From this journal on the above publication date —
Related material — Geometric Group Theory in this journal.
From this journal on the above publication date —
Related material — Geometric Group Theory in this journal.
The December 2012 Notices of the American
Mathematical Society has an ad on page 1564
(in a review of two books on vulgarized mathematics)
for three workshops next year on "Lowdimensional
Topology, Geometry, and Dynamics"—
(Only the top part of the ad is shown; for further details
see an ICERM page.)
(ICERM stands for Institute for Computational
and Experimental Research in Mathematics.)
The ICERM logo displays seven subcubes of
a 2x2x2 eightcube array with one cube missing—
The logo, apparently a stylized image of the architecture
of the Providence building housing ICERM, is not unlike
a picture of Froebel's Third Gift—
Photo by Norman Brosterman from the Inventing Kindergarten
exhibit at The Institute for Figuring (cofounded by Margaret Wertheim)
The eighth cube, missing in the ICERM logo and detached in the
Froebel Cubes photo, may be regarded as representing the origin
(0,0,0) in a coordinatized version of the 2x2x2 array—
in other words the cube invariant under linear , as opposed to
more general affine , permutations of the cubes in the array.
These cubes are not without relevance to the workshops' topics—
lowdimensional exotic geometric structures, group theory, and dynamics.
See The Eightfold Cube, A Simple Reflection Group of Order 168, and
The Quaternion Group Acting on an Eightfold Cube.
Those who insist on vulgarizing their mathematics may regard linear
and affine group actions on the eight cubes as the dance of
Snow White (representing (0,0,0)) and the Seven Dwarfs—
.
A Google search today yielded no results
for the phrase "congruent group actions."
Places where this phrase might prove useful include—
The phrase "Blue Dream" in the previous post
suggests a Web search for Traumnovelle .
That search yields an interesting weblog post
from 2014 commemorating the 1999 dies natalis
(birth into heaven) of St. Stanley Kubrick.
Related material from March 7, 2014,
in this journal —
That 2014 post was titled "Kummer Varieties." It is now tagged
"Kummerhenge." For some backstory, see other posts so tagged.
A New, Improved Version of Quantum Suffering !
Background for group actions on the eightfold cube —
See also other posts now tagged Quantum Suffering
as well as — related to the image above of the Great Wall —
The title is from this morning's online New York Times review
of a new Jackie Chan film.
Click the image below for some related posts.
The previous two posts dealt, rather indirectly, with
the notion of "cube bricks" (Cullinane, 1984) —
Group actions on partitions —
Cube Bricks 1984 —
Another mathematical remark from 1984 —
For further details, see Triangles Are Square.
From the Log24 post "A Point of Identity" (August 8, 2016) —
A logo that may be interpreted as oneeighth of a 2x2x2 array
of cubes —
The figure in white above may be viewed as a subcube representing,
when the eightcube array is coordinatized, the identity (i.e., (0, 0, 0)).
Recent remarks related to the July 29 death of Landon T. Clay
suggest a review of a notable figure associated with Clay.
From a 2006 obituary of mathematician George Mackey —
"A deep thinker whose work in representation theory,
group actions, and functional analysis helped
bring closer together the fields of math and physics,
Dr. Mackey died March 15 of complications from
pneumonia. He was 90, had lived in Cambridge, and
was Landon T. Clay professor emeritus at Harvard University."
— Bryan Marquard, Boston Globe, April 28, 2006
See also this journal on the date of Mackey's death (posts now tagged
Ides of March 2006).
Continuing the previous post's theme …
Group actions on partitions —
Cube Bricks 1984 —
Related material — Posts now tagged Device Narratives.
The above 1985 note was an attempt to view the diamond theorem
in a more general context. I know no more about the note now than
I did in 1985. The only item in the search results above that is not
by me (the seventh) seems of little relevance.
The contraction of the title is from group actions on
the ninefold square (with the center subsquare fixed)
to group actions on the eightfold cube.
From a post of June 4, 2014 …
At math.stackexchange.com on March 112, 2013:
“Is there a geometric realization of the Quaternion group?” —
The above illustration, though neatly drawn, appeared under the
cloak of anonymity. No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note “GL(2,3) actions on a cube” of April 5, 1985).
"And as the characters in the meme twitch into the abyss
that is the sky, this meme will disappear into whatever
internet abyss swallowed MySpace."
—Staff writer Kamila Czachorowski, Harvard Crimson today
From Log24 posts tagged Art Space —
From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
“The Universal Kummer Threefold,” by
Qingchun Ren, Steven V Sam, Gus Schrader, and
Bernd Sturmfels —
Two such considerations —
From mathematician Izabella Laba today —
From Harry T. Antrim’s 1967 thesis on Eliot —
“That words can be made to reach across the void
left by the disappearance of God (and hence of all
Absolutes) and thereby reestablish some basis of
relation with forms existing outside the subjective
and egocentered self has been one of the chief
concerns of the first half of the twentieth century.”
… And then there is the Snow White void —
A logo that may be interpreted as oneeighth of a 2x2x2 array
of cubes —
The figure in white above may be viewed as a subcube representing,
when the eightcube array is coordinatized, the identity (i.e., (0, 0, 0)).
See posts tagged SpiegelSpiel.
"Mirror, Mirror …." —
A logo that may be interpreted as oneeighth of
a 2x2x2 array of cubes —
The figure in white above may be viewed as a subcube representing,
when the eightcube array is coordinatized, the identity (i.e., (0, 0, 0)).
For a Monkey Grammarian (Viennese Version)
"At the point of convergence by Octavio Paz, translated by Helen Lane

A logo that may be interpreted as oneeighth of a 2x2x2 array
of cubes —
The figure in white above may be viewed as a subcube representing,
when the eightcube array is coordinatized, the identity (i.e., (0, 0, 0)).
Shown below are a few variations on the figure by VCQ,
the Vienna Center for Quantum Science and Technology —
(Click image to enlarge.)
In this post, "Omega" denotes a generic 4element set.
For instance … Cullinane's
or Schmeikal's
.
The mathematics appropriate for describing
group actions on such a set is not Schmeikal's
Clifford algebra, but rather Galois's finite fields.
Three notes on local symmetries
that induce global symmetries
From July 1, 2011 —
From November 5, 1981 —
From December 24, 1981 —
Foreword by Sir Michael Atiyah —
"Poincaré said that science is no more a collection of facts
than a house is a collection of bricks. The facts have to be
ordered or structured, they have to fit a theory, a construct
(often mathematical) in the human mind. . . .
… Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to
explain how art fits into our subject and what we mean by beauty.
In attempting to bridge this divide I have always found that
architecture is the best of the arts to compare with mathematics.
The analogy between the two subjects is not hard to describe
and enables abstract ideas to be exemplified by bricks and mortar,
in the spirit of the Poincaré quotation I used earlier."
— Sir Michael Atiyah, "The Art of Mathematics"
in the AMS Notices , January 2010
Judy Bass, Los Angeles Times , March 12, 1989 —
"Like Rubik's Cube, The Eight demands to be pondered."
As does a figure from 1984, Cullinane's Cube —
For natural group actions on the Cullinane cube,
see "The Eightfold Cube" and
"A Simple Reflection Group of Order 168."
See also the recent post Cube Bricks 1984 —
Related remark from the literature —
Note that only the static structure is described by Felsner, not the
168 group actions discussed by Cullinane. For remarks on such
group actions in the literature, see "Cube Space, 19842003."
(From Anatomy of a Cube, Sept. 18, 2011.)
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 M_{24},
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?
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.
See Found Symbol in this journal.
See also the Imperial College theorem symbol
and a page from Imperial College about
group actions on a space Ω —
For Han Solo, some less imperial symbology —
Detail of a CKEditor plugin screenshot:
horizontal line, smiley, special characters,
and iframe area.
The title refers to a Scientific American weblog item
discussed here on May 31, 2014:
Some closely related material appeared here on
Dec. 30, 2011:
A version of the above quaternion actions appeared
at math.stackexchange.com on March 12, 2013:
"Is there a geometric realization of Quaternion group?" —
The above illustration, though neatly drawn, appeared under the
cloak of anonymity. No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note "GL(2,3) actions on a cube" of April 5, 1985).
Galois matrices, the subject of the previous post,
are of course not new. See, for instance, Steinberg in 1951:
The American Mathematical Society reports that Steinberg died
on May 25, 2014.
As the above 1951 paper indicates, Steinberg was well acquainted with
what Weyl called “the devil of abstract algebra.” In this journal, however,
Steinberg himself appears rather as an angel of geometry.
The Dream of the Expanded Field continues…
From Klein's 1893 Lectures on Mathematics —
"The varieties introduced by Wirtinger may be called Kummer varieties…."
— E. Spanier, 1956
From this journal on March 10, 2013 —
From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
"The Universal Kummer Threefold," by
Qingchun Ren, Steven V Sam, Gus Schrader, and Bernd Sturmfels —
Two such considerations —
Update of 10 PM ET March 7, 2014 —
The following slides by one of the "Kummer Threefold" authors give
some background related to the above 64point vector space and
to the Weyl group of type E_{7}, W (E_{7}):
The Cayley reference is to "Algorithm for the characteristics of the
triple ϑfunctions," Journal für die Reine und Angewandte
Mathematik 87 (1879): 165169. <http://eudml.org/doc/148412>.
To read this in the context of Cayley's other work, see pp. 441445
of Volume 10 of his Collected Mathematical Papers .
See a Haaretz story commemorating the Feb. 14,
1917, birthday of a crystallographer.
Related material in this journal —
At the Still Point (June 15, 2013):
The illustration is for those who, like Andy Magid and
Steven Strogatz in the March 2014 AMS Notices,
enjoy the vulgarization of mathematics.
Backstory: Group Actions (November 14, 2012).
In "Notes on Finite Group Theory"
by Peter J. Cameron (October 2013),
http://www.maths.qmul.ac.uk/~pjc/notes/gt.pdf,
some parts are particularly related to the mathematics of
the 4×4 square (viewable in various ways as four quartets)—
Cameron is the author of Parallelisms of Complete Designs ,
a book notable in part for its chapter epigraphs from T.S. Eliot's
Four Quartets . These epigraphs, if not the text proper, seem
appropriate for All Saints' Day.
But note also Log24 posts tagged Not Theology.
Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo—
Compare to an image of Vril muse Maria Orsitsch.
From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —
Josefine Lyche
Keywords (to help place my artwork in the (See also the original catalog page.) 
Clearly most of this (the nonhighlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.
For some background, see (for instance)
Conspiracy Theories and Secret Societies for Dummies .
This journal on July 5, 2007 —
“It is not clear why MySpace China will be successful."
— The Chinese magazine Caijing in 2007, quoted in
Asia Sentinel on July 12, 2011
This journal on that same date, July 12, 2011 —
See also the eightfold cube and kindergarten blocks
at finitegeometry.org/sc.
Friedrich Froebel, Froebel's Chief Writings on Education ,
Part II, "The Kindergarten," Ch. III, "The Third Play":
"The little ones, who always long for novelty and change,
love this simple plaything in its unvarying form and in its
constant number, even as they love their fairy tales with
the everrecurring dwarfs…."
This journal, Group Actions, Nov. 14, 2012:
"Those who insist on vulgarizing their mathematics
may regard linear and affine group actions on the eight
cubes as the dance of Snow White (representing (0,0,0))
and the Seven Dwarfs—
Detail from the video in the previous post:
For other permutations of points in the
order3 affine plane—
See Quaternions in an Affine Galois Plane
and Group Actions, 19842009.
See, too, the Mathematics and Narrative post
from April 28, 2013, and last night's
For Indiana Spielberg.
The concept of "deep structure," once a popular meme,
has long been abandoned by Chomskians.
It still applies, however, to the 1976 mathematics, diamond theory ,
underlying the formal patterns discussed in a Royal Society paper
this year.
A review of deep structure, from the Wikipedia article Cartesian linguistics—
[Numbers in parentheses refer to pages in the original 1966 Harper edition of Chomsky's book Cartesian Linguistics .] Deep structure vs. surface structure "Pursuing the fundamental distinction between body and mind, Cartesian linguistics characteristically assumes that language has two aspects" (32). These are namely the sound/character of a linguistic sign and its significance (32). Semantic interpretation or phonetic interpretation may not be identical in Cartesian linguistics (32). Deep structures are often only represented in the mind (a mirror of thought), as opposed to surface structures, which are not. Deep structures vary less between languages than surface structures. For instance, the transformational operations to derive surface forms of Latin and French may obscure common features of their deep structures (39). Chomsky proposes, "In many respects, it seems to me quite accurate, then, to regard the theory of transformational generative grammar, as it is developing in current work, as essentially a modern and more explicit version of the PortRoyal theory" (39). Summary of Port Royal Grammar The Port Royal Grammar is an often cited reference in Cartesian Linguistics and is considered by Chomsky to be a more than suitable example of Cartesian linguistic philosophy. "A sentence has an inner mental aspect (a deep structure that conveys its meaning) and an outer, physical aspect as a sound sequence"***** This theory of deep and surface structures, developed in Port Royal linguistics, meets the formal requirements of language theory. Chomsky describes it in modern terms as "a base system that generates deep structures and a transformational system that maps these into surface structures", essentially a form of transformational grammar akin to modern studies (42). 
The corresponding concepts from diamond theory are…
"Deep structure"— The line diagrams indicating the underlying
structure of varying patterns
"A base system that generates deep structures"—
Group actions on square arrays… for instance, on the 4×4 square
"A transformational system"— The decomposition theorem
that maps deep structure into surface structure (and viceversa)
Some webpages at finitegeometry.org discuss
group actions on Sylvester’s duads and synthemes.
Those pages are based on the square model of
PG(3,2) described in the 1980’s by Steven H. Cullinane.
A rival tetrahedral model of PG(3,2) was described
in the 1990’s by Burkard Polster.
Polster’s tetrahedral model appears, notably, in
a Mathematics Magazine article from April 2009—
Click for a pdf of the article.
Related material:
“The Religion of Cubism” (May 9, 2003) and “Art and Lies”
(Nov. 16, 2008).
This post was suggested by following the link in yesterday’s
Sunday School post to High White Noon, and the link from
there to A Study in Art Education, which mentions the date of
Rudolf Arnheim‘s death, June 9, 2007. This journal
on that date—
The FinkGuy article was announced in a Mathematical
Association of America newsletter dated April 15, 2009.
Those who prefer narrative to mathematics may consult
a Log24 post from a few days earlier, “Where Entertainment is God”
(April 12, 2009), and, for some backstory, The Judas Seat
(February 16, 2007).
The second Logos figure in the previous post
summarized affine group actions on partitions
that generate a group of about 1.3 trillion
permutations of a 4x4x4 cube (shown below)—
Click for further details.
"Poetry is an illumination of a surface…."
— Wallace Stevens
Some poetic remarks related to a different surface, Klein's Quartic—
This link between the Klein map κ and the Mathieu group M_{24}
is a source of great delight to the author. Both objects were
found in the 1870s, but no connection between them was
known. Indeed, the class of maximal subgroups of M_{24}
isomorphic to the simple group of order 168 (often known,
especially to geometers, as the Klein group; see Baker [8])
remained undiscovered until the 1960s. That generators for
the group can be read off so easily from the map is
immensely pleasing.
— R. T. Curtis, Symmetric Generation of Groups ,
Cambridge University Press, 2007, page 39
Other poetic remarks related to the simple group of order 168—
In memory of William S. Knowles, chiral chemist, who died last Wednesday (June 13, 2012)—
Detail from the Harvard Divinity School 1910 bookplate in yesterday morning's post—
"ANDOVER–HARVARD THEOLOGICAL LIBRARY"
Detail from Knowles's obituary in this morning's New York Times—
William Standish Knowles was born in Taunton, Mass., on June 1, 1917. He graduated a year early from the Berkshire School, a boarding school in western Massachusetts, and was admitted to Harvard. But after being strongly advised that he was not socially mature enough for college, he did a second senior year of high school at another boarding school, Phillips Academy in Andover, N.H.
Dr. Knowles graduated from Harvard with a bachelor’s degree in chemistry in 1939….
"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
From Pilate Goes to Kindergarten—
The six congruent quaternion actions illustrated above are based on the following coordinatization of the eightfold cube—
Problem: Is there a different coordinatization
that yields greater symmetry in the pictures of
quaternion group actions?
A paper written in a somewhat similar spirit—
"Chiral Tetrahedrons as Unitary Quaternions"—
ABSTRACT: Chiral tetrahedral molecules can be dealt [with] under the standard of quaternionic algebra. Specifically, noncommutativity of quaternions is a feature directly related to the chirality of molecules….
Two papers suggested by Google searches tonight—
[PDF] PAPERS HELD OVER FROM THEME ISSUE ON ALGEBRA AND …
ajse.kfupm.edu.sa/articles/271A_08p.pdf
File Format: PDF/Adobe Acrobat – View as HTML by RT Curtis – 2001 – Related articles This paper is based on a talk given at the Scottish Algebra Day 1998 in Edinburgh. …… 
Curtis discusses the exceptional outer automorphism of S_{6}
as arising from group actions of PGL(2,5).
See also Cameron and Galois on PGL(2,5)—
[PDF] ON GROUPS OF DEGREE n AND n1, AND HIGHLYSYMMETRIC …

Illustration from Cameron (1973)—
A post by Gowers today on group actions suggests a review.
See Window, Window Continued, and The Galois Window.
R.D. Carmichael's seminal 1931 paper on tactical configurations suggests
a search for later material relating such configurations to block designs.
Such a search yields the following—
"… it seems that the relationship between
BIB [balanced incomplete block ] designs
and tactical configurations, and in particular,
the Steiner system, has been overlooked."
— D. A. Sprott, U. of Toronto, 1955
The figure by Cullinane included above shows a way to visualize Sprott's remarks.
For the group actions described by Cullinane, see "The Eightfold Cube" and
"A Simple Reflection Group of Order 168."
Update of 7:42 PM Sept. 18, 2011—
From a Summer 2011 course on discrete structures at a Berlin website—
A different illustration of the eightfold cube as the Steiner system S(3, 4, 8)—
Note that only the static structure is described by Felsner, not the
168 group actions discussed (as above) by Cullinane. For remarks on
such group actions in the literature, see "Cube Space, 19842003."
Yesterday's midday post, borrowing a phrase from the theology of Marvel Comics,
offered Rubik's mechanical contrivance as a rather absurd "Cosmic Cube."
A simpler candidate for the "Cube" part of that phrase:
The Eightfold Cube
As noted elsewhere, a simple reflection group* of order 168 acts naturally on this structure.
"Because of their truly fundamental role in mathematics,
even the simplest diagrams concerning finite reflection groups
(or finite mirror systems, or root systems—
the languages are equivalent) have interpretations
of cosmological proportions."
— Alexandre V. Borovik in "Coxeter Theory: The Cognitive Aspects"
Borovik has a such a diagram—
The planes in Borovik's figure are those separating the parts of the eightfold cube above.
In Coxeter theory, these are Euclidean hyperplanes. In the eightfold cube, they represent three of seven projective points that are permuted by the above group of order 168.
In light of Borovik's remarks, the eightfold cube might serve to illustrate the "Cosmic" part of the Marvel Comics phrase.
For some related theological remarks, see Cube Trinity in this journal.
Happy St. Augustine's Day.
* I.e., one generated by reflections : group actions that fix a hyperplane pointwise. In the eightfold cube, viewed as a vector space of 3 dimensions over the 2element Galois field, these hyperplanes are certain sets of four subcubes.
The title refers not to numbers of the form p^{ 3}, p prime, but to geometric cubes with p ^{3} subcubes.
Such cubes are natural models for the finite vector spaces acted upon by general linear groups viewed as permutation groups of degree (not order ) p^{ 3}.
For the case p =2, see The Eightfold Cube.
For the case p =3, see the "External links" section of the Nov. 30, 2009, version of Wikipedia article "General Linear Group." (That is the version just prior to the Dec. 14, 2009, revision by anonymous user "Greenfernglade.")
For symmetries of group actions for larger primes, see the related 1985 remark* on two dimensional linear groups—
"Actions of GL(2,p ) on a p ×p coordinatearray
have the same sorts of symmetries,
where p is any odd prime."
The LA Times on last weekend's film "Thor"—
"… the film… attempts to bridge director Kenneth Branagh's highminded Shakespearean intentions with Marvel Entertainment's bottomlineoriented need to crank out entertainment product."
Those averse to Nordic religion may contemplate a different approach to entertainment (such as Taymor's recent approach to SpiderMan).
A highminded— if not Shakespearean— nonNordic approach to groups acting—
"What was wrong? I had taken almost four semesters of algebra in college. I had read every page of Herstein, tried every exercise. Somehow, a message had been lost on me. Groups act . The elements of a group do not have to just sit there, abstract and implacable; they can do things, they can 'produce changes.' In particular, groups arise naturally as the symmetries of a set with structure. And if a group is given abstractly, such as the fundamental group of a simplical complex or a presentation in terms of generators and relators, then it might be a good idea to find something for the group to act on, such as the universal covering space or a graph."
— Thomas W. Tucker, review of Lyndon's Groups and Geometry in The American Mathematical Monthly , Vol. 94, No. 4 (April 1987), pp. 392394
"Groups act "… For some examples, see
Related entertainment—
Highminded— Many Dimensions—
Not so highminded— The Cosmic Cube—
One way of blending high and low—
The highminded Charles Williams tells a story
in his novel Many Dimensions about a cosmically
significant cube inscribed with the Tetragrammaton—
the name, in Hebrew, of God.
The following figure can be interpreted as
the Hebrew letter Aleph inscribed in a 3×3 square—
The above illustration is from undated software by Ed Pegg Jr.
For mathematical background, see a 1985 note, "Visualizing GL(2,p)."
For entertainment purposes, that note can be generalized from square to cube
(as Pegg does with his "GL(3,3)" software button).
For the Nordicaverse, some background on the Hebrew connection—
In a nutshell —
Epigraph to "The Aleph," a 1945 story by Borges:
O God! I could be bounded in a nutshell,
and count myself a King of infinite space…
— Hamlet, II, 2
The story in book form, 1949
A 2006 biography of geometer H.S.M. Coxeter:
The Aleph (implicit in a 1950 article by Coxeter):
The details:
Related material: Group Actions, 19842009.
Apollo's 13: A Group Theory Narrative —
I. At Wikipedia —
II. Here —
See Cube Spaces and Cubist Geometries.
The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
subcubes in the 27part (Galois) 3×3×3 cube–
A note from 1985 describing group actions on a 3×3 plane array—
Undated software by Ed Pegg Jr. displays
group actions on a 3×3×3 cube that extend the
3×3 group actions from 1985 described above—
Pegg gives no reference to the 1985 work on group actions.
"Examples are the stainedglass
windows of knowledge." — Nabokov
Related material:
Cubic models of finite geometries
display an interplay between
Euclidean and Galois geometry.
Example 1— The 2×2×2 Cube—
also known as the eightfold cube—
Group actions on the eightfold cube, 1984—
Version by Laszlo Lovasz et al., 2003—
Lovasz et al. go on to describe the same group actions
as in the 1984 note, without attribution.
Example 2— The 3×3×3 Cube
A note from 1985 describing group actions on a 3×3 plane array—
Undated software by Ed Pegg Jr. displays
group actions on a 3×3×3 cube that extend the
3×3 group actions from 1985 described above—
Pegg gives no reference to the 1985 work on group actions.
Example 3— The 4×4×4 Cube
A note from 27 years ago today—
As far as I know, this version of the
groupactions theorem has not yet been ripped off.
"The cube has…13 axes of symmetry:
6 C_{2} (axes joining midpoints of opposite edges),
4 C_{3} (space diagonals), and
3C_{4} (axes joining opposite face centroids)."
–Wolfram MathWorld article on the cube
These 13 symmetry axes can be used to illustrate the interplay between Euclidean and Galois geometry in a cubic model of the 13point Galois plane.
The geometer's 3×3×3 cube–
27 separate subcubes unconnected
by any Rubiklike mechanism–
The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
subcubes in the (Galois) 3×3×3 cube–
A closely related structure–
the finite projective plane
with 13 points and 13 lines–
A later version of the 13point plane
by Ed Pegg Jr.–
A group action on the 3×3×3 cube
as illustrated by a Wolfram program
by Ed Pegg Jr. (undated, but closely
related to a March 26, 1985 note
by Steven H. Cullinane)–
The above images tell a story of sorts.
The moral of the story–
Galois projective geometries can be viewed
in the context of the larger affine geometries
from which they are derived.
The standard definition of points in a Galois projective plane is that they are lines through the (arbitrarily chosen) origin in a corresponding affine 3space converted to a vector 3space.
If we choose the origin as the center cube in coordinatizing the 3×3×3 cube (See Weyl's relativity problem ), then the cube's 13 axes of symmetry can, if the other 26 cubes have properly (Weyl's "objectively") chosen coordinates, illustrate nicely the 13 projective points derived from the 27 affine points in the cube model.
The 13 lines of the resulting Galois projective plane may be derived from Euclidean planes through the cube's center point that are perpendicular to the cube's 13 Euclidean symmetry axes.
The above standard definition of points in a Galois projective plane may of course also be used in a simpler structure– the eightfold cube.
(The eightfold cube also allows a less standard way to picture projective points that is related to the symmetries of "diamond" patterns formed by group actions on graphic designs.)
See also Ed Pegg Jr. on finite geometry on May 30, 2006
at the Mathematical Association of America.
From a 1984 book review:
"After three decades of intensive research by hundreds of group theorists, the century old problem of the classification of the finite simple groups has been solved and the whole field has been drastically changed. A few years ago the one focus of attention was the program for the classification; now there are many active areas including the study of the connections between groups and geometries, sporadic groups and, especially, the representation theory. A spate of books on finite groups, of different breadths and on a variety of topics, has appeared, and it is a good time for this to happen. Moreover, the classification means that the view of the subject is quite different; even the most elementary treatment of groups should be modified, as we now know that all finite groups are made up of groups which, for the most part, are imitations of Lie groups using finite fields instead of the reals and complexes. The typical example of a finite group is
— Jonathan L. Alperin,
review of books on group theory,
Bulletin (New Series) of the American
Mathematical Society 10 (1984) 121, doi:
10.1090/S027309791984152108
The same example
at Wolfram.com:
Citation data from Wolfram.com:
"GL(2,p) and GL(3,3) Acting on Points"
from The Wolfram Demonstrations Project,
http://demonstrations.wolfram.com/GL2PAndGL33ActingOnPoints/,
Contributed by: Ed Pegg Jr"
As well as displaying Cullinane's 48 pictures of group actions from 1985, the Pegg program displays many, many more actions of small finite general linear groups over finite fields. It illustrates Cullinane's 1985 statement:
"Actions of GL(2,p) on a p×p coordinatearray have the same sorts of symmetries, where p is any odd prime."
Pegg's program also illustrates actions on a cubical array– a 3×3×3 array acted on by GL(3,3). For some other actions on cubical arrays, see Cullinane's Finite Geometry of the Square and Cube.
Laura A. Smit, Calvin College, “Towards an Aesthetic Teleology: Romantic Love, Imagination and the Beautiful in the Thought of Simone Weil and Charles Williams“–
“My work is motivated by a hope that there may be a way to recapture the ancient and medieval vision of both Beauty and purpose in a way which is relevant to our own century. I even dare to hope that the two ideas may be related, that Beauty is actually part of the meaning and purpose of life.”
“The Reverend T. P. Kirkman knew in 1862 that there exists a group of degree 16 and order 322560 with a normal, elementary abelian, subgroup of order 16 [1, p. 108]. Frobenius identified this group in 1904 as a subgroup of the Mathieu group M_{24} [4, p. 570]….”
1. Biggs N.L., “T. P. Kirkman, Mathematician,” Bulletin of the London Mathematical Society 13, 97–120 (1981).
4. Frobenius G., “Über die Charaktere der mehrfach transitiven Gruppen,” Sitzungsber. Königl. Preuss. Akad. Wiss. zu Berlin, 558–571 (1904). Reprinted in Frobenius, Gesammelte Abhandlungen III (J.P. Serre, editor), pp. 335–348. Springer, Berlin (1968).
Olli Pottonen, “Classification of Steiner Quadruple Systems” (Master’s thesis, Helsinki, 2005)–
“The concept of group actions is very useful in the study of isomorphisms of combinatorial structures.”
“Simplify, simplify.”
— Thoreau
“Beauty is bound up
with symmetry.”
— Weyl
Pottonen’s thesis is
dated Nov. 16, 2005.
For some remarks on
images and theology,
see Log24 on that date.
Click on the above image
for some further details.
On Spekkens’ toy system and finite geometry
Background–
On finite geometry:
The actions of permutations on a 4 × 4 square in Spekkens’ paper (quantph/0401052), and Leifer’s suggestion of the need for a “generalized framework,” suggest that finite geometry might supply such a framework. The geometry in the webpage John cited is that of the affine 4space over the twoelement field.
Related material:
See also arXiv:0707.0074v1 [quantph], June 30, 2007:
A fully epistemic model for a local hidden variable emulation of quantum dynamics,
by Michael Skotiniotis, Aidan Roy, and Barry C. Sanders, Institute for Quantum Information Science, University of Calgary. Abstract: "In this article we consider an augmentation of Spekkens’ toy model for the epistemic view of quantum states [1]…."
Hypercube from the Skotiniotis paper:
Reference:
Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5 (Received 11 October 2005; revised 2 November 2006; published 19 March 2007.)
Serious
“I don’t think the ‘diamond theorem’ is anything serious, so I started with blitzing that.”
— Charles Matthews at Wikipedia, Oct. 2, 2006
“The ‘seriousness’ of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is ‘significant’ if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas.”
— G. H. Hardy, A Mathematician’s Apology
Matthews yesterday deleted references to the diamond theorem and related material in the following Wikipedia articles:
Affine group
Reflection group
Symmetry in mathematics
Incidence structure
Invariant (mathematics)
Symmetry
Finite geometry
Group action
History of geometry
This would appear to be a fairly large complex of mathematical ideas.
See also the following “large complex” cited, following the above words of Hardy, in Diamond Theory:
Affine geometry, affine planes, affine spaces, automorphisms, binary codes, block designs, classical groups, codes, coding theory, collineations, combinatorial, combinatorics, conjugacy classes, the Conwell correspondence, correlations, design theory, duads, duality, error correcting codes, exceptional groups, finite fields, finite geometry, finite groups, finite rings, Galois fields, generalized quadrangles, generators, geometry, GF(2), GF(4), the (24,12) Golay code, group actions, group theory, Hadamard matrices, hypercube, hyperplanes, hyperspace, incidence structures, invariance, Karnaugh maps, Kirkman’s schoolgirls problem, Latin squares, Leech lattice, linear groups, linear spaces, linear transformations, Mathieu groups, matrix theory, Meno, Miracle Octad Generator, MOG, multiply transitive groups, octads, the octahedral group, orthogonal arrays, outer automorphisms, parallelisms, partial geometries, permutation groups, PG(3,2), polarities, PolyaBurnside theorem, projective geometry, projective planes, projective spaces, projectivities, ReedMuller codes, the relativity problem, Singer cycle, skew lines, sporadic simple groups, Steiner systems, symmetric, symmetry, symplectic, synthemes, synthematic, tesseract, transvections, Walsh functions, Witt designs.
"The symmetric group S_{6} of permutations of 6 objects is the only symmetric group with an outer automorphism….
This outer automorphism can be regarded as the seed from which grow about half of the sporadic simple groups…."
This "seed" may be pictured as
within what Burkard Polster has called "the smallest perfect universe"– PG(3,2), the projective 3space over the 2element field.
Related material: yesterday's entry for Sylvester's birthday.
Matrix group actions,
March 26, 1985
"We symbolize logical necessity
with the box
and logical possibility
with the diamond
— Keith Allen Korcz,
(Log24.net, 1/25/05)
And what do we
symbolize by _{} ?
"The possibilia that exist,
and out of which
the Universe arose,
are located in
a necessary being…."
— Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)
Trinity and Counterpoint
Today's Roman Catholic meditation is from Gerry Adams, leader of Sinn Fein, the political arm of the Irish Republican Army:
"I certainly regret what happened and I make no bones about that," Adams said on the 30th anniversary of pub bombings that killed 21 on Nov. 21, 1974, in Birmingham, England.
Those who care what Roman Catholics think of the Trinity may read the remarks of St. Bonaventure at math16.com.
That site also offers a less holy but more intelligible trinity based on the irrefutable fact that
For a Protestant view of this trinity, see a website at the University of Birmingham in England.
That site's home page links to Birmingham's City Evangelical Church.
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