) block designs, of which only one has an automorphism
group of order 322,560. This is the affine 4-space over GF(2)."

Thursday, August 15, 2019

On Steiner Quadruple Systems of Order 16

Filed under: General — Tags: , — m759 @ 4:11 AM

An image from a Log24 post of March 5, 2019

Cullinane's 1978  square model of PG(3,2)

The following paragraph from the above image remains unchanged
as of this morning at Wikipedia:

"A 3-(16,4,1) block design has 140 blocks of size 4 on 16 points,
such that each triplet of points is covered exactly once. Pick any
single point, take only the 35 blocks containing that point, and
delete that point. The 35 blocks of size 3 that remain comprise
a PG(3,2) on the 15 remaining points."

Exercise —

Prove or disprove the above assertion about a general "3-(16,4,1) 
block design," a structure also known as a Steiner quadruple system
(as I pointed out in the March 5 post).

Relevant literature —

A paper from Helsinki in 2005* says there are more than a million
3-(16,4,1) block designs, of which only one has an automorphism
group of order 322,560. This is the affine 4-space over GF(2),
from which PG(3,2) can be derived using the well-known process
from finite geometry described in the above Wikipedia paragraph.

* "The Steiner quadruple systems of order 16," by Kaski et al.,
   Journal of Combinatorial Theory Series A  
Volume 113, Issue 8, 
   November 2006, pages 1764-1770.

Thursday, June 27, 2019

Group Actions on the 4x4x4 Cube

Filed under: General — Tags: — m759 @ 6:23 AM

For affine  group actions, see Ex Fano Appollinis  (June 24)
and Solomon's Cube.

For one approach to Mathieu  group actions on a 24-cube subset
of the 4x4x4 cube, see . . .

For a different sort of Mathieu cube, see Aitchison.

Friday, May 3, 2019

“As a Chinese jar” — T. S. Eliot

Filed under: General — Tags: , , — m759 @ 1:06 PM

 

Tuesday, March 5, 2019

A Block Design 3-(16,4,1) as a Steiner Quadruple System:

Filed under: General — Tags: , — m759 @ 11:19 AM

A Midrash for Wikipedia 

Midrash —

Related material —


________________________________________________________________________________

The Miracle Octad Generator (MOG), the affine 4-space over GF(2), and the Cullinane diamond theorem

Friday, March 1, 2019

Wikipedia Scholarship (Continued)

Filed under: General — Tags: , , — m759 @ 12:45 PM

This post continues a post from yesterday on the square model of
PG(3,2) that apparently first appeared (presented as such*) in . . .

Cullinane, "Symmetry invariance in a diamond ring,"
Notices of the AMS , pp. A193-194, Feb. 1979.

The Cullinane diamond theorem, AMS Notices, Feb. 1979, pp. A-193-194

Yesterday's Wikipedia presentation of the square model was today
revised by yet another anonymous author —

Revision history accounting for the above change from yesterday —

The jargon "rm OR" means "remove original research."

The added verbiage about block designs is a smokescreen having
nothing to do with the subject, which is square  representation
of the 35 points and lines.

* The 35 squares, each consisting of four 4-element subsets, appeared earlier
   in the Miracle Octad Generator (MOG) of R. T. Curtis (published in 1976).
  They were not at that time  presented as constituting a finite geometry, 
  either affine (AG(4,2)) or projective (PG(3,2)).

Friday, February 22, 2019

Back Issues of AMS Notices

Filed under: General — m759 @ 3:04 PM

From the online home page of the new March issue —

Feb. 22, 2019 — AMS Notices back issues now available.

For instance . . .

The Cullinane diamond theorem, AMS Notices, Feb. 1979, pp. A-193-194

Related material now at Wikipedia

The Miracle Octad Generator (MOG), the affine 4-space over GF(2), and the Cullinane diamond theorem

Friday, February 15, 2019

The Gifts Reserved for Age

Filed under: General — m759 @ 3:30 PM

"But, as the passage now presents no hindrance
     To the spirit unappeased and peregrine
     Between two worlds become much like each other…."

T. S. Eliot, Four Quartets

Related obituary:

https://www.nytimes.com/2019/02/15/obituaries/tom-cade-dead.html

Related date:

"as of Feb. 6, 2019" (from a post at 12 AM ET Feb. 7) —

"There is  such a thing as a four-dimensional finite affine space."
— Saying adapted from a 1962 young-adult novel by Madeleine L'Engle

Thursday, February 7, 2019

Geometry of the 4×4 Square: The Kummer Configuration

Filed under: General — Tags: , — m759 @ 12:00 AM

From the series of posts tagged Kummerhenge

A Wikipedia article relating the above 4×4 square to the work of Kummer —

A somewhat more interesting aspect of the geometry of the 4×4 square
is its relationship to the 4×6 grid underlying the Miracle Octad Generator
(MOG) of R. T. Curtis.  Hudson's 1905 classic Kummer's Quartic Surface
deals with the Kummer properties above and also foreshadows, without
explicitly describing, the finite-geometry properties of the 4×4 square as
a finite affine 4-space — properties that are of use in studying the Mathieu
group M24  with the aid of the MOG.

Tuesday, December 25, 2018

Simply

Filed under: General — m759 @ 11:42 AM
 

"So to obtain the isomorphism from L2(7) onto L3(2) we simply
multiply any given permutation of L2(7) by the affine translation
that restores to its rightful place."

— Sphere Packings, Lattices and Groups ,
by John H. Conway and N. J. A. Sloane.
First edition, 1988, published by Springer-Verlag New York, Inc.
Chapter 11 (by J. H. Conway), "The Golay Codes and the Mathieu Groups," 
Section 12, "The trio group 26:(S3×L2(7))"

Compare and contrast —

Why PSL(2,7) is isomorphic to GL(3.2)

This post was suggested by a New York Times  headline today —

Sunday, December 9, 2018

Quaternions in a Small Space

Filed under: G-Notes,General,Geometry — Tags: , , — m759 @ 2:00 PM

The previous post, on the 3×3 square in ancient China,
suggests a review of group actions on that square
that include the quaternion group.

Click to enlarge

Three links from the above finitegeometry.org webpage on the
quaternion group —

Related material —

Iain Aitchison on the 'symmetric generation' of R. T. Curtis

See as well the two Log24 posts of December 1st, 2018 —

Character and In Memoriam.

Sunday, September 16, 2018

Husserl for Beginners

Filed under: General — m759 @ 10:31 PM

http://www.log24.com/log/pix18/180916-Hustvedt-Auster-covers.jpg

See also Trivial + Affine and Eidetic Reduction.

Friday, September 14, 2018

Denkraum

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 1:00 AM

http://www.log24.com/log/pix18/180914-Warburg_Denkraum-Google-result.jpg

I Ching Geometry search result

Underlying the I Ching  structure  is the finite affine space
of six dimensions over the Galois field with two elements.

In this field,  "1 + 1 = 0,"  as noted here Wednesday.

See also other posts now tagged  Interstice.

http://www.log24.com/log/pix18/180914-Warburg-Wikipedia.jpg

Saturday, September 8, 2018

Review

Filed under: General,Geometry — m759 @ 2:00 AM

Affine groups on small binary spaces

Friday, June 29, 2018

Triangles in the Eightfold Cube

Filed under: General,Geometry — m759 @ 9:10 PM

From a post of July 25, 2008, "56 Triangles," on the Klein quartic
and the eightfold cube

"Baez's discussion says that the Klein quartic's 56 triangles
can be partitioned into 7 eight-triangle Egan 'cubes' that
correspond to the 7 points of the Fano plane in such a way
that automorphisms of the Klein quartic correspond to
automorphisms of the Fano plane. Show that the
56 triangles within the eightfold cube can also be partitioned
into 7 eight-triangle sets that correspond to the 7 points of the
Fano plane in such a way that (affine) transformations of the
eightfold cube induce (projective) automorphisms of the Fano plane."

Related material from 1975 —

More recently

Friday, February 16, 2018

Two Kinds of Symmetry

Filed under: General,Geometry — m759 @ 11:29 PM

The Institute for Advanced Study (IAS) at Princeton in its Fall 2015 Letter 
revived "Beautiful Mathematics" as a title:

This ugly phrase was earlier used by Truman State University
professor Martin Erickson as a book title. See below. 

In the same IAS Fall 2015 Letter appear the following remarks
by Freeman Dyson —

". . . a special case of a much deeper connection that Ian Macdonald 
discovered between two kinds of symmetry which we call modular and affine.
The two kinds of symmetry were originally found in separate parts of science,
modular in pure mathematics and affine in physics. Modular symmetry is
displayed for everyone to see in the drawings of flying angels and devils
by the artist Maurits Escher. Escher understood the mathematics and got the
details right. Affine symmetry is displayed in the peculiar groupings of particles
created by physicists with high-energy accelerators. The mathematician
Robert Langlands was the first to conjecture a connection between these and
other kinds of symmetry. . . ." (Wikipedia link added.)

The adjective "modular"  might aptly be applied to . . .

The adjective "affine"  might aptly be applied to . . .

From 'Beautiful Mathematics,' by Martin Erickson, an excerpt on the Cullinane diamond theorem (with source not mentioned)

The geometry of the 4×4 square combines modular symmetry
(i.e., related to theta functions) with the affine symmetry above.

Hudson's 1905 discussion of modular symmetry (that of Rosenhain
tetrads and Göpel tetrads) in the 4×4 square used a parametrization
of that square by the digit 0 and the fifteen 2-subsets of a 6-set, but 
did not discuss the 4×4 square as an affine space.

For the connection of the 15 Kummer modular 2-subsets with the 16-
element affine space over the two-element Galois field GF(2), see my note
of May 26, 1986, "The 2-subsets of a 6-set are the points of a PG(3,2)" —

— and the affine structure in the 1979 AMS abstract
"Symmetry invariance in a diamond ring" —

For some historical background on the symmetry investigations by
Dyson and Macdonald, see Dyson's 1972 article "MIssed Opportunities."

For Macdonald's own  use of the words "modular" and "affine," see
Macdonald, I. G., "Affine Lie algebras and modular forms," 
Séminaire N. Bourbaki , Vol. 23 (1980-1981), Talk no. 577, pp. 258-276.

Saturday, January 6, 2018

Yale News

Filed under: General,Geometry — Tags: — m759 @ 5:24 AM

The Yale of the title is not the university, but rather the
mathematician Paul B. Yale. Yale's illustration of the Fano
plane is below.

IMAGE- Triangular models of the 4-point affine plane A and 7-point projective plane PA

A different illustration from a mathematician named Greenberg —

This illustration of the ominous phrase "line at infinity"
may serve as a sort of Deathly Hallows  for Greenberg.
According to the AMS website yesterday, he died on
December 12, 2017:

A search of this  journal for Greenberg yields no mention of
the dead mathematician, but does yield some remarks
on art that are pehaps less bleak than the above illustration.

For instance —

Art adapted from the Google search screen. Discuss.

Wednesday, October 18, 2017

Dürer for St. Luke’s Day

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 1:00 PM

Structure of the Dürer magic square 

16   3   2  13
 5  10  11   8   decreased by 1 is …
 9   6   7  12
 4  15  14   1

15   2   1  12
 4   9  10   7
 8   5   6  11
 3  14  13   0 .

Base 4 —

33  02  01  30
10  21  22  13
20  11  12  23 
03  32  31  00 .

Two-part decomposition of base-4 array
as two (non-Latin) orthogonal arrays

3 0 0 3     3 2 1 0
1 2 2 1     0 1 2 3
2 1 1 2     0 1 2 3
0 3 3 0     3 2 1 0 .

Base 2 –

1111  0010  0001  1100
0100  1001  1010  0111
1000  0101  0110  1011
0011  1110  1101  0000 .

Four-part decomposition of base-2 array
as four affine hyperplanes over GF(2) —

1001  1001  1100  1010
0110  1001  0011  0101
1001  0110  0011  0101
0110  0110  1100  1010 .

— Steven H. Cullinane,
  October 18, 2017

See also recent related analyses of
noted 3×3 and 5×5 magic squares.

Friday, April 14, 2017

Hudson and Finite Geometry

Filed under: General,Geometry — Tags: , — m759 @ 3:00 AM

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

The above four-element sets of black subsquares of a 4×4 square array 
are 15 of the 60 Göpel tetrads , and 20 of the 80 Rosenhain tetrads , defined
by R. W. H. T. Hudson in his 1905 classic Kummer's Quartic Surface .

Hudson did not  view these 35 tetrads as planes through the origin in a finite
affine 4-space (or, equivalently, as lines in the corresponding finite projective
3-space).

In order to view them in this way, one can view the tetrads as derived,
via the 15 two-element subsets of a six-element set, from the 16 elements
of the binary Galois affine space pictured above at top left.

This space is formed by taking symmetric-difference (Galois binary)
sums of the 15 two-element subsets, and identifying any resulting four-
element (or, summing three disjoint two-element subsets, six-element)
subsets with their complements.  This process was described in my note
"The 2-subsets of a 6-set are the points of a PG(3,2)" of May 26, 1986.

The space was later described in the following —

IMAGE- Dolgachev and Keum, coordinatization of the 4x4 array in 'Birational Automorphisms of Quartic Hessian Surfaces,' AMS Transactions, 2002

Wednesday, March 29, 2017

The Crimson Abyss

Filed under: General,Geometry — Tags: , — m759 @ 3:19 PM

"And as the characters in the meme twitch into the abyss
that is the sky, this meme will disappear into whatever
internet abyss swallowed MySpace."

—Staff writer Kamila Czachorowski, Harvard Crimson , March 29

1984

IMAGE- 'Affine Groups on Small Binary Spaces,' illustration

2010

Logo design for Stack Exchange Math by Jin Yang
 

Recent posts now tagged Crimson Abyss suggest
the above logo be viewed in light of a certain page 29

"… as if into a crimson abyss …." —

Update of 9 PM ET March 29, 2017:

Prospero's Children  was first published by HarperCollins,
London, in 1999. A statement by the publisher provides
an instance of the famous "much-needed gap." —

"This is English fantasy at its finest. Prospero’s Children 
steps into the gap that exists between The Lion, the Witch
and the Wardrobe
  and Clive Barker’s Weaveworld , and
is destined to become a modern classic."

Related imagery —

See also "Hexagram 64 in Context" (Log24, March 16, 2017).

Art Space Illustrated

Filed under: General,Geometry — Tags: , — m759 @ 10:45 AM

Another view of the previous post's art space  —

IMAGE by Cullinane- 'Solomon's Cube' with 64 identical, but variously oriented, subcubes, and six partitions of these 64 subcubes

More generally, see Solomon's Cube in Log24.

See also a remark from Stack Exchange in yesterday's post Backstory,
and the Stack Exchange math logo below, which recalls the above 
cube arrangement from "Affine groups on small binary spaces" (1984).

IMAGE- Current math.stackexchange.com logo and a 1984 figure from 'Notes on Groups and Geometry, 1978-1986'

Monday, January 2, 2017

Sitcom Theology

Filed under: General,Geometry — Tags: — m759 @ 1:20 PM

The Hollywood Reporter

"William Christopher, best known for playing Father Mulcahy
on the hit sitcom M*A*S*H , died Saturday [Dec. 31, 2016] of
lung cancer, his agent confirmed to The Hollywood Reporter.
He was 84.

Christopher died at his home in Pasadena, with his wife by
his bedside, at 5:10 a.m. on New Year's Eve, according to a
statement from his agent."

— 5:59 PM PST 12/31/2016 by Meena Jang

Image reshown in this journal on the midnight (Eastern time)
preceding Christopher's death —

IMAGE- Triangular models of the 4-point affine plane A and 7-point projective plane PA

Related material —

From a Log24 search for "Deathly Hallows" —

Mathematics

http://www.log24.com/log/pix11A/110505-WikipediaFanoPlane.jpg

The Fano plane block design

Magic

http://www.log24.com/log/pix11A/110505-DeathlyHallows.jpg

The Deathly Hallows symbol—
Two blocks short of  a design.

Those who prefer Latin with their theology
may search this journal for "In Nomine Patris."

Friday, December 23, 2016

Requiem for a Mathematician

Filed under: General,Geometry — m759 @ 2:10 PM

From a Dec. 21 obituary posted by the
University of Tennessee at Knoxville —

"Wade was ordained as a pastor and served
at Oakwood Baptist Church in Knoxville."

Other information —

In a Log24 post, "Seeing the Finite Structure,"
of August 16, 2008, Wade appeared as a co-author
of the Walsh series book mentioned above —

Walsh Series: An Introduction to Dyadic Harmonic Analysis, by F. Schipp et. al.

Walsh Series: An Introduction
to Dyadic Harmonic Analysis
,
by F. Schipp et al.,
Taylor & Francis, 1990

From the 2008 post —

The patterns on the faces of the cube on the cover
of Walsh Series above illustrate both the 
Walsh functions of order 3 and the same structure
in a different guise, subspaces of the affine 3-space 
over the binary field. For a note on the relationship
of Walsh functions to finite geometry, see 
Symmetry of Walsh Functions.

Friday, December 16, 2016

Memory, History, Geometry

Filed under: General,Geometry — Tags: — m759 @ 9:48 AM

These are Rothko's Swamps .

See a Log24 search for related meditations.

For all three topics combined, see Coxeter —

" There is a pleasantly discursive treatment 
of Pontius Pilate’s unanswered question
‘What is truth?’ "

— Coxeter, 1987, introduction to Trudeau’s
     The Non-Euclidean Revolution

Update of 10 AM ET —  Related material, with an elementary example:

Posts tagged "Defining Form." The example —

IMAGE- Triangular models of the 4-point affine plane A and 7-point projective plane PA

Monday, September 12, 2016

The Kummer Lattice

Filed under: General,Geometry — Tags: , , , — m759 @ 2:00 PM

The previous post quoted Tom Wolfe on Chomsky's use of
the word "array." 

An example of particular interest is the 4×4  array
(whether of dots or of unit squares) —

      .

Some context for the 4×4 array —

The following definition indicates that the 4×4 array, when
suitably coordinatized, underlies the Kummer lattice .

Further background on the Kummer lattice:

Alice Garbagnati and Alessandra Sarti, 
"Kummer Surfaces and K3 surfaces
with $(Z/2Z)^4$ symplectic action." 
To appear in Rocky Mountain J. Math.

The above article is written from the viewpoint of traditional
algebraic geometry. For a less traditional view of the underlying
affine 4-space from finite  geometry, see the website
Finite Geometry of the Square and Cube.

Some further context

"To our knowledge, the relation of the Golay code
to the Kummer lattice is a new observation."

— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of
Kummer surfaces in the Mathieu group M24 
"

As noted earlier, Taormina and Wendland seem not to be aware of
R. W. H. T. Hudson's use of the (uncoordinatized*) 4×4 array in his
1905 book Kummer's Quartic Surface.  The array was coordinatized,
i.e. given a "vector space structure," by Cullinane eight years prior to
the cited remarks of Curtis.

* Update of Sept. 14: "Uncoordinatized," but parametrized  by 0 and
the 15 two-subsets of a six-set. See the post of Sept. 13.

Monday, August 1, 2016

Cube

Filed under: General,Geometry — m759 @ 10:28 PM

From this journal —

See (for instance) Sacred Order, July 18, 2006 —

The finite Galois affine space with 64 points

From a novel published July 26, 2016, and reviewed
in yesterday's (print) New York Times Book Review —

The doors open slowly. I step into a hangar. From the rafters high above, lights blaze down, illuminating a twelve-foot cube the color of gunmetal. My pulse rate kicks up. I can’t believe what I’m looking at. Leighton must sense my awe, because he says, “Beautiful, isn’t it?” It is exquisitely beautiful. At first, I think the hum inside the hangar is coming from the lights, but it can’t be. It’s so deep I can feel it at the base of my spine, like the ultralow-frequency vibration of a massive engine. I drift toward the box, mesmerized.

— Crouch, Blake. Dark Matter: A Novel
(Kindle Locations 2004-2010).
Crown/Archetype. Kindle Edition. 

See also Log24 on the publication date of Dark Matter .

Thursday, July 28, 2016

The Giglmayr Foldings

Filed under: General,Geometry — m759 @ 1:44 PM

Giglmayr's transformations (a), (c), and (e) convert
his starting pattern

  1    2   5   6
  3    4   7   8
  9  10 13 14
11  12 15 16

to three length-16 sequences. Putting these resulting
sequences back into the 4×4 array in normal reading
order, we have

  1    2    3    4        1   2   4   3          1    4   2   3
  5    6    7    8        5   6   8   7          7    6   8   5 
  9  10  11  12      13 14 16 15       15 14 16 13
13  14  15  16       9  10 12 11        9  12 10 11

         (a)                         (c)                      (e)

Four length-16 basis vectors for a Galois 4-space consisting
of the origin and 15 weight-8 vectors over GF(2):

0 0 0 0       0 0 0 0       0 0 1 1       0 1 0 1
0 0 0 0       1 1 1 1       0 0 1 1       0 1 0 1 
1 1 1 1       0 0 0 0       0 0 1 1       0 1 0 1
1 1 1 1       1 1 1 1       0 0 1 1       0 1 0 1 .

(See "Finite Relativity" at finitegeometry.org/sc.)

The actions of Giglmayr's transformations on the above
four basis vectors indicate the transformations are part of
the affine group (of order 322,560) on the affine space
corresponding to the above vector space.

For a description of such transformations as "foldings,"
see a search for Zarin + Folded in this journal.

Wednesday, May 4, 2016

Golomb and Symmetry

Filed under: General,Geometry — m759 @ 12:00 PM

From the webpage Diamond Theory Bibliography

Golomb, Solomon W. 
Shift register sequences  (Revised edition)
Aegean Park Press, Laguna Hills, CA, 1982
   The fifteen "stencils" in Golomb's Fig. VIII-8, page 219,
   are the same as the fifteen affine hyperplanes that
   account for patterns' symmetry in diamond theory.
   This figure occurs in a discussion of Rademacher-
   Walsh functions.

Elsewhere

Wednesday, April 20, 2016

Symmetric Generation of a Simple Group

Filed under: General,Geometry — Tags: , — m759 @ 9:00 PM

The reference in the previous post to the work of Guitart and 
The Road to Universal Logic  suggests a fiction involving
the symmetric generation of the simple group of order 168.

See The Diamond Archetype and a fictional account of the road to Hell 

'PyrE' in Bester's 'The Stars My Destination'

The cover illustration below has been adapted to
replace the flames of PyrE with the eightfold cube.

IMAGE- 'The Stars My Destination' (with cover slightly changed)

For related symmetric generation of a much larger group, see Solomon's Cube.

Friday, April 8, 2016

Space Cross

Filed under: General,Geometry — Tags: — m759 @ 11:00 PM

For George Orwell

Illustration from a book on mathematics —

This illustrates the Galois space  AG(4,2).

For some related spaces, see a note from 1984.

"There is  such a thing as a space cross."
— Saying adapted from a young-adult novel

Tuesday, December 15, 2015

Square Triangles

Filed under: General,Geometry — m759 @ 3:57 PM

Click image for some background.

Exercise:  Note that, modulo color-interchange, the set of 15 two-color
patterns above is invariant under the group of six symmetries of the
equilateral triangle. Are there any other such sets of 15 two-color triangular
patterns that are closed as sets , modulo color-interchange, under the six
triangle symmetries and  under the 322,560 permutations of the 16
subtriangles induced by actions of the affine group AGL(4,2)
on the 16 subtriangles' centers , given a suitable coordinatization?

Friday, August 28, 2015

Art and Space

Filed under: General,Geometry — Tags: , — m759 @ 10:00 AM

IMAGE- Spielfeld (1982-83), by Wolf Barth
 

            Observatory scene from "Magic in the Moonlight"

"The sixteen nodes… can be parametrized
by the sixteen points in affine four-space
over the tiny field F2 with two elements."

Wolf Barth

Friday, August 14, 2015

Discrete Space

Filed under: General,Geometry — Tags: — m759 @ 7:24 AM

(A review)

Galois space:

Image-- examples from Galois affine geometry

Counting symmetries of  Galois space:
IMAGE - The Diamond Theorem

The reason for these graphic symmetries in affine Galois space —

symmetries of the underlying projective Galois space:

Monday, June 15, 2015

Omega Matrix

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

See that phrase in this journal.

See also last night's post.

The Greek letter Ω is customarily used to
denote a set that is acted upon by a group.
If the group is the affine group of 322,560
transformations of the four-dimensional
affine space over the two-element Galois
field, the appropriate Ω is the 4×4 grid above.

See the Cullinane diamond theorem.

Thursday, June 11, 2015

Omega

Filed under: General,Geometry — m759 @ 12:00 PM

Omega is a Greek letter, Ω , used in mathematics to denote
a set on which a group acts. 

For instance, the affine group AGL(3,2) is a group of 1,344
actions on the eight elements of the vector 3-space over the
two-element Galois field GF(2), or, if you prefer, on the Galois
field  Ω = GF(8).

Related fiction:  The Eight , by Katherine Neville.

Related non-fiction:  A remark by Werner Heisenberg
in this journal on Saturday, June 6, 2015, the eightfold cube ,
and the illustrations below —

Mathematics

http://www.log24.com/log/pix11A/110505-WikipediaFanoPlane.jpg

The Fano plane block design

Magic

http://www.log24.com/log/pix11A/110505-DeathlyHallows.jpg

The Deathly Hallows symbol—
Two blocks short of  a design.

Wednesday, May 13, 2015

Space

Filed under: General,Geometry — Tags: — m759 @ 2:00 PM

Notes on space for day 13 of May, 2015 —

The 13 symmetry axes of the cube may be viewed as
the 13 points of the Galois projective space PG(2,3).
This space (a plane) may also be viewed as the nine points
of the Galois affine space AG(2,3) plus the four points on
an added "line at infinity."

Related poetic material:

The ninefold square and Apollo, as well as 

http://www.log24.com/log/pix11A/110426-ApolloAndDionysus.jpg

Tuesday, March 24, 2015

Brouwer on the Galois Tesseract

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

Yesterday's post suggests a review of the following —

Andries Brouwer, preprint, 1982:

"The Witt designs, Golay codes and Mathieu groups"
(unpublished as of 2013)

Pages 8-9:

Substructures of S(5, 8, 24)

An octad is a block of S(5, 8, 24).

Theorem 5.1

Let B0 be a fixed octad. The 30 octads disjoint from B0
form a self-complementary 3-(16,8,3) design, namely 

the design of the points and affine hyperplanes in AG(4, 2),
the 4-dimensional affine space over F2.

Proof….

… (iv) We have AG(4, 2).

(Proof: invoke your favorite characterization of AG(4, 2) 
or PG(3, 2), say 
Dembowski-Wagner or Veblen & Young. 

An explicit construction of the vector space is also easy….)

Related material:  Posts tagged Priority.

Saturday, March 14, 2015

Unicode Diamonds

Filed under: General,Geometry — m759 @ 9:16 PM

The following figure, intended to display as
a black diamond, was produced with
HTML and Unicode characters. Depending
on the technology used to view it, the figure
may contain gaps or overlaps.

◢◣
◥◤

Some variations:

◤◥
◣◢

◤◥
◢◣

◤◣
◢◥

◤◣
◥◢

Such combined Unicode characters —

◢  black lower right triangle,
◣  black lower left triangle,
᭘  black upper left triangle,
᭙  black upper right triangle 

— might be used for a text-only version of the Diamond 16 Puzzle
that is more easily programmed than the current version.

The tricky part would be coding the letter-spacing and
line-height to avoid gaps or overlaps within the figures in
a variety of browsers. The w3.org visual formatting model
may or may not be helpful here.

Update of 11:20 PM ET March 15, 2015 — 
Seekers of simplicity should note that there is
a simple program in the Processing.js  language, not  using
such Unicode characters, that shows many random affine
permutations of a 4×4 diamond-theorem array when the
display window is clicked.

Friday, January 16, 2015

A versus PA

Filed under: General,Geometry — Tags: — m759 @ 8:48 PM

"Reality is the beginning not the end,
Naked Alpha, not the hierophant Omega,
of dense investiture, with luminous vassals."

— “An Ordinary Evening in New Haven” VI

From the series of posts tagged "Defining Form" —

The 4-point affine plane A  and
the 7-point projective plane PA  —

IMAGE- Triangular models of the 4-point affine plane A and 7-point projective plane PA

The circle-in-triangle of Yale's Figure 30b (PA ) may,
if one likes, be seen as having an occult meaning.

For the mathematical  meaning of the circle in PA
see a search for "line at infinity."

A different, cubic, model of PA  is perhaps more perspicuous.

Friday, October 31, 2014

Structure

Filed under: General,Geometry — m759 @ 3:00 AM

On Devil’s Night

Introducing a group of 322,560 affine transformations of Dürer’s ‘Magic’ Square

IMAGE- Introduction to 322,560 Affine Transformations of Dürer's 'Magic' Square

The four vector-space substructures of digits in 1st, 2nd, 3rd, 4th place,
together with the diamond theorem, indicate that Dürer’s square “minus one”
can be transformed by permutations of rows, columns, and quadrants to a
square with (decimal) digits in the usual numerical order, increasing from
top left to bottom right. Such permutations form a group of order 322,560.

(Continued from Vector Addition in a Finite Field, Twelfth Night, 2013.)

Saturday, October 25, 2014

Foundation Square

Filed under: General,Geometry — Tags: — m759 @ 2:56 PM

In the above illustration of the 3-4-5 Pythagorean triangle,
the grids on each side may be regarded as figures of
Euclidean  geometry or of Galois  geometry.

In Euclidean geometry, these grids illustrate a property of
the inner triangle.

In elementary Galois geometry, ignoring the connection with
the inner triangle, the grids may be regarded instead as
illustrating vector spaces over finite (i.e., Galois) fields.
Previous posts in this journal have dealt with properties of
the 3×3 and 4×4 grids.  This suggests a look at properties of
the next larger grid, the 5×5 array, viewed as a picture of the
two-dimensional vector space (or affine plane) over the finite
Galois field GF(5) (also known as ℤ5).

The 5×5 array may be coordinatized in a natural way, as illustrated
in (for instance) Matters Mathematical , by I.N. Herstein and
Irving Kaplansky, 2nd ed., Chelsea Publishing, 1978, p. 171:

See Herstein and Kaplansky for the elementary Galois geometry of
the 5×5 array.

For 5×5 geometry that is not so elementary, see…

Hafner's abstract:

We describe the Hoffman-Singleton graph geometrically, showing that
it is closely related to the incidence graph of the affine plane over ℤ5.
This allows us to construct all automorphisms of the graph.

The remarks of Brouwer on graphs connect the 5×5-related geometry discussed
by Hafner with the 4×4 geometry related to the Steiner system S(5,8,24).
(See the Miracle Octad Generator of R. T. Curtis and the related coordinatization
by Cullinane of the 4×4 array as a four-dimensional vector space over GF(2).)

Sunday, September 14, 2014

Sensibility

Filed under: General,Geometry — Tags: , — m759 @ 9:26 AM

Structured gray matter:

Graphic symmetries of Galois space:
IMAGE - The Diamond Theorem

The reason for these graphic symmetries in affine  Galois space —

symmetries of the underlying projective  Galois space:

Thursday, July 17, 2014

Paradigm Shift:

Filed under: General,Geometry — Tags: — m759 @ 11:01 AM
 

Continuous Euclidean space to discrete Galois space*

Euclidean space:

Point, line, square, cube, tesseract

From a page by Bryan Clair

Counting symmetries in Euclidean space:

Galois space:

Image-- examples from Galois affine geometry

Counting symmetries of  Galois space:
IMAGE - The Diamond Theorem

The reason for these graphic symmetries in affine Galois space —

symmetries of the underlying projective Galois space:

* For related remarks, see posts of May 26-28, 2012.

Thursday, March 27, 2014

Diamond Space

Filed under: General,Geometry — Tags: — m759 @ 2:28 PM

(Continued)

Definition:  A diamond space  — informal phrase denoting
a subspace of AG(6, 2), the six-dimensional affine space
over the two-element Galois field.

The reason for the name:

IMAGE - The Diamond Theorem, including the 4x4x4 'Solomon's Cube' case

Click to enlarge.

Friday, March 21, 2014

Three Constructions of the Miracle Octad Generator

Filed under: General,Geometry — Tags: , , — m759 @ 12:24 PM

IMAGE- Two constructions, by Turyn/Curtis, and by Conway, of the Miracle Octad Generator

See also a Log24 post on this subject from Dec. 14, 2013,
especially (scroll down) the update of March 9, 2014.

Related material on the Turyn-Curtis construction
from the University of Cambridge —

— Slide by “Dr. Parker” — Apparently Richard A. Parker —
Lecture 4, “Discovering M24,” in slides for lectures 1-8 from lectures
at Cambridge in 2010-2011 on “Sporadic and Related Groups.”
See also the Parker lectures of 2012-2013 on the same topic.

A third construction of Curtis’s 35  4×6  1976 MOG arrays would use
Cullinane’s analysis of the 4×4 subarrays’ affine and projective structure,
and point out the fact that Conwell’s 1910 correspondence of the 35
4+4-partitions of an 8-set with the 35 lines of the projective 3-space
over the 2-element field, PG(3, 2), is essentially the same correspondence
as that constituting Curtis’s 1976 MOG.

See The Diamond Theorem,  Finite RelativityGalois Space,
Generating the Octad Generator, and The Klein Correspondence.

Update of March 22-March 23 —

Adding together as (0,1)-matrices over GF(2) the black parts (black
squares as 1’s, all other squares as 0’s) of the 35  4×6 arrays of the 1976
Curtis MOG would then reveal*  the symmetric role played in octads
by what Curtis called the heavy brick , and so reveal also the action of
S3 on the three Curtis bricks that leaves invariant the set of all 759
octads of the S(5, 8, 24) constructed from the 35  MOG arrays.  For more
details of this “by-hand” construction, see Geometry of the 4×4 Square.
For the mathematical properties of the S(5, 8, 24), it is convenient to
have a separate construction,  not  by hand (such as Turyn’s), of the
extended binary Golay code. See the Brouwer preprint quoted above.

* “Then a miracle occurs,” as in the classic 1977 Sidney Harris cartoon.

Illustration of array addition from March 23 —

IMAGE- Discovery of the S_3 action on bricks in the Conwell-Cullinane 'by-hand' approach to octad-building

Friday, January 17, 2014

The 4×4 Relativity Problem

Filed under: General,Geometry — Tags: , , — m759 @ 11:00 PM

The sixteen-dot square array in yesterday’s noon post suggests
the following remarks.

“This is the relativity problem:  to fix objectively a class of
equivalent coordinatizations and to ascertain the group of
transformations S mediating between them.”

— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16

The Galois tesseract  appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

The 1977 matrix Q is echoed in the following from 2002—

IMAGE- Dolgachev and Keum, coordinatization of the 4x4 array in 'Birational Automorphisms of Quartic Hessian Surfaces,' AMS Transactions, 2002

A different representation of Cullinane’s 1977 square model of the
16-point affine geometry over the two-element Galois field GF(2)
is supplied by Conway and Sloane in Sphere Packings, Lattices and Groups   
(first published in 1988) :

IMAGE- The Galois tesseract as a four-dimensional vector space, from a diagram by Conway and Sloane in 'Sphere Packings, Lattices, and Groups'

Here a, b, c, d   are basis vectors in the vector 4-space over GF(2).
(For a 1979 version of this vector space, see AMS Abstract 79T-A37.)

See also a 2011 publication of the Mathematical Association of America —

From 'Beautiful Mathematics,' by Martin Erickson, an excerpt on the Cullinane diamond theorem (with source not mentioned)

Saturday, September 21, 2013

Mathematics and Narrative (continued)

Filed under: General,Geometry — Tags: , — m759 @ 1:00 AM

Mathematics:

A review of posts from earlier this month —

Wednesday, September 4, 2013

Moonshine

Filed under: Uncategorized — m759 @ 4:00 PM

Unexpected connections between areas of mathematics
previously thought to be unrelated are sometimes referred
to as "moonshine."  An example—  the apparent connections
between parts of complex analysis and groups related to the
large Mathieu group M24. Some recent work on such apparent
connections, by Anne Taormina and Katrin Wendland, among
others (for instance, Miranda C.N. Cheng and John F.R. Duncan),
involves structures related to Kummer surfaces .
In a classic book, Kummer's Quartic Surface  (1905),
R.W.H.T. Hudson pictured a set of 140 structures, the 80
Rosenhain tetrads and the 60 Göpel tetrads, as 4-element
subsets of a 16-element 4×4 array.  It turns out that these
140 structures are the planes of the finite affine geometry
AG(4,2) of four dimensions over the two-element Galois field.
(See Diamond Theory in 1937.)

Thursday, September 5, 2013

Moonshine II

Filed under: Uncategorized — Tags:  — m759 @ 10:31 AM

(Continued from yesterday)

The foreword by Wolf Barth in the 1990 Cambridge U. Press
reissue of Hudson's 1905 classic Kummer's Quartic Surface
covers some of the material in yesterday's post Moonshine.

The distinction that Barth described in 1990 was also described, and illustrated,
in my 1986 note "Picturing the smallest projective 3-space."  The affine 4-space
over the the finite Galois field GF(2) that Barth describes was earlier described—
within a 4×4 array like that pictured by Hudson in 1905— in a 1979 American
Mathematical Society abstract, "Symmetry invariance in a diamond ring."

"The distinction between Rosenhain and Goepel tetrads
is nothing but the distinction between isotropic and
non-isotropic planes in this affine space over the finite field."

The 1990 paragraph of Barth quoted above may be viewed as a summary
of these facts, and also of my March 17, 2013, note "Rosenhain and Göpel
Tetrads in PG(3,2)
."

Narrative:

Aooo.

Happy birthday to Stephen King.

Thursday, September 5, 2013

Moonshine II

Filed under: General,Geometry — Tags: , , , , , — m759 @ 10:31 AM

(Continued from yesterday)

The foreword by Wolf Barth in the 1990 Cambridge U. Press
reissue of Hudson's 1905 classic Kummer's Quartic Surface
covers some of the material in yesterday's post Moonshine.

The distinction that Barth described in 1990 was also described, and illustrated,
in my 1986 note "Picturing the smallest projective 3-space."  The affine 4-space
over the the finite Galois field GF(2) that Barth describes was earlier described—
within a 4×4 array like that pictured by Hudson in 1905— in a 1979 American
Mathematical Society abstract, "Symmetry invariance in a diamond ring."

"The distinction between Rosenhain and Goepel tetrads
is nothing but the distinction between isotropic and
non-isotropic planes in this affine space over the finite field."

The 1990 paragraph of Barth quoted above may be viewed as a summary
of these facts, and also of my March 17, 2013, note "Rosenhain and Göpel
Tetrads in PG(3,2)
."

Wednesday, September 4, 2013

Moonshine

Filed under: General,Geometry — Tags: , , — m759 @ 4:00 PM

Unexpected connections between areas of mathematics
previously thought to be unrelated are sometimes referred
to as "moonshine."  An example—  the apparent connections
between parts of complex analysis and groups related to the 
large Mathieu group M24. Some recent work on such apparent
connections, by Anne Taormina and Katrin Wendland, among
others (for instance, Miranda C.N. Cheng and John F.R. Duncan),
involves structures related to Kummer surfaces .
In a classic book, Kummer's Quartic Surface  (1905),
R.W.H.T. Hudson pictured a set of 140 structures, the 80
Rosenhain tetrads and the 60 Göpel tetrads, as 4-element
subsets of a 16-element 4×4 array.  It turns out that these
140 structures are the planes of the finite affine geometry
AG(4,2) of four dimensions over the two-element Galois field.
(See Diamond Theory in 1937.) 

A Google search documents the moonshine
relating Rosenhain's and Göpel's 19th-century work
in complex analysis to M24  via the book of Hudson and
the geometry of the 4×4 square.

Tuesday, July 9, 2013

Vril Chick

Filed under: General,Geometry — m759 @ 4:30 AM

Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo

Profile picture for "Jo Lyxe" (Josefine Lyche) at Vimeo

Compare to an image of Vril muse Maria Orsitsch.

From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —

Josefine Lyche
Born in 1973 in Bergen, Norway.
Lives and works in Oslo and Berlin.

Keywords (to help place my artwork in the
proper context): Aliens, affine geometry, affine
planes, affine spaces, automorphisms, binary
codes, block designs, classical groups, codes,
coding theory, collineations, combinatorial,
combinatorics, conjugacy classes, the Conwell
correspondence, correlations, Cullinane,
R. T. Curtis, design theory, the diamond theorem,
diamond theory, duads, duality, error correcting
codes, esoteric, exceptional groups,
extraterrestrials, finite fields, finite geometry, finite
groups, finite rings, Galois fields, generalized
quadrangles, generators, geometry, GF(2),
GF(4), the (24,12) Golay code, group actions,
group theory, Hadamard matrices, hypercube,
hyperplanes, hyperspace, incidence structures,
invariance, Karnaugh maps, Kirkman’s schoolgirls
problem, Latin squares, Leech lattice, linear
groups, linear spaces, linear transformations,
Magick, Mathieu groups, matrix theory, Meno,
Miracle Octad Generator, MOG, multiply transitive
groups, occultism, octahedron, the octahedral
group, Orsic, orthogonal arrays, outer automorphisms,
parallelisms, partial geometries,
permutation groups, PG(3,2), Plato, Platonic
solids, polarities, Polya-Burnside theorem, projective
geometry, projective planes, projective
spaces, projectivities, Pythagoras, reincarnation,
Reed-Muller codes, the relativity problem,
reverse engineering, sacred geometry, Singer
cycle, skew lines, Socrates, sporadic simple
groups, Steiner systems, Sylvester, symmetric,
symmetry, symplectic, synthemes, synthematic,
Theosophical Society tesseract, Tessla, transvections,
Venn diagrams, Vril society, Walsh
functions, Witt designs.

(See also the original catalog page.)

Clearly most of this (the non-highlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.

For some background, see (for instance) 
Conspiracy Theories and Secret Societies for Dummies .

Friday, June 14, 2013

Object of Beauty

Filed under: General,Geometry — Tags: , — m759 @ 2:01 AM

This journal on July 5, 2007 —

The Eightfold Cube and its Inner Structure

“It is not clear why MySpace China will be successful."

— The Chinese magazine Caijing  in 2007, quoted in
Asia Sentinel  on July 12, 2011

This  journal on that same date,  July 12, 2011 —

http://www.log24.com/log/pix11B/110712-ObjectOfBeauty.jpg

See also the eightfold cube and kindergarten blocks
at finitegeometry.org/sc.

Friedrich Froebel, Froebel's Chief Writings on Education ,
Part II, "The Kindergarten," Ch. III, "The Third Play":

"The little ones, who always long for novelty and change,
love this simple plaything in its unvarying form and in its
constant number, even as they love their fairy tales with
the ever-recurring dwarfs…."

This journal, Group Actions, Nov. 14, 2012:

"Those who insist on vulgarizing their mathematics
may regard linear and affine group actions on the eight
cubes as the dance of  Snow White (representing (0,0,0))
and the Seven Dwarfs—

  ."

Edwin M. Knowles Fine China Company, 1991

Monday, June 10, 2013

Galois Coordinates

Filed under: General,Geometry — Tags: , — m759 @ 10:30 PM

Today's previous post on coordinate systems
suggests a look at the phrase "Galois coordinates."

A search shows that the phrase, though natural,
has apparently not been used before 2011* for solutions
to what Hermann Weyl called "the relativity problem."

A thorough historical essay on Galois coordinatization
in this sense would require more academic resources
than I have available. It would likely describe a number
of applications of Galois-field coordinates to square
(and perhaps to cubical) arrays that were studied before
1976, the date of my Diamond Theory  monograph.

But such a survey might not  find any such pre-1976
coordinatization of a 4×4 array  by the 16 elements
of the vector 4-space  over the Galois field with two
elements, GF(2).

Such coordinatizations are important because of their
close relationship to the Mathieu group 24 .

See a preprint by Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group 24 ," with its remark
denying knowledge of any such coordinatization
prior to a 1989 paper by R. T. Curtis.

Related material: 

Some images related to Galois coordinates, excerpted
from a Google search today (click to enlarge)—

*  A rather abstract  2011 paper that uses the phrase
   "Galois coordinates" may have some implications 
   for the naive form of the relativity problem
   related to square and cubical arrays.

Tuesday, June 4, 2013

Cover Acts

Filed under: General,Geometry — Tags: — m759 @ 11:00 AM

The Daily Princetonian  today:

IMAGE- 'How Jay White, a Neil Diamond cover act, duped Princeton'

A different cover act, discussed here  Saturday:

IMAGE- The diamond theorem affine group of order 322,560, published without acknowledgment of its source by the Mathematical Association of America in 2011

See also, in this journal, the Galois tesseract and the Crosswicks Curse.

"There is  such a thing as a tesseract." — Crosswicks saying

Saturday, June 1, 2013

Permanence

Filed under: General,Geometry — Tags: , , — m759 @ 4:00 PM

"What we do may be small, but it has
  a certain character of permanence."

— G. H. Hardy, A Mathematician's Apology

The diamond theorem  group, published without acknowledgment
of its source by the Mathematical Association of America in 2011—

IMAGE- The diamond-theorem affine group of order 322,560, published without acknowledgment of its source by the Mathematical Association of America in 2011

Tuesday, May 14, 2013

Snakes on a Plane

Filed under: General,Geometry — m759 @ 7:27 AM

Continued.

The order-3 affine plane:

Detail from the video in the previous post:

For other permutations of points in the
order-3 affine plane—

See Quaternions in an Affine Galois Plane
and Group Actions, 1984-2009.

See, too, the Mathematics and Narrative post 
from April 28, 2013, and last night's
For Indiana Spielberg.

Sunday, March 10, 2013

Galois Space

Filed under: General,Geometry — Tags: — m759 @ 5:30 PM

(Continued)

The 16-point affine Galois space:

Further properties of this space:

In Configurations and Squares, see the
discusssion of the Kummer 166 configuration.

Some closely related material:

  • Wolfgang Kühnel,
    "Minimal Triangulations of Kummer Varieties,"
    Abh. Math. Sem. Univ. Hamburg 57, 7-20 (1986).

    For the first two pages, click here.

  • Jonathan Spreer and Wolfgang Kühnel,
    "Combinatorial Properties of the 3 Surface:
    Simplicial Blowups and Slicings,"
    preprint, 26 pages. (2009/10) (pdf).
    (Published in Experimental Math. 20,
    issue 2, 201–216 (2011).)

Saturday, January 5, 2013

Vector Addition in a Finite Field

Filed under: General,Geometry — Tags: , — m759 @ 10:18 AM

The finite (i.e., Galois) field GF(16),
according to J. J. Seidel in 1974—

The same field according to Steven H. Cullinane in 1986,
in its guise as the affine 4-space over GF(2)—


The same field, again disguised as an affine 4-space,
according to John H. Conway and N.J.A. Sloane in
Sphere Packings, Lattices, and Groups , first published in 1988—

The above figure by Conway and Sloane summarizes, using
a 4×4 array, the additive vector-space structure of the finite
field GF(16).

This structure embodies what in Euclidean space is called
the parallelogram rule for vector addition—

(Thanks to June Lester for the 3D (uvw) part of the above figure.)

For the transition from this colored Euclidean hypercube
(used above to illustrate the parallelogram rule) to the
4×4 Galois space (illustrated by Cullinane in 1979 and
Conway and Sloane in 1988— or later… I do not have
their book’s first edition), see Diamond Theory in 1937,
Vertex Adjacency in a Tesseract and in a 4×4 Array,
Spaces as Hypercubes, and The Galois Tesseract.

For some related narrative, see tesseract  in this journal.

(This post has been added to finitegeometry.org.)

Update of August 9, 2013—

Coordinates for hypercube vertices derived from the
parallelogram rule in four dimensions were better
illustrated by Jürgen Köller in a web page archived in 2002.

Update of August 13, 2013—

The four basis vectors in the 2002 Köller hypercube figure
are also visible at the bottom of the hypercube figure on
page 7 of “Diamond Theory,” excerpts from a 1976 preprint
in Computer Graphics and Art , Vol. 2, No. 1, February 1977.
A predecessor:  Coxeter’s 1950 hypercube figure from
Self-Dual Configurations and Regular Graphs.”

Thursday, November 22, 2012

Finite Relativity

Filed under: General,Geometry — m759 @ 10:48 PM

(Continued from 1986)

S. H. Cullinane
The relativity problem in finite geometry.
Feb. 20, 1986.

This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.

— H. Weyl, The Classical Groups ,
Princeton Univ. Pr., 1946, p. 16

In finite geometry "points" are often defined as ordered n-tuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such n-tuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S.

This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame.

The frame: A 4×4 array.

The invariant structure:

The following set of 15 partitions of the frame into two 8-sets.

Fifteen partitions of a 4x4 array into two 8-sets

A representative coordinatization:

 

0000  0001  0010  0011
0100  0101  0110  0111
1000  1001  1010  1011
1100  1101  1110  1111

 

The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2).

S. H. Cullinane
The relativity problem in finite geometry.
Nov. 22, 2012.

This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.

— H. Weyl, The Classical Groups ,
Princeton Univ. Pr., 1946, p. 16

In finite geometry "points" are often defined as ordered n-tuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such n-tuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S.

This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame.

The frame: An array of 16 congruent equilateral subtriangles that make up a larger equilateral triangle.

The invariant structure:

The following set of 15 partitions of the frame into two 8-sets.


Fifteen partitions of an array of 16 triangles into two 8-sets


A representative coordinatization:

Coordinates for a triangular finite geometry

The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2).

For some background on the triangular version,
see the Square-Triangle Theorem,
noting particularly the linked-to coordinatization picture.

Wednesday, November 14, 2012

Group Actions

Filed under: General,Geometry — Tags: — m759 @ 4:30 PM

The December 2012 Notices of the American
Mathematical Society  
has an ad on page 1564
(in a review of two books on vulgarized mathematics)
for three workshops next year on "Low-dimensional
Topology, Geometry, and Dynamics"—

(Only the top part of the ad is shown; for further details
see an ICERM page.)

(ICERM stands for Institute for Computational
and Experimental Research in Mathematics.)

The ICERM logo displays seven subcubes of
a 2x2x2 eight-cube array with one cube missing—

The logo, apparently a stylized image of the architecture 
of the Providence building housing ICERM, is not unlike
a picture of Froebel's Third Gift—

 

Froebel's third gift, the eightfold cube

© 2005 The Institute for Figuring

Photo by Norman Brosterman from the Inventing Kindergarten
exhibit at The Institute for Figuring (co-founded by Margaret Wertheim)

The eighth cube, missing in the ICERM logo and detached in the
Froebel Cubes photo, may be regarded as representing the origin
(0,0,0) in a coordinatized version of the 2x2x2 array—
in other words the cube invariant under linear , as opposed to
more general affine , permutations of the cubes in the array.

These cubes are not without relevance to the workshops' topics—
low-dimensional exotic geometric structures, group theory, and dynamics.

See The Eightfold Cube, A Simple Reflection Group of Order 168, and 
The Quaternion Group Acting on an Eightfold Cube.

Those who insist on vulgarizing their mathematics may regard linear
and affine group actions on the eight cubes as the dance of
Snow White (representing (0,0,0)) and the Seven Dwarfs—

.

Sunday, August 5, 2012

Cube Partitions

Filed under: General,Geometry — Tags: — m759 @ 7:59 AM

The second Logos  figure in the previous post
summarized affine group actions on partitions
that generate a group of about 1.3 trillion
permutations of a 4x4x4 cube (shown below)—

IMAGE by Cullinane- 'Solomon's Cube' with 64 identical, but variously oriented, subcubes, and six partitions of these 64 subcubes

Click for further details.

Thursday, August 2, 2012

Logos

Filed under: General,Geometry — m759 @ 12:00 PM

(Continued from December 26th, 2011)

IMAGE- Current math.stackexchange.com logo and a 1984 figure from 'Notes on Groups and Geometry, 1978-1986'

Some material at math.stackexchange.com related to
yesterday evening's post on Elementary Finite Geometry

Questions on this topic have recently been
discussed at Affine plane of order 4? and at
Turning affine planes into projective planes.

(For a better discussion of the affine plane of order 4,
see Affine Planes and Mutually Orthogonal Latin Squares
at the website of William Cherowitzo, professor at UC Denver.)

Wednesday, August 1, 2012

Elementary Finite Geometry

Filed under: General,Geometry — Tags: , — m759 @ 7:16 PM

I. General finite geometry (without coordinates):

A finite affine plane of order has n^2 points.

A finite projective plane of order n  has n^2 + n + 1 

points because it is formed from an order-n finite affine 

plane by adding a line at infinity  that contains n + 1 points.

Examples—

Affine plane of order 3

Projective plane of order 3

II. Galois finite geometry (with coordinates over a Galois field):

A finite projective Galois plane of order n has n^2 + n + 1

points because it is formed from a finite affine Galois 3-space

of order n with n^3 points by discarding the point (0,0,0) and 

identifying the points whose coordinates are multiples of the

(n-1) nonzero scalars.

Note: The resulting Galois plane of order n has 

(n^3-1)/(n-1)= (n^2 + n + 1) points because 

(n^2 + n + 1)(n – 1) =

(n^3 + n^2 + n – n^2 – n – 1) = (n^3 – 1) .
 

III. Related art:

Another version of a 1994 picture that accompanied a New Yorker
article, "Atheists with Attitude," in the issue dated May 21, 2007:

IMAGE- 'Four Gods,' by Jonathan Borofsky

The Four Gods  of Borofsky correspond to the four axes of 
symmetry
  of a square and to the four points on a line at infinity 
in an order-3 projective plane as described in Part I above.

Those who prefer literature to mathematics may, if they like,
view the Borofsky work as depicting

"Blake's Four Zoas, which represent four aspects
of the Almighty God" —Wikipedia

Thursday, July 26, 2012

Solomon’s Seal

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM

(Mathematics and Narrative, continued)

Narrative—

The Ring and The Stone from yesterday's post, and…

"In Medieval Jewish, Christian and Islamic legends,
the Seal of Solomon was a magical signet ring
said to have been possessed by King Solomon…."

— Wikipedia article, Seal of Solomon

Mathematics—

IMAGE- Eric Temple Bell on the mathematics of 'Solomon's Seal' (in his 'Development of Mathematics')

A fact related to the mathematical
"Solomon's seal" described above by Bell:

IMAGE- J.W.P. Hirschfeld on the mathematics of 'Solomon's Seal', with reference to Edge on the same topic

The reference to Edge is as follows—

[3] Edge, W. L., Quadrics over GF(2) and
their relevance for the cubic surface group
,
Canadian J. Maths. 11 (1959) ….

(This reference relates Hirschfeld's remarks
quoted above to the 64-point affine space
illustrated below (via the associated
63-point projective  space PG (5, 2)).

As for the narrative's Stone… 

See Solomon's Cube.

IMAGE- 'Solomon's Cube'

Monday, July 16, 2012

Mapping Problem continued

Filed under: General,Geometry — m759 @ 2:56 AM

Another approach to the square-to-triangle
mapping problem (see also previous post)—

IMAGE- Triangular analogs of the hyperplanes in the square model of PG(3,2)

For the square model referred to in the above picture, see (for instance)

Coordinates for the 16 points in the triangular arrays 
of the corresponding affine space may be deduced
from the patterns in the projective-hyperplanes array above.

This should solve the inverse problem of mapping,
in a natural way, the triangular array of 16 points 
to the square array of 16 points.

Update of 9:35 AM ET July 16, 2012:

Note that the square model's 15 hyperplanes S 
and the triangular model's 15 hyperplanes T —

— share the following vector-space structure —

   0     c     d   c + d
   a   a + c   a + d a + c + d
   b   b + c   b + d b + c + d
a + b a + b + c a + b + d   a + b + 
  c + d

   (This vector-space a b c d  diagram is from
   Chapter 11 of   Sphere Packings, Lattices
   and Groups
, by   John Horton Conway and
   N. J. A. Sloane, first published by Springer
   in 1988.)

Sunday, June 17, 2012

Congruent Group Actions

Filed under: General,Geometry — Tags: , , — m759 @ 9:00 PM

A Google search today yielded no results
for the phrase "congruent group actions."

Places where this phrase might prove useful include—

Sunday, March 18, 2012

Square-Triangle Diamond

Filed under: General,Geometry — Tags: — m759 @ 5:01 AM

The diamond shape of yesterday's noon post
is not wholly without mathematical interest …

The square-triangle theorem

"Every triangle is an n -replica" is true
if and only if n  is a square.

IMAGE- Square-to-diamond (rhombus) shear in proof of square-triangle theorem

The 16 subdiamonds of the above figure clearly
may be mapped by an affine transformation
to 16 subsquares of a square array.

(See the diamond lattice  in Weyl's Symmetry .)

Similarly for any square n , not just 16.

There is a group of 322,560 natural transformations
that permute the centers  of the 16 subsquares
in a 16-part square array. The same group may be
viewed as permuting the centers  of the 16 subtriangles
in a 16-part triangular array.

(Updated March 29, 2012, to correct wording and add Weyl link.)

Saturday, February 18, 2012

Symmetry

Filed under: General,Geometry — m759 @ 7:35 PM

From the current Wikipedia article "Symmetry (physics)"—

"In physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are 'unchanged', according to a particular observation. A symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is 'preserved' under some change.

A family of particular transformations may be continuous  (such as rotation of a circle) or discrete  (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group)."….

"A discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance."

Note the confusion here between continuous (or discontinuous) transformations  and "continuous" (or "discontinuous," i.e. "discrete") groups .

This confusion may impede efforts to think clearly about some pure mathematics related to current physics— in particular, about the geometry of spaces made up of individual units ("points") that are not joined together in a continuous manifold.

For an attempt to forestall such confusion, see Noncontinuous Groups.

For related material, see Erlanger and Galois as well as the opening paragraphs of Diamond Theory

Symmetry is often described as invariance under a group of transformations. An unspoken assumption about symmetry in Euclidean 3-space is that the transformations involved are continuous.

Diamond theory rejects this assumption, and in so doing reveals that Euclidean symmetry may itself  be invariant under rather interesting groups of non-continuous (and a-symmetric) transformations. (These might be called noncontinuous  groups, as opposed to so-called discontinuous  (or discrete ) symmetry groups. See Weyl's Symmetry .)

For example, the affine group A on the 4-space over the 2-element field has a natural noncontinuous and asymmetric but symmetry-preserving action on the elements of a 4×4 array. (Details)

(Version first archived on March 27, 2002)

Update of Sunday, February 19, 2012—

The abuse of language by the anonymous authors
of the above Wikipedia article occurs also in more
reputable sources. For instance—

IMAGE- Brading and Castellani, 'Symmetries in Physics'- Four main sections include 'Continuous Symmetries' and 'Discrete Symmetries.'

Some transformations referred to by Brading and Castellani
and their editees as "discrete symmetries" are, in fact, as
linear transformations of continuous spaces, themselves
continuous  transformations.

This unfortunate abuse of language is at least made explicit
in a 2003 text, Mathematical Perspectives on Theoretical
Physics 
(Nirmala Prakash, Imperial College Press)—

"… associated[*] with any given symmetry there always exists
a continuous or a discrete group of transformations….
A symmetry whose associated group is continuous (discrete)
is called a continuous  (discrete ) symmetry ." — Pp. 235, 236

[* Associated how?]

Monday, January 30, 2012

Enda’s Game*

Filed under: General — Tags: — m759 @ 7:00 PM

The following passage by Tolkien was suggested by a copy of next Sunday's New York Times Book Review  that arrived in the mail today. (See Orson Scott Card's remarks on page 26— "Uncle Orson"— and the Review 's concluding essay "Grand Allusion.")

"Lastly, tengwesta  [system or code of signs] has also become an impediment. It is in Incarnates clearer and more precise than their direct reception of thought. By it also they can communicate easily with others, when no strength is added to their thought: as, for example, when strangers first meet. And, as we have seen, the use of 'language' soon becomes habitual, so that the practice of ósanwe  (interchange of thought) is neglected and becomes more difficult. Thus we see that the Incarnate tend more and more to use or to endeavour to use ósanwe  only in great need and urgency, and especially when lambe  is unavailing. As when the voice cannot be heard, which comes most often because of distance. For distance in itself offers no impediment whatever to ósanwe . But those who by affinity might well use ósanwe  will use lambe  when in proximity, by habit or preference. Yet we may mark also how the 'affine' may more quickly understand the lambe  that they use between them, and indeed all that they would say is not put into words. With fewer words they come swifter to a better understanding. There can be no doubt that here ósanwe  is also often taking place; for the will to converse in lambe  is a will to communicate thought, and lays the minds open. It may be, of course, that the two that converse know already part of the matter and the thought of the other upon it, so that only allusions dark to the stranger need be made; but this is not always so. The affine** will reach an understanding more swiftly than strangers upon matters that neither have before discussed, and they will more quickly perceive the import of words that, however numerous, well-chosen, and precise, must remain inadequate."

* "If a poem catches a student's interest at all, he or she should damned well be able to look up an unfamiliar word in the dictionary…."

   — Elizabeth Bishop, quoted in the essay "Grand Allusion" mentioned above. For a brief dictionary of most of the unfamiliar words in this post's title and in the above passage, see Vinyar Tengwar  39 (July 1998). This is copyrighted but freely available on the Web.

** The word "affine" has connotations not intended by Tolkien. See that word in this journal. See also page 5 of next Sunday's Times Book Review , which contains a full-page ad for the 50th anniversary edition of A Wrinkle in Time . "There is  such a thing as a tesseract."

Monday, January 23, 2012

How It Works

Filed under: General,Geometry — Tags: , — m759 @ 7:59 PM

(Continued)

J. H. Conway in 1971 discussed the role of an elementary abelian group
of order 16 in the Mathieu group M24. His approach at that time was
purely algebraic, not geometric—

IMAGE- J. H. Conway in 1971 discussed the role of the elementary abelian group of order 16 in the Mathieu group M24. His approach then was purely algebraic, not geometric.

For earlier (and later) discussions of the geometry  (not the algebra )
of that order-16 group (i.e., the group of translations of the affine space
of 4 dimensions over the 2-element field), see The Galois Tesseract.

Friday, December 30, 2011

Quaternions on a Cube

Filed under: General,Geometry — Tags: , , — m759 @ 5:48 AM

The following picture provides a new visual approach to
the order-8 quaternion  group's automorphisms.

IMAGE- Quaternion group acting on an eightfold cube

Click the above image for some context.

Here the cube is called "eightfold" because the eight vertices,
like the eight subcubes of a 2×2×2 cube,* are thought of as
independently movable. See The Eightfold Cube.

See also…

Related material: Robin Chapman and Karen E. Smith
on the quaternion group's automorphisms.

* See Margaret Wertheim's Christmas Eve remarks on mathematics
and the following eightfold cube from an institute she co-founded—

Froebel's third gift, the eightfold cube
© 2005 The Institute for Figuring

Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)

Tuesday, December 27, 2011

Getting with the Program

Filed under: General,Geometry — Tags: — m759 @ 4:28 AM

Stanley Fish in The New York Times  yesterday evening—

IMAGE- Stanley Fish, 'The Old Order Changeth,' Boxing Day, 2011

From the MLA program Fish discussed—

IMAGE- MLA session, 'Defining Form,' chaired by Colleen Rosenfeld of Pomona College

Above: An MLA session, "Defining Form," led
by Colleen Rosenfeld of Pomona College

An example from Pomona College in 1968—

IMAGE- Triangular models of small affine and projective finite geometries

The same underlying geometries (i.e., "form") may be modeled with
a square figure and a cubical figure rather than with the triangular
figures of 1968 shown above.

See Finite Geometry of the Square and Cube.

Those who prefer a literary approach to form may enjoy the recent post As Is.
(For some context, see Game of Shadows.)

Monday, December 26, 2011

Logos

Filed under: General,Geometry — m759 @ 5:09 PM

IMAGE- Logo for math.stackexchange.com

IMAGE- 'Affine Groups on Small Binary Spaces,' illustration

  Click images for context.

Wednesday, September 7, 2011

The Most Important Configuration

Filed under: General,Geometry — m759 @ 6:30 PM

A search for some background on Gian-Carlo Rota's remarks
in Indiscrete Thoughts * on a geometric configuration
leads to the following passages in Hilbert and Cohn-Vossen's
classic Geometry and the Imagination

http://www.log24.com/log/pix11B/110907-HCV-BPconfigSm.jpg

These authors describe the Brianchon-Pascal configuration
of 9 points and 9 lines, with 3 points on each line
and 3 lines through each point, as being
"the most important configuration of all geometry."

Thus it seems worthwhile to relate it to the web page
on square configurations referenced here Tuesday.

The Encyclopaedia of Mathematics , ed. by Michiel Hazewinkel,
supplies a summary of the configuration apparently
derived from Hilbert and Cohn-Vossen

http://www.log24.com/log/pix11B/110907-HazewEnc-Brianchon-Pascal-Annot3Sm.jpg

My own annotation at right above shows one way to picture the
Brianchon-Pascal points and lines— regarded as those of a finite,
purely combinatorial , configuration— as subsets of the nine-point
square array discussed in Configurations and Squares. The
rearrangement of points in the square yields lines that are in
accord with those in the usual square picture of the 9-point
affine plane.

A more explicit picture—

http://www.log24.com/log/pix11B/110907-AG23lines500w.jpg

The Brianchon-Pascal configuration is better known as Pappus's  configuration,
and a search under that name will give an idea of its importance in geometry.

* Birkhäuser Boston, 1998 2nd printing, p. 145

Saturday, September 3, 2011

The Galois Tesseract (continued)

Filed under: General,Geometry — Tags: , — m759 @ 1:00 PM

A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
two-thirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79T-A37, Notices , Feb. 1979).

Here is some supporting material—

http://www.log24.com/log/pix11B/110903-Carmichael-Conway-Curtis.jpg

The passage from Carmichael above emphasizes the importance of
the 4×4 square within the MOG.

The passage from Conway and Sloane, in a book whose first edition
was published in 1988, makes explicit the structure of the MOG's
4×4 square as the affine 4-space over the 2-element Galois field.

The passage from Curtis (1974, published in 1976) describes 35 sets
of four "special tetrads" within the 4×4 square of the MOG. These
correspond to the 35 sets of four parallel 4-point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 "special tetrads" rather by the parity
of their intersections with the square's rows and columns.

The affine structure appears in the 1979 abstract mentioned above—

IMAGE- An AMS abstract from 1979 showing how the affine group AGL(4,2) of 322,560 transformations acts on a 4x4 square

The "35 structures" of the abstract were listed, with an application to
Latin-square orthogonality, in a note from December 1978

IMAGE- Projective-space structure and Latin-square orthogonality in a set of 35 square arrays

See also a 1987 article by R. T. Curtis—

Further elementary techniques using the miracle octad generator
, by R. T. Curtis. Abstract:

“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”

(Received July 20 1987)

Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353

* For instance:

Algebraic structure in the 4x4 square, by Cullinane (1985) and Curtis (1987)

Update of Sept. 4— This post is now a page at finitegeometry.org.

Sunday, June 26, 2011

Paradigms Lost

Filed under: General,Geometry — m759 @ 7:20 AM

Continued from March 10, 2011 — A post that says

"If Galois geometry is thought of as a paradigm shift
from Euclidean geometry, both… the Kuhn cover
and the nine-point affine plane may be viewed…
as illustrating the shift."

Yesterday's posts The Fano Entity and Theology for Antichristmas,
together with this morning's New York Times  obituaries (below)—

http://www.log24.com/log/pix11A/110626-NYTobits.jpg

—suggest a Sunday School review from last year's
    Devil's Night (October 30-31, 2010)

Sunday, October 31, 2010

ART WARS

m759 @ 2:00 AM

                                …    There is a Cave
Within the Mount of God, fast by his Throne,
Where light and darkness in perpetual round
Lodge and dislodge by turns, which makes through Heav'n
Grateful vicissitude, like Day and Night….

Paradise Lost , by John Milton

http://www.log24.com/log/pix09A/091024-RayFigure.jpg

Click on figure for details.

http://www.log24.com/log/pix10B/101031-Pacino.jpg

Al Pacino in Devil's Advocate
as attorney John Milton

See also Ash Wednesday Surprise and Geometry for Jews.

Friday, April 22, 2011

Romancing the Hyperspace

Filed under: General,Geometry — m759 @ 7:59 PM

For the title, see Palm Sunday.

"There is a pleasantly discursive treatment of
Pontius Pilate's unanswered question 'What is truth?'" — H. S. M. Coxeter, 1987

From this date (April 22) last year—

Image-- examples from Galois affine geometry

Richard J. Trudeau in The Non-Euclidean Revolution , chapter on "Geometry and the Diamond Theory of Truth"–

"… Plato and Kant, and most of the philosophers and scientists in the 2200-year interval between them, did share the following general presumptions:

(1) Diamonds– informative, certain truths about the world– exist.
(2) The theorems of Euclidean geometry are diamonds.

Presumption (1) is what I referred to earlier as the 'Diamond Theory' of truth. It is far, far older than deductive geometry."

Trudeau's book was published in 1987. The non-Euclidean* figures above illustrate concepts from a 1976 monograph, also called "Diamond Theory."

Although non-Euclidean,* the theorems of the 1976 "Diamond Theory" are also, in Trudeau's terminology, diamonds.

* "Non-Euclidean" here means merely "other than  Euclidean." No violation of Euclid's parallel postulate is implied.

Trudeau comes to reject what he calls the "Diamond Theory" of truth. The trouble with his argument is the phrase "about the world."

Geometry, a part of pure mathematics, is not  about the world. See G. H. Hardy, A Mathematician's Apology .

Tuesday, March 29, 2011

Diamond Star

Filed under: General,Geometry — m759 @ 4:03 PM

From last night's note on finite geometry—

"The (83, 83) Möbius-Kantor configuration here described by Coxeter is of course part of the larger (94, 123) Hesse configuration. Simply add the center point of the 3×3 Galois affine plane and the four lines (1 horizontal, 1 vertical, 2 diagonal) through the center point." An illustration—

http://www.log24.com/log/pix11/110329-DiamondStar.jpg
This suggests a search for "diamond+star."

Friday, March 18, 2011

Defining Configurations*

Filed under: General,Geometry — m759 @ 7:00 PM

The On-Line Encyclopedia of Integer Sequences has an article titled "Number of combinatorial configurations of type (n_3)," by N.J.A. Sloane and D. Glynn.

From that article:

  • DEFINITION: A combinatorial configuration of type (n_3) consists of an (abstract) set of n points together with a set of n triples of points, called lines, such that each point belongs to 3 lines and each line contains 3 points.
  • EXAMPLE: The unique (8_3) configuration consists of the triples 125, 148, 167, 236, 278, 347, 358, 456.

The following corrects the word "unique" in the example.

http://www.log24.com/log/pix11/110320-MoebiusKantorConfig500w.jpg

* This post corrects an earlier post, also numbered 14660 and dated 7 PM March 18, 2011, that was in error.
   The correction was made at about 11:50 AM on March 20, 2011.

_____________________________________________________________

Update of March 21

The problem here is of course with the definition. Sloane and Glynn failed to include in their definition a condition that is common in other definitions of configurations, even abstract or purely "combinatorial" configurations. See, for instance, Configurations of Points and Lines , by Branko Grunbaum (American Mathematical Society, 2009), p. 17—

In the most general sense we shall consider combinatorial (or abstract) configurations; we shall use the term set-configurations as well. In this setting "points" are interpreted as any symbols (usually letters or integers), and "lines" are families of such symbols; "incidence" means that a "point" is an element of a "line". It follows that combinatorial configurations are special kinds of general incidence structures. Occasionally, in order to simplify and clarify the language, for "points" we shall use the term marks, and for "lines" we shall use blocks. The main property of geometric configurations that is preserved in the generalization to set-configurations (and that characterizes such configurations) is that two marks are incident with at most one block, and two blocks with at most one mark.

Whether or not omitting this "at most one" condition from the definition is aesthetically the best choice, it dramatically changes the number  of configurations in the resulting theory, as the above (8_3) examples show.

Update of March 22 (itself updated on March 25)

For further background on configurations, see Dolgachev—

http://www.log24.com/log/pix11/110322-DolgachevIntro.gif

Note that the two examples Dolgachev mentions here, with 16 points and 9 points, are not unrelated to the geometry of 4×4 and 3×3 square arrays. For the Kummer and related 16-point configurations, see section 10.3, "The Three Biplanes of Order 4," in Burkard Polster's A Geometrical Picture Book  (Springer, 1998). See also the 4×4 array described by Gordon Royle in an undated web page and in 1980 by Assmus and Sardi. For the Hesse configuration, see (for instance) the passage from Coxeter quoted in Quaternions in an Affine Galois Plane.

Update of March 27

See the above link to the (16,6) 4×4 array and the (16,6) exercises using this array in R.D. Carmichael's classic Introduction to the Theory of Groups of Finite Order  (1937), pp. 42-43. For a connection of this sort of 4×4 geometry to the geometry of the diamond theorem, read "The 2-subsets of a 6-set are the points of a PG(3,2)" (a note from 1986) in light of R.W.H.T. Hudson's 1905 classic Kummer's Quartic Surface , pages 8-9, 16-17, 44-45, 76-77, 78-79, and 80.

Thursday, March 10, 2011

Paradigms Lost

Filed under: General,Geometry — Tags: — m759 @ 5:48 PM

(Continued from February 19)

The cover of the April 1, 1970 second edition of The Structure of Scientific Revolutions , by Thomas S. Kuhn—

http://www.log24.com/log/pix11/110310-KuhnCover.jpg

This journal on January 19, 2011

IMAGE- A Galois cube: model of the 27-point affine 3-space

If Galois geometry is thought of as a paradigm shift from Euclidean geometry,
both images above— the Kuhn cover and the nine-point affine plane—
may be viewed, taken together, as illustrating the shift. The nine subcubes
of the Euclidean  3×3 cube on the Kuhn cover do not  form an affine plane
in the coordinate system of the Galois  cube in the second image, but they
at least suggest  such a plane. Similarly, transformations of a
non-mathematical object, the 1974 Rubik  cube, are not Galois  transformations,
but they at least suggest  such transformations.

See also today's online Harvard Crimson  illustration of problems of translation
not unrelated to the problems of commensurability  discussed by Kuhn.

http://www.log24.com/log/pix11/110310-CrimsonSm.jpg

Wednesday, January 19, 2011

Intermediate Cubism

Filed under: General,Geometry — Tags: — m759 @ 2:22 PM

The following is a new illustration for Cubist Geometries

IMAGE- A Galois cube: model of the 27-point affine 3-space

(For elementary cubism, see Pilate Goes to Kindergarten and The Eightfold Cube.
 For advanced, see Solomon's Cube and Geometry of the I Ching .)

Cézanne's Greetings.

Wednesday, November 17, 2010

Church Narrative

Filed under: General,Geometry — m759 @ 2:22 AM

Thanks to David Lavery for the following dialogue on the word "narrative" in politics—

"It's like – does this fit into narrative?
It's like, wait, wait, what about a platform? What about, like, ideas?
What about, you know, these truths we hold to be self-evident?
No, it's the narrative."

"Is narrative a fancy word for spin?"

Related material —

Church Logic (Log24, October 29) —

  What sort of geometry
    is the following?

IMAGE- The four-point, six-line geometry

 

"What about, you know, these truths we hold to be self-evident?"

Some background from Cambridge University Press in 1976 —

http://www.log24.com/log/pix10B/101117-CameronIntro2.jpg

Commentary —

The Church Logic post argues that Cameron's implicit definition of "non-Euclidean" is incorrect.

The four-point, six-line geometry has as lines "all subsets of the point set" which have cardinality 2.

It clearly satisfies Euclid's parallel postulate.  Is it, then, not  non-Euclidean?

That would, according to the principle of the excluded middle (cf. Church), make it Euclidean.

A definition from Wikipedia that is still essentially the same as it was when written on July 14, 2003

"Finite geometry describes any geometric system that has only a finite number of points. Euclidean geometry, for example, is not finite, because a Euclidean line contains infinitely many points…."

This definition would seem to imply that a finite geometry (such as the four-point geometry above) should be called non -Euclidean whether or not  it violates Euclid's parallel postulate. (The definition's author, unlike many at Wikipedia, is not  anonymous.)

See also the rest  of Little Gidding.

Friday, October 29, 2010

Church Logic

Filed under: General,Geometry — m759 @ 1:23 PM

"The law of excluded middle is the logical principle in
accordance with which every proposition is either true or
false. This principle is used, in particular, whenever a proof
is made by the method of reductio ad absurdum . And it is
this principle, also, which enables us to say that the denial of
the denial of a proposition is equivalent to the assertion of
the proposition."

Alonzo Church, "On the Law of Excluded Middle,"
    Bulletin of the American Mathematical Society ,
    Vol. 34, No. 1 (Jan.–Feb. 1928), pp. 75–78

It seems reasonable to define a Euclidean  geometry as one describing what mathematicians now call a Euclidean  space.

    What sort of geometry
    is the following?

http://www.log24.com/log/pix10B/101029-AffinePlane.bmp

   Four points and six lines,
   with parallel lines indicated
   by being colored alike.

Consider the proposition "The finite geometry with four points and six lines is non-Euclidean."
Consider its negation. Absurd? Of course.

"Non-Euclidean," therefore, does not apply only  to geometries that violate Euclid's parallel postulate.

The problem here is not with geometry, but with writings about  geometry.

A pop-science website

"In the plainest terms, non-Euclidean geometry
 took something that was rather simple and straightforward
 (Euclidean geometry) and made it endlessly more difficult."

Had the Greeks investigated finite  geometry before Euclid came along, the reverse would be true.

Friday, September 17, 2010

The Galois Window

Filed under: General,Geometry — Tags: , — m759 @ 5:01 AM

Yesterday's excerpt from von Balthasar supplies some Catholic aesthetic background for Galois geometry.

That approach will appeal to few mathematicians, so here is another.

Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace  is a book by Leonard Mlodinow published in 2002.

More recently, Mlodinow is the co-author, with Stephen Hawking, of The Grand Design  (published on September 7, 2010).

A review of Mlodinow's book on geometry—

"This is a shallow book on deep matters, about which the author knows next to nothing."
— Robert P. Langlands, Notices of the American Mathematical Society,  May 2002

The Langlands remark is an apt introduction to Mlodinow's more recent work.

It also applies to Martin Gardner's comments on Galois in 2007 and, posthumously, in 2010.

For the latter, see a Google search done this morning—

http://www.log24.com/log/pix10B/100917-GardnerGalois.jpg

Here, for future reference, is a copy of the current Google cache of this journal's "paged=4" page.

Note the link at the bottom of the page in the May 5, 2010, post to Peter J. Cameron's web journal. Following the link, we find…

For n=4, there is only one factorisation, which we can write concisely as 12|34, 13|24, 14|23. Its automorphism group is the symmetric group S4, and acts as S3 on the set of three partitions, as we saw last time; the group of strong automorphisms is the Klein group.

This example generalises, by taking the factorisation to consist of the parallel classes of lines in an affine space over GF(2). The automorphism group is the affine group, and the group of strong automorphisms is its translation subgroup.

See also, in this  journal, Window and Window, continued (July 5 and 6, 2010).

Gardner scoffs at the importance of Galois's last letter —

"Galois had written several articles on group theory, and was
  merely annotating and correcting those earlier published papers."
Last Recreations, page 156

For refutations, see the Bulletin of the American Mathematical Society  in March 1899 and February 1909.

Friday, August 20, 2010

The Moore Correspondence

Filed under: General,Geometry — m759 @ 5:01 PM

There is a remarkable correspondence between the 35 partitions of an eight-element set H into two four-element sets and the 35 partitions of the affine 4-space L over GF(2) into four parallel four-point planes. Under this correspondence, two of the H-partitions have a common refinement into 2-sets if and only if the same is true of the corresponding L-partitions (Peter J. Cameron, Parallelisms of Complete Designs, Cambridge U. Press, 1976, p. 60). The correspondence underlies the isomorphism* of the group A8 with the projective general linear group PGL(4,2) and plays an important role in the structure of the large Mathieu group M24.

A 1954 paper by W.L. Edge suggests the correspondence should be named after E.H. Moore. Hence the title of this note.

Edge says that

It is natural to ask what, if any, are the 8 objects which undergo
permutation. This question was discussed at length by Moore…**.
But, while there is no thought either of controverting Moore's claim to
have answered it or of disputing his priority, the question is primarily
a geometrical one….

Excerpts from the Edge paper—

http://www.log24.com/log/pix10B/100820-Edge-Geometry-1col.gif

Excerpts from the Moore paper—

Pages 432, 433, 434, and 435, as well as the section mentioned above by Edge— pp. 438 and 439

* J.W.P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Oxford U. Press, 1985, p. 72

** Edge cited "E.H. Moore, Math. Annalen, 51 (1899), 417-44." A more complete citation from "The Scientific Work of Eliakim Hastings Moore," by G.A. Bliss,  Bull. Amer. Math. Soc. Volume 40, Number 7 (1934), 501-514— E.H. Moore, "Concerning the General Equations of the Seventh and Eighth Degrees," Annalen, vol. 51 (1899), pp. 417-444.

Saturday, August 7, 2010

The Matrix Reloaded

Filed under: General,Geometry — m759 @ 12:00 AM

   For aficionados of mathematics and narrative

Illustration from
"The Galois Quaternion— A Story"

The Galois Quaternion

This resembles an attempt by Coxeter in 1950 to represent
a Galois geometry in the Euclidean plane—
Coxeter's 1950 representation in the Euclidean plane of the 9-point affine plane over GF(3)

The quaternion illustration above shows a more natural way to picture this geometry—
not with dots representing points in the Euclidean  plane, but rather with unit squares
representing points in a finite Galois  affine plane. The use of unit squares to
represent points in Galois space allows, in at least some cases, the actions
of finite groups to be represented more naturally than in Euclidean space.

See Galois Geometry, Geometry Simplified, and
Finite Geometry of the Square and Cube.

Thursday, July 15, 2010

Brightness at Noon, continued

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

"What exactly was Point Omega?"

This is Robert Wright in Nonzero: The Logic of Human Destiny.

Wright is discussing not the novel Point Omega  by Don DeLillo,
but rather a (related) concept of  the Jesuit philosopher Pierre Teilhard de Chardin.

My own idiosyncratic version of a personal "point omega"—

Image- Josefine Lyche work (with 1986 figures by Cullinane) in a 2009 exhibition in Oslo

Click for further details.

The circular sculpture in the foreground
is called by the artist "The Omega Point."
This has been described as
"a portal that leads in or out of time and space."

For some other sorts of points, see the drawings
on the wall and Geometry Simplified

Image-- The trivial two-point affine space and the trivial one-point projective space, visualized

The two points of the trivial affine space are represented by squares,
and the one point of the trivial projective space is represented by
a line segment separating the affine-space squares.

For related darkness  at noon, see Derrida on différance
as a version of Plato's khôra

(Click to enlarge.)

Image-- Fordham University Press on Derrida, differance, and khora

The above excerpts are from a work on and by Derrida
published in 1997 by Fordham University,
a Jesuit institutionDeconstruction in a Nutshell

Image-- A Catholic view of Derrida

For an alternative to the Villanova view of Derrida,
see Angels in the Architecture.

Tuesday, July 6, 2010

Window, continued

Filed under: General,Geometry — Tags: , — m759 @ 10:31 AM

"Simplicity, simplicity, simplicity!  I say, let your affairs
be as two or three,
and not a hundred or a thousand;
instead of a million count half a dozen,
and keep your accounts on your thumb-nail."
— Henry David Thoreau, Walden

This quotation is the epigraph to Section 1.1 of
Alexandre V. Borovik's
Mathematics Under the Microscope:

Notes on Cognitive Aspects of Mathematical Practice
(American Mathematical Society, Jan. 15, 2010, 317 pages).

From Peter J. Cameron's review notes for
his new course in group theory

http://www.log24.com/log/pix10A/100705-CameronExample.jpg

From Log24 on June 24

Geometry Simplified

Image-- The Four-Point Plane: A Finite Affine Space
(an affine  space with subsquares as points
and sets  of subsquares as hyperplanes)

Image-- The Three-Point Line: A Finite Projective Space
(a projective  space with, as points, sets
  of line segments that separate subsquares)

Exercise

Show that the above geometry is a model
for the algebra discussed by Cameron.

Thursday, June 24, 2010

Midsummer Noon

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

Geometry Simplified

Image-- The Three-Point Line: A Finite Projective Space
(a projective space)

The above finite projective space
is the simplest nontrivial example
of a Galois geometry (i.e., a finite
geometry with coordinates in a
finite (that is, Galois) field.)

The vertical (Euclidean) line represents a
 (Galois) point, as does the horizontal line
and also the vertical-and-horizontal
cross that represents the first two points'
binary sum (i.e., symmetric difference,
if the lines are regarded as sets).

Homogeneous coordinates for the
points of this line —

(1,0), (0,1), (1,1).

Here 0 and 1 stand for the elements
of the two-element Galois field GF(2).

The 3-point line is the projective space
corresponding to the affine space
(a plane, not a line) with four points —

http://www.log24.com/log/pix10A/100624-The4PointPlane.bmp
(an affine space)

The (Galois) points of this affine plane are
  not the single and combined (Euclidean)
line segments that play the role of
  points in the 3-point projective line,
but rather the four subsquares
that the line segments separate.

For further details, see Galois Geometry.

There are, of course, also the trivial
two-point affine space and the corresponding
trivial one-point projective space —

http://www.log24.com/log/pix10A/100624-TrivialSpaces.bmp

Here again, the points of the affine space are
represented by squares, and the point of the
projective space is represented by a line segment
separating the affine-space squares.

Monday, June 21, 2010

Cube Spaces

Filed under: General,Geometry — Tags: — m759 @ 11:30 AM

Cubic models of finite geometries
display an interplay between
Euclidean and Galois geometry.

Example 1— The 2×2×2 Cube—

also known as the eightfold  cube

2x2x2 cube

Group actions on the eightfold cube, 1984—

http://www.log24.com/log/pix10A/100621-diandwh-detail.GIF

Version by Laszlo Lovasz et al., 2003—

http://www.log24.com/log/pix10A/100621-LovaszCubeSpace.gif

Lovasz et al. go on to describe the same group actions
as in the 1984 note, without attribution.
 

Example 2— The 3×3×3 Cube

A note from 1985 describing group actions on a 3×3 plane array—

http://www.log24.com/log/pix10A/100621-VisualizingDetail.gif

Undated software by Ed Pegg Jr. displays
group actions on a 3×3×3 cube that extend the
  3×3 group actions from 1985 described above—

Ed Pegg Jr.'s program at Wolfram demonstrating concepts of a 1985 
note by Cullinane

Pegg gives no reference to the 1985 work on group actions.
 

Example 3— The 4×4×4 Cube

A note from 27 years ago today—

http://www.log24.com/log/pix10A/100621-Cube830621.gif

As far as I know, this version of the
group-actions theorem has not yet been ripped off.

Wednesday, June 16, 2010

Geometry of Language

Filed under: General,Geometry — m759 @ 10:31 AM

(Continued from April 23, 2009, and February 13, 2010.)

Paul Valéry as quoted in yesterday’s post:

“The S[elf] is invariant, origin, locus or field, it’s a functional property of consciousness” (Cahiers, 15:170 [2: 315])

The geometric example discussed here yesterday as a Self symbol may seem too small to be really impressive. Here is a larger example from the Chinese, rather than European, tradition. It may be regarded as a way of representing the Galois field GF(64). (“Field” is a rather ambiguous term; here it does not, of course, mean what it did in the Valéry quotation.)

From Geometry of the I Ching

Image-- The 64 hexagrams of the I Ching

The above 64 hexagrams may also be regarded as
the finite affine space AG(6,2)— a larger version
of the finite affine space AG(4,2) in yesterday’s post.
That smaller space has a group of 322,560 symmetries.
The larger hexagram  space has a group of
1,290,157,424,640 affine symmetries.

From a paper on GL(6,2), the symmetry group
of the corresponding projective  space PG(5,2),*
which has 1/64 as many symmetries—

(Click to enlarge.)

Image-- Classes of the Group GL(6,)

For some narrative in the European  tradition
related to this geometry, see Solomon’s Cube.

* Update of July 29, 2011: The “PG(5,2)” above is a correction from an earlier error.

Wednesday, May 5, 2010

Symmetry and Parallelisms

Filed under: General,Geometry — m759 @ 10:00 AM

From a post of Peter J. Cameron today —

"… I want to consider the question: What is the role of the symmetric group in mathematics? "

Cameron's examples include, notably, parallelisms of lines in affine spaces over GF(2).

Monday, April 26, 2010

Types of Ambiguity

Filed under: General,Geometry — Tags: — m759 @ 10:31 AM

From Ursula K. Le Guin’s novel
The Dispossessed: An Ambiguous Utopia
(1974)—

Chapter One

“There was a wall. It did not look important. It was built of uncut rocks roughly mortared. An adult could look right over it, and even a child could climb it. Where it crossed the roadway, instead of having a gate it degenerated into mere geometry, a line, an idea of boundary. But the idea was real. It was important. For seven generations there had been nothing in the world more important than that wall.

Like all walls it was ambiguous, two-faced. What was inside it and what was outside it depended upon which side of it you were on.”

Note—

“We note that the phrase ‘instead of having a gate it degenerated into mere geometry’ is mere fatuousness. If there is an idea here, degenerate, mere, and geometry  in concert do not fix it. They bat at it like a kitten at a piece of loose thread.”

— Samuel R. Delany, The Jewel-Hinged Jaw: Notes on the Language of Science Fiction  (Dragon Press, 1977), page 110 of revised edition, Wesleyan University Press, 2009

(For the phrase mere geometry  elsewhere, see a note of April 22. The apparently flat figures in that note’s illustration “Galois Affine Geometry” may be regarded as degenerate  views of cubes.)

Later in the Le Guin novel—

“… The Terrans had been intellectual imperialists, jealous wall builders. Even Ainsetain, the originator of the theory, had felt compelled to give warning that his physics embraced no mode but the physical and should not be taken as implying the metaphysical, the philosophical, or the ethical. Which, of course, was superficially true; and yet he had used number, the bridge between the rational and the perceived, between psyche and matter, ‘Number the Indisputable,’ as the ancient founders of the Noble Science had called it. To employ mathematics in this sense was to employ the mode that preceded and led to all other modes. Ainsetain had known that; with endearing caution he had admitted that he believed his physics did, indeed, describe reality.

Strangeness and familiarity: in every movement of the Terran’s thought Shevek caught this combination, was constantly intrigued. And sympathetic: for Ainsetain, too, had been after a unifying field theory. Having explained the force of gravity as a function of the geometry of spacetime, he had sought to extend the synthesis to include electromagnetic forces. He had not succeeded. Even during his lifetime, and for many decades after his death, the physicists of his own world had turned away from his effort and its failure, pursuing the magnificent incoherences of quantum theory with its high technological yields, at last concentrating on the technological mode so exclusively as to arrive at a dead end, a catastrophic failure of imagination. Yet their original intuition had been sound: at the point where they had been, progress had lain in the indeterminacy which old Ainsetain had refused to accept. And his refusal had been equally correct– in the long run. Only he had lacked the tools to prove it– the Saeba variables and the theories of infinite velocity and complex cause. His unified field existed, in Cetian physics, but it existed on terms which he might not have been willing to accept; for the velocity of light as a limiting factor had been essential to his great theories. Both his Theories of Relativity were as beautiful, as valid, and as useful as ever after these centuries, and yet both depended upon a hypothesis that could not be proved true and that could be and had been proved, in certain circumstances, false.

But was not a theory of which all the elements were provably true a simple tautology? In the region of the unprovable, or even the disprovable, lay the only chance for breaking out of the circle and going ahead.

In which case, did the unprovability of the hypothesis of real coexistence– the problem which Shevek had been pounding his head against desperately for these last three days. and indeed these last ten years– really matter?

He had been groping and grabbing after certainty, as if it were something he could possess. He had been demanding a security, a guarantee, which is not granted, and which, if granted, would become a prison. By simply assuming the validity of real coexistence he was left free to use the lovely geometries of relativity; and then it would be possible to go ahead. The next step was perfectly clear. The coexistence of succession could be handled by a Saeban transformation series; thus approached, successivity and presence offered no antithesis at all. The fundamental unity of the Sequency and Simultaneity points of view became plain; the concept of interval served to connect the static and the dynamic aspect of the universe. How could he have stared at reality for ten years and not seen it? There would be no trouble at all in going on. Indeed he had already gone on. He was there. He saw all that was to come in this first, seemingly casual glimpse of the method, given him by his understanding of a failure in the distant past. The wall was down. The vision was both clear and whole. What he saw was simple, simpler than anything else. It was simplicity: and contained in it all complexity, all promise. It was revelation. It was the way clear, the way home, the light.”

Related material—

Time Fold, Halloween 2005, and May and Zan.

See also The Devil and Wallace Stevens

“In a letter to Harriet Monroe, written December 23, 1926, Stevens refers to the Sapphic fragment that invokes the genius of evening: ‘Evening star that bringest back all that lightsome Dawn hath scattered afar, thou bringest the sheep, thou bringest the goat, thou bringest the child home to the mother.’ Christmas, writes Stevens, ‘is like Sappho’s evening: it brings us all home to the fold’ (Letters of Wallace Stevens, 248).”

— “The Archangel of Evening,” Chapter 5 of Wallace Stevens: The Intensest Rendezvous, by Barbara M. Fisher, The University Press of Virginia, 1990

Thursday, April 22, 2010

Mere Geometry

Filed under: General,Geometry — Tags: — m759 @ 1:00 PM

Image-- semeion estin ou meros outhen

Image-- Euclid's definition of 'point'

Stanford Encyclopedia of Philosophy

Mereology (from the Greek μερος, ‘part’) is the theory of parthood relations: of the relations of part to whole and the relations of part to part within a whole. Its roots can be traced back to the early days of philosophy, beginning with the Presocratics….”

A non-Euclidean* approach to parts–

Image-- examples from Galois affine geometry

Corresponding non-Euclidean*
projective points —

Image-- The smallest Galois geometries

Richard J. Trudeau in The Non-Euclidean Revolution, chapter on “Geometry and the Diamond Theory of Truth”–

“… Plato and Kant, and most of the philosophers and scientists in the 2200-year interval between them, did share the following general presumptions:

(1) Diamonds– informative, certain truths about the world– exist.
(2) The theorems of Euclidean geometry are diamonds.

Presumption (1) is what I referred to earlier as the ‘Diamond Theory’ of truth. It is far, far older than deductive geometry.”

Trudeau’s book was published in 1987. The non-Euclidean* figures above illustrate concepts from a 1976 monograph, also called “Diamond Theory.”

Although non-Euclidean,* the theorems of the 1976 “Diamond Theory” are also, in Trudeau’s terminology, diamonds.

* “Non-Euclidean” here means merely “other than  Euclidean.” No violation of Euclid’s parallel postulate is implied.

Saturday, February 27, 2010

Cubist Geometries

Filed under: General,Geometry — Tags: , — m759 @ 2:01 PM

"The cube has…13 axes of symmetry:
  6 C2 (axes joining midpoints of opposite edges),
4 C3 (space diagonals), and
3C4 (axes joining opposite face centroids)."
–Wolfram MathWorld article on the cube

These 13 symmetry axes can be used to illustrate the interplay between Euclidean and Galois geometry in a cubic model of the 13-point Galois plane.

The geometer's 3×3×3 cube–
27 separate subcubes unconnected
by any Rubik-like mechanism–

The 3x3x3 geometer's cube, with coordinates

The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
  subcubes in the (Galois) 3×3×3 cube–

The 13 symmetry axes of the cube

A closely related structure–
the finite projective plane
with 13 points and 13 lines–

Oxley's 2004 drawing of the 13-point projective plane

A later version of the 13-point plane
by Ed Pegg Jr.–

Ed Pegg Jr.'s 2007 drawing of the 13-point projective plane

A group action on the 3×3×3 cube
as illustrated by a Wolfram program
by Ed Pegg Jr. (undated, but closely
related to a March 26, 1985 note
by Steven H. Cullinane)–

Ed Pegg Jr.'s program at Wolfram demonstrating concepts of a 1985 note by Cullinane

The above images tell a story of sorts.
The moral of the story–

Galois projective geometries can be viewed
in the context of the larger affine geometries
from which they are derived.

The standard definition of points in a Galois projective plane is that they are lines through the (arbitrarily chosen) origin in a corresponding affine 3-space converted to a vector 3-space.

If we choose the origin as the center cube in coordinatizing the 3×3×3 cube (See Weyl's relativity problem ), then the cube's 13 axes of symmetry can, if the other 26 cubes have properly (Weyl's "objectively") chosen coordinates, illustrate nicely the 13 projective points derived from the 27 affine points in the cube model.

The 13 lines of the resulting Galois projective plane may be derived from Euclidean planes  through the cube's center point that are perpendicular to the cube's 13 Euclidean symmetry axes.

The above standard definition of points in a Galois projective plane may of course also be used in a simpler structure– the eightfold cube.

(The eightfold cube also allows a less standard way to picture projective points that is related to the symmetries of "diamond" patterns formed by group actions on graphic designs.)

See also Ed Pegg Jr. on finite geometry on May 30, 2006
at the Mathematical Association of America.

Saturday, November 14, 2009

Mathematics and Narrative, continued:

Filed under: General,Geometry — Tags: , — m759 @ 10:10 PM

A graphic novel reviewed in the current Washington Post  features Alfred North Whitehead and Bertrand Russell–

Whitehead and Russell, 'Logicomix' page 181

Related material:

Whitehead on Fano’s finite projective three-space:

“This is proved by the consideration of a three dimensional geometry in which there are only fifteen points.”

The Axioms of Projective Geometry , Cambridge University Press, 1906

A related affine six-space:

Grey cube, 4x4x4

Further reading:

See Solomon’s Cube and the link at the end of today’s previous entry, then compare and contrast the above portraits of Whitehead and Russell with Charles Williams’s portraits of Sir Giles Tumulty and Lord Arglay in the novel Many Dimensions .

It was a dark and stormy night….

Tuesday, February 24, 2009

Tuesday February 24, 2009

Filed under: General,Geometry — Tags: — m759 @ 1:00 PM
 
Hollywood Nihilism
Meets
Pantheistic Solipsism

Tina Fey to Steve Martin
at the Oscars:
"Oh, Steve, no one wants
 to hear about our religion
… that we made up."

Tina Fey and Steve Martin at the 2009 Oscars

From Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, p. 117:

… in 'The Pediment of Appearance,' a slight narrative poem in Transport to Summer

 A group of young men enter some woods 'Hunting for the great ornament, The pediment of appearance.' Though moving through the natural world, the young men seek the artificial, or pure form, believing that in discovering this pediment, this distillation of the real, they will also discover the 'savage transparence,' the rude source of human life. In Stevens's world, such a search is futile, since it is only through observing nature that one reaches beyond it to pure form. As if to demonstrate the degree to which the young men's search is misaligned, Stevens says of them that 'they go crying/The world is myself, life is myself,' believing that what surrounds them is immaterial. Such a proclamation is a cardinal violation of Stevens's principles of the imagination.


Superficially the young men's philosophy seems to resemble what Wikipedia calls "pantheistic solipsism"– noting, however, that "This article has multiple issues."

As, indeed, does pantheistic solipsism– a philosophy (properly called "eschatological pantheistic multiple-ego solipsism") devised, with tongue in cheek, by science-fiction writer Robert A. Heinlein.

Despite their preoccupation with solipsism, Heinlein and Stevens point, each in his own poetic way, to a highly non-solipsistic topic from pure mathematics that is, unlike the religion of Martin and Fey, not made up– namely, the properties of space.

Heinlein:

"Sharpie, we have condensed six dimensions into four, then we either work by analogy into six, or we have to use math that apparently nobody but Jake and my cousin Ed understands. Unless you can think of some way to project six dimensions into three– you seem to be smart at such projections."
    I closed my eyes and thought hard. "Zebbie, I don't think it can be done. Maybe Escher could have done it."

Stevens:

A discussion of Stevens's late poem "The Rock" (1954) in Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, p. 120:

For Stevens, the poem "makes meanings of the rock." In the mind, "its barrenness becomes a thousand things/And so exists no more." In fact, in a peculiar irony that only a poet with Stevens's particular notion of the imagination's function could develop, the rock becomes the mind itself, shattered into such diamond-faceted brilliance that it encompasses all possibilities for human thought:

The rock is the gray particular of man's life,
The stone from which he rises, up—and—ho,
The step to the bleaker depths of his descents ...

The rock is the stern particular of the air,
The mirror of the planets, one by one,
But through man's eye, their silent rhapsodist,

Turquoise the rock, at odious evening bright
With redness that sticks fast to evil dreams;
The difficult rightness of half-risen day.

The rock is the habitation of the whole,
Its strength and measure, that which is near,
     point A
In a perspective that begins again

At B: the origin of the mango's rind.

                    (Collected Poems, 528)

Stevens's rock is associated with empty space, a concept that suggests "nothingness" to one literary critic:

B. J. Leggett, "Stevens's Late Poetry" in The Cambridge Companion to Wallace Stevens— On the poem "The Rock":

"… the barren rock of the title is Stevens's symbol for the nothingness that underlies all existence, 'That in which space itself is contained'….  Its subject is its speaker's sense of nothingness and his need to be cured of it."

This interpretation might appeal to Joan Didion, who, as author of the classic novel Play It As It Lays, is perhaps the world's leading expert on Hollywood nihilism.

More positively…

Space is, of course, also a topic
in pure mathematics…
For instance, the 6-dimensional
affine space
(or the corresponding
5-dimensional projective space)

The 4x4x4 cube

over the two-element Galois field
can be viewed as an illustration of
Stevens's metaphor in "The Rock."

Heinlein should perhaps have had in mind the Klein correspondence when he discussed "some way to project six dimensions into three." While such a projection is of course trivial for anyone who has taken an undergraduate course in linear algebra, the following remarks by Philippe Cara present a much more meaningful mapping, using the Klein correspondence, of structures in six (affine) dimensions to structures in three.

Cara:

Philippe Cara on the Klein correspondence
Here the 6-dimensional affine
space contains the 63 points
of PG(5, 2), plus the origin, and
the 3-dimensional affine
space contains as its 8 points
Conwell's eight "heptads," as in
Generating the Octad Generator.

Thursday, February 5, 2009

Thursday February 5, 2009

Filed under: General,Geometry — Tags: , , — m759 @ 1:00 PM

Through the
Looking Glass:

A Sort of Eternity

From the new president's inaugural address:

"… in the words of Scripture, the time has come to set aside childish things."

The words of Scripture:

9 For we know in part, and we prophesy in part.
10 But when that which is perfect is come, then that which is in part shall be done away.
11 When I was a child, I spake as a child, I understood as a child, I thought as a child: but when I became a man, I put away childish things.
12 For now we see through a glass, darkly, but then face to face: now I know in part; but then shall I know even as also I am known.

 

First Corinthians 13

"through a glass"

[di’ esoptrou].
By means of
a mirror [esoptron]
.

Childish things:

Froebel's third gift, the eightfold cube
© 2005 The Institute for Figuring
 
Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)

 

Not-so-childish:

Three planes through
the center of a cube
that split it into
eight subcubes:
Cube subdivided into 8 subcubes by planes through the center
Through a glass, darkly:

A group of 8 transformations is
generated by affine reflections
in the above three planes.
Shown below is a pattern on
the faces of the 2x2x2 cube
 that is symmetric under one of
these 8 transformations–
a 180-degree rotation:

Design Cube 2x2x2 for demonstrating Galois geometry

(Click on image
for further details.)

But then face to face:

A larger group of 1344,
rather than 8, transformations
of the 2x2x2 cube
is generated by a different
sort of affine reflections– not
in the infinite Euclidean 3-space
over the field of real numbers,
but rather in the finite Galois
3-space over the 2-element field.

Galois age fifteen, drawn by a classmate.

Galois age fifteen,
drawn by a classmate.

These transformations
in the Galois space with
finitely many points
produce a set of 168 patterns
like the one above.
For each such pattern,
at least one nontrivial
transformation in the group of 8
described above is a symmetry
in the Euclidean space with
infinitely many points.

For some generalizations,
see Galois Geometry.

Related material:

The central aim of Western religion–

 

"Each of us has something to offer the Creator...
the bridging of
 masculine and feminine,
 life and death.
It's redemption.... nothing else matters."
-- Martha Cooley in The Archivist (1998)

The central aim of Western philosophy–

 Dualities of Pythagoras
 as reconstructed by Aristotle:
  Limited Unlimited
  Odd Even
  Male Female
  Light Dark
  Straight Curved
  ... and so on ....

"Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy."

— Jamie James in The Music of the Spheres (1993)

"In the garden of Adding
live Even and Odd…
And the song of love's recision
is the music of the spheres."

— The Midrash Jazz Quartet in City of God, by E. L. Doctorow (2000)

A quotation today at art critic Carol Kino's website, slightly expanded:

"Art inherited from the old religion
the power of consecrating things
and endowing them with
a sort of eternity;
museums are our temples,
and the objects displayed in them
are beyond history."

— Octavio Paz,"Seeing and Using: Art and Craftsmanship," in Convergences: Essays on Art and Literature (New York: Harcourt Brace Jovanovich 1987), 52 

From Brian O'Doherty's 1976 Artforum essays– not on museums, but rather on gallery space:

"Inside the White Cube"

"We have now reached
a point where we see
not the art but the space first….
An image comes to mind
of a white, ideal space
that, more than any single picture,
may be the archetypal image
of 20th-century art."

http://www.log24.com/log/pix09/090205-cube2x2x2.gif

"Space: what you
damn well have to see."

— James Joyce, Ulysses  

Monday, January 5, 2009

Monday January 5, 2009

Filed under: General,Geometry — Tags: — m759 @ 9:00 PM
A Wealth of
Algebraic Structure

A 4x4 array (part of chessboard)

A 1987 article by R. T. Curtis on the geometry of his Miracle Octad Generator (MOG) as it relates to the geometry of the 4×4 square is now available online ($20):

Further elementary techniques using the miracle octad generator
, by R. T. Curtis. Abstract:

"In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an 'octad generator'; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code."

 

(Received July 20 1987)

Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353, doi:10.1017/S0013091500004600.

(Published online by Cambridge University Press 19 Dec 2008.)

In the above article, Curtis explains how two-thirds of his 4×6 MOG array may be viewed as the 4×4 model of the four-dimensional affine space over GF(2).  (His earlier 1974 paper (below) defining the MOG discussed the 4×4 structure in a purely combinatorial, not geometric, way.)

For further details, see The Miracle Octad Generator as well as Geometry of the 4×4 Square and Curtis's original 1974 article, which is now also available online ($20):

A new combinatorial approach to M24, by R. T. Curtis. Abstract:

"In this paper, we define M24 from scratch as the subgroup of S24 preserving a Steiner system S(5, 8, 24). The Steiner system is produced and proved to be unique and the group emerges naturally with many of its properties apparent."

 

(Received June 15 1974)

Mathematical Proceedings of the Cambridge Philosophical Society (1976), 79: 25-42, doi:10.1017/S0305004100052075.

(Published online by Cambridge University Press 24 Oct 2008.)
 

* For instance:

Algebraic structure in the 4x4 square, by Cullinane (1985) and Curtis (1987)

Click for details.
 

Sunday, December 21, 2008

Sunday December 21, 2008

Filed under: General,Geometry — m759 @ 1:06 PM
Interpretive Grids

Projective points as grids interpreting the structure of an affine space

The 15 grids in the picture at right above may be regarded as interpreting the structure of the space at left above.

This pair of pictures was suggested by yesterday’s entry at Ars Mathematica containing the phrase “a dramatic extension of the notion of points.”

For other uses of the phrase “interpretive grid,” see today’s previous entry.

Saturday, December 6, 2008

Saturday December 6, 2008

Filed under: General,Geometry — m759 @ 2:01 PM
Another Opening,
Another Show

“While feasts of Saint Nicholas are not observed nationally, cities with strong German influences like Milwaukee, Cincinnati, and St. Louis celebrate St. Nick’s Day on a scale similar to the German custom.” —Wikipedia

A footprint from Germany:

Germany
Python-urllib
/504856559/item.html 12/6/2008
1:21 PM

The link in the above footprint leads
to an entry of July 5, 2006.

The access method:

The urllib Module

“The Python urllib module implements a fairly high-level abstraction for making any web object with a URL act like a Python file: i.e., you open it, and get back an object….”

For more pictures and discussion
of the object fetched by Python,
see Anti-Christmas 2007.

For a larger and more sophisticated
relative of that object,
 see Solomon’s Cube and
the related three presents
from the German link’s target:

Spellbound: A trinity of Christmas presents

1. Many Dimensions
2. Boggle
3. My Space

Friday, October 24, 2008

Friday October 24, 2008

Filed under: General,Geometry — m759 @ 8:08 AM
The Cube Space” is a name given to the eightfold cube in a vulgarized mathematics text, Discrete Mathematics: Elementary and Beyond, by Laszlo Lovasz et al., published by Springer in 2003. The identification in a natural way of the eight points of the linear 3-space over the 2-element field GF(2) with the eight vertices of a cube is an elementary and rather obvious construction, doubtless found in a number of discussions of discrete mathematics. But the less-obvious generation of the affine group AGL(3,2) of order 1344 by permutations of parallel edges in such a cube may (or may not) have originated with me. For descriptions of this process I wrote in 1984, see Diamonds and Whirls and Binary Coordinate Systems. For a vulgarized description of this process by Lovasz, without any acknowledgement of his sources, see an excerpt from his book.

Saturday, August 16, 2008

Saturday August 16, 2008

Filed under: General,Geometry — m759 @ 8:00 AM

Seeing the Finite Structure

The following supplies some context for remarks of Halmos on combinatorics.

From Paul Halmos: Celebrating 50 years of Mathematics, by John H. Ewing, Paul Richard Halmos, Frederick W. Gehring, published by Springer, 1991–

Interviews with Halmos, “Paul Halmos by Parts,” by Donald J. Albers–

“Part II: In Touch with God*“– on pp. 27-28:

The Root of All Deep Mathematics

Albers. In the conclusion of ‘Fifty Years of Linear Algebra,’ you wrote: ‘I am inclined to believe that at the root of all deep mathematics there is a combinatorial insight… I think that in this subject (in every subject?) the really original, really deep insights are always combinatorial, and I think for the new discoveries that we need– the pendulum needs– to swing back, and will swing back in the combinatorial direction.’ I always thought of you as an analyst.

Halmos: People call me an analyst, but I think I’m a born algebraist, and I mean the same thing, analytic versus combinatorial-algebraic. I think the finite case illustrates and guides and simplifies the infinite.

Some people called me full of baloney when I asserted that the deep problems of operator theory could all be solved if we knew the answer to every finite dimensional matrix question. I still have this religion that if you knew the answer to every matrix question, somehow you could answer every operator question. But the ‘somehow’ would require genius. The problem is not, given an operator question, to ask the same question in finite dimensions– that’s silly. The problem is– the genius is– given an infinite question, to think of the right finite question to ask. Once you thought of the finite answer, then you would know the right answer to the infinite question.

Combinatorics, the finite case, is where the genuine, deep insight is. Generalizing, making it infinite, is sometimes intricate and sometimes difficult, and I might even be willing to say that it’s sometimes deep, but it is nowhere near as fundamental as seeing the finite structure.”

Finite Structure
 on a Book Cover:

Walsh Series: An Introduction to Dyadic Harmonic Analysis, by F. Schipp et. al.

Walsh Series: An Introduction
to Dyadic Harmonic Analysis
,
by F. Schipp et al.,
Taylor & Francis, 1990

Halmos’s above remarks on combinatorics as a source of “deep mathematics” were in the context of operator theory. For connections between operator theory and harmonic analysis, see (for instance) H.S. Shapiro, “Operator Theory and Harmonic Analysis,” pp. 31-56 in Twentieth Century Harmonic Analysis– A Celebration, ed. by J.S. Byrnes, published by Springer, 2001.


Walsh Series
states that Walsh functions provide “the simplest non-trivial model for harmonic analysis.”

The patterns on the faces of the cube on the cover of Walsh Series above illustrate both the Walsh functions of order 3 and the same structure in a different guise, subspaces of the affine 3-space over the binary field. For a note on the relationship of Walsh functions to finite geometry, see Symmetry of Walsh Functions.

Whether the above sketch of the passage from operator theory to harmonic analysis to Walsh functions to finite geometry can ever help find “the right finite question to ask,” I do not know. It at least suggests that finite geometry (and my own work on models in finite geometry) may not be completely irrelevant to mathematics generally regarded as more deep.

* See the Log24 entries following Halmos’s death.

Monday, August 11, 2008

Monday August 11, 2008

Filed under: General,Geometry — m759 @ 9:00 PM
 New Illustration
for the four-color
decomposition theorem:

Four-color decompostion applied to the 8-point binary affine space

Sunday, August 3, 2008

Sunday August 3, 2008

Filed under: General,Geometry — Tags: , — m759 @ 3:00 PM
Kindergarten
Geometry

Preview of a Tom Stoppard play presented at Town Hall in Manhattan on March 14, 2008 (Pi Day and Einstein's birthday):

The play's title, "Every Good Boy Deserves Favour," is a mnemonic for the notes of the treble clef EGBDF.

The place, Town Hall, West 43rd Street. The time, 8 p.m., Friday, March 14. One single performance only, to the tinkle– or the clang?– of a triangle. Echoing perhaps the clang-clack of Warsaw Pact tanks muscling into Prague in August 1968.

The “u” in favour is the British way, the Stoppard way, "EGBDF" being "a Play for Actors and Orchestra" by Tom Stoppard (words) and André Previn (music).

And what a play!– as luminescent as always where Stoppard is concerned. The music component of the one-nighter at Town Hall– a showcase for the Boston University College of Fine Arts– is by a 47-piece live orchestra, the significant instrument being, well, a triangle.

When, in 1974, André Previn, then principal conductor of the London Symphony, invited Stoppard "to write something which had the need of a live full-time orchestra onstage," the 36-year-old playwright jumped at the chance.

One hitch: Stoppard at the time knew "very little about 'serious' music… My qualifications for writing about an orchestra," he says in his introduction to the 1978 Grove Press edition of "EGBDF," "amounted to a spell as a triangle player in a kindergarten percussion band."

Jerry Tallmer in The Villager, March 12-18, 2008

Review of the same play as presented at Chautauqua Institution on July 24, 2008:

"Stoppard's modus operandi– to teasingly introduce numerous clever tidbits designed to challenge the audience."

Jane Vranish, Pittsburgh Post-Gazette, Saturday, August 2, 2008

"The leader of the band is tired
And his eyes are growing old
But his blood runs through
My instrument
And his song is in my soul."

— Dan Fogelberg

"He's watching us all the time."

Lucia Joyce

 

Finnegans Wake,
Book II, Episode 2, pp. 296-297:

I'll make you to see figuratleavely the whome of your eternal geomater. And if you flung her headdress on her from under her highlows you'd wheeze whyse Salmonson set his seel on a hexengown.1 Hissss!, Arrah, go on! Fin for fun!

1 The chape of Doña Speranza of the Nacion.

 

Log 24, Sept. 3, 2003:
 
Reciprocity

From my entry of Sept. 1, 2003:

"…the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity….

… E. M. Forster famously advised his readers, 'Only connect.' 'Reciprocity' would be Michael Kruger's succinct philosophy, with all that the word implies."

— William Boyd, review of Himmelfarb, a novel by Michael Kruger, in The New York Times Book Review, October 30, 1994

Last year's entry on this date: 

 

Today's birthday:
James Joseph Sylvester

"Mathematics is the music of reason."
— J. J. Sylvester

Sylvester, a nineteenth-century mathematician, coined the phrase "synthematic totals" to describe some structures based on 6-element sets that R. T. Curtis has called "rather unwieldy objects." See Curtis's abstract, Symmetric Generation of Finite Groups, John Baez's essay, Some Thoughts on the Number 6, and my website, Diamond Theory.

 

The picture above is of the complete graph K6  Six points with an edge connecting every pair of points… Fifteen edges in all.

Diamond theory describes how the 15 two-element subsets of a six-element set (represented by edges in the picture above) may be arranged as 15 of the 16 parts of a 4×4 array, and how such an array relates to group-theoretic concepts, including Sylvester's synthematic totals as they relate to constructions of the Mathieu group M24.

If diamond theory illustrates any general philosophical principle, it is probably the interplay of opposites….  "Reciprocity" in the sense of Lao Tzu.  See

Reciprocity and Reversal in Lao Tzu.

For a sense of "reciprocity" more closely related to Michael Kruger's alleged philosophy, see the Confucian concept of Shu (Analects 15:23 or 24) described in

Shu: Reciprocity.

Kruger's novel is in part about a Jew: the quintessential Jewish symbol, the star of David, embedded in the K6 graph above, expresses the reciprocity of male and female, as my May 2003 archives illustrate.  The star of David also appears as part of a graphic design for cubes that illustrate the concepts of diamond theory:

Click on the design for details.

Those who prefer a Jewish approach to physics can find the star of David, in the form of K6, applied to the sixteen 4×4 Dirac matrices, in

A Graphical Representation
of the Dirac Algebra
.

The star of David also appears, if only as a heuristic arrangement, in a note that shows generating partitions of the affine group on 64 points arranged in two opposing triplets.

Having thus, as the New York Times advises, paid tribute to a Jewish symbol, we may note, in closing, a much more sophisticated and subtle concept of reciprocity due to Euler, Legendre, and Gauss.  See

The Jewel of Arithmetic and


FinnegansWiki:

Salmonson set his seel:

"Finn MacCool ate the Salmon of Knowledge."

Wikipedia:

"George Salmon spent his boyhood in Cork City, Ireland. His father was a linen merchant. He graduated from Trinity College Dublin at the age of 19 with exceptionally high honours in mathematics. In 1841 at age 21 he was appointed to a position in the mathematics department at Trinity College Dublin. In 1845 he was appointed concurrently to a position in the theology department at Trinity College Dublin, having been confirmed in that year as an Anglican priest."

Related material:

Kindergarten Theology,

Kindergarten Relativity,

Arrangements for
56 Triangles
.

For more on the
arrangement of
triangles discussed
in Finnegans Wake,
see Log24 on Pi Day,
March 14, 2008.

Happy birthday,
Martin Sheen.
 

Friday, July 25, 2008

Friday July 25, 2008

Filed under: General,Geometry — m759 @ 6:01 PM
56 Triangles

Greg Egan's drawing of the 56 triangles on the Klein quartic 3-hole torus

John Baez on
  Klein's quartic:

"This wonderful picture was drawn by Greg Egan with the help of ideas from Mike Stay and Gerard Westendorp. It's probably the best way for a nonmathematician to appreciate the symmetry of Klein's quartic. It's a 3-holed torus, but drawn in a way that emphasizes the tetrahedral symmetry lurking in this surface! You can see there are 56 triangles: 2 for each of the tetrahedron's 4 corners, and 8 for each of its 6 edges."

Exercise:

The Eightfold Cube: The Beauty of Klein's Simple Group

Click on image for further details.

Note that if eight points are arranged
in a cube (like the centers of the
eight subcubes in the figure above),
there are 56 triangles formed by
the 8 points taken 3 at a time.

Baez's discussion says that the Klein quartic's 56 triangles can be partitioned into 7 eight-triangle Egan "cubes" that correspond to the 7 points of the Fano plane in such a way that automorphisms of the Klein quartic correspond to automorphisms of the Fano plane. Show that the 56 triangles within the eightfold cube can also be partitioned into 7 eight-triangle sets that correspond to the 7 points of the Fano plane in such a way that (affine) transformations of the eightfold cube induce (projective) automorphisms of the Fano plane.

Saturday, February 23, 2008

Saturday February 23, 2008

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM
Jumpers

"An acute study of the links
between word and fact"
Nina daVinci Nichols

 
Thanks to a Virginia reader for a reminder:
 
Virginia /391062427/item.html? 2/22/2008 7:37 PM
 
The link is to a Log24 entry
that begins as follows…

An Exercise

of Power

Johnny Cash:
"And behold,
a white horse."

Springer logo - A chess knight
Chess Knight
(in German, Springer)

This, along with the "jumper" theme in the previous two entries, suggests a search on springer jumper.That search yields a German sports phrase, "Springer kommt!"  A search on that phrase yields the following:
"Liebe Frau vBayern,
mich würde interessieren wie man
mit diesem Hintergrund
(vonbayern.de/german/anna.html)
zu Springer kommt?"

Background of "Frau vBayern" from thePeerage.com:

Anna-Natascha Prinzessin zu Sayn-Wittgenstein-Berleburg 

F, #64640, b. 15 March 1978Last Edited=20 Oct 2005

     Anna-Natascha Prinzessin zu Sayn-Wittgenstein-Berleburg was born on 15 March 1978. She is the daughter of Ludwig Ferdinand Prinz zu Sayn-Wittgenstein-Berleburg and Countess Yvonne Wachtmeister af Johannishus. She married Manuel Maria Alexander Leopold Jerg Prinz von Bayern, son of Leopold Prinz von Bayern and Ursula Mohlenkamp, on 6 August 2005 at Nykøping, Södermanland, Sweden.

 

The date of the above "Liebe Frau vBayern" inquiry, Feb. 1, 2007, suggests the following:

From Log24 on
St. Bridget's Day, 2007:

The quotation
"Science is a Faustian bargain"
and the following figure–

Change

The 63 yang-containing hexagrams of the I Ching as a Singer 63-cycle

From a short story by
the above Princess:

"'I don't even think she would have wanted to change you. But she for sure did not want to change herself. And her values were simply a part of her.' It was true, too. I would even go so far as to say that they were her basis, if you think about her as a geometrical body. That's what they couldn't understand, because in this age of the full understanding for stretches of values in favor of self-realization of any kind, it was a completely foreign concept."

To make this excellent metaphor mathematically correct,
change "geometrical body" to "space"… as in

"For Princeton's Class of 2007"

Review of a 2004 production of a 1972 Tom Stoppard play, "Jumpers"–

John Lahr on Tom Stoppard's play Jumpers

Related material:

Knight Moves (Log24, Jan. 16),
Kindergarten Theology (St. Bridget's Day, 2008),
and

The image “My space -(the affine space of six dimensions over the two-element field
(Click on image for details.)

Sunday, September 2, 2007

Sunday September 2, 2007

Filed under: General,Geometry — Tags: — m759 @ 5:11 PM

Comment at the
n-Category Cafe

Re: This Week’s Finds in Mathematical Physics (Week 251)

On Spekkens’ toy system and finite geometry

Background–

  • In “Week 251” (May 5, 2007), John wrote:
    “Since Spekkens’ toy system resembles a qubit, he calls it a “toy bit”. He goes on to study systems of several toy bits – and the charming combinatorial geometry I just described gets even more interesting. Alas, I don’t really understand it well: I feel there must be some mathematically elegant way to describe it all, but I don’t know what it is…. All this is fascinating. It would be nice to find the mathematical structure that underlies this toy theory, much as the category of Hilbert spaces underlies honest quantum mechanics.”
  • In the n-Category Cafe ( May 12, 2007, 12:26 AM, ) Matt Leifer wrote:
    “It’s crucial to Spekkens’ constructions, and particularly to the analog of superposition, that the state-space is discrete. Finding a good mathematical formalism for his theory (I suspect finite fields may be the way to go) and placing it within a comprehensive framework for generalized theories would be very interesting.”
  • In the n-category Cafe ( May 12, 2007, 6:25 AM) John Baez wrote:
    “Spekkens and I spent an afternoon trying to think about his theory as quantum mechanics over some finite field, but failed — we almost came close to proving it couldnt’ work.”

On finite geometry:

The actions of permutations on a 4 × 4 square in Spekkens’ paper (quant-ph/0401052), and Leifer’s suggestion of the need for a “generalized framework,” suggest that finite geometry might supply such a framework. The geometry in the webpage John cited is that of the affine 4-space over the two-element field.

Related material:

Update of
Sept. 5, 2007

See also arXiv:0707.0074v1 [quant-ph], June 30, 2007:

A fully epistemic model for a local hidden variable emulation of quantum dynamics,

by Michael Skotiniotis, Aidan Roy, and Barry C. Sanders, Institute for Quantum Information Science, University of Calgary. Abstract: "In this article we consider an augmentation of Spekkens’ toy model for the epistemic view of quantum states [1]…."
 

Skotiniotis et al. note that the group actions on the 4×4 square described in Spekkens' paper [1] may be viewed (as in Geometry of the 4×4 Square and Geometry of Logic) in the context of a hypercube, or tesseract, a structure in which adjacency is isomorphic to adjacency in the 4 × 4 square (on a torus).

Hypercube from the Skotiniotis paper:

Hypercube

Reference:

[1] Robert W. Spekkens, Phys. Rev. A 75, 032110 (2007),

Evidence for the epistemic view of quantum states: A toy theory
,

Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5 (Received 11 October 2005; revised 2 November 2006; published 19 March 2007.)

"There is such a thing
as a tesseract."
A Wrinkle in Time  
 

Sunday, August 12, 2007

Sunday August 12, 2007

Filed under: General,Geometry — Tags: — m759 @ 9:00 AM

The Geometry of Qubits

In the context of quantum information theory, the following structure seems to be of interest–

"… the full two-by-two matrix ring with entries in GF(2), M2(GF(2))– the unique simple non-commutative ring of order 16 featuring six units (invertible elements) and ten zero-divisors."

— "Geometry of Two-Qubits," by Metod Saniga (pdf, 17 pp.), Jan. 25, 2007

A 16-element affine space and a corresponding 16-element matrix ring

This ring is another way of looking at the 16 elements of the affine space A4(GF(2)) over the 2-element field.  (Arrange the four coordinates of each element– 1's and 0's– into a square instead of a straight line, and regard the resulting squares as matrices.)  (For more on A4(GF(2)), see Finite Relativity and related notes at Finite Geometry of the Square and Cube.)  Using the above ring, Saniga constructs a system of 35 objects (not unlike the 35 lines of the finite geometry PG(3,2)) that he calls a "projective line" over the ring.  This system of 35 objects has a subconfiguration isomorphic to the (2,2) generalized quadrangle W2 (which occurs naturally as a subconfiguration of PG(3,2)– see Inscapes.)

Saniga concludes:
 

"We have demonstrated that the basic properties of a system of two interacting spin-1/2 particles are uniquely embodied in the (sub)geometry of a particular projective line, found to be equivalent to the generalized quadrangle of order two. As such systems are the simplest ones exhibiting phenomena like quantum entanglement and quantum non-locality and play, therefore, a crucial role in numerous applications like quantum cryptography, quantum coding, quantum cloning/teleportation and/or quantum computing to mention the most salient ones, our discovery thus

  • not only offers a principally new geometrically-underlined insight into their intrinsic nature,
  • but also gives their applications a wholly new perspective
  • and opens up rather unexpected vistas for an algebraic geometrical modelling of their higher-dimensional counterparts."
It would seem that my own
study of pure mathematics
for instance, of the following
"diamond ring"–
 
The image “http://www.log24.com/theory/images/FourD.gif” cannot be displayed, because it contains errors.
 
is not without relevance to
the physics of quantum theory.

Thursday, June 21, 2007

Thursday June 21, 2007

Filed under: General,Geometry — m759 @ 12:07 PM
Let No Man
Write My Epigraph

(See entries of June 19th.)

“His graceful accounts of the Bach Suites for Unaccompanied Cello illuminated the works’ structural logic as well as their inner spirituality.”

Allan Kozinn on Mstislav Rostropovich in The New York Times, quoted in Log24 on April 29, 2007

“At that instant he saw, in one blaze of light, an image of unutterable conviction…. the core of life, the essential pattern whence all other things proceed, the kernel of eternity.”

— Thomas Wolfe, Of Time and the River, quoted in Log24 on June 9, 2005

“… the stabiliser of an octad preserves the affine space structure on its complement, and (from the construction) induces AGL(4,2) on it. (It induces A8 on the octad, the kernel of this action being the translation group of the affine space.)”

— Peter J. Cameron, “The Geometry of the Mathieu Groups” (pdf)

“… donc Dieu existe, réponse!

— Attributed, some say falsely,
to Leonhard Euler


“Only gradually did I discover
what the mandala really is:
‘Formation, Transformation,
Eternal Mind’s eternal recreation'”

(Faust, Part Two, as
quoted by Jung in
Memories, Dreams, Reflections)

Wolfgang Pauli as Mephistopheles

“Pauli as Mephistopheles
in a 1932 parody of
Goethe’s Faust at Niels Bohr’s
institute in Copenhagen.
The drawing is one of
many by George Gamow
illustrating the script.”
Physics Today

“Borja dropped the mutilated book on the floor with the others. He was looking at the nine engravings and at the circle, checking strange correspondences between them.

‘To meet someone’ was his enigmatic answer. ‘To search for the stone that the Great Architect rejected, the philosopher’s stone, the basis of the philosophical work. The stone of power. The devil likes metamorphoses, Corso.'”

The Club Dumas, basis for the Roman Polanski film “The Ninth Gate” (See 12/24/05.)

“Pauli linked this symbolism
with the concept of automorphism.”

The Innermost Kernel
 (previous entry)

And from
Symmetry in Mathematics
and Mathematics of Symmetry

(pdf), by Peter J. Cameron,
a paper presented at the
International Symmetry Conference,
Edinburgh, Jan. 14-17, 2007,
we have

The Epigraph–

Weyl on automorphisms
(Here “whatever” should
of course be “whenever.”)

Also from the
Cameron paper:

Local or global?

Among other (mostly more vague) definitions of symmetry, the dictionary will typically list two, something like this:

• exact correspondence of parts;
• remaining unchanged by transformation.

Mathematicians typically consider the second, global, notion, but what about the first, local, notion, and what is the relationship between them?  A structure M is homogeneous if every isomorphism between finite substructures of M can be extended to an automorphism of M; in other words, “any local symmetry is global.”

Some Log24 entries
related to the above politically
(women in mathematics)–

Global and Local:
One Small Step

and mathematically–

Structural Logic continued:
Structure and Logic
(4/30/07):

This entry cites
Alice Devillers of Brussels–

Alice Devillers

“The aim of this thesis
is to classify certain structures
which are, from a certain
point of view, as homogeneous
as possible, that is which have
  as many symmetries as possible.”

“There is such a thing
as a tesseract.”

Madeleine L’Engle 

Wednesday, June 20, 2007

Wednesday June 20, 2007

Filed under: General,Geometry — Tags: — m759 @ 1:06 AM

Kernel

Mathematical Reviews citation:

MR2163497 (2006g:81002) 81-03 (81P05)
Gieser, Suzanne The innermost kernel. Depth psychology and quantum physics. Wolfgang Pauli's dialogue with C. G. Jung. Springer-Verlag, Berlin, 2005. xiv+378 pp. ISBN: 3-540-20856-9

A quote from MR at Amazon.com:

"This revised translation of a Swedish Ph. D. thesis in philosophy offers far more than a discussion of Wolfgang Pauli's encounters with the psychoanalyst Carl Gustav Jung…. Here the book explains very well how Pauli attempted to extend his understanding beyond superficial esotericism and spiritism…. To understand Pauli one needs books like this one, which… seems to open a path to a fuller understanding of Pauli, who was seeking to solve a quest even deeper than quantum physics." (Arne Schirrmacher, Mathematical Reviews, Issue 2006g)
 

An excerpt:

 

The image “http://www.log24.com/theory/images/PauliSquare.gif” cannot be displayed, because it contains errors.

I do not yet know what Gieser means by "the innermost kernel." The following is my version of a "kernel" of sorts– a diagram well-known to students of anthropologist Claude Levi-Strauss and art theorist Rosalind Krauss:

The four-group is also known as the Vierergruppe or Klein group.  It appears, notably, as the translation subgroup of A, the group of 24 automorphisms of the affine plane over the 2-element field, and therefore as the kernel of the homomorphism taking A to the group of 6 automorphisms of the projective line over the 2-element field. (See Finite Geometry of the Square and Cube.)

Related material:

The "chessboard" of
   Nov. 7, 2006
(as revised Nov. 7, 2012)–

I Ching chessboard. Previous version replaced on Nov. 7, 2012, by original 1989 chessboard arrangement

I Ching chessboard

None of this material really has much to do with the history of physics, except for its relation to the life and thought of physicist Wolfgang Pauli— the "Mephistopheles" of the new book Faust in Copenhagen. (See previous entry.)

"Only gradually did I discover
what the mandala really is:
'Formation, Transformation,
Eternal Mind's eternal recreation'"

(Faust, Part Two, as
quoted by Jung in
Memories, Dreams, Reflections)
 

Monday, May 28, 2007

Monday May 28, 2007

Filed under: General,Geometry — Tags: — m759 @ 5:00 PM
Space-Time
and a Finite Model

Notes by Steven H. Cullinane
May 28, 2007

Part I: A Model of Space-Time

The following paper includes a figure illustrating Penrose's model of  "complexified, compactified Minkowski space-time as the Klein quadric in complex projective 5-space."
 
The image “http://www.log24.com/log/pix07/070528-Twistor.jpg” cannot be displayed, because it contains errors.
Click on picture to enlarge.

For some background on the Klein quadric and space-time, see Roger Penrose, "On the Origins of Twistor Theory," from Gravitation and Geometry: A Volume in Honor of Ivor Robinson, Bibliopolis, 1987.


Part II: A Corresponding Finite Model

The Klein quadric also occurs in a finite model of projective 5-space.  See a 1910 paper:

G. M. Conwell, The 3-space PG(3,2) and its group, Ann. of Math. 11, 60-76.

Conwell discusses the quadric, and the related Klein correspondence, in detail.  This is noted in a more recent paper by Philippe Cara:

The image “http://www.log24.com/log/pix07/070528-Quadric.jpg” cannot be displayed, because it contains errors.

As Cara goes on to explain, the Klein correspondence underlies Conwell's discussion of eight heptads.  These play an important role in another correspondence, illustrated in the Miracle Octad Generator of R. T. Curtis, that may be used to picture actions of the large Mathieu group M24.


Related material:

 

The projective space PG(5,2), home of the Klein quadric in the finite model, may be viewed as the set of 64 points of the affine space AG(6,2), minus the origin.

The 64 points of this affine space may in turn be viewed as the 64 hexagrams of the Classic of Transformation, China's I Ching.

There is a natural correspondence between the 64 hexagrams and the 64 subcubes of a 4x4x4 cube.  This correspondence leads to a natural way to generate the affine group AGL(6,2).  This may in turn be viewed as a group of over a trillion natural transformations of the 64 hexagrams.

Geometry of the I Ching.
 
"Once Knecht confessed to his teacher that he wished to learn enough to be able to incorporate the system of the I Ching into the Glass Bead Game.  Elder Brother laughed.  'Go ahead and try,' he exclaimed.  'You'll see how it turns out.  Anyone can create a pretty little bamboo garden in the world.  But I doubt that the gardener would succeed in incorporating the world in his bamboo grove.'"
 
— Hermann Hesse, The Glass Bead Game,
  translated by Richard and Clara Winston
 

Wednesday, March 21, 2007

Wednesday March 21, 2007

Filed under: General,Geometry — Tags: — m759 @ 3:18 PM
Finite Relativity
continued

This afternoon I added a paragraph to The Geometry of Logic that makes it, in a way, a sequel to the webpage Finite Relativity:

"As noted previously, in Figure 2 viewed as a lattice the 16 digital labels 0000, 0001, etc., may be interpreted as naming the 16 subsets of a 4-set; in this case the partial ordering in the lattice is the structure preserved by the lattice's group of 24 automorphisms– the same automorphism group as that of the 16 Boolean connectives.  If, however, these 16 digital labels are interpreted as naming the 16 functions from a 4-set to a 2-set  (of two truth values, of two colors, of two finite-field elements, and so forth), it is not obvious that the notion of partial order is relevant.  For such a set of 16 functions, the relevant group of automorphisms may be the affine group of A mentioned above.  One might argue that each Venn diagram in Fig. 3 constitutes such a function– specifically, a mapping of four nonoverlapping regions within a rectangle to a set of two colors– and that the diagrams, considered simply as a set of two-color mappings, have an automorphism group of order larger than 24… in fact, of order 322,560.  Whether such a group can be regarded as forming part of a 'geometry of logic' is open to debate."

The epigraph to "Finite Relativity" is:

"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."

— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16

The added paragraph seems to fit this description.

Sunday, March 18, 2007

Sunday March 18, 2007

Filed under: General,Geometry — m759 @ 2:20 PM

Update to
The Geometry of Logic:

A detailed description of a group of 16 “logical automorphisms” of the 16 binary connectives has been given in the paper “Simetria y Logica: La notacion de Peirce para los 16 conectivos binarios,” by Mireya Garcia, Jhon Fredy Gomez, and Arnold Oostra. (Published in the Memorias del XII Encuentro de Geometria y sus Aplicaciones, Universidad Pedagogica Nacional, Bogota, June 2001; on the Web at http://www.unav.es/gep/Articulos/SimetriaYLogica.pdf.) The authors do not identify this group as a subgroup of the affine group of A (the finite affine geometry of four dimensions over the two-element field); this can serve as an exercise.  Another exercise: determining whether the authors’ order-16 group includes all transformations that might reasonably be called “logical automorphisms” of the 16 binary connectives.

Friday, November 24, 2006

Friday November 24, 2006

Filed under: General,Geometry — m759 @ 1:06 PM
Galois’s Window:

Geometry
from Point
to Hyperspace


by Steven H. Cullinane

  Euclid is “the most famous
geometer ever known
and for good reason:
  for millennia it has been
his window
  that people first look through
when they view geometry.”

  Euclid’s Window:
The Story of Geometry
from Parallel Lines
to Hyperspace
,
by Leonard Mlodinow

“…the source of
all great mathematics
is the special case,
the concrete example.
It is frequent in mathematics
that every instance of a
  concept of seemingly
great generality is
in essence the same as
a small and concrete
special case.”

— Paul Halmos in
I Want To Be a Mathematician

Euclid’s geometry deals with affine
spaces of 1, 2, and 3 dimensions
definable over the field
of real numbers.

Each of these spaces
has infinitely many points.

Some simpler spaces are those
defined over a finite field–
i.e., a “Galois” field–
for instance, the field
which has only two
elements, 0 and 1, with
addition and multiplication
as follows:

+ 0 1
0 0 1
1 1 0
* 0 1
0 0 0
1 0 1
We may picture the smallest
affine spaces over this simplest
field by using square or cubic
cells as “points”:
Galois affine spaces

From these five finite spaces,
we may, in accordance with
Halmos’s advice,
select as “a small and
concrete special case”
the 4-point affine plane,
which we may call

Galois's Window

Galois’s Window.

The interior lines of the picture
are by no means irrelevant to
the space’s structure, as may be
seen by examining the cases of
the above Galois affine 3-space
and Galois affine hyperplane
in greater detail.

For more on these cases, see

The Eightfold Cube,
Finite Relativity,
The Smallest Projective Space,
Latin-Square Geometry, and
Geometry of the 4×4 Square.

(These documents assume that
the reader is familar with the
distinction between affine and
projective geometry.)

These 8- and 16-point spaces
may be used to
illustrate the action of Klein’s
simple group of order 168
and the action of
a subgroup of 322,560 elements
within the large Mathieu group.

The view from Galois’s window
also includes aspects of
quantum information theory.
For links to some papers
in this area, see
  Elements of Finite Geometry.

Tuesday, October 3, 2006

Tuesday October 3, 2006

Filed under: General,Geometry — m759 @ 9:26 AM

Serious

"I don't think the 'diamond theorem' is anything serious, so I started with blitzing that."

Charles Matthews at Wikipedia, Oct. 2, 2006

"The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas."

— G. H. Hardy, A Mathematician's Apology

Matthews yesterday deleted references to the diamond theorem and related material in the following Wikipedia articles:

Affine group‎
Reflection group‎
Symmetry in mathematics‎
Incidence structure‎
Invariant (mathematics)‎
Symmetry‎
Finite geometry‎
Group action‎
History of geometry‎

This would appear to be a fairly large complex of mathematical ideas.

See also the following "large complex" cited, following the above words of Hardy, in Diamond Theory:

Affine geometry, affine planes, affine spaces, automorphisms, binary codes, block designs, classical groups, codes, coding theory, collineations, combinatorial, combinatorics, conjugacy classes, the Conwell correspondence, correlations, design theory, duads, duality, error correcting codes, exceptional groups, finite fields, finite geometry, finite groups, finite rings, Galois fields, generalized quadrangles, generators, geometry, GF(2), GF(4), the (24,12) Golay code, group actions, group theory, Hadamard matrices, hypercube, hyperplanes, hyperspace, incidence structures, invariance, Karnaugh maps, Kirkman's schoolgirl problem, Latin squares, Leech lattice, linear groups, linear spaces, linear transformations, Mathieu groups, matrix theory, Meno, Miracle Octad Generator, MOG, multiply transitive groups, octads, the octahedral group, orthogonal arrays, outer automorphisms, parallelisms, partial geometries, permutation groups, PG(3,2), polarities, Polya-Burnside theorem, projective geometry, projective planes, projective spaces, projectivities, Reed-Muller codes, the relativity problem, Singer cycle, skew lines,  sporadic simple groups, Steiner systems, symmetric, symmetry, symplectic, synthemes, synthematic, tesseract, transvections, Walsh functions, Witt designs.

Sunday, October 1, 2006

Sunday October 1, 2006

Filed under: General,Geometry — m759 @ 8:00 AM
Tales of Philosophy:

Recipe for Disaster
 
according to Jerome Kagan,
Harvard psychologist emeritus
 

From Log24 —
 

The Line

The image “http://www.log24.com/theory/images/GridCube165C3.jpg” cannot be displayed, because it contains errors.

The Cube

From Harvard's
Jerome Kagan —
"'Humans demand that there be a clear right and wrong,' he said. 'You've got to believe that the track you've taken is the right track. You get depressed if you're not certain as to what it is you're supposed to be doing or what's right and wrong in the world.'" "People need to divide the world into good and evil, us and them, Kagan continued. To do otherwise– to entertain the possibility that life is not black and white, but variously shaded in gray– is perhaps more honest, rational and decent. But it's also, psychically, a recipe for disaster."
The image “http://www.log24.com/log/pix06A/061001-epi3-w156.jpg” cannot be displayed, because it contains errors.

Black and White:

Log24 in
May 2005

The image “http://www.log24.com/log/pix06A/061001-Grays.jpg” cannot be displayed, because it contains errors.

Shades of Gray:

An affine space
and 
Harvard's
Jerome Kagan

 

The above Kagan quotes are taken
from a New York Times essay by
Judith Warner as transcribed by
Mark Finkelstein on Sept. 29.

See also Log24 on
Sept. 29 and 30.

Related material:

Kagan's book

The image “http://www.log24.com/log/pix06A/SurpriseUncertainty.jpg” cannot be displayed, because it contains errors.

Surprise, Uncertainty,
and Mental Structures

(Harvard U. Press, April 2002)

and Werner Heisenberg–
discoverer of the
uncertainty principle
as Anakin Skywalker
being tempted by
the Dark Side:

The image “http://www.log24.com/log/pix05/050519-Anakin.jpg” cannot be displayed, because it contains errors.

(From Log24, May 2005)
 
George Lucas, who has profited
enormously from public depictions
of the clash between
good and evil, light and dark,
may in private life be inclined
to agree with Hercule Poirot:
 
"It is the brain, the little gray cells
on which one must rely.
One must seek the truth
within– not without."
 
(This is another version of the
"Descartes before dehors" principle–
See "A Table," Sept. 28.)
 

Wednesday, September 20, 2006

Wednesday September 20, 2006

Filed under: General,Geometry — m759 @ 8:00 AM

Public Space

“… the Danish cartoons crisis last March showed ‘two world views colliding in public space with no common point of reference.'”

George Carey, Archbishop of Canterbury from 1991 to 2002, quoted in today’s London Times.

Related material:

Geometry and Christianity
   (Google search yielding
    “about 1,540,000” results)

Geometry and Islam
   (Google search yielding
    “about 1,580,000” results)

MySpace.com/affine

The image “http://www.log24.com/log/pix06A/060828-Cube.jpg” cannot be displayed, because it contains errors.

A Public Space

The image “http://www.log24.com/log/pix06A/060920-Motto.jpg” cannot be displayed, because it contains errors.

— Motto of 
Plato’s Academy

Background from
Log24 on Feb. 15, 2006:

  

Hellmut Wilhelm on the Tao

If we replace the Chinese word “I” (change, transformation) with the word “permutation,” the relevance of Western mathematics (which some might call “the Logos“) to the I Ching (“Changes Classic”) beomes apparent.

For the relevance of Plato to
Islam, see David Wade’s
Pattern in Islamic Art
and a Google search on
Plato and Islam
(“about 1,680,000” results).

“We should let ourselves be guided by what is common to all. Yet although the Logos is common to all, most men live as if each had a private intelligence of his own.”

Heraclitus of Ephesus, about 500 B.C.

Saturday, July 29, 2006

Saturday July 29, 2006

Filed under: General,Geometry — Tags: — m759 @ 2:02 PM

Big Rock

Thanks to Ars Mathematicaa link to everything2.com:

"In mathematics, a big rock is a result which is vastly more powerful than is needed to solve the problem being considered. Often it has a difficult, technical proof whose methods are not related to those of the field in which it is applied. You say 'I'm going to hit this problem with a big rock.' Sard's theorem is a good example of a big rock."

Another example:

Properties of the Monster Group of R. L. Griess, Jr., may be investigated with the aid of the Miracle Octad Generator, or MOG, of R. T. Curtis.  See the MOG on the cover of a book by Griess about some of the 20 sporadic groups involved in the Monster:
 

The image “http://www.log24.com/theory/images/TwelveSG.jpg” cannot be displayed, because it contains errors.

The MOG, in turn, illustrates (via Abstract 79T-A37, Notices of the American Mathematical Society, February 1979) the fact that the group of automorphisms of the affine space of four dimensions over the two-element field is also the natural group of automorphisms of an arbitrary 4×4 array.

This affine group, of order 322,560, is also the natural group of automorphisms of a family of graphic designs similar to those on traditional American quilts.  (See the diamond theorem.)

This top-down approach to the diamond theorem may serve as an illustration of the "big rock" in mathematics.

For a somewhat simpler, bottom-up, approach to the theorem, see Theme and Variations.

For related literary material, see Mathematics and Narrative and The Diamond as Big as the Monster.
 

"The rock cannot be broken.
It is the truth."

Wallace Stevens,
"Credences of Summer"

Thursday, July 13, 2006

Thursday July 13, 2006

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM

Today's birthday:
Harrison Ford

"The forest here at the bottom of the canyon is mostly pine, with a few aspen and broad-leafed shrubs. Steep canyon walls rise way above us on both sides. Occasionally the trail opens into a patch of sunlight and grass that edges the canyon stream, but soon it reenters the deep shade of the pines. The earth of the trail is covered with a soft springy duff of pine needles. It is very quiet here.

Mountains like these and travelers in the mountains and events that happen to them here are found not only in Zen literature but in the tales of every major religion."– Robert Pirsig

Related material:
"Canyon Breeze" as played at
myspace.com/montanaskies

"… a point of common understanding between the classic and romantic worlds. Quality, the cleavage term between hip and square, seemed to be it. Both worlds used the term. Both knew what it was. It was just that the romantic left it alone and appreciated it for what it was and the classic tried to turn it into a set of intellectual building blocks for other purposes."– Robert Pirsig

 

For such building blocks, see
myspace.com/affine.

The image “http://www.log24.com/theory/images/MySpace.jpg” cannot be displayed, because it contains errors.
The background music there
is the same, by Montana Skies.

Wednesday, July 5, 2006

Wednesday July 5, 2006

Filed under: General,Geometry — m759 @ 12:25 PM

And now, from
the author of Sphere

CUBE

He beomes aware of something else… some other presence.
“Anybody here?” he says.
I am here.
He almost jumps, it is so loud. Or it seems loud. Then he wonders if he has heard anything at all.
“Did you speak?”
No.
How are we communicating? he wonders.
The way everything communicates with everything else.
Which way is that?
Why do you ask if you already know the answer?

Sphere, by Michael Crichton, Harvard ’64

“… when I went to Princeton things were completely different. This chapel, for instance– I remember when it was just a clearing, cordoned off with sharp sticks.  Prayer was compulsory back then, and you couldn’t just fake it by moving your lips; you had to know the words, and really mean them.  I’m dating myself, but this was before Jesus Christ.”

Baccalaureate address at Princeton, Pentecost 2006, reprinted in The New Yorker, edited by David Remnick, Princeton ’81

Related figures:

The image “http://www.log24.com/log/pix06A/060617-Spellbound.jpg” cannot be displayed, because it contains errors.

For further details,
see Solomon’s Cube
and myspace.com/affine.

The image “http://www.log24.com/log/pix06A/060705-Cube.jpg” cannot be displayed, because it contains errors.

For further details,
see Jews on Buddhism
and
Adventures in Group Theory.

“In this way we are offered
a formidable lesson
for every Christian community.”

Pope Benedict XVI
on Pentecost,
June 4, 2006,
St. Peter’s Square
.

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