Cube Space « Log24

Monday, May 18, 2026

Affine Groups over GF(2) from S₄ Actions

Filed under: General — Tags: Cube Space, Halma — m759 @ 3:06 am

In the square . . .
The affine group over GF(2) of order 24 acting on
the 4 vertices of a square
is generated by (and is) S₄ actions
on the set of 4 vertices.

In the cube . . .
The affine group over GF(2) of order 1344 acting on
the 8 vertices of a cube
is generated by combining S₄ actions
on each of the 3 sets of 4 parallel edges.

In the hypercube . . .
The affine group over GF(2) of order 322,560 acting on
the 16 vertices of a hypercube
is generated by combining S₄ actions
on each of the 6 sets of 4 parallel faces.

Exercise . . . To what extent can these results be generalized?

(Specifically, is it true or false that the general n-dimensional
vector space over GF(2) is made up of sets of 4 parallel
(n-2)-dimensional subspaces, and that combining arbitrary
S₄ permutations within these subspace-sets yields the
affine group on the full n-dimensional space?)

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Friday, February 13, 2026

Cube Space

Filed under: General — Tags: Affine Cube, Cube School, Cube Space, Planes of Reality — m759 @ 3:37 am

Theorem:

Some large natural symmetry groups of the sets of 8, 16, 32, or 64 points
in Euclidean space that are located at the vertices of a cube in 3, 4, 5. or 6
dimensions are generated by, respectively, arbitrary permutations of
parallel edges or parallel faces or parallel cubes or parallel hypercubes .

(For an example, see Diamond Theory in 1937.)

Illustration of related group actions:

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Monday, April 4, 2016

Cube for Berlin

Filed under: General,Geometry — Tags: Brick Space, Cube Brick, Cube School, Cube Space, Cubehenge, Cullinane Cube, Eightfold Cube, Symmetric Generation — 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 —

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 —

Related remark from the literature —

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.)

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Thursday, August 28, 2014

Source of the Finite

Filed under: General,Geometry — Tags: Bauhaus Cube, Cube School, Cube Space, Eightfold Cube — m759 @ 10:20 am

"Die Unendlichkeit ist die uranfängliche Tatsache: es wäre nur
zu erklären, woher das Endliche stamme…."

— Friedrich Nietzsche, Das Philosophenbuch/Le livre du philosophe
(Paris: Aubier-Flammarion, 1969), fragment 120, p. 118

Cited as above, and translated as "Infinity is the original fact;
what has to be explained is the source of the finite…." in
The Production of Space , by Henri Lefebvre. (Oxford: Blackwell,
1991 (1974)), p. 181.

This quotation was suggested by the Bauhaus-related phrase
"the laws of cubical space" (see yesterday's Schau der Gestalt )
and by the laws of cubical space discussed in the webpage
Cube Space, 1984-2003.

For a less rigorous approach to space at the Harvard Graduate
School of Design, see earlier references to Lefebvre in this journal.

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Tuesday, June 26, 2012

Bright Black

Filed under: General,Geometry — Tags: Cube School, Cube Space, Looking Deeply — m759 @ 12:12 am

"'In the dictionary next to [the] word "bright," you should see Paula’s picture,' he said. 'She was super smart, with a sparkling wit. … She had a beautiful sense of style and color.'"

— Elinor J. Brecher in The Miami Herald on June 8, quoting Palm Beach Post writer John Lantigua on the late art historian Paula Hays Harper

This journal on the date of her death—

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

For some simpleminded commentary, see László Lovász on the cube space.

Some less simpleminded commentary—

"Was ist Raum, wie können wir ihn
erfassen und gestalten?"

Walter Gropius,

The Theory and
Organization of the
Bauhaus (1923)

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Sunday, September 18, 2011

Anatomy of a Cube

Filed under: General,Geometry — Tags: Cube School, Cube Space, Cubehenge, Eightfold Cube — m759 @ 12:00 pm

R.D. Carmichael's seminal 1931 paper on tactical configurations suggests
a search for later material relating such configurations to block designs.
Such a search yields the following—

"… it seems that the relationship between
BIB [balanced incomplete block ] designs
and tactical configurations, and in particular,
the Steiner system, has been overlooked."
— D. A. Sprott, U. of Toronto, 1955

The figure by Cullinane included above shows a way to visualize Sprott's remarks.

For the group actions described by Cullinane, see "The Eightfold Cube" and
"A Simple Reflection Group of Order 168."

Update of 7:42 PM Sept. 18, 2011—

From a Summer 2011 course on discrete structures at a Berlin website—

A different illustration of the eightfold cube as the Steiner system S(3, 4, 8)—

Note that only the static structure is described by Felsner, not the
168 group actions discussed (as above) by Cullinane. For remarks on
such group actions in the literature, see "Cube Space, 1984-2003."

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Monday, June 21, 2010

Cube Spaces

Filed under: General,Geometry — Tags: Affine Cube, Appropriation, Change in China, Cube School, Cube Space, Eightfold Cube, Groups and Spaces, Interplay — m759 @ 11:30 am

Cubic models of finite geometries
display an interplay between
Euclidean and Galois geometry.

Example 1— The 2×2×2 Cube—

also known as the eightfold cube—

2x2x2 cube

Group actions on the eightfold cube, 1984—

Version by Laszlo Lovasz et al., 2003—

Lovasz et al. go on to describe the same group actions
as in the 1984 note, without attribution.

Example 2— The 3×3×3 Cube

A note from 1985 describing group actions on a 3×3 plane array—

Undated software by Ed Pegg Jr. displays
group actions on a 3×3×3 cube that extend the
3×3 group actions from 1985 described above—

Pegg gives no reference to the 1985 work on group actions.

Example 3— The 4×4×4 Cube

A note from 27 years ago today—

As far as I know, this version of the
group-actions theorem has not yet been ripped off.

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Thursday, October 30, 2008

Thursday October 30, 2008

Filed under: General,Geometry — Tags: Cube School, Cube Space — m759 @ 12:25 pm

Readings for
Devil's Night

Pope Benedict XVI, formerly the modern equivalent of The Grand Inquisitor

1. Today's New York Times review
of Peter Brook's production of
"The Grand Inquisitor"
2. Mathematics and Theology
3. Christmas, 2005
4. Cube Space, 1984-2003

"I love those Bavarians…
so meticulous."

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Friday, October 24, 2008

Friday October 24, 2008

Filed under: General,Geometry — Tags: Affine Cube, Cube School, Cube Space, Diamonds and Whirls, Eightfold Cube, Springer — m759 @ 8:08 am

"The Cube Space" is a name given to the eightfold cube in a vulgarized mathematics text, Discrete Mathematics: Elementary and Beyond, by Laszlo Lovasz et al., published by Springer in 2003. The identification in a natural way of the eight points of the linear 3-space over the 2-element field GF(2) with the eight vertices of a cube is an elementary and rather obvious construction, doubtless found in a number of discussions of discrete mathematics. But the less-obvious generation of the affine group AGL(3,2) of order 1344 by permutations of parallel edges in such a cube may (or may not) have originated with me. For descriptions of this process I wrote in 1984, see Diamonds and Whirls and Binary Coordinate Systems. For a vulgarized description of this process by Lovasz, without any acknowledgement of his sources, see an excerpt from his book.

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