Related material: Theodore Sturgeon's novel The Dreaming Jewels
and his story "What Dead Men Tell" . . .
Related material: Theodore Sturgeon's novel The Dreaming Jewels
and his story "What Dead Men Tell" . . .
About the author of the above —
A related questionable "proof of concept" :
Aitchison at Hiroshima in this journal — a scholar's 2018 investigation
of M24 actions on a cuboctahedon — and . . .
Two examples from the Wikipedia article "Archimedean solid" —
Iain Aitchison said in a 2018 talk at Hiroshima that
the Mathieu group M24 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 S4 in a permutahedron —
Can M24 be represented as permuting naturally
the 24 vertices of the truncated octahedron?
Related material from the day Orson Welles and Yul Brynner died —
* See other posts tagged Aitchison in this journal.
Shown below is an illustration from "The Puzzle Layout Problem" —
Exercise: Using the above numerals 1 through 24
(with 23 as 0 and 24 as ∞) to represent the points
∞, 0, 1, 2, 3 … 22 of the projective line over GF(23),
reposition the labels 1 through 24 in the above illustration
so that they appropriately* illustrate the cube-parts discussed
by Iain Aitchison in his March 2018 Hiroshima slides on
cube-part permutations by the Mathieu group M24.
A note for Northrop Frye —
Interpenetration in the eightfold cube — the three midplanes —
A deeper example of interpenetration:
Aitchison has shown that the Mathieu group M24 has a natural
action on the 24 center points of the subsquares on the eightfold
cube's six faces (four such points on each of the six faces). Thus
the 759 octads of the Steiner system S(5, 8, 24) interpenetrate
on the surface of the cube.
* "Appropriately" — I.e. , so that the Aitchison cube octads correspond
exactly, via the projective-point labels, to the Curtis MOG octads.
The title is from a post of January 10, 2019.
A figure from this journal on June 1, 2019 —
The following figure may help relate labelings of the
truncated octahedron ("permutahedron") to labelings
of its fellow Archimedean solid, the cuboctahedron.
See as well other posts tagged Aitchison.
For fans of "The Zero Theorem" —
The 24 permutations of S4 arranged on a cube
by Cristi Stoica of Bucharest at
http://www.unitaryflow.com/2009/06/polyhedra-and-groups.html:
The 759 octads of the Steiner system S(5,8,24) are displayed
rather neatly in the Miracle Octad Generator of R. T. Curtis.
A March 9, 2018, construction by Iain Aitchison* pictures the
759 octads on the faces of a cube , with octad elements the
24 edges of a cuboctahedron :
The Curtis octads are related to symmetries of the square.
See my webpage "Geometry of the 4×4 square" from March 2004.
Aitchison's p. 42 slide includes an illustration from that page —
Aitchison's octads are instead related to symmetries of the cube.
Note that essentially the same model as Aitchison's can be pictured
by using, instead of the 24 edges of a cuboctahedron, the 24 outer
faces of subcubes in the eightfold cube .
Image from Christmas Day 2005.
* http://www.math.sci.hiroshima-u.ac.jp/branched/files/2018/
presentations/Aitchison-Hiroshima-2-2018.pdf.
See also Aitchison in this journal.
Click to enlarge.
Shown below are Aitchison's March 2018 M24 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 mu0-mu6 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.
Saturday evening's post Diamond Globe suggests a review of …
Iain Aitchison on symmetric generation of M24 —
* A Greek version for the late John SImon:
«Ἀνερρίφθω κύβος».
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 . . .
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 —
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.
For affine group actions, see Ex Fano Appollinis (June 24)
and Solomon's Cube.
For one approach to Mathieu group actions on a 24-cube subset
of the 4x4x4 cube, see . . .
For a different sort of Mathieu cube, see Aitchison.
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 —"
The above sketch indicates one way to apply the elements of S4
to the Aitchison cuboctahedron . It is a rough sketch illustrating a
correspondence between four edge-hexagons and four label-sets.
The labeling is not as neat as that of a permutahedron by S4
shown below, but can perhaps be improved.
Permutahedron labeled by S4 .
Update of 9 PM EDT June 1, 2019 —
. . . And then of course there is the obvious labeling derived from
the above permutahedron —
The title refers to Calabi-Yau 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 12-dimensional space over GF(2).
Related material from Plato and R. T. Curtis —
A related Calabi-Yau "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/K3-pix.pdf
"Here is a recipe for preparing a copy of the Mathieu group M24.
The main ingredient is a genus-3 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 3-dimensional space."
— "How to Make the Mathieu Group M24 ," 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.
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).
Two examples from the Wikipedia article "Archimedean solid" —
Iain Aitchison said in a talk last year at Hiroshima that
the Mathieu group M24 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 S4 in a permutahedron —
Can M24 be represented as permuting naturally
the 24 vertices of the truncated octahedron?
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.
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 —
Related material —
See as well the two Log24 posts of December 1st, 2018 —
Character and In Memoriam.
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.
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 . . .
Image from Christmas Day 2005.
“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.
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.hiroshima-u.ac.jp/ branched/files/2018/abstract/Aitchison.txt
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 2-fold 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 Horrocks-Mumford bundle. Poincare's homology 3-sphere, 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 so-called `Arnol'd Trinities'. Motivated originally by the `Eightfold Way' sculpture at MSRI in Berkeley, we discuss inter-relationships 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 inter-connectedness 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 graphic-illustrated 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 8-component 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.
"What we do may be small, but it has
a certain character of permanence."
— G. H. Hardy,
A Mathematician's Apology
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.
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/behind-the-glitter/
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 hexagon-inside-a-cube in my "Diamonds and Whirls" note (1984).
"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?
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 square-triangle 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 24-triangle-hexagon
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.
(Continued from last Sunday)
For some background, see Permutahedron in this journal.
See also…
* 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 disco-era movie.)
“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 11557-11572)
See also Many Dimensions in this journal and Solomon’s Cube.
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 7-point 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
A footnote was added to Finite Relativity—
Background:
Weyl on what he calls 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
Click the above image for some background.
Related material:
Skateboard legend Andy Kessler,
this morning's The Gleaming,
and But Sometimes I Hit London.
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