Log24

Sunday, March 15, 2020

The “Octad Group

Filed under: General — Tags: , — m759 @ 4:17 PM

The phrase “octad group” discussed here in a post
of March 7 is now a domain name, “octad.group,”
that leads to that post. Remarks by Conway and
Sloane now quoted there indicate how the group
that I defined in 1979 is embedded in the large
Mathieu group M24.

Related literary notes — Watson + Embedding.

Saturday, March 7, 2020

The “Octad Group” as Symmetries of the 4×4 Square

Filed under: General — Tags: — m759 @ 6:32 PM

From Mathieu Moonshine and Symmetry Surfing” —

(Submitted on 29 Sep 2016, last revised 22 Jan 2018)
by Matthias R. Gaberdiel (1), Christoph A. Keller (2),
and Hynek Paul (1)

(1)  Institute for Theoretical Physics, ETH Zurich
(2)  Department of Mathematics, ETH Zurich

https://arxiv.org/abs/1609.09302v2 —

“This presentation of the symmetry groups Gi  is
particularly well-adapted for the symmetry surfing
philosophy. In particular it is straightforward to
combine them into an overarching symmetry group G
by combining all the generators. The resulting group is
the so-called octad group

G = (Z2)4  A8 .

It can be described as a maximal subgroup of M24
obtained by the setwise stabilizer of a particular
‘reference octad’ in the Golay code, which we take
to be O= {3,5,6,9,15,19,23,24} ∈ 𝒢24. The octad
subgroup is of order 322560, and its index in M24
is 759, which is precisely the number of
different reference octads one can choose.”

This “octad group” is in fact the symmetry group of the affine 4-space over GF(2),
so described in 1979 in connection not with the Golay code but with the geometry
of the 4×4 square.* Its nature as an affine group acting on the Golay code was
known long before 1979, but its description as an affine group acting on
the 4×4 square may first have been published in connection with the
Cullinane diamond theorem and Abstract 79T-A37, “Symmetry invariance in a
diamond ring
,” by Steven H. Cullinane in Notices of the American Mathematical
Society
, February 1979, pages A-193, 194.

* The Galois tesseract .

Update of March 15, 2020 —

Conway and Sloane on the “octad group” in 1993 —

Friday, December 20, 2019

Triangles, Spreads, Mathieu

Filed under: General — Tags: , — m759 @ 1:38 AM

Continued.

An addendum for the post “Triangles, Spreads, Mathieu” of Oct. 29:

Friday, November 22, 2019

Triangles, Spreads, Mathieu

Filed under: General — Tags: , — m759 @ 4:39 PM

Continued from October 29, 2019.

More illustrations (click to enlarge) —

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 —

Thursday, December 6, 2018

The Mathieu Cube of Iain Aitchison

This journal ten years ago today —

Surprise Package

Santa and a cube
From a talk by a Melbourne mathematician on March 9, 2018 —

The Mathieu group cube of Iain Aitchison (2018, Hiroshima)

The source — Talk II below —

Search Results

pdf of talk I  (March 8, 2018)

www.math.sci.hiroshima-u.ac.jp/branched/…/Aitchison-Hiroshima-2018-Talk1-2.pdf

Iain Aitchison. Hiroshima  University March 2018 … Immediate: Talk given last year at Hiroshima  (originally Caltech 2010).

pdf of talk II  (March 9, 2018)  (with model for M24)

www.math.sci.hiroshima-u.ac.jp/branched/files/…/Aitchison-Hiroshima-2-2018.pdf

Iain Aitchison. Hiroshima  University March 2018. (IRA: Hiroshima  03-2018). Highly symmetric objects II.

Abstract

www.math.sci.hiroshima-u.ac.jp/branched/files/2018/abstract/Aitchison.txt

Iain AITCHISON  Title: Construction of highly symmetric Riemann surfaces , related manifolds, and some exceptional objects, I, II Abstract: Since antiquity, some …

Related material — 

The 56 triangles of  the eightfold cube . . .

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

   Image from Christmas Day 2005.

Thursday, December 3, 2020

Brick Joke

Filed under: General — Tags: , , — m759 @ 10:56 AM

The “bricks” in posts tagged Octad Group suggest some remarks
from last year’s HBO “Watchmen” series —

Related material — The two  bricks constituting a 4×4 array, and . . .

“(this is the famous Kummer abstract configuration )”
Igor Dolgachev, ArXiv, 16 October 2019.

As is this

.

The phrase “octad group” does not, as one might reasonably
suppose, refer to symmetries of an octad (a “brick”), but
instead to symmetries of the above 4×4 array.

A related Broomsday event for the Church of Synchronology

Tuesday, October 6, 2020

Spreads via the Knight Cycle

Filed under: General — Tags: — m759 @ 2:10 AM

A Graphic Construction of the 56 Spreads of PG(3,2)

(An error in Fig. 4 was corrected at about
10:25 AM ET on Tuesday, Oct. 6, 2020.)

Friday, July 17, 2020

Poetic as Well as Prosaic

Filed under: General — Tags: , — m759 @ 9:51 AM

Prosaic —

Structure and Mutability

Poetic —

Crystal and Dragon

 

Prosaic —

These devices may have some
theoretical as well as practical value.

Poetic —

Counting symmetries with the orbit-stabilizer theorem

Thursday, April 23, 2020

Octads and Geometry

Filed under: General — Tags: , , , — m759 @ 10:11 PM

See the web pages octad.group and octad.us.

Related geometry (not the 759 octads, but closely related to them) —


The 4×6 rectangle of R. T. Curtis
illustrates the geometry of octads —

Counting symmetries with the orbit-stabilizer theorem

Curtis splits the 4×6 rectangle into three 4×2 “bricks” —

.

“In fact the construction enables us to describe the octads
in a very revealing manner. It shows that each octad,
other than Λ1, Λ2, Λ3, intersects at least one of these ‘ bricks’ —
the ‘heavy brick’ – in just four points.” . . . .

— R. T. Curtis (1976). “new combinatorial approach to M24,”
Mathematical Proceedings of the Cambridge Philosophical Society ,
79, pp 25-42.

Monday, December 23, 2019

Orbit

Filed under: General — Tags: , — m759 @ 7:34 PM

"December 22, the birth anniversary of India’s famed mathematician
Srinivasa Ramanujan, is celebrated as National Mathematics Day."
Indian Express  yesterday

"Orbits and stabilizers are closely related." — Wikipedia

Symmetries by Plato and R. T. Curtis —

Counting symmetries with the orbit-stabilizer theorem

In the above, 322,560 is the order 
of the octad stabilizer group .

Sunday, December 22, 2019

M24 from the Eightfold Cube

Filed under: General — Tags: , — m759 @ 12:01 PM

Exercise:  Use the Guitart 7-cycles below to relate the 56 triples
in an 8-set (such as the eightfold cube) to the 56 triangles in
a well-known Klein-quartic hyperbolic-plane tiling. Then use
the correspondence of the triples with the 56 spreads of PG(3,2)
to construct M24.

Click image below to download a Guitart PowerPoint presentation.

See as well earlier posts also tagged Triangles, Spreads, Mathieu.

Thursday, October 31, 2019

56 Triangles

Filed under: General — Tags: , — m759 @ 8:09 AM

The post “Triangles, Spreads, Mathieu” of October 29 has been
updated with an illustration from the Curtis Miracle Octad Generator.

Related material — A search in this journal for “56 Triangles.”

Saturday, May 4, 2019

The Chinese Jars of Shing-Tung Yau

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

The title refers to Calabi-Yau spaces.

T. S. Eliot —

Four Quartets

. . . Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.

A less "cosmic" but still noteworthy code — The Golay code.

This resides in a 12-dimensional space over GF(2).

Related material from Plato and R. T. Curtis

Counting symmetries with the orbit-stabilizer theorem

A related Calabi-Yau "Chinese jar" first described in detail in 1905

Illustration of K3 surface related to Mathieu moonshine

A figure that may or may not be related to the 4x4x4 cube that
holds the classical  Chinese "cosmic code" — the I Ching

ftp://ftp.cs.indiana.edu/pub/hanson/forSha/AK3/old/K3-pix.pdf

Thursday, March 7, 2019

In Reality

Filed under: General — m759 @ 11:45 AM

The previous post, quoting a characterization of the R. T. Curtis
Miracle Octad Generator , describes it as a "hand calculator ."

Other views 
 

A "natural diagram " —


 

A geometric object

Counting symmetries with the orbit-stabilizer theorem.

Wednesday, September 5, 2018

Multifaceted Narrative

Filed under: General,Geometry — m759 @ 8:19 PM

http://www.log24.com/log/pix18/180905-To_build_the_narrative-Galerie_St_Etienne.gif

http://www.log24.com/log/pix18/180905-Messier-Objects.gif

See also, in this  journal, 23-cycle.

Update of Sept. 6, 2018, 9:05 AM ET:  "The Cubist Method"

Multifaceted narrative by James Joyce —

http://www.log24.com/log/pix18/180819-Joyce-Possible_Permutations-Cambridge_Companion-2004-p168.gif

Multifaceted structures in pure mathematics, from Plato and R. T. Curtis  —

Counting symmetries with the orbit-stabilizer theorem

Saturday, August 4, 2018

Manifestations of Exquisite Geometry

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

An alleged manifestation in physics, from Scientific American  —

http://www.log24.com/log/pix18/180804-Exquisite_Geometry-subhead-Sciam-500w.jpg

Manifestations in pure mathematics, from Plato and R. T. Curtis  —

Counting symmetries with the orbit-stabilizer theorem

For some entertaining literary  manifestations, see Wrinkle.

Wednesday, July 18, 2018

Doodle

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

From "The Educated Imagination: A Website Dedicated
to Northrop Frye
" —

"In one of the notebooks for his first Bible book Frye writes,

'For at least 25 years I’ve been preoccupied by
the notion of a key to all mythologies.' . . . .

Frye made a valiant effort to provide a key to all mythology,
trying to fit everything into what he called the Great Doodle. . . ."

From a different page at the same website —

Here Frye provides a diagram of four sextets.

I prefer the Miracle Octad Generator of R. T. Curtis —

Counting symmetries with the orbit-stabilizer theorem.

Sunday, July 1, 2018

Deutsche Ordnung

The title is from a phrase spoken, notably, by Yul Brynner
to Christopher Plummer in the 1966 film “Triple Cross.”

Related structures —

Greg Egan’s animated image of the Klein quartic —

For a smaller tetrahedral arrangement, within the Steiner quadruple
system of order 8 modeled by the eightfold cube, see a book chapter
by Michael Huber of Tübingen

Steiner quadruple system in eightfold cube

For further details, see the June 29 post Triangles in the Eightfold Cube.

See also, from an April 2013 philosophical conference:

Abstract for a talk at the City University of New York:

The Experience of Meaning
Jan Zwicky, University of Victoria
09:00-09:40 Friday, April 5, 2013

Once the question of truth is settled, and often prior to it, what we value in a mathematical proof or conjecture is what we value in a work of lyric art: potency of meaning. An absence of clutter is a feature of such artifacts: they possess a resonant clarity that allows their meaning to break on our inner eye like light. But this absence of clutter is not tantamount to ‘being simple’: consider Eliot’s Four Quartets  or Mozart’s late symphonies. Some truths are complex, and they are simplified  at the cost of distortion, at the cost of ceasing to be  truths. Nonetheless, it’s often possible to express a complex truth in a way that precipitates a powerful experience of meaning. It is that experience we seek — not simplicity per se , but the flash of insight, the sense we’ve seen into the heart of things. I’ll first try to say something about what is involved in such recognitions; and then something about why an absence of clutter matters to them.

For the talk itself, see a YouTube video.

The conference talks also appear in a book.

The book begins with an epigraph by Hilbert

Friday, June 29, 2018

Triangles in the Eightfold Cube

Filed under: General,Geometry — Tags: , — 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

Thursday, June 28, 2018

All in Plato

Filed under: General — m759 @ 12:32 AM

"It's all in Plato" — C. S. Lewis

See too Platonic in this journal —

Counting symmetries with the orbit-stabilizer theorem

Sunday, May 6, 2018

The Osterman Omega

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

From "The Osterman Weekend" (1983) —

Counting symmetries of the R. T. Curtis Omega:

An Illustration from Shakespeare's birthday

Counting symmetries with the orbit-stabilizer theorem

Monday, April 23, 2018

Facets

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

Counting symmetries with the orbit-stabilizer theorem

See also the Feb. 17, 2017, post on Bertram Kostant
as well as "Mathieu Moonshine and Symmetry Surfing."

Monday, September 28, 2015

Cracker Jack Prize

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

From a post of July 24, 2011

Mira Sorvino in 'The Last Templar'

A review —

“The story, involving the Knights Templar, the Vatican, sunken treasure,
the fate of Christianity and a decoding device that looks as if it came out of 
a really big box of medieval Cracker Jack, is the latest attempt to combine
Indiana Jones derring-do with ‘Da Vinci Code’ mysticism.”

— The New York Times

A feeble attempt at a purely mathematical "decoding device"
from this journal earlier this month

Image that may or may not be related to the extended binary Golay code and the large Witt design

For some background, see a question by John Baez at Math Overflow
on Aug. 20, 2015.

The nonexistence of a 24-cycle in the large Mathieu group
might discourage anyone hoping for deep new insights from
the above figure.

See Marston Conder's "Symmetric Genus of the Mathieu Groups" —

Saturday, September 19, 2015

Geometry of the 24-Point Circle

Filed under: General,Geometry — Tags: — m759 @ 1:13 AM

The latest Visual Insight  post at the American Mathematical
Society website discusses group actions on the McGee graph,
pictured as 24 points arranged in a circle that are connected
by 36 symmetrically arranged edges.

Wikipedia remarks that

"The automorphism group of the McGee graph
is of order 32 and doesn't act transitively upon
its vertices: there are two vertex orbits of lengths
8 and 16."

The partition into 8 and 16 points suggests, for those familiar
with the Miracle Octad Generator and the Mathieu group M24,
the following exercise:

Arrange the 24 points of the projective line
over GF(23) in a circle in the natural cyclic order
, 1, 2, 3,  , 22, 0 ).  Can the McGee graph be
modeled by constructing edges in any natural way?

Image that may or may not be related to the extended binary Golay code and the large Witt design

In other words, if the above set of edges has no
"natural" connection with the 24 points of the
projective line over GF(23), does some other 
set of edges in an isomorphic McGee graph
have such a connection?

Update of 9:20 PM ET Sept. 20, 2015:

Backstory: A related question by John Baez
at Math Overflow on August 20.

Tuesday, June 17, 2014

Finite Relativity

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

Continued.

Anyone tackling the Raumproblem  described here
on Feb. 21, 2014 should know the history of coordinatizations
of the 4×6 Miracle Octad Generator (MOG) array by R. T. Curtis
and J. H. Conway. Some documentation:

The above two images seem to contradict a statement by R. T. Curtis
in a 1989 paper.  Curtis seemed in that paper to be saying, falsely, that
his original 1973 and 1976 MOG coordinates were those in array M below—

This seemingly false statement involved John H. Conway's supposedly
definitive and natural canonical coordinatization of the 4×6 MOG
array by the symbols for the 24 points of the projective line over GF(23)—
{∞, 0, 1, 2, 3… , 21, 22}:

An explanation of the apparent falsity in Curtis's 1989 paper:

By "two versions of the MOG" Curtis seems to have meant merely that the
octads , and not the projective-line coordinates , in his earlier papers were
mirror images of the octads  that resulted later from the Conway coordinates,
as in the images below.

Friday, February 21, 2014

Raumproblem*

Filed under: General,Geometry — Tags: , , — m759 @ 7:01 PM

Despite the blocking of Doodles on my Google Search
screen, some messages get through.

Today, for instance —

"Your idea just might change the world.
Enter Google Science Fair 2014"

Clicking the link yields a page with the following image—

IMAGE- The 24-triangle hexagon

Clearly there is a problem here analogous to
the square-triangle coordinatization problem,
but with the 4×6 rectangle of the R. T. Curtis
Miracle Octad Generator playing the role of
the square.

I once studied this 24-triangle-hexagon
coordinatization problem, but was unable to
obtain any results of interest. Perhaps
someone else will have better luck.

* For a rather different use of this word,
see Hermann Weyl in the Stanford
Encyclopedia of Philosophy.

Friday, July 25, 2008

Friday July 25, 2008

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

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