Log24

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)

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 —

 
 

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

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?

 
 

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 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.

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.

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.

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.

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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