Log24

Friday, March 13, 2015

Mathieu Moonshine

Filed under: General — m759 @ 1:26 PM

(Continued from yesterday's "earlier references" link.)

Yesterday at the Simons Foundation's Quanta Magazine :

See also earlier Log24 references to Mathieu moonshine .
I do not know the origin of this succinct phrase, taken from
an undated web page of Anne Taormina.

Wednesday, September 28, 2011

Mathieu Moonshine

Filed under: General — m759 @ 2:56 PM

The properties of the Mathieu group M 24 have recently interested some physicists

(Click to enlarge.)

IMAGE- July 2011 conference on Mathieu moonshine

For some related papers, see Mendeley.com.

Saturday, March 7, 2020

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

Filed under: General — 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 â‹Š 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} ∈ đť’˘24. 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 —

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

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."

Friday, September 25, 2015

A Great Moonshine

Filed under: General — Tags: — m759 @ 8:45 PM

Pope's 'have seen a great light' homily on 9/25/15

Saturday, March 14, 2015

Introduction to Yau

Filed under: Uncategorized — m759 @ 12:00 PM 

Home page of Doctor Yau

This is related somewhat distantly to Mathieu moonshine.

A note on the somewhat distant relation —

Illustration of K3 surface related to Mathieu moonshine

See also Kummer in this journal.

Saturday, March 14, 2015

Introduction to Yau

Filed under: General — m759 @ 12:00 PM

Home page of Doctor Yau

This is related somewhat distantly to Mathieu moonshine.

Monday, June 10, 2013

Galois Coordinates

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

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

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

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

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

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

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

Related material: 

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

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

Tuesday, April 30, 2013

Logline

Filed under: General,Geometry — m759 @ 9:29 AM

Found this morning in a search:

logline  is a one-sentence summary of your script.
www.scriptologist.com/Magazine/Tips/Logline/logline.html
It's the short blurb in TV guides that tells you what a movie
is about and helps you decide if you're interested 

The search was suggested by a screenwriting weblog post,
"Loglines: WHAT are you doing?".

What is your story about?
No, seriously, WHAT are you writing about?
Who are the characters? What happens to them?
Where does it take place? What’s the theme?
What’s the style? There are nearly a million
little questions to answer when you set out
to tell a story. But it all starts with one
super, overarching question.
What are you writing about? This is the first
big idea that we pull out of the ether, sometimes
before we even have any characters.
What is your story about?

The screenwriting post was found in an earlier search for
the highlighted phrase.

The screenwriting post was dated December 15, 2009.

What I am doing now  is checking for synchronicity.

This  weblog on December 15, 2009, had a post
titled A Christmas Carol. That post referred to my 1976
monograph titled Diamond Theory .

I guess the script I'm summarizing right now is about
the heart of that theory, a group of 322,560 permutations
that preserve the symmetry of a family of graphic designs.

For that group in action, see the Diamond 16 Puzzle.

The "super overarching" phrase was used to describe
this same group in a different context:

IMAGE- Anne Taormina on 'Mathieu Moonshine' and the 'super overarching symmetry group'

This is from "Mathieu Moonshine," a webpage by Anne Taormina.

A logline summarizing my  approach to that group:

Finite projective geometry explains
the surprising symmetry properties
of some simple graphic designs— 
found, for instance, in quilts.

The story thus summarized is perhaps not destined for movie greatness.

Saturday, April 27, 2013

Mark and Remark

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

“Fact and fiction weave in and out of novels like a shell game.” —R.B. Kitaj

Not just novels.

Fact: 

IMAGE- Anne Taormina on 'Mathieu Moonshine' and the 'super overarching symmetry group'

The mark preceding A8  in the above denotes the semidirect product.

Symbol from the box-style
I Ching  (Cullinane, 1/6/89).
This is Hexagram 55,
“Abundance [Fullness].”

The mathematical quote, from last evening’s Symmetry, is from Anne Taormina.

The I Ching  remark is not.

Another version of Abbondanza —

IMAGE- Taormina sunset from inabbondanza.com on June 22, 2009

Fiction:

Found in Translation and the giorno  June 22, 2009, here.

Friday, April 26, 2013

Symmetry

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

Anne Taormina on Mathieu Moonshine —

IMAGE- Anne Taormina on 'Mathieu Moonshine' and the 'super overarching symmetry group'

This is, of course, the same group (of order 322,560) underlying the Diamond 16 Puzzle.

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