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
, February 1979, pages A-193, 194.

* The Galois tesseract .

Update of March 15, 2020 —

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

Thursday, March 7, 2013

Ten Years After

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

Rock guitarist Alvin Lee, a founder of
the band Ten Years After , died
on March 6, 2013 (Michelangelo's
birthday). In his memory, a figure
from a post Ten Years Before —

Plato's reported motto for his Academy:
"Let no one ignorant of geometry enter."

For visual commentary by an artist ignorant
of geometry, see a work by Sol LeWitt.

For verbal commentary by an art critic  ignorant
of geometry, see a review of LeWitt by
Robert Hughes—

"A Beauty Really Bare" (TIME, Feb. 6, 2001).

See also Ten Years Group and Four Gods.

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.

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)


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)


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



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.

Tuesday, September 20, 2011

Relativity Problem Revisited

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

A footnote was added to Finite Relativity


Weyl on what he calls the relativity problem

IMAGE- Weyl in 1949 on the relativity problem

“The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time.”

– Hermann Weyl, 1949, “Relativity Theory as a Stimulus in Mathematical Research

“This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.”

– Hermann Weyl, 1946, The Classical Groups , Princeton University Press, p. 16

…. A note of Feb. 20, 1986, supplied an example of such coordinatizations in finite geometry. In that note, the group of mediating transformations acted directly on  coordinates within a 4×4 array. When the 4×4 array is embedded in a 4×6 array, a larger and more interesting group, M 24 (containing the original group), acts on the larger array.  There is no obvious solution to Weyl’s relativity problem for M 24.  That is, there is no obvious way* to apply exactly 24 distinct transformable coordinate-sets (or symbol-strings ) to the 24 array elements in such a way that the natural group of mediating transformations of the 24 symbol-strings is M 24. ….

Footnote of Sept. 20, 2011:

* R.T. Curtis has, it seems, a non-obvious way that involves strings of seven symbols.  His abstract for a 1990 paper says that in his construction “The generators of M 24 are defined… as permutations of twenty-four 7-cycles in the action of PSL2(7) on seven letters….”

See “Geometric Interpretations of the ‘Natural’ Generators of the Mathieu groups,” by R.T. Curtis,  Mathematical Proceedings of the Cambridge Philosophical Society  (1990), Vol. 107, Issue 01, pp. 19-26. (Rec. Jan. 3, 1989, revised Feb. 3, 1989.) This paper was published online on Oct. 24, 2008.

Some related articles by Curtis:

R.T. Curtis, “Natural Constructions of the Mathieu groups,” Math. Proc. Cambridge Philos. Soc.  (1989), Vol. 106, pp. 423-429

R.T. Curtis. “Symmetric Presentations I: Introduction, with Particular Reference to the Mathieu groups M 12  and M 24” In Proceedings of 1990 LMS Durham Conference ‘Groups, Combinatorics and Geometry’  (eds. M. W. Liebeck and J. Saxl),  London Math. Soc. Lecture Note Series 165, Cambridge University Press, 1992, pp. 380–396

R.T. Curtis, “A Survey of Symmetric Generation of Sporadic Simple Groups,” in The Atlas of Finite Groups: Ten Years On , (eds. R.T. Curtis and R.A. Wilson), London Math. Soc. Lecture Note Series 249, Cambridge University Press, 1998, pp. 39–57

Thursday, December 2, 2010


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

The Dreidel Is Cast

The Nietzschean phrase "ruling and Caesarian spirits" occurred in yesterday morning's post "Novel Ending."

That post was followed yesterday morning by a post marking, instead, a beginning— that of Hanukkah 2010. That Jewish holiday, whose name means "dedication," commemorates the (re)dedication of the Temple in Jerusalem in 165 BC.

The holiday is celebrated with, among other things, the Jewish version of a die—  the dreidel . Note the similarity of the dreidel  to an illustration of The Stone*  on the cover of the 2001 Eerdmans edition of  Charles Williams's 1931 novel Many Dimensions


For mathematics related to the dreidel , see Ivars Peterson's column on this date fourteen years ago.
For mathematics related (if only poetically) to The Stone , see "Solomon's Cube" in this journal.

Here is the opening of Many Dimensions


For a fanciful linkage of the dreidel 's concept of chance to The Stone 's concept of invariant law, note that the New York Lottery yesterday evening (the beginning of Hanukkah) was 840. See also the number 840 in the final post (July 20, 2002) of the "Solomon's Cube" search.

Some further holiday meditations on a beginning—

Today, on the first full day of Hanukkah, we may or may not choose to mark another beginning— that of George Frederick James Temple, who was born in London on this date in 1901. Temple, a mathematician, was President of the London Mathematical Society in 1951-1953. From his MacTutor biography

"In 1981 (at the age of 80) he published a book on the history of mathematics. This book 100 years of mathematics (1981) took him ten years to write and deals with, in his own words:-

those branches of mathematics in which I had been personally involved.

He declared that it was his last mathematics book, and entered the Benedictine Order as a monk. He was ordained in 1983 and entered Quarr Abbey on the Isle of Wight. However he could not stop doing mathematics and when he died he left a manuscript on the foundations of mathematics. He claims:-

The purpose of this investigation is to carry out the primary part of Hilbert's programme, i.e. to establish the consistency of set theory, abstract arithmetic and propositional logic and the method used is to construct a new and fundamental theory from which these theories can be deduced."

For a brief review of Temple's last work, see the note by Martin Hyland in "Fundamental Mathematical Theories," by George Temple, Philosophical Transactions of the Royal Society, A, Vol. 354, No. 1714 (Aug. 15, 1996), pp. 1941-1967.

The following remarks by Hyland are of more general interest—

"… one might crudely distinguish between philosophical and mathematical motivation. In the first case one tries to convince with a telling conceptual story; in the second one relies more on the elegance of some emergent mathematical structure. If there is a tradition in logic it favours the former, but I have a sneaking affection for the latter. Of course the distinction is not so clear cut. Elegant mathematics will of itself tell a tale, and one with the merit of simplicity. This may carry philosophical weight. But that cannot be guaranteed: in the end one cannot escape the need to form a judgement of significance."

— J. M. E. Hyland. "Proof Theory in the Abstract." (pdf)
Annals of Pure and Applied Logic 114, 2002, 43-78.

Here Hyland appears to be discussing semantic ("philosophical," or conceptual) and syntactic ("mathematical," or structural) approaches to proof theory. Some other remarks along these lines, from the late Gian-Carlo Rota


    (Click to enlarge.)

See also "Galois Connections" at alpheccar.org and "The Galois Connection Between Syntax and Semantics" at logicmatters.net.

* Williams's novel says the letters of The Stone  are those of the Tetragrammaton— i.e., Yod, He, Vau, He  (cf. p. 26 of the 2001 Eerdmans edition). But the letters on the 2001 edition's cover Stone  include the three-pronged letter Shin , also found on the dreidel .  What esoteric religious meaning is implied by this, I do not know.

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