Click to enlarge.
Tuesday, December 17, 2019
Friday, November 29, 2019
Verifying Aitchison’s Cuboctahedral Generation of M_{24}
Shown below are Aitchison's March 2018 M_{24} permutations
and their relabeling, with digits only, for MAGMA checking.
In the versions below, r g b stand for red, green, blue.
Infinity has been replaced by 7 (because a digit was needed,
and the position of the infinity symbol in the Aitchison cube
was suited to the digit 7).
(r7,r1)(b2,g4)(r3,r5)(r6,g0)
mu0= (g7,g2)(r4,b1)(g6,g3)(g5,b0)
(b7,b4)(g1,r2)(b5,b6)(b3,r0)
mu1 = (r7,r2,)(b3,g5)(r4,r6)(r0,g1)
(g7,g3)(r5,b2)(g0,g4)(g6,b1)
(b7,b5)(g2,r3)(b6,b0)(b4,r1)
mu2 = (r7,r3)(b4,g6)(r5,r0)(r1,g2)
(g7,g4)(r6,b3)(g1,g5)(g0,b2)
(b7,b6)(g3,r4)(b0,b1)(b5,r2)
mu3 = (r7,r4)(b5,g0)(r6,r1)(r2,g3)
(g7,g5)(r0,b4)(g2,g6)(g1,b3)
(b7,b0)(g4,r5)(b1,b2)(b6,r3)
mu4 = (r7,r5)(b6,g1)(r0,r2)(r3,g4)
(g7,g6)(r1,b5)(g3,g0)(g2,b4)
(b7,b1)(g5,r6)(b2,b3)(b0,r4)
mu5 = (r7,r6)(b0,g2)(r1,r3)(r4,g5)
(g7,g0)(r2,b6)(g4,g1)(g3,b5)
(b7,b2)(g6,r0)(b3,b4)(b1,r5)
mu6 = (r7,r0)(b1,g3)(r2,r4)(r5,g6)
(g7,g1)(r3,b0)(g5,g2)(g4,b6)
(b7,b3)(g0,r1)(b4,b5)(b2,r6)
Table 1 —
0 1 2 3 4 5 6 7
r 1 2 3 4 5 6 7 8
g 9 10 11 12 13 14 15 16
b 17 18 19 20 21 22 23 24
The wReplace program was used with Table 1 above
to rewrite mu0mu6 for MAGMA.
The resulting code for MAGMA —
G := sub< Sym(24) 
(8,3)(20,14)(5,7)(1,10)
(8,4)(21,15)(6,1)(2,11)
(8,5)(22,9)(7,2)(3,12)
(8,6)(23,10)(1,3)(4,13)
(8,7)(17,11)(2,4)(5,14)
(8,1)(18,12)(3,5)(6,15)
G; 
The Aitchison generators passed the MAGMA test.
Tuesday, November 26, 2019
Alea Iacta Est*
Saturday evening's post Diamond Globe suggests a review of …
Iain Aitchison on symmetric generation of M_{24} —
* A Greek version for the late John SImon:
«Ἀνερρίφθω κύβος».
Sunday, November 24, 2019
Hyperbolic Memorial
From "Back to the Saddle," a post of Nov. 23, 2010 —
"A characteristic property of hyperbolic geometry
is that the angles of a triangle add to less
than a straight angle (half circle)." — Wikipedia
See as well . . .
Saturday, November 23, 2019
Diamond Globe
An image from All Souls' Day 2010 —
This is from earlier posts tagged Permutahedron.
See also
Wallace Stevens:
A World of Transforming Shapes.
From that book (click to enlarge) —
"Before time began, there was the Cube."
— Optimus Prime.
Also from earlier posts tagged Permutahedron —
Thursday, October 3, 2019
Apocalypse* Note
For a first look at octad.space, see that domain.
For a second look, see octad.design.
For some other versions, see Aitchison in this journal.
Thursday, June 27, 2019
Group Actions on the 4x4x4 Cube
For affine group actions, see Ex Fano Appollinis (June 24)
and Solomon's Cube.
For one approach to Mathieu group actions on a 24cube subset
of the 4x4x4 cube, see . . .
For a different sort of Mathieu cube, see Aitchison.
Friday, June 21, 2019
Cube Tales for Solstice Day
Saturday, June 1, 2019
Cuboctahedron Labeling Update
See this evening's update to the May 31 post
"Working Sketch of Aitchison’s Mathieu Cuboctahedron" —
". . . And then of course there is the obvious labeling derived from
the … permutahedron —"
Friday, May 31, 2019
Working Sketch of Aitchison’s Mathieu Cuboctahedron
The above sketch indicates one way to apply the elements of S_{4}
to the Aitchison cuboctahedron . It is a rough sketch illustrating a
correspondence between four edgehexagons and four labelsets.
The labeling is not as neat as that of a permutahedron by S_{4}
shown below, but can perhaps be improved.
Permutahedron labeled by S_{4} .
Update of 9 PM EDT June 1, 2019 —
. . . And then of course there is the obvious labeling derived from
the above permutahedron —
Saturday, May 4, 2019
The Chinese Jars of ShingTung Yau
The title refers to CalabiYau spaces.
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 12dimensional space over GF(2).
Related material from Plato and R. T. Curtis —
A related CalabiYau "Chinese jar" first described in detail in 1905 —
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/K3pix.pdf
Wednesday, January 16, 2019
The Dreaming Polyhedron
"Here is a recipe for preparing a copy of the Mathieu group M_{24}.
The main ingredient is a genus3 regular polyhedron X
with 56 triangular faces, 84 edges, and 24 vertices.
The most delicate part of this recipe is to hold the polyhedron
by the 24 vertices and immerse the rest of it in 3dimensional space."
— "How to Make the Mathieu Group M_{24} ," undated webpage
by David A. Richter, Western Michigan University
Illustration from that page —
"Another model of the (universal cover of the) polyhedron X"
Related fiction —
Cover of a 1971 British paperback edition of The Dreaming Jewels,
a story by Theodore Sturgeon (first version published in 1950):
Discuss Richter's model and the Sturgeon tale
in the context of posts tagged Aitchison.
Tuesday, January 15, 2019
Sunday, January 13, 2019
Sunday the Thirteenth (Revisited)
Friday, January 11, 2019
Permutations at Oslo
See also yesterday's Archimedes at Hiroshima and the
above 24 graphic permutations on All Souls' Day 2010.
For some backstory, see Narrative Line (November 10, 2014).
Thursday, January 10, 2019
Archimedes at Hiroshima
Two examples from the Wikipedia article "Archimedean solid" —
Iain Aitchison said in a talk last year at Hiroshima that
the Mathieu group M_{24 }can be represented as permuting
naturally the 24 edges of the cuboctahedron.
The 24 vertices of the truncated octahedron are labeled
naturally by the 24 elements of S_{4} in a permutahedron —
Can M_{24} be represented as permuting naturally
the 24 vertices of the truncated octahedron?
Tuesday, January 1, 2019
Sunday, December 30, 2018
Also Sprach Aitchison
The New Yorker reviewing "Bumblebee" —
"There is one reliable source for superhero sublimity,
and it’s all the more surprising that it’s a franchise with
no sacred inspiration whatsoever but, rather, of purely
and unabashedly mercantile origins: the 'Transformers'
series, based on a set of toys, in which Michael Bay’s
exhilarating filmmaking offers phantasmagorical textures
of an uncanny unconscious resonance."
— Richard Brody on December 29, 2018
"Before time began, there was the Cube."
— Optimus Prime
Some backstory — A Riddle for Davos, Jan. 22, 2014.
Sunday, December 9, 2018
Quaternions in a Small Space
The previous post, on the 3×3 square in ancient China,
suggests a review of group actions on that square
that include the quaternion group.
Click to enlarge —
Three links from the above finitegeometry.org webpage on the
quaternion group —

Visualizing GL(2,p) — A 1985 note illustrating group actions
on the 3×3 (ninefold) square. 
Another 1985 note showing group actions on the 3×3 square
transferred to the 2x2x2 (eightfold) cube.  Quaternions in an Affine Galois Plane — A webpage from 2010.
Related material —
See as well the two Log24 posts of December 1st, 2018 —
Character and In Memoriam.
Friday, December 7, 2018
An Ark for Hanukkah
From religionnews.com —
"The word 'Hanukkah' means dedication.
It commemorates the rededicating of the
ancient Temple in Jerusalem in 165 B.C. . . . ."
From The New York Times this morning —
Related material —
From this journal on Wednesday, December 5, 2018 —
Megan Fox in "Transformers" (2007) —
From a Google image search this morning —
The image search was suggested by recent posts tagged Aitchison
and by this morning's previous post.
Thursday, December 6, 2018
The Mathieu Cube of Iain Aitchison
This journal ten years ago today —
Surprise Package
From a talk by a Melbourne mathematician on March 9, 2018 —
The source — Talk II below —
Search Results

Related material —
The 56 triangles of the eightfold cube . . .
 in Aitchison's March 9, 2018, talk (slides 3234), and
 in this journal on July 25, 2008, and later.
Image from Christmas Day 2005.
Wednesday, December 5, 2018
Caesarian (continued)
Tuesday, December 4, 2018
Melbourne Noir
Monday, December 3, 2018
The Relativity Problem at Hiroshima
“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, The Classical Groups ,
Princeton University Press, 1946, p. 16
See also Relativity Problem and Diamonds and Whirls.
Sunday, December 2, 2018
Symmetric Generation …
Symmetry at Hiroshima
A search this morning for articles mentioning the Miracle Octad Generator
of R. T. Curtis within the last year yielded an abstract for two talks given
at Hiroshima on March 8 and 9, 2018 —
http://www.math.sci.hiroshimau.ac.jp/ Iain AITCHISON Title: Construction of highly symmetric Riemann surfaces, related manifolds, and some exceptional objects, I, II Abstract: Since antiquity, some mathematical objects have played a special role, underpinning new mathematics as understanding deepened. Perhaps archetypal are the Platonic polyhedra, subsequently related to Platonic idealism, and the contentious notion of existence of mathematical reality independent of human consciousness. Exceptional or unique objects are often associated with symmetry – manifest or hidden. In topology and geometry, we have natural base points for the moduli spaces of closed genus 2 and 3 surfaces (arising from the 2fold branched cover of the sphere over the 6 vertices of the octahedron, and Klein's quartic curve, respectively), and Bring's genus 4 curve arises in Klein's description of the solution of polynomial equations of degree greater than 4, as well as in the construction of the HorrocksMumford bundle. Poincare's homology 3sphere, and Kummer's surface in real dimension 4 also play special roles. In other areas: we have the exceptional Lie algebras such as E8; the sporadic finite simple groups; the division algebras: Golay's binary and ternary codes; the Steiner triple systems S(5,6,12) and S(5,8,24); the Leech lattice; the outer automorphisms of the symmetric group S6; the triality map in dimension 8; and so on. We also note such as: the 27 lines on a cubic, the 28 bitangents of a quartic curve, the 120 tritangents of a sextic curve, and so on, related to Galois' exceptional finite groups PSL2(p) (for p= 5,7,11), and various other socalled `Arnol'd Trinities'. Motivated originally by the `Eightfold Way' sculpture at MSRI in Berkeley, we discuss interrelationships between a selection of these objects, illustrating connections arising via highly symmetric Riemann surface patterns. These are constructed starting with a labeled polygon and an involution on its label set. Necessarily, in two lectures, we will neither delve deeply into, nor describe in full, contexts within which exceptional objects arise. We will, however, give sufficient definition and detail to illustrate essential interconnectedness of those exceptional objects considered. Our starting point will be simplistic, arising from ancient Greek ideas underlying atomism, and Plato's concepts of space. There will be some overlap with a previous talk on this material, but we will illustrate with some different examples, and from a different philosophical perspective. Some new results arising from this work will also be given, such as an alternative graphicillustrated MOG (Miracle Octad Generator) for the Steiner system S(5,8,24), and an alternative to Singerman – Jones' genus 70 Riemann surface previously proposed as a completion of an Arnol'd Trinity. Our alternative candidate also completes a Trinity whose two other elements are Thurston's highly symmetric 6 and 8component links, the latter related by Thurston to Klein's quartic curve. 
See also yesterday morning's post, "Character."
Update: For a followup, see the next Log24 post.
Saturday, December 1, 2018
Character
"What we do may be small, but it has
a certain character of permanence."
— G. H. Hardy,
A Mathematician's Apology
Saturday, March 24, 2018
Sure, Whatever.
The search for Langlands in the previous post
yields the following Toronto Star illustration —
From a review of the recent film "Justice League" —
"Now all they need is to resurrect Superman (Henry Cavill),
stop Steppenwolf from reuniting his three Mother Cubes
(sure, whatever) and wrap things up in under two cinematic
hours (God bless)."
For other cubic adventures, see yesterday's post on A Piece of Justice
and the block patterns in posts tagged Design Cube.
Friday, March 23, 2018
From the Personal to the Platonic
On the Oslo artist Josefine Lyche —
"Josefine has taken me through beautiful stories,
ranging from the personal to the platonic
explaining the extensive use of geometry in her art.
I now know that she bursts into laughter when reading
Dostoyevsky, and that she has a weird connection
with a retired mathematician."
— Ann Cathrin Andersen,
http://bryggmagasin.no/2017/behindtheglitter/
Personal —
The Rushkoff Logo
— From a 2016 graphic novel by Douglas Rushkoff.
See also Rushkoff and Talisman in this journal.
Platonic —
Compare and contrast the shifting hexagon logo in the Rushkoff novel above
with the hexagoninsideacube in my "Diamonds and Whirls" note (1984).
Monday, November 10, 2014
Narrative Line
"We live entirely, especially if we are writers, by the imposition
of a narrative line upon disparate images…." — Joan Didion
Narrative Line:
Disparate images:
Exercise:
Can the above narrative line be imposed in any sensible way
upon the above disparate images?
Friday, February 21, 2014
Raumproblem*
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—
Clearly there is a problem here analogous to
the squaretriangle 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 24trianglehexagon
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.
Sunday, October 6, 2013
Church with Josefine*
(Continued from last Sunday)
For some background, see Permutahedron in this journal.
See also…
 The Saturday evening post from yesterday
 The recent Arcade Fire "Reflektor" mirror ball video
 Malcolm Lowry on All Souls' Day, 2010
* Jews may prefer to retitle this post "Sunday Shul with Josefine"
and stage it as a SNL sketch, "Norwegian Disco," with
The Sunshine Girls. (For the Norwegian part, see Kristen Wiig,
of Norwegian ancestry. For the disco part, see Amy Adams,
who stars in a new discoera movie.)
Wednesday, January 11, 2012
Cuber
"Examples galore of this feeling must have arisen in the minds of the people who extended the Magic Cube concept to other polyhedra, other dimensions, other ways of slicing. And once you have made or acquired a new 'cube'… you will want to know how to export a known algorithm , broken up into its fundamental operators , from a familiar cube. What is the essence of each operator? One senses a deep invariant lying somehow 'down underneath' it all, something that one can’t quite verbalize but that one recognizes so clearly and unmistakably in each new example, even though that example might violate some feature one had thought necessary up to that very moment. In fact, sometimes that violation is what makes you sure you’re seeing the same thing , because it reveals slippabilities you hadn’t sensed up till that time….
… example: There is clearly only one sensible 4 × 4 × 4 Magic Cube. It is the answer; it simply has the right spirit ."
— Douglas R. Hofstadter, 1985, Metamagical Themas: Questing for the Essence of Mind and Pattern (Kindle edition, locations 1155711572)
See also Many Dimensions in this journal and Solomon's Cube.
Wednesday, September 21, 2011
Symmetric Generation
Suggested by yesterday's Relativity Problem Revisited and by Cassirer on Objectivity—
From Symmetric Generation of Groups , by R.T. Curtis (Cambridge U. Press, 2007)—
"… we are saying much more than that
some set of seven involutions, which would be a very weak
requirement. We are asserting that M_{ 24} is generated by a set
of seven involutions which possesses all the symmetries of
acting on the points of the 7point projective plane…."
— Symmetric Generation , p. 41
"It turns out that this approach is particularly revealing and that
many simple groups, both sporadic and classical, have surprisingly
simple definitions of this type."
— Symmetric Generation , p. 42
See also (click to enlarge)—
Cassirer's remarks connect the concept of objectivity with that of object .
The above quotations perhaps indicate how the Mathieu group
"This is the moment which I call epiphany. First we recognise that the object is one integral thing, then we recognise that it is an organised composite structure, a thing in fact: finally, when the relation of the parts is exquisite, when the parts are adjusted to the special point, we recognise that it is that thing which it is. Its soul, its whatness, leaps to us from the vestment of its appearance. The soul of the commonest object, the structure of which is so adjusted, seems to us radiant. The object achieves its epiphany."
— James Joyce, Stephen Hero
For a simpler object "which possesses all the symmetries of
For symmetric generation of
Friday, June 24, 2011
The Cube
Click the above image for some background.
Related material:
Skateboard legend Andy Kessler,
this morning's The Gleaming,
and But Sometimes I Hit London.