Log24

Wednesday, February 19, 2020

Aitchison’s Octads

Filed under: General — Tags: , — m759 @ 11:36 AM

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 .

The Eightfold Cube: The Beauty of Klein's Simple Group

   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.

 
 

Tuesday, December 17, 2019

Picturing Aitchison’s Mathieu Generators

Filed under: General — Tags: — m759 @ 11:07 AM

Click to enlarge.

Friday, November 29, 2019

Verifying Aitchison’s Cuboctahedral Generation of M24

Filed under: General — Tags: — m759 @ 1:06 AM

Iain Aitchison on symmetric generation of M24

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Infinity has been replaced by 7 (because a digit was needed,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) |
(8,2)(19,13)(4,6)(7,9)
(16,11)(5,18)(15,12)(14,17)
(24,21)(10,3)(22,23)(20,1),

(8,3)(20,14)(5,7)(1,10)
(16,12)(6,19)(9,13)(15,18)
(24,22)(11,4)(23,17)(21,2),

(8,4)(21,15)(6,1)(2,11)
(16,13)(7,20)(10,14)(9,19)
(24,23)(12,5)(17,18)(22,3),

(8,5)(22,9)(7,2)(3,12)
(16,14)(1,21)(11,15)(10,20)
(24,17)(13,6)(18,19)(23,4),

(8,6)(23,10)(1,3)(4,13)
(16,15)(2,22)(12,9)(11,21)
(24,18)(14,7)(19,20)(17,5),

(8,7)(17,11)(2,4)(5,14)
(16,9)(3,23)(13,10)(12,22)
(24,19)(15,1)(20,21)(18,6),

(8,1)(18,12)(3,5)(6,15)
(16,10)(4,17)(14,11)(13,23)
(24,20)(9,2)(21,22)(19,7)>;

G;
Order(G);
CompositionFactors(G);

The Aitchison generators passed the MAGMA test.

Friday, May 31, 2019

Working Sketch of Aitchison’s Mathieu Cuboctahedron

Filed under: General — Tags: , , — m759 @ 5:33 AM

Cuboctahedron with its 24 edges labeled by the 24 permutations of a 4-set. By Cullinane on 5/31/19.

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 —

 
 

Sunday, December 30, 2018

Also Sprach Aitchison

Filed under: General — Tags: , — m759 @ 2:48 PM

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

Iain Aitchison on symmetric generation of M24

Some backstory — A Riddle for Davos,  Jan. 22, 2014.

Thursday, December 6, 2018

The Mathieu Cube of Iain Aitchison

Filed under: General — Tags: , , — m759 @ 12:00 PM

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)

www.math.sci.hiroshima-u.ac.jp/branched/…/Aitchison-Hiroshima-2018-Talk1-2.pdf

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)

www.math.sci.hiroshima-u.ac.jp/branched/files/…/Aitchison-Hiroshima-2-2018.pdf

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

Abstract

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 …

Related material — 

The 56 triangles of  the eightfold cube . . .

The Eightfold Cube: The Beauty of Klein's Simple Group

   Image from Christmas Day 2005.

Monday, September 7, 2020

Remedial Reading

Filed under: General — m759 @ 1:14 AM

'A Discovery of Witches' S1 E2 0:33:45

Hopkins at Hiroshima

See also Archimedes at Hiroshima
and, more generally, Aitchison.

Wednesday, September 2, 2020

Space Wars:  Sith Pyramid vs. Jedi Cube

Filed under: General — m759 @ 11:18 AM

For the Sith Pyramid, see posts tagged Pyramid Game.
For the Jedi Cube, see posts tagged Enigma Cube
and cube-related remarks by Aitchison at Hiroshima.

This  post was suggested by two events of May 16, 2019 —
A weblog post by Frans Marcelis on the Miracle Octad
Generator of R. T. Curtis (illustrated with a pyramid),
and the death of I. M. Pei, architect of the Louvre pyramid.

That these events occurred on the same date is, of course,
completely coincidental.

Perhaps Dan Brown can write a tune to commemorate
the coincidence.

Saturday, August 22, 2020

Magic for Liars* . . .

Filed under: General — m759 @ 6:27 PM

From a web page

From YouTube, for the Church of Synchronology

Meanwhile, elsewhere . . .

* See that book title in this journal.

Saturday, May 23, 2020

Structure for Linguists

Filed under: General — Tags: — m759 @ 11:34 AM

“MIT professor of linguistics Wayne O’Neil died on March 22
at his home in Somerville, Massachusetts.”

MIT Linguistics, May 1, 2020

The “deep  structure” above is the plane cutting the cube in a hexagon
(as in my note Diamonds and Whirls of September 1984).

See also . . .

IMAGE- Redefining the cube's symmetry planes: 13 planes, not 9.

Friday, May 22, 2020

Annals of Crystalline Beauty

Filed under: General — m759 @ 4:58 PM

The phrase “laborious cerebration” quoted in the previous post,
Sombre Figuration, suggests . . .

For an example of such cerebration, see Aitchison’s Octads.

Sunday, May 17, 2020

“The Ultimate Epistemological Fact”

Filed under: General — m759 @ 11:49 PM

“Let me say this about that.” — Richard Nixon

Interpenetration in Weyl’s epistemology —

Interpenetration in Mazzola’s music theory —

Interpenetration in the eightfold cube — the three midplanes —

IMAGE- The Trinity Cube (three interpenetrating planes that split the eightfold cube into its eight subcubes)

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.

Tuesday, March 17, 2020

Geometric Theology

Filed under: General — Tags: — m759 @ 12:00 AM

“Before time began” — Optimus Prime

IMAGE- The Trinity of Max Black (a 3-set, with its eight subsets arranged in a Hasse diagram that is also a cube)

See also posts tagged Aitchison.

 

Tuesday, March 10, 2020

Labeling a Cuboctahedron

Filed under: General — Tags: , , — m759 @ 12:01 PM

The above arrangement of graphic images on  cube faces is purely
decorative and static, and of  little mathematical interest. 

(A less static, but structurally chaotic, artifact might be made by
pasting the above 24 graphic images in the "Cosets in S4" picture
above onto the 24 faces of a 2x2x2 Rubik cube. This suggests the
reflection below on the poet Wallace Stevens, whose "Connoisseur
of Chaos" first appeared on page 90 of Twentieth Century Verse ,
Numbers 12-13, October 1938.)

If mathematically interesting  permutations of the graphic images
are to be done, the images should be imagined as situated on 
parallel  planes, as in the permutahedron below —

IMAGE- 'Permutahedron of Opposites'-- 24 graphic patterns arranged in space as 12 pairs of opposites

Click the above permutahedron for an analysis of its structure.

Monday, March 9, 2020

“Archimedes at Hiroshima” Continues.

Filed under: General — Tags: , , — m759 @ 7:34 PM

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.

The Bucharest Cross

Filed under: General — Tags: , , — m759 @ 12:12 AM

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:

Wednesday, January 1, 2020

Le Mot Juste

Filed under: General — Tags: — m759 @ 3:23 PM

Related art

Friday, November 29, 2019

Symmetry in Practice

Filed under: General — Tags: — m759 @ 11:03 AM

Some  background for the previous post

Tuesday, November 26, 2019

Alea Iacta Est*

Filed under: General — Tags: , — m759 @ 11:11 AM

Saturday evening's post Diamond Globe suggests a review of

Iain Aitchison on symmetric generation of M24 —

Iain Aitchison on symmetric generation of M24

     * A Greek version for the late John SImon:

«Ἀνερρίφθω κύβος».

Sunday, November 24, 2019

Hyperbolic Memorial

Filed under: General — Tags: — m759 @ 8:46 AM

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

http://www.log24.com/log/pix10B/101123-Saddle.jpg

See as well . . .

Saturday, November 23, 2019

Diamond Globe

Filed under: General — Tags: , , — m759 @ 7:23 PM

An image from All Souls' Day 2010 —

IMAGE- 'Permutahedron of Opposites'-- 24 graphic patterns arranged in space as 12 pairs of opposites

This is from earlier posts tagged Permutahedron.

See also
Wallace Stevens:
A World of Transforming Shapes
.

From that book (click to enlarge) —

http://www.log24.com/log/pix11C/111224-Perlis-500w.jpg

"Before time began, there was the Cube."
— Optimus Prime.

Also from earlier posts tagged Permutahedron

The Mathieu group cube of Iain Aitchison (2018, Hiroshima)

Sunday, October 27, 2019

Friday Night Lights

Filed under: General — m759 @ 2:05 PM

Entertainment from NBC on Friday night —

The above question, and Saturday morning's post on a film director
from Melbourne, suggest an image from December's Melbourne Noir

 (March 8, 2018, was the date of death for Melbourne author Peter Temple.)

Thursday, October 3, 2019

Apocalypse* Note

Filed under: General — Tags: — m759 @ 7:00 PM

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.

* The X-Men character.

Thursday, June 27, 2019

Group Actions on the 4x4x4 Cube

Filed under: General — Tags: — m759 @ 6:23 AM

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.

Friday, June 21, 2019

Cube Tales for Solstice Day

Filed under: General — Tags: , — m759 @ 3:45 PM

See also "Six-Set" in this journal
and "Cube Geometry Continues."

 
 

Saturday, June 1, 2019

Cuboctahedron Labeling Update

Filed under: General — Tags: , , — m759 @ 9:01 PM

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

Bulk Apperception

Filed under: General,Geometry — Tags: — m759 @ 10:38 PM

(Continued)

Tuesday, May 21, 2019

Cube Geometry Continues.

Filed under: General — Tags: — m759 @ 1:30 PM

An illustration from the April 20, 2016, post

Symmetric Generation of a Simple Group

IMAGE- Bester,'The Stars My Destination' (with cover slightly changed)


"The geometry of unit cubes is a meeting point
 of several different subjects in mathematics."
 — Chuanming ZongBulletin of the American
Mathematical Society 
, January 2005

Iain Aitchison on symmetric generation of M24

Saturday, May 4, 2019

The Chinese Jars of Shing-Tung Yau

Filed under: General — Tags: , , — m759 @ 11:00 AM

The title refers to Calabi-Yau spaces.

T. S. Eliot —

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

Counting symmetries with the orbit-stabilizer theorem

A related Calabi-Yau "Chinese jar" first described in detail in 1905

Illustration of K3 surface related to Mathieu moonshine

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

Saturday, March 9, 2019

The Hiroshima Model

Filed under: General — m759 @ 11:00 AM

Commemorating a talk given by Iain Aitchison
at Hiroshima a year ago today.

Wednesday, January 16, 2019

The Dreaming Polyhedron

Filed under: General — Tags: , — m759 @ 5:32 AM

"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 —

Illustration from a webpage by David A. Richter, Western Michigan University

"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

Finding

Filed under: General — Tags: — m759 @ 10:26 AM

Click to enlarge

Results of a 15 Jan. 2019 search for Aitchison + 'Mathieu group'

Sunday, January 13, 2019

Sunday the Thirteenth (Revisited)

Filed under: General — Tags: — m759 @ 10:00 PM

IMAGE- Redefining the cube's symmetry planes: 13 planes, not 9.

For some context, see "A Riddle for Davos."

Friday, January 11, 2019

Permutations at Oslo

Filed under: General — Tags: , , — m759 @ 8:45 PM

Webpage at Oslo of Josefine Lyche, 'Plato's Diamond'

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

Filed under: General,Geometry — Tags: , , , — m759 @ 7:35 PM

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?

 
 

Sunday, January 6, 2019

For Broom Bridge*

Filed under: General,Geometry — Tags: — m759 @ 11:00 AM

GL(2,3) is not unrelated to GL(3,2).

See Quaternion Automorphisms 
and Spinning in Infinity.

* See Wikipedia.

Tuesday, January 1, 2019

The Magnificent Seven

Filed under: General — Tags: , — m759 @ 12:00 AM

Brief introduction to the 'Symmetric Generation' of R. T. Curtis

Sunday, December 9, 2018

Quaternions in a Small Space

Filed under: G-Notes,General,Geometry — Tags: , , — m759 @ 2:00 PM

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 —

Iain Aitchison on the 'symmetric generation' of R. T. Curtis

See as well the two Log24 posts of December 1st, 2018 —

Character and In Memoriam.

Friday, December 7, 2018

An Ark for Hanukkah

Filed under: General — Tags: , — m759 @ 11:01 AM

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.

Wednesday, December 5, 2018

Caesarian (continued)

Filed under: General — Tags: — m759 @ 1:00 AM

Later editions of a book first published on New Year's Day 2002
by Bantam in Australia —

Tuesday, December 4, 2018

Melbourne Noir

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 11:30 AM

 March 8, 2018, was the date of death for Melbourne author Peter Temple.

Monday, December 3, 2018

The Relativity Problem at Hiroshima

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 6:21 PM

“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

Iain Aitchison's 'dice-labelled' cuboctahedron at Hiroshima, March 2018

See also Relativity Problem and Diamonds and Whirls.

Sunday, December 2, 2018

Symmetric Generation …

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 12:00 PM

Continued .   See as well a Log24 search for "Symmetric Generation."

Iain Aitchison on symmetric generation of M24

Iain Aitchison on symmetric generation of M24

Update of 2 PM ET —

Symmetry at Hiroshima

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 6:43 AM

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.

Saturday, December 1, 2018

Character

Filed under: General — Tags: — m759 @ 11:00 AM

"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.

Filed under: General,Geometry — Tags: , , — m759 @ 11:13 AM

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

Filed under: General,Geometry — Tags: , , — m759 @ 11:01 AM

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 —

The Diamond Cube.

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).

Monday, November 10, 2014

Narrative Line

Filed under: General,Geometry — Tags: , , — m759 @ 11:02 PM

"We live entirely, especially if we are writers, by the imposition
of a narrative line upon 
disparate images…." — Joan Didion

Narrative Line:

IMAGE- R. D. Carmichael's 1931 construction of the Steiner system S(5, 8, 24)

IMAGE- Harvard senior Jeremy Booher in 2010 discusses Carmichael's 1931 construction of S(5, 8, 24) without mentioning Carmichael.

Disparate images:

Exercise:

Can the above narrative line be imposed in any sensible way
upon the above disparate images?

Friday, February 21, 2014

Raumproblem*

Filed under: General,Geometry — Tags: , , — m759 @ 7:01 PM

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—

IMAGE- The 24-triangle hexagon

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.

Sunday, October 6, 2013

Church with Josefine*

Filed under: General — Tags: , , — m759 @ 10:10 AM

(Continued from last Sunday)

IMAGE- 'Permutahedron of Opposites'-- 24 graphic patterns arranged in space as 12 pairs of opposites

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.)

Wednesday, January 11, 2012

Cuber

Filed under: General,Geometry — Tags: , , , , — m759 @ 12:00 PM

“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.

Wednesday, September 21, 2011

Symmetric Generation

Filed under: General,Geometry — Tags: , , , , — m759 @ 2:00 PM

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 G M 24 is generated by
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 L3(2)
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)—

http://www.log24.com/log/pix11B/110921-CassirerOnObjectivity-400w.jpg

Cassirer's remarks connect the concept of objectivity  with that of object .

The above quotations perhaps indicate how the Mathieu group M 24 may be viewed as an object.

"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 L3(2) acting on the points of the 7-point projective plane…." see The Eightfold Cube.

For symmetric generation of L3(2) on that cube, see A Simple Reflection Group of Order 168.

Tuesday, September 20, 2011

Relativity Problem Revisited

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

A footnote was added to Finite Relativity

Background:

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

Friday, June 24, 2011

The Cube

Filed under: General — Tags: , , — m759 @ 12:00 PM

IMAGE- 'The Stars My Destination' (with cover slightly changed)

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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