Log24

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."

 
[addtoany]
 

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

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 —

 
[addtoany]
 

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 Zong, Bulletin of the American
Mathematical Society 
, January 2005

Iain Aitchison on symmetric generation of M24

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.

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 —

Saturday, September 22, 2018

Symmetric Generation, by Curtis

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 10:15 am

Norwegian artist Josefine Lyche —

Lyche's shirt honors the late Kurt Cobain.

"Here we are now, entertain us."

Symmetric Generation, by Netflix

Filed under: General — Tags: , , — m759 @ 5:05 am

Suggested by the previous post . . .

'Out of nothing' opening of 'Maniac' at Netflix

"The pattern is the pattern."

Friday, September 21, 2018

Symmetric Generation, by Nao

Filed under: General — Tags: , , — m759 @ 1:30 pm

"The creation of a new world
        starts now.
Once again I am tied
        to the logic of this
Hyper-symmetrical-dimension."

Bill Murray and Scarlett Johansson in 'Lost in Translation'

Saturday, August 25, 2018

“Waugh, Orwell. Orwell, Waugh.”

Suggested by a review of Curl on Modernism —

http://www.log24.com/log/pix18/180825-Ballard-on-Modernism.gif

Related material —

Waugh + Orwell in this journal and …

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168

Monday, June 4, 2018

The Trinity Stone Defined

“Unsheathe your dagger definitions.” — James Joyce, Ulysses

The “triple cross” link in the previous post referenced the eightfold cube
as a structure that might be called the trinity stone .

An Approach to Symmetric Generation of the Simple Group of Order 168

Some small Galois spaces (the Cullinane models)

Tuesday, January 2, 2018

Debs and Redhead

Filed under: General — Tags: — m759 @ 3:15 pm

Or:  Schoolgirl Problems

The above images were suggested in part by the birthdays
on Sept. 21, 2011, of Bill Murray and Stephen King.

More seriously, also in this journal on that date, from a post
titled Symmetric Generation —

Wednesday, September 13, 2017

Summer of 1984

The previous two posts dealt, rather indirectly, with
the notion of "cube bricks" (Cullinane, 1984) —

Group actions on partitions —

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168

Another mathematical remark from 1984 —

For further details, see Triangles Are Square.

Tuesday, June 20, 2017

Epic

Continuing the previous post's theme … 

Group actions on partitions —

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168

Related material — Posts now tagged Device Narratives.

Tuesday, March 28, 2017

Bit by Bit

Filed under: General,Geometry — Tags: , , , , — m759 @ 11:45 am

From Log24, "Cube Bricks 1984" —

An Approach to Symmetric Generation of the Simple Group of Order 168

Also on March 9, 2017 —

For those who prefer graphic  art —

Broken Symmetries  in  Diamond Space  —

Thursday, November 3, 2016

Triple Cross

(Continued … See the title in this journal, as well as Cube Bricks.)

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168
Related material —

Dirac and Geometry in this journal,
Kummer's Quartic Surface in this journal,
Nanavira Thera in this journal, and
The Razor's Edge  and Nanavira Thera.

See as well Bill Murray's 1984 film "The Razor's Edge" …

Movie poster from 1984 —

"A thin line separates
love from hate,
success from failure,
life from death."

Three other dualities, from Nanavira Thera in 1959 —

"I find that there are, in every situation,
three independent dualities…."

(Click to enlarge.)

Wednesday, October 5, 2016

Sources

Filed under: General,Geometry — Tags: , , , — m759 @ 9:00 am

From a Google image search yesterday —

Sources (left to right, top to bottom) —

Math Guy (July 16, 2014)
The Galois Tesseract (Sept. 1, 2011)
The Full Force of Roman Law (April 21, 2014)
A Great Moonshine (Sept. 25, 2015)
A Point of Identity (August 8, 2016)
Pascal via Curtis (April 6, 2013)
Correspondences (August 6, 2011)
Symmetric Generation (Sept. 21, 2011)

Thursday, July 14, 2016

Meditation from an April 1

Filed under: General,Geometry — Tags: , — m759 @ 9:38 am

Related material from the same day —

See also …

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168

The above bricks appeared in some earlier Log24 posts.

Wednesday, April 20, 2016

Symmetric Generation of a Simple Group

The reference in the previous post to the work of Guitart and
The Road to Universal Logic  suggests a fiction involving
the symmetric generation of the simple group of order 168.

See The Diamond Archetype and a fictional account of the road to Hell …

'PyrE' in Bester's 'The Stars My Destination'

The cover illustration below has been adapted to
replace the flames of PyrE with the eightfold cube.

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

For related symmetric generation of a much larger group, see Solomon's Cube.

Monday, April 4, 2016

Cube for Berlin

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 —

The Eightfold 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 —

An Approach to Symmetric Generation of the Simple Group of Order 168

Related remark from the literature —

http://www.log24.com/log/pix11B/110918-Felsner.jpg

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, 1984-2003."

(From Anatomy of a Cube, Sept. 18, 2011.)

Friday, March 4, 2016

Cube Bricks 1984

Filed under: General,Geometry — Tags: , , — m759 @ 12:06 pm

An Approach to Symmetric Generation of the Simple Group of Order 168

Related aesthetics —

"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

Monday, October 19, 2015

Symmetric Generation of the Simple Order-168 Group

Filed under: General,Geometry — Tags: , , , — m759 @ 8:48 pm

This post continues recent thoughts on the work of René Guitart.
A 2014 article by Guitart gives a great deal of detail on his
approach to symmetric generation of the simple group of order 168 —

“Hexagonal Logic of the Field F8 as a Boolean Logic
with Three Involutive Modalities,” pp. 191-220 in

The Road to Universal Logic:
Festschrift for 50th Birthday of
Jean-Yves Béziau, Volume I,

Editors: Arnold Koslow, Arthur Buchsbaum,
Birkhäuser Studies in Universal Logic, dated 2015
by publisher but Oct. 11, 2014, by Amazon.com.

See also the eightfold cube in this journal.

Friday, March 27, 2015

Pursuit of Gestalt*

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

The art above is by the Copenhagen studio
Hvass & Hannibal. For a photo of the artists,
see a webpage on Beijing Design Week 2011.

Hvass and Hannibal were apparently in Beijing
for the "open workshop," Sept. 17-23, 2011.

Gestalt-related material from this journal that week —

* Title suggested by that of a book by Quine.

Thursday, February 28, 2013

Paperweights

Filed under: General,Geometry — Tags: , — m759 @ 1:06 pm

A different dodecahedral space (Log24 on Oct. 3, 2011)—

R. T. Curtis, symmetric generation of M12 in a dodecahedron

Monday, June 18, 2012

Surface

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

"Poetry is an illumination of a surface…."

— Wallace Stevens

IMAGE- NY Times online front page, June 18, 2012- New Microsoft 'Surface' computer

Some poetic remarks related to a different surface, Klein's Quartic—

This link between the Klein map κ and the Mathieu group M24
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 M24
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—

Sunday, June 17, 2012

Congruent Group Actions

A Google search today yielded no results
for the phrase "congruent group actions."

Places where this phrase might prove useful include—

Monday, October 3, 2011

Mathieu Symmetry

Filed under: General,Geometry — Tags: , — m759 @ 7:08 am

The following may help show why R.T. Curtis calls his approach
to sporadic groups symmetric  generation—

(Click to enlarge.)

http://www.log24.com/log/pix11C/111003-Curtis10YrsOn-Dodecahedron-320w.jpg

Related material— Yesterday's Symmetric Generation Illustrated.

Sunday, October 2, 2011

Emerson on Mathematics

Filed under: General,Geometry — Tags: — m759 @ 9:29 pm

From "The Poet" (1844)—

If the imagination intoxicates the poet, it is not inactive in other men. The metamorphosis excites in the beholder an emotion of joy.

The use of symbols has a certain power of emancipation and exhilaration for all men. We seem to be touched by a wand, which makes us dance and run about happily, like children. We are like persons who come out of a cave or cellar into the open air. This is the effect on us of tropes, fables, oracles, and all poetic forms. Poets are thus liberating gods. Men have really got a new sense, and found within their world, another world or nest of worlds; for the metamorphosis once seen, we divine that it does not stop. I will not now consider how much this makes the charm of algebra and the mathematics, which also have their tropes, but it is felt in every definition….

… Here is the difference betwixt the poet and the mystic, that the last nails a symbol to one sense, which was a true sense for a moment, but soon becomes old and false…. Mysticism consists in the mistake of an accidental and individual symbol for an universal one….  And the mystic must be steadily told,— All that you say is just as true without the tedious use of that symbol as with it. Let us have a little algebra, instead of this trite rhetoric,— universal signs, instead of these village symbols,— and we shall both be gainers.

See also Weyl on the use of symbols (in coordinate systems) and today's previous posts Birth of a Poet and Symmetric Generation Illustrated.

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