Sunday, November 19, 2017

Galois Space

Filed under: General,Geometry — Tags: — m759 @ 8:00 PM

This is a sequel to yesterday's post Cube Space Continued.

Tuesday, May 31, 2016

Galois Space

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

A very brief introduction:

Seven is Heaven...

Tuesday, January 12, 2016

Harmonic Analysis and Galois Spaces

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

The above sketch indicates, in a vague, hand-waving, fashion,
a connection between Galois spaces and harmonic analysis.

For more details of the connection, see (for instance) yesterday
afternoon's post Space Oddity.

Sunday, March 10, 2013

Galois Space

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


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).)

Monday, March 4, 2013

Occupy Galois Space

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

Continued from February 27, the day Joseph Frank died

"Throughout the 1940s, he published essays
and criticism in literary journals, and one,
'Spatial Form in Modern Literature'—
a discussion of experimental treatments
of space and time by Eliot, Joyce, Proust,
Pound and others— published in
The Sewanee Review  in 1945, propelled him
to prominence as a theoretician."

— Bruce Weber in this morning's print copy
of The New York Times  (p. A15, NY edition)

That essay is reprinted in a 1991 collection
of Frank's work from Rutgers University Press:

See also Galois Space and Occupy Space in this journal.

Frank was best known as a biographer of Dostoevsky.
A very loosely related reference… in a recent Log24 post,
Freeman Dyson's praise of a book on the history of
mathematics and religion in Russia:

"The intellectual drama will attract readers
who are interested in mystical religion
and the foundations of mathematics.
The personal drama will attract readers
who are interested in a human tragedy
with characters who met their fates with
exceptional courage."

Frank is survived by, among others, his wife, a mathematician.

Thursday, February 21, 2013

Galois Space

Filed under: General,Geometry — Tags: — m759 @ 6:00 PM


The previous post suggests two sayings:

"There is  such a thing as a Galois space."

— Adapted from Madeleine L'Engle

"For every kind of vampire, there is a kind of cross."

Thomas Pynchon


(Click to enlarge.)

Thursday, July 12, 2012

Galois Space

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

An example of lines in a Galois space * —

The 35 lines in the 3-dimensional Galois projective space PG(3,2)—

(Click to enlarge.)

There are 15 different individual linear diagrams in the figure above.
These are the points of the Galois space PG(3,2).  Each 3-set of linear diagrams
represents the structure of one of the 35  4×4 arrays and also represents a line
of the projective space.

The symmetry of the linear diagrams accounts for the symmetry of the
840 possible images in the kaleidoscope puzzle.

* For further details on the phrase "Galois space," see
Beniamino Segre's "On Galois Geometries," Proceedings of the
International Congress of Mathematicians, 1958  
(Cambridge U. Press, 1960, 488-499.)

(Update of Jan. 5, 2013— This post has been added to finitegeometry.org.)

Monday, October 15, 2018

For Zingari Shoolerim*

Filed under: General,Geometry — Tags: , — m759 @ 12:19 PM

IMAGE- Site with keywords 'Galois space, Galois geometry, finite geometry' at DiamondSpace.net

The structure at top right is that of the
in the previous post.

* "Zingari shoolerim" is from
    Finnegans Wake .

Saturday, September 29, 2018

“Ikonologie des Zwischenraums”

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 9:29 AM

The title is from Warburg. The Zwischenraum  lines and shaded "cuts"
below are to be added together in characteristic two, i.e., via the
set-theoretic symmetric difference  operator.

Some small Galois spaces (the Cullinane models)

Monday, August 27, 2018

Geometry and Simplicity

Filed under: General,Geometry — m759 @ 9:27 PM


Thinking in Four Dimensions
By Dusa McDuff

"I’ve got the rather foolhardy idea of trying to explain
to you the kind of mathematics I do, and the kind of
ideas that seem simple to me. For me, the search
for simplicity is almost synonymous with the search
for structure.

I’m a geometer and topologist, which means that
I study the structure of space
. . . .

In each dimension there is a simplest space
called Euclidean space … "

— In Roman Kossak, ed.,
Simplicity:  Ideals of Practice in Mathematics and the Arts
(Kindle Locations 705-710, 735). Kindle Edition.


For some much simpler spaces of various
dimensions, see Galois Space in this journal.

Some small Galois spaces (the Cullinane models)

Monday, June 4, 2018

The Trinity Stone Defined

Filed under: General,Geometry — m759 @ 8:56 PM

"Unsheathe your dagger definitions." — James Joyce, Ulysses

The "triple cross" link in the previous post referenced the eightfold cube
as a structure that might be called the trinity stone .

An Approach to Symmetric Generation of the Simple Group of Order 168

Some small Galois spaces (the Cullinane models)

Sunday, March 4, 2018

The Square Inch Space: A Brief History

Filed under: General,Geometry — Tags: , — m759 @ 11:21 AM

1955  ("Blackboard Jungle") —

1976 —

2009 —

2016 —

 Some small Galois spaces (the Cullinane models)

Friday, September 15, 2017

Space Art

Filed under: General,Geometry — Tags: — m759 @ 2:05 PM

Silas in "Equals" (2015) —

Ever since we were kids it's been drilled into us that 
Our purpose is to explore the universe, you know.
Outer space is where we'll find 
…  the answers to why we're here and 
…  and where we come from.

Related material — 

'The Art of Space Art' in The Paris Review, Sept. 14, 2017

See also Galois Space  in this  journal.

Tuesday, May 2, 2017

Image Albums

Filed under: General,Geometry — Tags: — m759 @ 1:05 PM

Pinterest boards uploaded to the new m759.net/piwigo

Diamond Theorem 

Diamond Theorem Correlation

Miracle Octad Generator

The Eightfold Cube

Six-Set Geometry

Diamond Theory Cover

Update of May 2 —

Four-Color Decomposition

Binary Galois Spaces

The Galois Tesseract

Update of May 3 —

Desargues via Galois

The Tetrahedral Model

Solomon's Cube

Update of May 8 —

Art Space board created at Pinterest

Saturday, June 18, 2016

Midnight in Herald Square

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

In memory of New Yorker  artist Anatol Kovarsky,
who reportedly died at 97 on June 1.

Note the Santa, a figure associated with Macy's at Herald Square.

See also posts tagged Herald Square, as well as the following
figure from this journal on the day preceding Kovarsky's death.

A note related both to Galois space and to
the "Herald Square"-tagged posts —

"There is  such a thing as a length-16 sequence."
— Saying adapted from a young-adult novel.

Sunday, May 8, 2016

The Three Solomons

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

Earlier posts have dealt with Solomon Marcus and Solomon Golomb,
both of whom died this year — Marcus on Saint Patrick's Day, and
Golomb on Orthodox Easter Sunday. This suggests a review of
Solomon LeWitt, who died on Catholic Easter Sunday, 2007.

A quote from LeWitt indicates the depth of the word "conceptual"
in his approach to "conceptual art."

From Sol LeWitt: A Retrospective , edited by Gary Garrels, Yale University Press, 2000, p. 376:

by Sol LeWitt

"The best that can be said for either the square or the cube is that they are relatively uninteresting in themselves. Being basic representations of two- and three-dimensional form, they lack the expressive force of other more interesting forms and shapes. They are standard and universally recognized, no initiation being required of the viewer; it is immediately evident that a square is a square and a cube a cube. Released from the necessity of being significant in themselves, they can be better used as grammatical devices from which the work may proceed."

"Reprinted from Lucy R. Lippard et al ., “Homage to the Square,” Art in America  55, No. 4 (July-August 1967): 54. (LeWitt’s contribution was originally untitled.)"

See also the Cullinane models of some small Galois spaces

 Some small Galois spaces (the Cullinane models)

Friday, May 6, 2016


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

 Some small Galois spaces (the Cullinane models)

Monday, January 11, 2016

Space Oddity

Filed under: General,Geometry — Tags: — m759 @ 3:15 PM

It is an odd fact that the close relationship between some
small Galois spaces and small Boolean spaces has gone 
unremarked by mathematicians.  

A Google search today for "Galois spaces" + "Boolean spaces"
yielded, apart from merely terminological sources, only some
introductory material I have put on the Web myself. 

Some more sophisticated searches, however led to a few
documents from the years 1971 – 1981 …

"Harmonic Analysis of Switching Functions" ,
by Robert J. Lechner, Ch. 5 in A. Mukhopadhyay, editor,
Recent Developments in Switching Theory , Academic Press, 1971.

"Galois Switching Functions and Their Applications,"
by B. Benjauthrit and I. S. Reed,
JPL Deep Space Network Progress Report 42-27 , 1975

D.K. Pradhan, “A Theory of Galois Switching Functions,”
IEEE Trans. Computers , vol. 27, no. 3, pp. 239-249, Mar. 1978

"Switching functions constructed by Galois extension fields,"
by Iwaro Takahashi, Information and Control ,
Volume 48, Issue 2, pp. 95–108, February 1981

An illustration from the Lechner paper above —

"There is  such a thing as harmonic analysis of switching functions."

— Saying adapted from a young-adult novel

Wednesday, October 21, 2015

Algebra and Space

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

"Perhaps an insane conceit …."    Perhaps.

Related remarks on algebra and space —

"The Quality Without a Name" (Log24, August 26, 2015).

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, January 5, 2015


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

Wednesday, March 13, 2013

Blackboard Jungle

Filed under: Uncategorized — m759 @ 8:00 AM 

From a review in the April 2013 issue of
Notices of the American Mathematical Society

"The author clearly is passionate about mathematics
as an art, as a creative process. In reading this book,
one can easily get the impression that mathematics
instruction should be more like an unfettered journey
into a jungle where an individual can make his or her
own way through that terrain."

From the book under review—

"Every morning you take your machete into the jungle
and explore and make observations, and every day
you fall more in love with the richness and splendor 
of the place."

— Lockhart, Paul (2009-04-01). 
A Mathematician's Lament:
How School Cheats Us Out of Our Most Fascinating
and Imaginative Art Form 
 (p. 92).
Bellevue Literary Press. Kindle Edition. 

Related material: Blackboard Jungle in this journal.

See also Galois Space and Solomon's Mines.

"I pondered deeply, then, over the
adventures of the jungle. And after
some work with a colored pencil
I succeeded in making my first drawing.
My Drawing Number One.
It looked something like this:

I showed my masterpiece to the
grown-ups, and asked them whether
the drawing frightened them.

But they answered: 'Why should
anyone be frightened by a hat?'"

The Little Prince

* For the title, see Plato Thanks the Academy (Jan. 3).

Monday, September 22, 2014


Filed under: General,Geometry — Tags: — m759 @ 11:17 AM

Review of an image from a post of May 6, 2009:

Galois space of six dimensions represented in Euclidean spaces of three and of two dimensions

Sunday, September 14, 2014


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:

Sunday, August 31, 2014

Sunday School

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

The Folding

Cynthia Zarin in The New Yorker , issue dated April 12, 2004—

“Time, for L’Engle, is accordion-pleated. She elaborated,
‘When you bring a sheet off the line, you can’t handle it
until it’s folded, and in a sense, I think, the universe can’t
exist until it’s folded — or it’s a story without a book.’”

The geometry of the 4×4 square array is that of the
3-dimensional projective Galois space PG(3,2).

This space occurs, notably, in the Miracle Octad Generator (MOG)
of R. T. Curtis (submitted to Math. Proc. Camb. Phil. Soc.  on
15 June 1974).  Curtis did not, however, describe its geometric
properties. For these, see the Cullinane diamond theorem.

Some history: 

Curtis seems to have obtained the 4×4 space by permuting,
then “folding” 1×8 binary sequences into 4×2 binary arrays.
The original 1×8 sequences came from the method of Turyn
(1967) described by van Lint in his book Coding Theory
(Springer Lecture Notes in Mathematics, No. 201 , first edition
published in 1971). Two 4×2 arrays form each 4×4 square array
within the MOG. This construction did not suggest any discussion
of the geometric properties of the square arrays.

[Rewritten for clarity on Sept. 3, 2014.]

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.

Wednesday, July 16, 2014

Finite Jest

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

(Continued from a private post of May 27, 2012)

Wednesday, May 21, 2014

Through the Vanishing Point*

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

Marshall McLuhan in "Annie Hall" —

"You know nothing of my work."

Related material — 

"I need a photo opportunity
I want a shot at redemption
Don't want to end up a cartoon
In a cartoon graveyard"

— Paul Simon

It was a dark and stormy night…


— Page 180, Logicomix

A photo opportunity for Whitehead
(from Romancing the Cube, April 20, 2011)—

IMAGE- Whitehead on Fano's construction of the 15-point projective Galois space over GF(2)

See also Absolute Ambition (Nov. 19, 2010).

* For the title, see Vanishing Point in this journal.

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, March 7, 2014

Kummer Varieties

Filed under: General,Geometry — Tags: , , — m759 @ 11:20 AM

The Dream of the Expanded Field continues

Image-- The Dream of the Expanded Field

From Klein's 1893 Lectures on Mathematics —

"The varieties introduced by Wirtinger may be called Kummer varieties…."
E. Spanier, 1956

From this journal on March 10, 2013 —

From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
"The Universal Kummer Threefold," by
Qingchun Ren, Steven V Sam, Gus Schrader, and Bernd Sturmfels —

IMAGE- 'Consider the 6-dimensional vector space over the 2-element field,' from 'The Universal Kummer Threefold'

Two such considerations —

IMAGE- 'American Hustle' and Art Cube

IMAGE- Cube for study of I Ching group actions, with Jackie Chan and Nicole Kidman 

Update of 10 PM ET March 7, 2014 —

The following slides by one of the "Kummer Threefold" authors give
some background related to the above 64-point vector space and
to the Weyl group of type E7(E7):

The Cayley reference is to "Algorithm for the characteristics of the
triple ϑ-functions," Journal für die Reine und Angewandte
Mathematik  87 (1879): 165-169. <http://eudml.org/doc/148412>.
To read this in the context of Cayley's other work, see pp. 441-445
of Volume 10 of his Collected Mathematical Papers .

Wednesday, February 5, 2014

Mystery Box II

Filed under: General,Geometry — Tags: — m759 @ 4:07 PM

Continued from previous post and from Sept. 8, 2009.

Box containing Froebel's Third Gift-- The Eightfold Cube

Examination of the box's contents does not solve
the contents' real mystery. That requires knowledge
of the non-Euclidean geometry of Galois space.

In this case, without that knowledge, prattle (as in
today's online New York Times ) about creativity and
"thinking outside the box" is pointless.

Tuesday, December 3, 2013

Diamond Space

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

A new website illustrates its URL.
See DiamondSpace.net.

IMAGE- Site with keywords 'Galois space, Galois geometry, finite geometry' at DiamondSpace.net

Thursday, November 7, 2013

Pattern Grammar

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

Yesterday afternoon's post linked to efforts by
the late Robert de Marrais to defend a mathematical  
approach to structuralism and kaleidoscopic patterns. 

Two examples of non-mathematical discourse on
such patterns:

1.  A Royal Society paper from 2012—

Click the above image for related material in this journal.

2.  A book by Junichi Toyota from 2009—

Kaleidoscopic Grammar: Investigation into the Nature of Binarism

I find such non-mathematical approaches much less interesting
than those based on the mathematics of reflection groups . 

De Marrais described the approaches of Vladimir Arnold and,
earlier, of H. S. M. Coxeter, to such groups. These approaches
dealt only with groups of reflections in Euclidean  spaces.
My own interest is in groups of reflections in Galois  spaces.
See, for instance, A Simple Reflection Group of Order 168

Galois spaces over fields of characteristic 2  are particularly
relevant to what Toyota calls binarism .

Thursday, July 4, 2013

Declaration of Independent

Filed under: General,Geometry — Tags: — m759 @ 2:21 PM

"Classical Geometry in Light of Galois Geometry"
is now available at independent.academia.edu.

Related commentary Yesterday's post Vision 
and a post of February 21, 2013:  Galois Space.

Saturday, March 16, 2013

The Crosswicks Curse

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


From the prologue to the new Joyce Carol Oates
novel Accursed

"This journey I undertake with such anticipation
is not one of geographical space but one of Time—
for it is the year 1905 that is my destination.

1905!—the very year of the Curse."

Today's previous post supplied a fanciful link
between the Crosswicks Curse of Oates and
the Crosswicks tesseract  of Madeleine L'Engle.

The Crosswicks Curse according to L'Engle
in her classic 1962 novel A Wrinkle in Time —

"There is  such a thing as a tesseract."

A tesseract is a 4-dimensional hypercube that
(as pointed out by Coxeter in 1950) may also 
be viewed as a 4×4 array (with opposite edges

Meanwhile, back in 1905

For more details, see how the Rosenhain and Göpel tetrads occur naturally
in the diamond theorem model of the 35 lines of the 15-point projective
Galois space PG(3,2).

See also Conwell in this journal and George Macfeely Conwell in the
honors list of the Princeton Class of 1905.

Wednesday, March 13, 2013

Blackboard Jungle

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

From a review in the April 2013 issue of
Notices of the American Mathematical Society

"The author clearly is passionate about mathematics
as an art, as a creative process. In reading this book,
one can easily get the impression that mathematics
instruction should be more like an unfettered journey
into a jungle where an individual can make his or her
own way through that terrain."

From the book under review—

"Every morning you take your machete into the jungle
and explore and make observations, and every day
you fall more in love with the richness and splendor
of the place."

— Lockhart, Paul (2009-04-01). A Mathematician's Lament:
How School Cheats Us Out of Our Most Fascinating and
Imaginative Art Form 
(p. 92). Bellevue Literary Press.
Kindle Edition. 

Related material: Blackboard Jungle in this journal.

See also Galois Space and Solomon's Mines.

Wednesday, March 6, 2013

Midnight in Pynchon*

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

"It is almost as though Pynchon wishes to
repeat the grand gesture of Joyce’s Ulysses…."

Vladimir Tasic on Pynchon's Against the Day

Related material:

Tasic's Mathematics and the Roots of Postmodern Thought  
and Michael Harris's "'Why Mathematics?' You Might Ask"

*See also Occupy Galois Space and Midnight in Dostoevsky.

Tuesday, February 19, 2013


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

Yesterday's post Permanence dealt with the cube
as a symmetric model of the finite projective plane
PG(2,3), which has 13 points and 13 lines. The points
and lines of the finite geometry occur in the cube as
the 13 axes of symmetry and the 13 planes through
the center perpendicular to those axes. If the three
axes lying in  a plane that cuts the cube in a hexagon
are supplemented by the axis perpendicular  to that
plane, each plane is associated with four axes and,
dually, each axis is associated with four planes.

My web page on this topic, Cubist Geometries, was
written on February 27, 2010, and first saved to the
Internet Archive on Oct. 4, 2010

For a more recent treatment of this topic that makes
exactly the same points as the 2010 page, see p. 218
of Configurations from a Graphical Viewpoint , by
Tomaž Pisanski and Brigitte Servatius, published by
Springer on Sept. 23, 2012 (date from both Google
and Amazon.com):

For a similar 1998 treatment of the topic, see Burkard Polster's 
A Geometrical Picture Book  (Springer, 1998), pp. 103-104.

The Pisanski-Servatius book reinforces my argument of Jan. 13, 2013,
that the 13 planes through the cube's center that are perpendicular
to the 13 axes of symmetry of the cube should be called the cube's 
symmetry planes , contradicting the usual use of of that term.

That argument concerns the interplay  between Euclidean and
Galois geometry. Pisanski and Servatius (and, in 1998, Polster)
emphasize the Euclidean square and cube as guides* to
describing the structure of a Galois space. My Jan. 13 argument
uses Galois  structures as a guide to re-describing those of Euclid .
(For a similar strategy at a much more sophisticated level,
see a recent Harvard Math Table.)

Related material:  Remarks on configurations in this journal
during the month that saw publication of the Pisanski-Servatius book.

* Earlier guides: the diamond theorem (1978), similar theorems for
  2x2x2 (1984) and 4x4x4 cubes (1983), and Visualizing GL(2,p)
  (1985). See also Spaces as Hypercubes (2012).

Wednesday, January 16, 2013


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


IMAGE- Golomb and Mazur awarded National Medals of Science


IMAGE- The Leibniz medal

Click medal for some background. The medal may be regarded
as illustrating the 16-point Galois space. (See previous post.)

Related material: Jews in Hyperspace.

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.”

Monday, August 29, 2011

Many = Six.

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

A comment today on yesterday's New York Times  philosophy column "The Stone"
notes that "Augustine… incorporated Greek ideas of perfection into Christianity."

Yesterday's post here  for the Feast of St. Augustine discussed the 2×2×2 cube.

Today's Augustine comment in the Times  reflects (through a glass darkly)
a Log24 post  from Augustine's Day, 2006, that discusses the larger 4×4×4 cube.

For related material, those who prefer narrative to philosophy may consult
Charles Williams's 1931 novel Many Dimensions . Those who prefer mathematics
to either may consult an interpretation in which Many = Six.

Galois space of six dimensions represented in Euclidean spaces of three and of two dimensions

Click image for some background.

Tuesday, May 10, 2011

Groups Acting

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

The LA Times  on last weekend's film "Thor"—

"… the film… attempts to bridge director Kenneth Branagh's high-minded Shakespearean intentions with Marvel Entertainment's bottom-line-oriented need to crank out entertainment product."

Those averse to Nordic religion may contemplate a different approach to entertainment (such as Taymor's recent approach to Spider-Man).

A high-minded— if not Shakespearean— non-Nordic approach to groups acting—

"What was wrong? I had taken almost four semesters of algebra in college. I had read every page of Herstein, tried every exercise. Somehow, a message had been lost on me. Groups act . The elements of a group do not have to just sit there, abstract and implacable; they can do  things, they can 'produce changes.' In particular, groups arise naturally as the symmetries of a set with structure. And if a group is given abstractly, such as the fundamental group of a simplical complex or a presentation in terms of generators and relators, then it might be a good idea to find something for the group to act on, such as the universal covering space or a graph."

— Thomas W. Tucker, review of Lyndon's Groups and Geometry  in The American Mathematical Monthly , Vol. 94, No. 4 (April 1987), pp. 392-394

"Groups act "… For some examples, see

Related entertainment—

High-minded— Many Dimensions

Not so high-minded— The Cosmic Cube


One way of blending high and low—

The high-minded Charles Williams tells a story
in his novel Many Dimensions about a cosmically
significant cube inscribed with the Tetragrammaton—
the name, in Hebrew, of God.

The following figure can be interpreted as
the Hebrew letter Aleph inscribed in a 3×3 square—


The above illustration is from undated software by Ed Pegg Jr.

For mathematical background, see a 1985 note, "Visualizing GL(2,p)."

For entertainment purposes, that note can be generalized from square to cube
(as Pegg does with his "GL(3,3)" software button).

For the Nordic-averse, some background on the Hebrew connection—

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.

Saturday, June 19, 2010

Imago Creationis

Filed under: General,Geometry — Tags: , , , , — m759 @ 6:00 PM

Image-- The Four-Diamond Tesseract

In the above view, four of the tesseract's 16
vertices are overlaid by other vertices.
For views that are more complete and
moveable, see Smith's tesseract page.

Four-Part Tesseract Divisions


The above figure shows how four-part partitions
of the 16 vertices  of a tesseract in an infinite
Euclidean  space are related to four-part partitions
of the 16 points  in a finite Galois  space

Euclidean spaces versus Galois spaces
in a larger context—



Infinite versus Finite

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)

Another picture related to philosophy and religion—

Jung's Four-Diamond Figure from Aion


This figure was devised by Jung
to represent the Self. Compare the
remarks of Paul Valéry on the Self—

Flight from Eden: The Origins of Modern Literary Criticism and Theory, by Steven Cassedy, U. of California Press, 1990, pages 156-157—



Valéry saw the mind as essentially a relational system whose operation he attempted to describe in the language of group mathematics. "Every act of understanding is based on a group," he says (C, 1:331). "My specialty— reducing everything to the study of a system closed on itself and finite" (C, 19: 645). The transformation model came into play, too. At each moment of mental life the mind is like a group, or relational system, but since mental life is continuous over time, one "group" undergoes a "transformation" and becomes a different group in the next moment. If the mind is constantly being transformed, how do we account for the continuity of the self? Simple; by invoking the notion of the invariant. And so we find passages like this one: "The S[elf] is invariant, origin, locus or field, it's a functional property of consciousness" (C, 15:170 [2:315]). Just as in transformational geometry, something remains fixed in all the projective transformations of the mind's momentary systems, and that something is the Self (le Moi, or just M, as Valéry notates it so that it will look like an algebraic variable). Transformation theory is all over the place. "Mathematical science…  reduced to algebra, that is, to the analysis of the transformations of a purely differential being made up of homogeneous elements, is the most faithful document of the properties of grouping, disjunction, and variation in the mind" (O, 1:36). "Psychology is a theory of transformations, we just need to isolate the invariants and the groups" (C, 1:915). "Man is a system that transforms itself" (C, 2:896).


  Paul Valéry, Oeuvres  (Paris: Pléiade, 1957-60)

C   Valéry, Cahiers, 29 vols. (Paris: Centre National de le Recherche Scientifique, 1957-61)

Note also the remarks of George David Birkhoff at Rice University
in 1940 (pdf) on Galois's theory of groups and the related
"theory of ambiguity" in Galois's testamentary letter—

… metaphysical reasoning always relies on the Principle of Sufficient Reason, and… the true meaning of this Principle is to be found in the “Theory of Ambiguity” and in the associated mathematical “Theory of Groups.”

If I were a Leibnizian mystic, believing in his “preestablished harmony,” and the “best possible world” so satirized by Voltaire in “Candide,” I would say that the metaphysical importance of the Principle of Sufficient Reason and the cognate Theory of Groups arises from the fact that God thinks multi-dimensionally* whereas men can only think in linear syllogistic series, and the Theory of Groups is the appropriate instrument of thought to remedy our deficiency in this respect.

* That is, uses multi-dimensional symbols beyond our grasp.

Related material:

Imago Creationis

A medal designed by Leibniz to show how
binary arithmetic mirrors the creation by God
of something (1) from nothing (0).


Another array of 16 strings of 0's and 1's, this time
regarded as coordinates rather than binary numbers—

Frame of Reference


The Diamond Theorem


Some context by a British mathematician —



by Wallace Stevens

Who can pick up the weight of Britain, 
Who can move the German load 
Or say to the French here is France again? 
Imago. Imago. Imago. 

It is nothing, no great thing, nor man 
Of ten brilliancies of battered gold 
And fortunate stone. It moves its parade 
Of motions in the mind and heart, 

A gorgeous fortitude. Medium man 
In February hears the imagination's hymns 
And sees its images, its motions 
And multitudes of motions 

And feels the imagination's mercies, 
In a season more than sun and south wind, 
Something returning from a deeper quarter, 
A glacier running through delirium, 

Making this heavy rock a place, 
Which is not of our lives composed . . . 
Lightly and lightly, O my land, 
Move lightly through the air again.

Wednesday, May 6, 2009

Wednesday May 6, 2009

Filed under: General,Geometry — m759 @ 11:07 AM

“My pursuits are a joke
in that the universe is a joke.
One has to reflect
the universe faithfully.”

John Frederick Michell
Feb. 9, 1933 –
April 24, 2009 

“I laugh because I dare not cry.
This is a crazy world and
the only way to enjoy it
is to treat it as a joke.”

— Robert A. Heinlein,
The Number of the Beast

For Marisa Tomei
  (born Dec. 4, 1964) —
on the day that
   Bob Seger turns 64 —

A Joke:
Points All Her Own

Points All Her Own,
Part I:

(For the backstory, see
the Log24 entries and links
on Marisa Tomei’s birthday
last year.)

Ad for a movie of the book 'Flatland'

Points All Her Own,

Part II:

(For the backstory, see
Galois Geometry:
The Simplest Examples

Galois geometry: the simplest examples

Points All Her Own,

Part III:

(For the backstory, see
Geometry of the I Ching
and the history of
Chinese philosophy.)

Galois space of six dimensions represented in Euclidean spaces of three and of two dimensions

In simpler terms:


Garfield on May 6, 2009: Smackdown!

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