Overarching Group

Sunday, March 6, 2022

Overarching Symmetries

Filed under: General — Tags: Kummerhenge, Overarching Group — m759 @ 1:47 pm

By the Daniel J. Peterson whose Swarthmore honors thesis was quoted
here last night —

"What, then, is the relationship between theory-relative symmetries
(physical symmetries) and theory-independent symmetries
(overarching symmetries)? My statement of this problem is
a bit abstract, so let’s look at an example: classical Newtonian gravity
and classical electromagnetism . . . ."

— Prospects for a New Account of Time Reversal
by Daniel J. Peterson, Ph.D. dissertation, U. Mich., 2013, p. 16.

Another 2013 approach to the word "overarching" and sytmmetries —

Other terms of interest: Tenet , Nolanism , and Magic for Liars .

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Saturday, January 29, 2022

On the Diamond-Theorem Group* of Order 322,560

Filed under: General — Tags: Change Arises, Embedding Change, Kummerhenge, Octad, Overarching Group, Permanence, Priority, Strange Change — m759 @ 2:06 am

Taormina and Wendland have often discussed this group, which they
call "overarching" within the context of their Mathieu-moonshine research.

This seems to be the first time they have attempted to explore its geometric
background as an affine group, apart from its role as "the octad group" in the
researches of R. T. Curtis and John Conway on the large Mathieu group M₂₄.

* See a Log24 post of June 1, 2013.

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Friday, October 8, 2021

Square Dance, 1979-2021

Filed under: General — Tags: Gaberdiel, Mathieu Moonshine, Octad, Octad Generator, Octad Group, Overarching, Overarching Group — m759 @ 9:45 am

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Thursday, December 3, 2020

Brick Joke

Filed under: General — Tags: Brick Symmetry, Kummerhenge, Octad, Octad Generator, Octad Group, Overarching Group, Synchronology, The Lindelof Watchmen — 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 —

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Thursday, April 23, 2020

Octads and Geometry

Filed under: General — Tags: Art Issue, Brick Space, Brick Symmetry, Brick Wall, Devs, Octad, Octad Generator, Octad Group, Overarching Group — 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 —

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). "A new combinatorial approach to M₂₄,"
Mathematical Proceedings of the Cambridge Philosophical Society ,
79, pp 25-42.

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Sunday, March 15, 2020

The “Octad Group”

Filed under: General — Tags: Octad, Octad Generator, Octad Group, Overarching Group, Priority — 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.

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Saturday, March 7, 2020

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

Filed under: General — Tags: Gaberdiel, Mathieu Moonshine, Octad, Octad Generator, Octad Group, Overarching, Overarching Group — 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 G_i 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 O₉= {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 —

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Monday, March 11, 2019

Overarching Metanarratives

Filed under: General — Tags: Overarching Group — m759 @ 4:15 am

See also "Overarching + Tesseract" in this journal. From the results
of that search, some context for the "inscape" of the previous post —

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Monday, April 23, 2018

Super Symmetry Surfing

Filed under: General — Tags: No Joke, Overarching Group — m759 @ 11:17 am

Midrash —

Backstory — Search this journal for Taormina.

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Monday, September 12, 2016

The Kummer Lattice

Filed under: General,Geometry — Tags: Clash of the Titans, Geometry, Kummerhenge, Mathieu Moonshine, Overarching, Overarching Group, Priority — m759 @ 2:00 pm

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

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

Some context for the 4×4 array —

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

Further background on the Kummer lattice:

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

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

Some further context …

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

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

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

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

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Friday, February 26, 2016

Overarching

Filed under: General,Geometry — Tags: For the Mann Brothers, Overarching Group — m759 @ 4:00 pm

(Continued)

"The study of social memory allows scholars to
understand how different memories form within
a collective group, thus exploring the societal
and ideological elements of disparate groups
that form the over-arching memory of Melkisedeq."

— The Melkisedeq Memoirs , by Cale Staley,
2015 master's thesis at the University of Iowa

Elements of groups that I prefer —

"Right through hell
there is a path…."
— Malcolm Lowry,
Under the Volcano

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Thursday, December 3, 2015

Overarching Symmetry

Filed under: General,Geometry — Tags: Foiled, Overarching Group, Priority, Springer — m759 @ 10:45 pm

(Continued)

From p. 34 of the preprint "Snapshots of Conformal Field Theory,"
by Katrin Wendland, arXiv, 11 April 2014 —

50. Gannon, T.: Much ado about Mathieu (arXiv:1211.5531 [math.RT])

85. Taormina, A., Wendland, K.: The overarching finite symmetry group
of Kummer surfaces in the Mathieu group M24. JHEP 08, 125 (2013)

86. Taormina, A., Wendland, K.: Symmetry-surfing the moduli space
of Kummer K3s (arXiv:1303.2931 [hep-th])

87. Taormina, A., Wendland, K.: A twist in the M24 moonshine story
(arXiv:1303.3221 [hep-th])

The Wendland paper was published on Jan. 7, 2015, in
Mathematical Aspects of Quantum Field Theories ,
edited by Damien Calaque and Thomas Strobl
(Springer Mathematical Physics Studies), pages 89-129.

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Monday, May 25, 2015

A Stitch in Time

Filed under: General,Geometry — Tags: May 19 Gestalt, Overarching Group, Priority — m759 @ 12:00 am

The most recent version of a passage
quoted in posts tagged "May 19 Gestalt" —

"You've got to pick up every stitch." — Donovan

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Monday, June 10, 2013

Galois Coordinates

Filed under: General,Geometry — Tags: Geometry, Overarching Group, Priority — 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 M ₂₄.

See a preprint by Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group M 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.

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Sunday, June 9, 2013

Sicilian Reflections

Filed under: General,Geometry — Tags: Overarching Group — m759 @ 9:00 am

(Continued from Sept. 22, 2011)

See Taormina in this journal, and the following photo of "Anne Newton"—

Click photo for context.

Related material:

"Super Overarching" in this journal,
a group of order 322,560, and…

See also the MAA Spectrum program —

— and an excerpt from the above book:

Backstory —

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Sunday, May 19, 2013

Priority Claim

Filed under: General,Geometry — Tags: Geometry, Mathieu Moonshine, May 19 Gestalt, Octad, Octad Generator, Overarching, Overarching Group, Priority, Rosenhain and Göpel — m759 @ 9:00 am

From an arXiv preprint submitted July 18, 2011,
and last revised on March 11, 2013 (version 4):

"By our construction, this vector space is the dual
of our hypercube F₂⁴ built on I \ O₉. The vector space
structure of the latter, to our knowledge, is first
mentioned by Curtis in [Cur89]. Hence altogether
our proposition 2.3.4 gives a novel geometric
meaning in terms of Kummer geometry to the known
vector space structure on I \ O₉."

[Cur89] reference:
R. T. Curtis, "Further elementary techniques using
the miracle octad generator," Proc. Edinburgh
Math. Soc. 32 (1989), 345-353 (received on
July 20, 1987).

— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group M ₂₄,"
arXiv.org > hep-th > arXiv:1107.3834

"First mentioned by Curtis…."

No. I claim that to the best of my knowledge, the
vector space structure was first mentioned by me,
Steven H. Cullinane, in an AMS abstract submitted
in October 1978, some nine years before the
Curtis article.

Update of the above paragraph on July 6, 2013—

No. The vector space structure was described by
(for instance) Peter J. Cameron in a 1976
Cambridge University Press book —
Parallelisms of Complete Designs .
See the proof of Theorem 3A.13 on pages 59 and 60.

The vector space structure as it occurs in a 4×4 array
of the sort that appears in the Curtis Miracle Octad
Generator may first have been pointed out by me,
Steven H. Cullinane, in an AMS abstract submitted in
October 1978, some nine years before the Curtis article.

See Notes on Finite Geometry for some background.

See in particular The Galois Tesseract.

For the relationship of the 1978 abstract to Kummer
geometry, see Rosenhain and Göpel Tetrads in PG(3,2).

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Wednesday, May 1, 2013

The Crosswicks Curse

Filed under: General,Geometry — Tags: Crosswicks Curse, Overarching Group — 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.

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Tuesday, April 30, 2013

Logline

Filed under: General,Geometry — Tags: Overarching Group, Stakeholder, Steinberg Deathday — m759 @ 9:29 am

Found this morning in a search:

A 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:

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.

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Saturday, April 27, 2013

Mark and Remark

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

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

Not just novels.

Fact:

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 —

Fiction:

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

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Friday, April 26, 2013

Symmetry

Filed under: General,Geometry — Tags: Overarching Group, Priority, Writing on Glass — m759 @ 7:00 pm

Anne Taormina on Mathieu Moonshine —

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

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