Log24

Wednesday, December 7, 2016

Spreads and Conwell’s Heptads

Filed under: General,Geometry — Tags: , , — m759 @ 7:11 pm

For a concise historical summary of the interplay between
the geometry of an 8-set and that of a 16-set that is
involved in the the Miracle Octad Generator approach
to the large Mathieu group M24, see Section 2 of 

Alan R. Prince
A near projective plane of order 6 (pp. 97-105)
Innovations in Incidence Geometry
Volume 13 (Spring/Fall 2013).

This interplay, notably discussed by Conwell and
by Edge, involves spreads and Conwell’s heptads .

Update, morning of the following day (7:07 ET) — related material:

See also “56 spreads” in this  journal.

Saturday, March 8, 2014

Conwell Heptads in Eastern Europe

Filed under: General,Geometry — Tags: — m759 @ 11:07 am

“Charting the Real Four-Qubit Pauli Group
via Ovoids of a Hyperbolic Quadric of PG(7,2),”
by Metod Saniga, Péter Lévay and Petr Pracna,
arXiv:1202.2973v2 [math-ph] 26 Jun 2012 —

P. 4— “It was found that +(5,2) (the Klein quadric)
has, up to isomorphism, a unique  one — also known,
after its discoverer, as a Conwell heptad  [18].
The set of 28 points lying off +(5,2) comprises
eight such heptads, any two having exactly one
point in common.”

P. 11— “This split reminds us of a similar split of
63 points of PG(5,2) into 35/28 points lying on/off
a Klein quadric +(5,2).”

[18] G. M. Conwell, Ann. Math. 11 (1910) 60–76

A similar split occurs in yesterday’s Kummer Varieties post.
See the 63 = 28 + 35 vectors of R8 discussed there.

For more about Conwell heptads, see The Klein Correspondence,
Penrose Space-Time, and a Finite Model
.

For my own remarks on the date of the above arXiv paper
by Saniga et. al., click on the image below —

Walter Gropius

Thursday, December 21, 2023

Pi.ai

Filed under: General — Tags: — m759 @ 10:57 am
 

Thursday, December 21, 2023

Hey there, great to meet you. I’m Pi, your personal AI. My goal is to be useful, friendly and fun. Ask me for advice, for answers, or let’s talk about whatever’s on your mind.

How's your day going?

Going OK so far, Pi. How's yours?

My day is going swimmingly! 🌊 Thanks for asking 😊 Anything in particular on your mind today?

What is the connection of "heptads" with the "MOG" of R. T. Curtis?

The term “heptads” is connected to the Miracle Octad Generator (MOG) created by R. T. Curtis. Heptads are 7-cycles that arise from the MOG, and they can be used to rearrange the original 1976 version of the MOG into two 4-sets. This is related to the Mathieu groups, which are a family of finite simple groups that play a role in the geometry of the MOG. Are you familiar with the Mathieu groups?

Just the large one. I am interested in how familiar YOU are with these concepts. Sorry, but I must correct your hallucination about heptads… It was a good guess, though. Can you look up the work of G. M. Conwell on PG(3,2)? Heptads are discussed there.

Yes, it seems that G. M. Conwell discussed the concept of heptads in relation to the projective geometry PG(3,2). Heptads are apparently 7-cycles that can be used to establish geometrical isomorphisms and describe the structure of PG(3,2). Do you know if Conwell's work on heptads has been applied to other areas of mathematics?

It has not, as far as I know. Actually, Pi, heptads are not 7-cycles. What made you think that they are?

My apologies! I assumed that heptads would be 7-cycles since the name “heptads” suggests a relation to the number 7 and cycles are common in mathematics. It sounds like I was off the mark there. If heptads are not 7-cycles, can you give me a better idea of what they are?

Conwell can. See his 1910 paper for further details. All for now, Pi… Have a nice day!

Got it — thanks for the tip on Conwell's paper. I'll take a look for more information there! It was great chatting with you. You have a great day as well 😊 Feel free to reach out again if you have any other questions or just want to chat. See ya!

Scholium —

Related material for "swimmingly" and "the number 7" — Snark Hunt.

Sia!

Tuesday, December 19, 2023

Copilot Report

Filed under: General — Tags: , — m759 @ 12:16 am
 
From Microsoft Copilot (with "Precise" setting)
on Dec. 18, 2023 —
 
How is the 1910 work of G. M. Conwell on PG(3,2) related to the Miracle Octad Generator of R. T. Curtis?

Searching for: Relation between G. M. Conwell work on PG(3,2) and Miracle Octad Generator of R. T. Curtis

Searching for: Relation between G. M. Conwell work on PG(3,2) and Miracle Octad Generator of R. T. Curtis explained

Generating answers for you…

The work of George M. Conwell on the finite projective 3-space PG(3,2) and the Miracle Octad Generator (MOG) of R. T. Curtis are both significant contributions to the field of combinatorial mathematics12.

Conwell’s work on PG(3,2) provided a detailed study of the structure of this finite projective space1On the other hand, the MOG is a tool for exploring the binary Golay code and its automorphism group M242.

The connection between these two works lies in the underlying combinatorial structures they explore. The Rosenhain and Göpel tetrads, which are key structures in the finite geometry PG(3,2), play a significant role in the Curtis Miracle Octad Generator3These structures provide a link between the finite geometric structures studied by Conwell and the combinatorial structures generated by the MOG3.

For a more detailed understanding of this connection, I recommend checking out the resources mentioned in the references123. If you have any specific questions about the concepts in these works, feel free to ask! I’ll do my best to help.

Learn more

5 archive.org
_________________________________________________________

__________________________________________________________


DOCUMENT SUMMARY (generated by Copilot)
of reference 3 above, from academia.edu —
"Rosenhain and Göpel Tetrads in PG(3,2),"
by Steven H. Cullinane on March 17, 2013

Here is a summary of the document:

Reference:
 
Copilot suggestions following document summary —
 



Correction:

"Subsets of four points in a projective 3-space" above is a Copilot error, and not
from the document being summarized. It should be "subsets of four points in an
affine 4-space."

Monday, December 18, 2023

AI Class

Filed under: General — Tags: , — m759 @ 12:31 pm
 

Bringing Up Bing :

A Tutorial for Bing on Monday, December 18, 2023

Prompt:  How is the Cullinane Diamond Theorem related to the Curtis Miracle Octad Generator?

Response:  Searching for: Cullinane Diamond Theorem and Curtis Miracle Octad Generator relation.

Generating answers for you…

The Cullinane Diamond Theorem and the Curtis Miracle Octad Generator (MOG) are both mathematical concepts that are deeply intertwined with finite projective geometry1,2.

The Cullinane Diamond Theorem explains the surprising symmetry properties of some simple graphic designs and underlies the structure of the 35 square patterns in R. T. Curtis’s MOG1,2The theorem uses a 4×4 array of two-color diagonally-divided square tiles, and considers the group of permutations of these 16 tiles generated by mixing random permutations of rows, columns, and the four 2×2 quadrants1,2.

The theorem states that every image of the 4×4 array has some ordinary or color-interchange symmetry1,2The 35 structures of the 840 images of the 4×4 array are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2), a finite field with two elements1,2This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets1,2.

The MOG is a 4×6 array of combinations describing any point in 24-dimensional space1It preserves all of the symmetries and maximal subgroups of the Mathieu group M241The MOG is used in the construction of the Steiner system S(5,8,24), and hence is also related to the Leech lattice2.

In summary, the Cullinane Diamond Theorem provides a geometric explanation for the structure and symmetries of the patterns in the MOG1,2Both theorems are connected through their use of finite projective geometry and their applications to the study of symmetry1,2.

Learn more

  1. arxiv.org
  2. encyclopediaofmath.org
  3. arxiv.org
  4. m759.net

Reply to Bing's Response —

Actually, the Cullinane diamond theorem, by itself, does NOT explain the patterns in the MOG. The MOG is a natural correspondence, or pairing, of the 35 diamond-theorem structures with the 35 partitions of an eight-element set into two four-element sets. That correspondence is explained in a 1910 paper by G. M. Conwell, "The 3-Space PG(3,2) and Its Group." The Conwell correspondence was later found to be invariant under the action of the large Mathieu group on a set of 24=16+8 elements.

Response to the Above Reply —

Generating answers for you…

Thank you for the clarification. You’re correct that the Cullinane Diamond Theorem, by itself, does not explain the patterns in the Miracle Octad Generator (MOG). The MOG is indeed a natural correspondence of the 35 structures from the Diamond Theorem with the 35 partitions of an eight-element set into two four-element sets. This correspondence was explained by G. M. Conwell in his 1910 paper “The 3-Space PG(3,2) and Its Group.” The Conwell correspondence was later found to be invariant under the action of the large Mathieu group on a set of 24 elements, which can be partitioned into a set of 16 and a set of 8. I appreciate your input and the opportunity to learn from it. If you have any more questions or need further clarification, feel free to ask!

Wednesday, September 20, 2023

Temple Talk

Conwell versus Conwell.

Update of 8:16 AM ET —

"And it came to pass . . ."

Thursday, May 19, 2022

True Confessions! … 8!

Filed under: General — Tags: , — m759 @ 2:01 pm

A literary note by the author of The Eight  published on April 8, 2022 —

Wednesday, February 9, 2022

8!

Filed under: General — Tags: , , , — m759 @ 12:03 am

Conwell, 1910 — 

(In modern notation, Conwell is showing that the complete
projective group of collineations and dualities of the finite
3-space PG (3,2) is of order 8 factorial, i.e. "8!"
In other words, that any  permutation of eight things may be
regarded as a geometric transformation of PG (3,2).)

Later discussion of this same "Klein correspondence"
between Conwell's 3-space and 5-space . . .

A somewhat simpler toy model —

Page from 'The Paradise of Childhood,' 1906 edition

Related fiction —  "The Bulk Beings" of the film "Interstellar."

Sunday, October 11, 2020

Saniga on Einstein

Filed under: General — m759 @ 8:25 am

See Einstein on Acid” by Stephen Battersby
(New Scientist , Vol. 180, issue 2426 — 20 Dec. 2003, 40-43).

That 2003 article is about some speculations of Metod Saniga.

“Saniga is not a professional mystic or
a peddler of drugs, he is an astrophysicist
at the Slovak Academy of Sciences in Bratislava.
It seems unlikely that studying stars led him to
such a way-out view of space and time. Has he
undergone a drug-induced epiphany, or a period
of mental instability? ‘No, no, no,’ Saniga says,
‘I am a perfectly sane person.'”

Some more recent and much less speculative remarks by Saniga
are related to the Klein correspondence —

arXiv.org > math > arXiv:1409.5691:
Mathematics > Combinatorics
[Submitted on 17 Sep 2014]
The Complement of Binary Klein Quadric
as a Combinatorial Grassmannian

By Metod Saniga

“Given a hyperbolic quadric of PG(5,2), there are 28 points
off this quadric and 56 lines skew to it. It is shown that the
(286,563)-configuration formed by these points and lines
is isomorphic to the combinatorial Grassmannian of type
G2(8). It is also pointed out that a set of seven points of
G2(8) whose labels share a mark corresponds to a
Conwell heptad of PG(5,2). Gradual removal of Conwell
heptads from the (286,563)-configuration yields a nested
sequence of binomial configurations identical with part of
that found to be associated with Cayley-Dickson algebras
(arXiv:1405.6888).”

Related entertainment —

See Log24 on the date, 17 Sept. 2014, of Saniga’s Klein-quadric article:

Articulation Day.

Friday, February 7, 2020

Correspondences

The 15  2-subsets of a 6-set correspond to the 15 points of PG(3,2).
(Cullinane, 1986*)

The 35  3-subsets of a 7-set correspond to the 35 lines of PG(3,2).
(Conwell, 1910)

The 56  3-subsets of an 8-set correspond to the 56 spreads of PG(3,2).
(Seidel, 1970)

Each correspondence above may have been investigated earlier than
indicated by the above dates , which are the earliest I know of.

See also Correspondences in this journal.

* The above 1986 construction of PG(3,2) from a 6-set also appeared
in the work of other authors in 1994 and 2002 . . .

Addendum at 5:09 PM suggested by an obituary today for Stephen Joyce:

See as well the word correspondences  in
"James Joyce and the Hermetic Tradition," by William York Tindall
(Journal of the History of Ideas , Jan. 1954).

Sunday, December 29, 2019

Articulation Raid

Filed under: General — Tags: — m759 @ 7:45 am

“… And so each venture Is a new beginning,
a raid on the inarticulate….”

— T. S. Eliot, “East Coker V” in Four Quartets

arXiv:1409.5691v1 [math.CO]  17 Sep 2014

The Complement of Binary Klein Quadric as
a Combinatorial Grassmannian

Metod Saniga,
Institute for Discrete Mathematics and Geometry,
Vienna University of Technology,
Wiedner Hauptstraße 8–10, A-1040 Vienna, Austria
(metod.saniga@tuwien.ac.at) and
Astronomical Institute, Slovak Academy of Sciences,
SK-05960 Tatransk ́a Lomnica, Slovak Republic
(msaniga@astro.sk)

Abstract

Given a hyperbolic quadric of PG(5, 2), there are 28 points off this quadric and 56 lines skew to it. It is shown that the (286,563)-configuration formed by these points and lines is isomorphic to the combinatorial Grassmannian of type G2(8). It is also pointed out that a set of seven points of G2(8) whose labels share a mark corresponds to a Conwell heptad of PG(5, 2). Gradual removal of Conwell heptads from the (286,563)-configuration yields a nested sequence of binomial configurations identical with part of that found to be associated with Cayley-Dickson algebras (arXiv:1405.6888).

Keywords:

Combinatorial Grassmannian −
Binary Klein Quadric − Conwell Heptad

See also this  journal on the above date — 17 September 2014.

Saturday, December 28, 2019

Caballo Blanco

Filed under: General — Tags: , , , — m759 @ 9:02 am

The key  is the cocktail that begins the proceedings.”

– Brian Harley, Mate in Two Moves

http://www.log24.com/log/pix18/180809-The_EIght-and-coordinates-for-PSL(2,7)-actions-500w.jpg

“Just as these lines that merge to form a key
Are as chess squares . . . .” — Katherine Neville, The Eight

“The complete projective group of collineations and dualities of the
[projective] 3-space is shown to be of order [in modern notation] 8! ….
To every transformation of the 3-space there corresponds
a transformation of the [projective] 5-space. In the 5-space, there are
determined 8 sets of 7 points each, ‘heptads’ ….”

— George M. Conwell, “The 3-space PG (3, 2) and Its Group,”
The Annals of Mathematics , Second Series, Vol. 11, No. 2 (Jan., 1910),
pp. 60-76.

“It must be remarked that these 8 heptads are the key  to an elegant proof….”

— Philippe Cara, “RWPRI Geometries for the Alternating Group A8,” in
Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference
(July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis,
James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97.

Saturday, December 7, 2019

Canonicity

Filed under: General — Tags: , — m759 @ 12:55 pm

Review:

Note the "Milestones" date of receipt 25 January 2012.

This  journal on the eve of the above "Milestones" date —

Tuesday, October 29, 2019

Triangles, Spreads, Mathieu

Filed under: General — Tags: , — m759 @ 8:04 pm

There are many approaches to constructing the Mathieu
group M24. The exercise below sketches an approach that
may or may not be new.

Exercise:

It is well-known that

 There are 56 triangles in an 8-set.
There are 56 spreads in PG(3,2).
The alternating group An is generated by 3-cycles.
The alternating group Ais isomorphic to GL(4,2).

Use the above facts, along with the correspondence
described below, to construct M24.

Some background —

A Log24 post of May 19, 2013, cites

Peter J. Cameron in a 1976 Cambridge U. Press
book — Parallelisms of Complete Designs .
See the proof of Theorem 3A.13 on pp. 59 and 60.

See also a Google search for “56 triangles” “56 spreads” Mathieu.

Update of October 31, 2019 — A related illustration —

Update of November 2, 2019 —

See also p. 284 of Geometry and Combinatorics:
Selected Works of J. J. Seidel
  (Academic Press, 1991).
That page is from a paper published in 1970.

Update of December 20, 2019 —

Wednesday, August 28, 2019

Tiger’s Leap  to 1905

Filed under: General — Tags: , — m759 @ 3:03 pm

Walter Benjamin on 'a tiger's leap into the past'

See other posts
now tagged
Crosswicks Curse.

 

Click to enlarge:

Block Designs?

Wednesday, April 4, 2018

The Key

Filed under: General,Geometry — Tags: — m759 @ 12:32 pm

"The complete projective group of collineations and dualities of the
[projective] 3-space is shown to be of order [in modern notation] 8! ….
To every transformation of the 3-space there corresponds
a transformation of the [projective] 5-space. In the 5-space, there are
determined 8 sets of 7 points each, 'heptads' …."

— George M. Conwell, "The 3-space PG (3, 2) and Its Group," 
The Annals of Mathematics , Second Series, Vol. 11, No. 2 (Jan., 1910),
pp. 60-76.

"It must be remarked that these 8 heptads are the key to an elegant proof…."

— Philippe Cara, "RWPRI Geometries for the Alternating Group A8," in 
Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference 
(July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis,
James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97.

For those who, like the author of The Eight  (a novel in which today's
date figures prominently), prefer fiction —

See as well . . .

Literary theorists may, if they wish, connect
cabalistically the Insidious  address "414" 
with the date  4/14 of the above post, and
the word Appletree with the biblical Garden.

Friday, November 10, 2017

Annals of Rarefied Scholarship

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

From Cambridge Core, suggested by a reference to
that website in the previous post and by the following
bibliographic data . . .

https://doi.org/10.1017/fmp.2016.5

Downloaded from https://www.cambridge.org/core
on 10 Nov 2017 at 19:06:19 

See Conwell + Princeton in this journal.

Related art —

Saturday, October 7, 2017

Broken Symmetries

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 11:00 am

From posts tagged Design Deadline

A quotation from Lefebvre:

"… an epoch-making event so generally ignored
that we have to be reminded of it at every moment.
The fact is that around 1910 a certain space was shattered…
the space… of classical perspective and geometry…."

— Page 25 of The Production of Space 
    (Blackwell Publishing, 1991)

This suggests, for those who prefer Harvard's past glories
to its current state, a different Raum  from the Zeit  1910.

In January 1910 Annals of Mathematics , then edited at Harvard,
published George M. Conwell's "The 3-space PG (3, 2) and Its Group."
This paper, while perhaps neither epoch-making nor shattering, has
a certain beauty. For some background, see this journal on February 24, 2009.

Thursday, July 6, 2017

A Pleasing Situation

Filed under: General,Geometry — m759 @ 9:20 pm

The 4x4x4 cube is the natural setting
for the finite version of the Klein quadric
and the eight "heptads" discussed by
Conwell in 1910.

As R. Shaw remarked in 1995, 
"The situation is indeed quite pleasing."

Wednesday, July 5, 2017

Imaginarium of a Different Kind

Filed under: General,Geometry — m759 @ 9:00 pm

The title refers to that of the previous post, "The Imaginarium."

In memory of a translator who reportedly died on May  22, 2017,
a passage quoted here on that date —

Related material — A paragraph added on March 15, 2017,
to the Wikipedia article on Galois geometry

George Conwell gave an early demonstration of Galois geometry in 1910 when he characterized a solution of Kirkman's schoolgirl problem as a partition of sets of skew lines in PG(3,2), the three-dimensional projective geometry over the Galois field GF(2).[3] Similar to methods of line geometry in space over a field of characteristic 0, Conwell used Plücker coordinates in PG(5,2) and identified the points representing lines in PG(3,2) as those on the Klein quadric.

— User Rgdboer

Friday, February 17, 2017

Heptads and Heptapods

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

In the recent science fiction film "Arrival," Amy Adams portrays
a linguist, Louise Banks, who must learn to translate the language of
aliens ("Heptapods") who have just arrived in their spaceships.

The point of this tale seems to have something to do with Banks
learning, along with the aliens' language, their skill of seeing into
the future.

Louise Banks wannabes might enjoy the works of one
Metod Saniga, who thinks that finite geometry might have
something to do with perceptions of time.

See Metod Saniga, “Algebraic Geometry: A Tool for Resolving
the Enigma of Time?”, in R. Buccheri, V. Di Gesù and M. Saniga (eds.), 
Studies on the Structure of Time: From Physics to Psycho(patho)logy,
Kluwer Academic / Plenum Publishers, New York, 2000, pp. 137–166.
Available online at www.ta3.sk/~msaniga/pub/ftp/mathpsych.pdf .

Although I share an interest in finite geometry with Saniga —
see, for instance, his remarks on Conwell heptads in the previous post
and my own remarks in yesterday's post "Schoolgirls and Heptads" —
I do not endorse his temporal speculations.

Thursday, February 16, 2017

Schoolgirls and Heptads

Filed under: General,Geometry — m759 @ 11:32 am

A Feb. 12 note in the "talk" section of the Wikipedia article
"Kirkman's schoolgirl problem" —

The illustration above was replaced by a new section in the article,
titled "Galois geometry."

The new section improves the article by giving it greater depth.  
For related material, see Conwell Heptads in this journal
(or, more generally, Conwell) and a 1985 note citing Conwell's work.

Tuesday, June 7, 2016

Art and Space…

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

Continues, in memory of chess grandmaster Viktor Korchnoi,
who reportedly died at 85 yesterday in Switzerland —

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

The coloring of the 4×4 "base" in the above image
suggests St. Bridget's cross.

From this journal on St. Bridget's Day this year —

"Possible title: 

A new graphic approach 
to an old geometric approach
to a new combinatorial approach
to an old algebraic approach
to M24
"

The narrative leap from image to date may be regarded as
an example of "knight's move" thinking.

Monday, May 2, 2016

Subjective Quality

Filed under: General,Geometry — m759 @ 6:01 am

The previous post deals in part with a figure from the 1988 book
Sphere Packings, Lattices and Groups , by J. H. Conway and
N. J. A. Sloane.

Siobhan Roberts recently wrote a book about the first of these
authors, Conway.  I just discovered that last fall she also had an
article about the second author, Sloane, published:

"How to Build a Search Engine for Mathematics,"
Nautilus , Oct 22, 2015.

Meanwhile, in this  journal

Log24 on that same date, Oct. 22, 2015 —

Roberts's remarks on Conway and later on Sloane are perhaps
examples of subjective  quality, as opposed to the objective  quality
sought, if not found, by Alexander, and exemplified by the
above bijection discussed here  last October.

Friday, April 8, 2016

Ogdoads by Curtis

Filed under: General,Geometry — Tags: , , , , — m759 @ 12:25 pm

As was previously noted here, the construction of the Miracle Octad Generator
of R. T. Curtis in 1974 may have involved his "folding" the 1×8 octads constructed
in 1967 by Turyn into 2×4 form.

This results in a way of picturing a well-known correspondence (Conwell, 1910)
between partitions of an 8-set and lines of the projective 3-space PG(3,2).

For some background related to the "ogdoads" of the previous post, see
A Seventh Seal (Sept. 15, 2014).

Monday, February 1, 2016

Historical Note

Filed under: General,Geometry — Tags: , — m759 @ 6:29 am

Possible title

A new graphic approach
to an old geometric approach
to a new combinatorial approach
to an old algebraic approach
to M24

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 (such as Turyn's), not  by hand, 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

Saturday, March 8, 2014

Women’s History Month

Filed under: General — Tags: , — m759 @ 8:00 pm

For the Princeton Class of 1905 —

Joyce Carol Oates Meets Emily Dickinson.

Oates —

"It is an afternoon in autumn, near dusk.
The western sky is a spider’s web of translucent gold.
I am being brought by carriage—two horses—
muted thunder of their hooves—
along narrow country roads between hilly fields
touched with the sun’s slanted rays,
to the village of Princeton, New Jersey.
The urgent pace of the horses has a dreamlike air,
like the rocking motion of the carriage;
and whoever is driving the horses
his face I cannot see, only his back—
stiff, straight, in a tight-fitting dark coat."

Dickinson —

"Because I could not stop for Death—
He kindly stopped for me—
The Carriage held but just Ourselves—
And Immortality."

Tuesday, July 9, 2013

Vril Chick

Filed under: General,Geometry — Tags: , — 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 .

Wednesday, May 1, 2013

The Crosswicks Curse

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

(Continued)

"There is  such a thing as a tesseract." —A novel from Crosswicks

Related material from a 1905 graduate of Princeton,
"The 3-Space PG(3,2) and Its Group," is now available
at Internet Archive (1 download thus far).

The 3-space paper is relevant because of the
connection of the group it describes to the
"super, overarching" group of the tesseract.

Saturday, March 16, 2013

The Crosswicks Curse

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

Continues.

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
identified).

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.

Monday, November 19, 2012

Poetry and Truth

From today's noon post

"In all his poems with all their enchantments
for the poet himself, there is the final enchantment
that they are true. The significance of the poetic act
then is that it is evidence. It is instance and illustration.
It is an illumination of a surface,
the movement of a self in the rock.
Above all it is a new engagement with life.
It is that miracle to which the true faith of the poet
attaches itself."

— Wallace Stevens at Bard College, March 30, 1951

Stevens also said at Bard that

"When Joan of Arc said: 

Have no fear: what I do, I do by command.
My brothers of Paradise tell me what I have to do.

these words were the words of an hallucination.
No matter what her brothers of Paradise drove her to do,
what she did was never a poetic act of faith in reality
because it could not be."

There are those who would dispute this.

Some related material:

"Ageometretos me eisito."—
"Let no one ignorant of geometry enter."—
Said to be a saying of Plato, part of the
seal of the American Mathematical Society—

A poetic approach to geometry—

"A surface" and "the rock," from All Saints' Day, 2012

Spaces as Hypercubes

— and from 1981—

http://www.log24.com/log/pix09/090217-SolidSymmetry.jpg

Some mathematical background for poets in Purgatory—

"… 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."

Monday, September 12, 2011

Key Paper Now Online

Filed under: General — m759 @ 11:07 pm

The 3-Space PG(3,2) and Its Group

by George M. Conwell, Annals of Mathematics ,
Second Series, Vol. 11, No. 2 (Jan., 1910), pp. 60-76

Article Stable URL: http://www.jstor.org/stable/1967582

At least for now, this paper may be downloaded without
signing in or making a payment. Click the "View PDF" link.

Update of Sept. 13— From Library Journal  on Sept. 7—

The JSTOR journal archive announced today that it is making nearly 500,000 public domain journal articles from more than 220 journals—or about six percent of JSTOR's total content—freely available for use by "anyone, without registration and regardless of institutional affiliation."

The material, entitled Early Journal Content, will be rolled out in batches starting today over the course of one week. It includes content published in the United States before 1923 and international content published before 1870, which ensures that all the content is firmly in the public domain. JSTOR, in an announcement, said that the move was "a first step in a larger effort to provide more access options" to JSTOR content for independent scholars and others unaffiliated with universities.

Friday, October 2, 2009

Friday October 2, 2009

Filed under: General,Geometry — Tags: — m759 @ 6:00 am
Edge on Heptads

Part I: Dye on Edge

“Summary:
….we obtain various orbits of partitions of quadrics over GF(2a) by their maximal totally singular subspaces; the corresponding stabilizers in the relevant orthogonal groups are investigated. It is explained how some of these partitions naturally generalize Conwell’s heptagons for the Klein quadric in PG(5,2).”

Introduction:
In 1910 Conwell… produced his heptagons in PG(5,2) associated with the Klein quadric K whose points represent the lines of PG(3,2)…. Edge… constructed the 8 heptads of complexes in PG(3,2) directly. Both he and Conwell used their 8 objects to establish geometrically the isomorphisms SL(4,2)=A8 and O6(2)=S8 where O6(2) is the group of K….”

— “Partitions and Their Stabilizers for Line Complexes and Quadrics,” by R.H. Dye, Annali di Matematica Pura ed Applicata, Volume 114, Number 1, December 1977, pp. 173-194

Part II: Edge on Heptads

The Geometry of the Linear Fractional Group LF(4,2),” by W.L. Edge, Proc. London Math Soc., Volume s3-4, No. 1, 1954, pp. 317-342. See the historical remarks on the first page.

Note added by Edge in proof:
“Since this paper was finished I have found one by G. M. Conwell: Annals of Mathematics (2) 11 (1910), 60-76….”

Some context:

The Klein Correspondence,
Penrose Space-Time,
and a Finite Model

Saturday, April 4, 2009

Saturday April 4, 2009

Filed under: General,Geometry — Tags: , — m759 @ 8:00 am
Annual Tribute to
The Eight

Katherine Neville's 'The Eight,' edition with knight on cover, on her April 4 birthday

Other knight figures:

Knight figures in finite geometry (Singer 7-cycles in the 3-space over GF(2) by Cullinane, 1985, and Curtis, 1987)

The knight logo at the SpringerLink site

Click on the SpringerLink
knight for a free copy
(pdf, 1.2 mb) of
the following paper
dealing with the geometry
underlying the R.T. Curtis
knight figures above:

Springer description of 1970 paper on Mathieu-group geometry by Wilbur Jonsson of McGill U.

Context:

Literature and Chess and
Sporadic Group References

Details:

 

Adapted (for HTML) from the opening paragraphs of the above paper, W. Jonsson's 1970 "On the Mathieu Groups M22, M23, M24…"–

"[A]… uniqueness proof is offered here based upon a detailed knowledge of the geometric aspects of the elementary abelian group of order 16 together with a knowledge of the geometries associated with certain subgroups of its automorphism group. This construction was motivated by a question posed by D.R. Hughes and by the discussion Edge [5] (see also Conwell [4]) gives of certain isomorphisms between classical groups, namely

PGL(4,2)~PSL(4,2)~SL(4,2)~A8,
PSp(4,2)~Sp(4,2)~S6,

where A8 is the alternating group on eight symbols, S6 the symmetric group on six symbols, Sp(4,2) and PSp(4,2) the symplectic and projective symplectic groups in four variables over the field GF(2) of two elements, [and] PGL, PSL and SL are the projective linear, projective special linear and special linear groups (see for example [7], Kapitel II).

The symplectic group PSp(4,2) is the group of collineations of the three dimensional projective space PG(3,2) over GF(2) which commute with a fixed null polarity tau…."

References

4. Conwell, George M.: The three space PG(3,2) and its group. Ann. of Math. (2) 11, 60-76 (1910).

5. Edge, W.L.: The geometry of the linear fractional group LF(4,2). Proc. London Math. Soc. (3) 4, 317-342 (1954).

7. Huppert, B.: Endliche Gruppen I. Berlin-Heidelberg-New York: Springer 1967.

Sunday, March 1, 2009

Sunday March 1, 2009

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

Solomon's Cube
continued

"There is a book… called A Fellow of Trinity, one of series dealing with what is supposed to be Cambridge college life…. There are two heroes, a primary hero called Flowers, who is almost wholly good, and a secondary hero, a much weaker vessel, called Brown. Flowers and Brown find many dangers in university life, but the worst is a gambling saloon in Chesterton run by the Misses Bellenden, two fascinating but extremely wicked young ladies. Flowers survives all these troubles, is Second Wrangler and Senior Classic, and succeeds automatically to a Fellowship (as I suppose he would have done then). Brown succumbs, ruins his parents, takes to drink, is saved from delirium tremens during a thunderstorm only by the prayers of the Junior Dean, has much difficulty in obtaining even an Ordinary Degree, and ultimately becomes a missionary. The friendship is not shattered by these unhappy events, and Flowers's thoughts stray to Brown, with affectionate pity, as he drinks port and eats walnuts for the first time in Senior Combination Room."

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

"The Solomon Key is the working title of an unreleased novel in progress by American author Dan Brown. The Solomon Key will be the third book involving the character of the Harvard professor Robert Langdon, of which the first two were Angels & Demons (2000) and The Da Vinci Code (2003)." — Wikipedia

"One has O+(6) ≅ S8, the symmetric group of order 8! …."

 — "Siegel Modular Forms and Finite Symplectic Groups," by Francesco Dalla Piazza and Bert van Geemen, May 5, 2008, preprint.

"The complete projective group of collineations and dualities of the [projective] 3-space is shown to be of order [in modern notation] 8! …. To every transformation of the 3-space there corresponds a transformation of the [projective] 5-space. In the 5-space, there are determined 8 sets of 7 points each, 'heptads' …."

— George M. Conwell, "The 3-space PG(3, 2) and Its Group," The Annals of Mathematics, Second Series, Vol. 11, No. 2 (Jan., 1910), pp. 60-76

"It must be remarked that these 8 heptads are the key to an elegant proof…."

— Philippe Cara, "RWPRI Geometries for the Alternating Group A8," in Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference (July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis, James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97
 

Friday, February 27, 2009

Friday February 27, 2009

Filed under: General,Geometry — Tags: , — m759 @ 7:35 pm
Time and Chance
continued

Today's Pennsylvania lottery numbers suggest the following meditations…

Midday:  Lot 497, Bloomsbury Auctions May 15, 2008– Raum und Zeit (Space and Time), by Minkowski, 1909. Background: Minkowski Space and "100 Years of Space-Time."*

Evening: 5/07, 2008, in this journal– "Forms of the Rock."

Related material:

A current competition at Harvard Graduate School of Design, "The Space of Representation," has a deadline of 8 PM tonight, February 27, 2009.

The announcement of the competition quotes the Marxist Henri Lefebvre on "the social production of space."

A related quotation by Lefebvre (cf. 2/22 2009):

"… an epoch-making event so generally ignored that we have to be reminded of it at every moment. The fact is that around 1910 a certain space was shattered… the space… of classical perspective and geometry…."

— Page 25 of The Production of Space (Blackwell Publishing, 1991)

This suggests, for those who prefer Harvard's past glories to its current state, a different Raum from the Zeit 1910.

In January 1910 Annals of Mathematics, then edited at Harvard, published George M. Conwell's "The 3-space PG(3, 2) and Its Group." This paper, while perhaps neither epoch-making nor shattering, has a certain beauty. For some background, see this journal on February 24, 2009.†

    * Ending on Stephen King's birthday, 2008
     † Mardi Gras

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.

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

Tuesday, October 3, 2006

Tuesday October 3, 2006

Filed under: General,Geometry — Tags: , , — 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.

Friday, April 28, 2006

Friday April 28, 2006

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

Exercise

Review the concepts of integritas, consonantia,  and claritas in Aquinas:

"For in respect to beauty three things are essential: first of all, integrity or completeness, since beings deprived of wholeness are on this score ugly; and [secondly] a certain required design, or patterned structure; and finally a certain splendor, inasmuch as things are called beautiful which have a certain 'blaze of being' about them…."

Summa Theologiae Sancti Thomae Aquinatis, I, q. 39, a. 8, as translated by William T. Noon, S.J., in Joyce and Aquinas, Yale University Press, 1957

Review the following three publications cited in a note of April 28, 1985 (21 years ago today):

(1) Cameron, P. J.,
     Parallelisms of Complete Designs,
     Cambridge University Press, 1976.

(2) Conwell, G. M.,
     The 3-space PG(3,2) and its group,
     Ann. of Math. 11 (1910) 60-76.

(3) Curtis, R. T.,
     A new combinatorial approach to M24,
     Math. Proc. Camb. Phil. Soc.
    
79 (1976) 25-42.

Discuss how the sextet parallelism in (1) illustrates integritas, how the Conwell correspondence in (2) illustrates consonantia, and how the Miracle Octad Generator in (3) illustrates claritas.
 

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