“Symmetric Generation” « Search Results

Friday, September 16, 2022

Symmetric Generation

Filed under: General — Tags: Bitspace, Cube-Motif Octads, Promises — m759 @ 2:20 pm

The above is about a subspace of the
24-dimensional vector space over GF(2)
. . . "An entire world of just 24 squares,"
to adapt a phrase from other Log24
posts tagged "Promises."

Update of 1:45 AM ET Sept. 18, 2022 —

It seems* from a Magma calculation that
the resemblance of the above extended
cube-motif code to the Golay code is only
superficial.

Without the highly symmetric generating codewords that were added
to extend its dimension from 8 to 12, the cube-motifs code apparently
does , like the Golay code, have nonzero weights of only 8, 12, 16, and 24 —

Perhaps someone can prove there is no way that adding more generating
codewords can turn the cube-motif code into the Golay code.

* The "seems" is because I have not yet encountered any of these
relatively rare (42 out of 4096) purported weight-4 codewords. Their
apparent existence may be due to an error in my typing of 0's and 1's.

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Sunday, December 2, 2018

Symmetric Generation …

Filed under: G-Notes,General,Geometry — Tags: Aitchison, Mathieu Cube, Symmetric Generation — m759 @ 12:00 pm

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

Iain Aitchison on symmetric generation of M24

Update of 2 PM ET —

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Saturday, September 22, 2018

Symmetric Generation, by Curtis

Filed under: G-Notes,General,Geometry — Tags: ABC Art 2018, Symmetric Generation — m759 @ 10:15 am

Norwegian artist Josefine Lyche —

Lyche's shirt honors the late Kurt Cobain.

"Here we are now, entertain us."

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Symmetric Generation, by Netflix

Filed under: General — Tags: ABC Art 2018, Maniac Series Begins, Symmetric Generation — m759 @ 5:05 am

Friday, September 21, 2018

Symmetric Generation, by Nao

Filed under: General — Tags: ABC Art 2018, Maniac Series Begins, Symmetric Generation — 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."

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Wednesday, April 20, 2016

Symmetric Generation of a Simple Group

Filed under: General,Geometry — Tags: Affine Cube, Cullinane Cube, Eightfold Cube, Geometric Theology, Road to Hell, Simple Group, Solomon's Cube, Springer, Symmetric Generation — m759 @ 9:00 pm

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.

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

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Monday, October 19, 2015

Symmetric Generation of the Simple Order-168 Group

Filed under: General,Geometry — Tags: Eightfold Cube, Geometric Theology, Springer, Symmetric Generation — 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.

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Sunday, October 2, 2011

Symmetric Generation Illustrated

Filed under: General,Geometry — Tags: Geometry, Symmetric Generation — m759 @ 7:20 pm

R.T. Curtis in a 1990 paper* discussed his method of "symmetric generation" of groups as applied to the Mathieu groups M₁₂ and M₂₄.

See Finite Relativity and the Log24 posts Relativity Problem Revisited (Sept. 20) and Symmetric Generation (Sept. 21).

Here is some exposition of how this works with M₁₂.

* "Geometric Interpretations of the ‘Natural’ Generators of the Mathieu groups," Mathematical Proceedings of the Cambridge Philosophical Society (1990), Vol. 107, Issue 01, pp. 19-26.

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Wednesday, September 21, 2011

Symmetric Generation

Filed under: General,Geometry — Tags: Aitchison, Crucible Raiders, Eightfold Cube, Generation, Geometry, Symmetric Generation — 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 ≅ M₂₄ is generated by
some set of seven involutions, which would be a very weak
requirement. We are asserting that M₂₄ 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

Saturday, July 11, 2026

The Three Cubes

Filed under: General — Tags: Affine Generation, The Cubes — m759 @ 8:32 am

Quote adapted from a famous young-adult novel . . .

"There is such a thing as affine group generation
by permutation of parallel subspaces."

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The Brautigan Reset

Filed under: General — Tags: Affine Generation, Resets, The Eureka Plot — m759 @ 12:23 am

Vide The Maxwell Enticement . . .

The Sept. 15, 1984, date in the image below was possibly
the death date of Hawkline Monster author Richard Brautigan.

Quote adapted from a famous young-adult novel . . .

"There is such a thing as affine group generation
by permutation of parallel subspaces."

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Sunday, March 22, 2026

Non-MOG Pattern Symmetry

Filed under: General — Tags: Cube-Motif Octads — m759 @ 8:36 am

The Miracle Octad Generator (MOG) of R. T. Curtis
greatly simplified the study of the 759 octads in the
Steiner system S(5, 8, 24).

The MOG arranges these octads very neatly in a 4×6 array
of square unit cells. There is, however, one aesthetic drawback
to the arrangement . . . It lacks symmetry under the natural
rotations and reflections of the entire 4×6 rectangular array.

A note of my own from 1981 may or may not lead eventually
to a rearrangement of the 759 octads, each within a 4×6 array,
that does have such overall symmetry under the symmetries
of a bare 4×6 rectangle . . . If, that is, such overall symmetry is
even possible, in light of purely group-theoretic considerations.

(Exercise: Would such symmetry imply the existence of a normal
subgroup in a group known to have no such subgroups?)

Previous posts in this journal have described approaches to the
above symmetric-rearrangement problem . . . a problem that
could be posed more generally, for binary patterns other than
those of the MOG.

The following Magma code shows that although the above space has
12 dimensions, it is NOT the Golay-code space.

// 260322 Magma Check March 22, 2026

> K := FiniteField(2);
> C := LinearCode<K, 24 |
> [0,0,0,0,0,1,0,1,0,0,1,1,1,0,0,1,0,1,1,1,0,1,1,1],
> [0,1,0,0,0,0,0,1,1,0,0,1,1,1,0,1,0,0,1,1,1,1,0,1],
> [0,1,1,0,1,0,0,0,1,0,0,0,1,1,1,1,1,0,1,0,1,1,0,0],
> [0,0,1,0,1,1,0,0,0,0,1,0,1,0,1,1,1,1,1,0,0,1,1,0],
> [1,0,0,0,0,0,1,1,0,0,1,0,1,0,1,0,0,1,1,1,1,0,1,1],
> [1,1,0,1,0,0,0,1,0,0,0,0,1,1,1,1,0,1,0,1,1,0,0,1],
> [0,1,0,1,1,0,0,0,0,1,0,0,0,1,1,1,1,1,0,0,1,1,0,1],
> [0,0,0,0,1,0,1,0,0,1,1,0,0,0,1,0,1,1,1,0,1,1,1,1],
> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0,0,1,1,0,1,0],
> [1,1,1,1,0,1,1,1,0,1,0,0,0,1,1,0,0,1,0,1,0,0,0,0],
> [1,1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,1,1,0,0,0,0,0,1],
> [1,0,0,1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0,0,1,0,1,1],
> [0,0,1,1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,1,0],
> [0,1,1,0,0,1,1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,1,0,0],
> [1,1,1,0,1,1,1,0,1,0,0,1,1,1,0,0,1,0,1,0,0,0,0,0],
> [1,0,1,1,1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0,0,0,1,0],
> [1,0,0,1,0,1,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,1],
> [0,0,0,1,0,0,0,0,1,1,0,1,0,1,0,1,1,0,0,1,1,1,1,1],
> [0,0,1,0,0,0,1,0,1,1,0,0,0,1,1,0,1,0,1,1,1,1,1,0],
> [1,0,1,0,0,1,1,0,0,0,0,0,1,1,1,0,1,1,1,1,0,0,1,0],
> [0,1,0,0,1,1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1],
> [1,1,0,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0,1,0,0,1],
> [1,1,1,1,1,0,0,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0,0,0],
> [0,1,1,1,1,1,0,1,0,1,1,0,0,0,1,1,0,1,0,0,0,1,0,0],
//
// Bricks
> [1,1,0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0],
> [0,0,1,1,0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,1,1,0,0],
> [0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,1,1],
// Top and bottom halves
> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0],
> [0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1],
// Left and right halves
> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0],
> [0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1]>;
//
> Dimension(C);
> MinimumWeight(C);
> WeightDistribution(C);

12
4
[ <0, 1>, <4, 42>, <8, 591>, <12, 2828>,
<16, 591>, <20, 42>, <24, 1> ]

Here are some earlier Log24 images related to this topic.

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Saturday, September 3, 2022

1984 Revisited

Filed under: General — Tags: Cube Brick, Galois Tesseract — m759 @ 2:46 pm

Cube Bricks 1984 —

Related material —

Note the three quadruplets of parallel edges in the 1984 figure above.

Further Reading —

The above Gates article appeared earlier, in the June 2010 issue of
Physics World , with bigger illustrations. For instance —

Exercise: Describe, without seeing the rest of the article,
the rule used for connecting the balls above.

Wikipedia offers a much clearer picture of a (non-adinkra) tesseract —

And then, more simply, there is the Galois tesseract —

For parts of my own world in June 2010, see this journal for that month.

The above Galois tesseract appears there as follows:

See also the Klein correspondence in a paper from 1968
in yesterday's 2:54 PM ET post.

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Tuesday, July 19, 2022

The Lost Message

Filed under: General — Tags: Groups Act, Rouge et Noir — m759 @ 12:10 pm

“Somehow, a message had been lost on me. Groups act .
The elements of a group do not have to just sit there,
abstract and implacable; they can do things, they can
‘produce changes.’ In particular, groups arise
naturally as the symmetries of a set with structure.”

— Thomas W. Tucker, review of Lyndon’s Groups and Geometry
in The American Mathematical Monthly , Vol. 94, No. 4
(April 1987), pp. 392-394.

"…groups are invariably best studied through their action on some structure…."

— R. T. Curtis, “Symmetric Generation of the Higman-Sims Group” in
Journal of Algebra 171 (1995), pp. 567-586.

Related material — Other posts now tagged Groups Act.

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Friday, February 11, 2022

For Space Groupies

Filed under: General — Tags: 1984 Cubes, Brick Space, Cube Brick, Deep Space — m759 @ 5:31 pm

A followup to Wednesday's post Deep Space —

Related material from this journal on July 9, 2019 —

Cube Bricks 1984 —

From "Tomorrowland" (2015) —

From other posts tagged 1984 Cubes —

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Wednesday, December 29, 2021

Throw Some Shapes

Filed under: General — Tags: Throwing Shapes — m759 @ 8:31 am

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Monday, January 4, 2021

The Purloined Title

Filed under: General — Tags: Art Title, Go Ask Alice, Promises, When Ken — m759 @ 10:02 am

"When logic and proportion have fallen sloppy dead . . . ."

The Mystery of 'Monomial Representations and Symmetric Presentations'

See as well "Symmetric Generation" in this journal.

"Feed your head." — Grace Slick

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Friday, November 29, 2019

Verifying Aitchison’s Cuboctahedral Generation of M₂₄

Filed under: General — Tags: Aitchison — m759 @ 1:06 am

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.

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Tuesday, November 26, 2019

Alea Iacta Est*

Filed under: General — Tags: Aitchison, Coup de Dés, Iacta Est — m759 @ 11:11 am

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».

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Monday, October 7, 2019

Berlekamp Garden vs. Kinder Garten

Filed under: General — Tags: Cube Brick, Eightfold Cube — m759 @ 11:00 pm

Stevens's Omega and Alpha (see previous post) suggest a review.

Omega — The Berlekamp Garden. See Misère Play (April 8, 2019).
Alpha — The Kinder Garten. See Eighfold Cube.

Illustrations —

The sculpture above illustrates Klein's order-168 simple group.
So does the sculpture below.

Cube Bricks 1984 —

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Tuesday, July 9, 2019

Schoolgirl Space: 1984 Revisited

Filed under: General — Tags: 1984 Cubes, Brick Space, Cube Brick, Cube School, Eightfold Cube, Ministry of Culture, Schoolgirl Space — m759 @ 9:24 pm

Cube Bricks 1984 —

From "Tomorrowland" (2015) —

From John Baez (2018) —

See also this morning's post Perception of Space
and yesterday's Exploring Schoolgirl Space.

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Tuesday, May 21, 2019

Cube Geometry Continues.

Filed under: General — Tags: Symmetric Generation — m759 @ 1:30 pm

An illustration from the April 20, 2016, post

Symmetric Generation of a Simple Group —

"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

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Tuesday, January 1, 2019

The Magnificent Seven

Filed under: General — Tags: Aitchison, Symmetric Generation — m759 @ 12:00 am

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Sunday, December 30, 2018

Also Sprach Aitchison

Filed under: General — Tags: Aitchison, Also Sprach, Symmetric Generation — 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

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

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Sunday, December 9, 2018

Quaternions in a Small Space

Filed under: G-Notes,General,Geometry — Tags: Affine Cube, Aitchison, Small Space, Symmetric Generation — 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 —

Visualizing GL(2,p) — A 1985 note illustrating group actions
on the 3×3 (ninefold) square.
Another 1985 note showing group actions on the 3×3 square
transferred to the 2x2x2 (eightfold) cube.
Quaternions in an Affine Galois Plane — A webpage from 2010.

Related material —