Log24

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, September 9, 2011

Galois vs. Rubik

Filed under: General,Geometry — Tags: , , , — m759 @ 2:56 PM

(Continued from Abel Prize, August 26)

IMAGE- Elementary Galois Geometry over GF(3)

The situation is rather different when the
underlying Galois field has two rather than
three elements… See Galois Geometry.

Image-- Sugar cube in coffee, from 'Bleu'

The coffee scene from “Bleu”

Related material from this journal:

The Dream of
the Expanded Field

Image-- 4x4 square and 4x4x4 cube

Monday, March 1, 2010

Visual Group Theory

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

The current article on group theory at Wikipedia has a Rubik's Cube as its logo– 

Wikipedia article 'Group theory' with Rubik Cube and quote from Nathan Carter-- 'What is symmetry?'

 

The article quotes Nathan C. Carter on the question "What is symmetry?"

This naturally suggests the question "Who is Nathan C. Carter?"

A search for the answer yields the following set of images…

Labelings of the eightfold cube

Click image for some historical background.

Carter turns out to be a mathematics professor at Bentley University.  His logo– an eightfold-cube labeling (in the guise of a Cayley graph)– is in much better taste than Wikipedia's.
 

Monday, August 27, 2018

Children of the Six Sides

Filed under: General,Geometry — Tags: — m759 @ 11:32 AM

http://www.log24.com/log/pix18/180827-Terminator-3-tx-arrival-publ-160917.jpg

http://www.log24.com/log/pix18/180827-Terminator-3-tx-arrival-publ-161018.jpg

From the former date above —

Saturday, September 17, 2016

A Box of Nothing

Filed under: Uncategorized — m759 @ 12:13 AM

(Continued)

"And six sides to bounce it all off of.

From the latter date above —

Tuesday, October 18, 2016

Parametrization

Filed under: Uncategorized — m759 @ 6:00 AM

The term "parametrization," as discussed in Wikipedia, seems useful for describing labelings that are not, at least at first glance, of a vector-space  nature.

Examples: The labelings of a 4×4 array by a blank space plus the 15 two-subsets of a six-set (Hudson, 1905) or by a blank plus the 5 elements and the 10 two-subsets of a five-set (derived in 2014 from a 1906 page by Whitehead), or by a blank plus the 15 line diagrams of the diamond theorem.

Thus "parametrization" is apparently more general than the word "coodinatization" used by Hermann Weyl —

“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

Note, however, that Weyl's definition of "coordinatization" is not limited to vector-space  coordinates. He describes it as simply a mapping to a set of reproducible symbols

(But Weyl does imply that these symbols should, like vector-space coordinates, admit a group of transformations among themselves that can be used to describe transformations of the point-space being coordinatized.)

From March 2018 —

http://www.log24.com/log/pix18/180827-MIT-Rubik-Robot.jpg

Wednesday, August 15, 2018

An Illusion of Brilliance

Filed under: General,Geometry — Tags: , — m759 @ 5:25 PM

" . . . the 3 by 3, the six-sided, three-layer configuration of
the original Rubik’s Cube, which bestows an illusion of brilliance
on those who can solve it."

— John Branch in the online New York Times  today,
     "Children of the Cube":

https://www.nytimes.com/2018/08/15/sports/
cubing-usa-nationals-max-park.html

Cube-solving, like other sports, allows for displays of
impressive and admirable skill, if not "brilliance."

The mathematics — group theory — that is sometimes associated
with Rubik's Cube is, however, not  a sport.  See Rubik + Group
in this journal.

http://www.log24.com/log/pix18/180815-Alperin-Bell-preface-1995.gif

Tuesday, March 27, 2018

Compare and Contrast

Filed under: General,Geometry — Tags: — m759 @ 4:28 PM

Weyl on symmetry, the eightfold cube, the Fano plane, and trigrams of the I Ching

Related material on automorphism groups

The "Eightfold Cube" structure shown above with Weyl
competes rather directly with the "Eightfold Way" sculpture 
shown above with Bryant. The structure and the sculpture
each illustrate Klein's order-168 simple group.

Perhaps in part because of this competition, fans of the Mathematical
Sciences Research Institute (MSRI, pronounced "Misery') are less likely
to enjoy, and discuss, the eight-cube mathematical structure  above
than they are an eight-cube mechanical puzzle  like the one below.

Note also the earlier (2006) "Design Cube 2x2x2" webpage
illustrating graphic designs on the eightfold cube. This is visually,
if not mathematically, related to the (2010) "Expert's Cube."

Monday, April 25, 2016

Peirce’s Accounts of the Universe

Filed under: General,Geometry — m759 @ 8:19 PM

Compare and contrast Peirce's seven systems of metaphysics with
the seven projective points in a post of March 1, 2010 —

Wikipedia article 'Group theory' with Rubik Cube and quote from Nathan Carter-- 'What is symmetry?'

From my commentary on Carter's question —

Labelings of the eightfold cube

Monday, April 4, 2016

Cube for Berlin

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

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.)

Sunday, February 8, 2015

Hungarian Phenomenon

Filed under: General — Tags: , , , — m759 @ 2:07 PM

For Autism Sunday —

Mathematician John von Neumann
reportedly died on this date.

“He belonged  to that so-called
Hungarian phenomenon….”

A webpage titled
“Von Neumann, Jewish Catholic”

Illustrations of another Hungarian phenomenon:

IMAGE- Anthony Hopkins exorcises a Rubik cube

Friday, June 13, 2014

It’s 10 PM

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

"The wind of change is blowing throughout the continent.
Whether we like it or not, this growth of national consciousness
is a political fact."— Prime Minister Harold Macmillan,
South Africa, 1960

"Lord knows when the cold wind blows
it'll turn your head around." — James Taylor

From a Log24 post of August 27, 2011:

IMAGE- 'Group Theory' Wikipedia article with Rubik's cube as main illustration and argument  by a cuber for the image's use

For related remarks on "national consciousness," see Frantz Fanon.

Thursday, June 5, 2014

Twisty Quaternion Symmetry

Filed under: General,Geometry — m759 @ 9:11 PM

The previous post told how user58512 at math.stackexchange.com
sought in 2013 a geometric representation of Q, the quaternion group.
He ended up displaying an illustration that very possibly was drawn,
without any acknowledgement of its source, from my own work.

On the date that user58512 published that illustration, he further
pursued his March 1, 2013, goal of a “twisty” quaternion model.

On March 12, 2013,  he suggested that the quaternion group might be
the symmetry group of the following twisty-cube coloring:

IMAGE- Twisty-cube coloring illustrated by Jim Belk

Illustration by Jim Belk

Here is part of a reply by Jim Belk from Nov. 11, 2013, elaborating on
that suggestion:

IMAGE- Jim Belk's proposed GAP construction of a 2x2x2 twisty-cube model of the quaternion group 

Belk argues that the colored cube is preserved under the group
of actions he describes. It is, however, also preserved under a
larger group.  (Consider, say, rotation of the entire cube by 180
degrees about the center of any one of its checkered faces.)  The
group Belk describes seems therefore to be a  symmetry group,
not the  symmetry group, of the colored cube.

I do not know if any combination puzzle has a coloring with
precisely  the quaternion group as its symmetry group.

(Updated at 12:15 AM June 6 to point out the larger symmetry group
and delete a comment about an arXiv paper on quaternion group models.)

Monday, April 28, 2014

Hymn

Filed under: General — Tags: — m759 @ 2:15 AM

Suggested by a Saturday death in Jersey City:

Somewhere, over the gray space

Gray Space, by Wagner

Filed under: General — Tags: — m759 @ 1:37 AM

(Orlin Wagner/Associated Press) – A vehicle tops a hill along
U.S. Route 56 as a severe thunderstorm moves through the area
near Baldwin City, Kansas, on Sunday, April 27, 2014.

See a related news story.

Sunday, April 27, 2014

Sunday School

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

Galois and Abel vs. Rubik

(Continued)

“Abel was done to death by poverty, Galois by stupidity.
In all the history of science there is no completer example
of the triumph of crass stupidity….”

— Eric Temple Bell,  Men of Mathematics

Gray Space  (Continued)

… For The Church of Plan 9.

Saturday, April 26, 2014

For Two Artists of Norway

Filed under: General — Tags: , — m759 @ 5:48 AM

IMAGE- Conclusion of introduction to Heinrich Zimmer's 'The King and the Corpse'

See also LYNX 760 , Rubik vs. Abel, and Toying.

Friday, April 25, 2014

Toying

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

IMAGE- 'Another instance of producers toying with artists' and a Rubik's Cube exhibition in Jersey City beginning Saturday, April 26

Related material: Quilt Geometry and Magical Realism Revisited.

Wednesday, July 11, 2012

Cuber

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

(Continued)

For Pete Rustan, space recon expert, who died on June 28—

(Click to enlarge.)

See also Galois vs. Rubik and Group Theory Template.

Sunday, August 28, 2011

The Cosmic Part

Filed under: General,Geometry — Tags: , — m759 @ 6:29 PM

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:

http://www.log24.com/log/pix10/100214-Cube2x2x2.gif

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—

http://www.log24.com/log/pix11B/110828-BorovikM.jpg

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 2-element Galois field, these hyperplanes are certain sets of four subcubes.

Saturday, August 27, 2011

Cosmic Cube*

Filed under: General,Geometry — Tags: , , , — m759 @ 12:25 PM

IMAGE- Anthony Hopkins exorcises a Rubik cube

Prequel (Click to enlarge)

IMAGE- Galois vs. Rubik: Posters for Abel Prize, Oslo, 2008

Background —

IMAGE- 'Group Theory' Wikipedia article with Rubik's cube as main illustration and argument by a cuber for the image's use

See also Rubik in this journal.

* For the title, see Groups Acting.

Thursday, May 26, 2011

For the Class of ’11

Filed under: General — Tags: — m759 @ 2:12 PM

IMAGE- Anthony Hopkins exorcises a Rubik cube

But leave the wise to wrangle, and with me
the quarrel of the universe let be;
and, in some corner of the hubbub couched,
make game of that which makes as much of thee.

John McKay at sci.math

Related material: Harvard Treasure, Favicon, and Crimson Tide.

Thursday, April 28, 2011

Crimson Tide…

Filed under: General — m759 @ 3:59 PM

A sequel to Wednesday afternoon's post on The Harvard Crimson ,
Atlas Shrugged (illustrated below) —

http://www.log24.com/log/pix11A/110427-CrimsonAtlas500w.jpg

Related material found today in Wikipedia—

A defense of Rubik by 'Pazouzou'

See also Savage Logic (Oct. 19, 2010), as well as
Stellan Skarsgård in Lie Groups for Holy Week (March 30, 2010)
and in Exorcist: The Beginning (2004).

Saturday, February 27, 2010

Cubist Geometries

Filed under: General,Geometry — Tags: , — m759 @ 2:01 PM

"The cube has…13 axes of symmetry:
  6 C2 (axes joining midpoints of opposite edges),
4 C3 (space diagonals), and
3C4 (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 13-point Galois plane.

The geometer's 3×3×3 cube–
27 separate subcubes unconnected
by any Rubik-like mechanism–

The 3x3x3 geometer's cube, with coordinates

The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
  subcubes in the (Galois) 3×3×3 cube–

The 13 symmetry axes of the cube

A closely related structure–
the finite projective plane
with 13 points and 13 lines–

Oxley's 2004 drawing of the 13-point projective plane

A later version of the 13-point plane
by Ed Pegg Jr.–

Ed Pegg Jr.'s 2007 drawing of the 13-point projective plane

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)–

Ed Pegg Jr.'s program at Wolfram demonstrating concepts of a 1985 note by 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 3-space converted to a vector 3-space.

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.

Friday, December 12, 2008

Friday December 12, 2008

Filed under: General,Geometry — m759 @ 3:09 PM
On the Symmetric Group S8

Wikipedia on Rubik's 2×2×2 "Pocket Cube"–
 

http://www.log24.com/log/pix08A/081212-PocketCube.jpg
 

"Any permutation of the 8 corner cubies is possible (8! positions)."

Some pages related to this claim–

Simple Groups at Play

Analyzing Rubik's Cube with GAP

Online JavaScript Pocket Cube.

The claim is of course trivially true for the unconnected subcubes of Froebel's Third Gift:
 

Froebel's third gift, the eightfold cube
© 2005 The Institute for Figuring

 

Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)

See also:

MoMA Goes to Kindergarten,

Tea Privileges
,

and

"Ad Reinhardt and Tony Smith:
A Dialogue,"
an exhibition opening today
at Pace Wildenstein.

For a different sort
of dialogue, click on the
artists' names above.

For a different
approach to S8,
see Symmetries.

"With humor, my dear Zilkov.
Always with a little humor."

-- The Manchurian Candidate

Wednesday, July 5, 2006

Wednesday July 5, 2006

Filed under: General,Geometry — Tags: — m759 @ 12:25 PM

And now, from
the author of Sphere

CUBE

He beomes aware of something else… some other presence.
“Anybody here?” he says.
I am here.
He almost jumps, it is so loud. Or it seems loud. Then he wonders if he has heard anything at all.
“Did you speak?”
No.
How are we communicating? he wonders.
The way everything communicates with everything else.
Which way is that?
Why do you ask if you already know the answer?

Sphere, by Michael Crichton, Harvard ’64

“… when I went to Princeton things were completely different. This chapel, for instance– I remember when it was just a clearing, cordoned off with sharp sticks.  Prayer was compulsory back then, and you couldn’t just fake it by moving your lips; you had to know the words, and really mean them.  I’m dating myself, but this was before Jesus Christ.”

Baccalaureate address at Princeton, Pentecost 2006, reprinted in The New Yorker, edited by David Remnick, Princeton ’81

Related figures:

The image “http://www.log24.com/log/pix06A/060617-Spellbound.jpg” cannot be displayed, because it contains errors.

For further details,
see Solomon’s Cube
and myspace.com/affine.

The image “http://www.log24.com/log/pix06A/060705-Cube.jpg” cannot be displayed, because it contains errors.

For further details,
see Jews on Buddhism
and
Adventures in Group Theory.

“In this way we are offered
a formidable lesson
for every Christian community.”

Pope Benedict XVI
on Pentecost,
June 4, 2006,
St. Peter’s Square
.

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