James Propp* *in the current *Math Horizons *on the eightfold cube —

For another puerile approach to the eightfold cube,

see **Cube Space, 1984-2003** (Oct. 24, 2008).

James Propp* *in the current *Math Horizons *on the eightfold cube —

For another puerile approach to the eightfold cube,

see **Cube Space, 1984-2003** (Oct. 24, 2008).

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**A sequel to this afternoon’s Rubik Quote:**

“The Cube was born in 1974 as a teaching tool

to help me and my students better understand

space and 3D. The Cube challenged us to find

order in chaos.”

— Professor Ernő Rubik at Chrome Cube Lab

(Click image below to enlarge.)

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For the late Cardinal Glemp of Poland,

who died yesterday, some links:

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**Cubic models of finite geometries
display an interplay between
Euclidean and **

**Example 1— The 2×2×2 Cube—**

also known as the *eightfold* 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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This is a sequel to yesterday's post Cube Space Continued.

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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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"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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**Continued from October 30 (Devil’s Night), 2013.**

“In a sense, we would see that change

arises from the structure of the object.”

— Theoretical physicist quoted in a

Simons Foundation article of Sept. 17, 2013

This suggests a review of mathematics and the

“*Classic of Change *,” the *I Ching* .

The physicist quoted above was discussing a rather

complicated object. His words apply to a much simpler

object, an embodiment of the eight trigrams underlying

the *I Ching* as the corners of a cube.

See also…

(Click for clearer image.)

The Cullinane image above illustrates the seven points of

the Fano plane as seven of the eight *I Ching* trigrams and as

seven natural ways of slicing the cube.

For a different approach to the mathematics of cube slices,

related to Gauss’s composition law for binary quadratic forms,

see the *Bhargava cube * in a post of April 9, 2012.

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“‘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—

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?”*

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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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**Apollo's 13: A Group Theory Narrative —**

I. At Wikipedia —

II. Here —

See Cube Spaces and Cubist Geometries.

**The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
subcubes in the 27-part (Galois) 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.

Comments Off on Mathematics and Narrative continued…

This journal’s 11 AM Sunday post was “Lovasz Wins Kyoto Prize.” This is now the top item on the American Mathematical Society online home page—

For more background on Lovasz, see today’s

previous Log24 post, Cube Spaces, and also

Cube Space, 1984-2003.

“If the Party could thrust its hand into the past and

say of this or that event, it never happened….”

— George Orwell, *1984*

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From a June 18 press release—

KYOTO, Japan, Jun 18, 2010 (BUSINESS WIRE) — The non-profit Inamori Foundation (President: Dr. Kazuo Inamori) today announced that Dr. Laszlo Lovasz will receive its 26th annual Kyoto Prize in Basic Sciences, which for 2010 focuses on the field of Mathematical Sciences. Dr. Lovasz, 62, a citizen of both Hungary and the United States, will receive the award for his outstanding contributions to the advancement of both the academic and technological possibilities of the mathematical sciences.

Dr. Lovasz currently serves as both director of the Mathematical Institute at Eotvos Lorand University in Budapest and as president of the International Mathematics Union. Among many positions held throughout his distinguished career, Dr. Lovasz also served as a senior research member at Microsoft Research Center and as a professor of computer science at Yale University.

Related material: Cube Space, 1984-2003.

See also “Kyoto Prize” in this journal—

The Kyoto Prize is “administered by the Inamori Foundation, whose president, Kazuo Inamori, is founder and chairman emeritus of Kyocera and KDDI Corporation, two Japanese telecommunications giants.”

— – *Montreal Gazette*, June 20, 2008

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The current (Feb. 2009) *Notices of the American Mathematical Society* has a written version of Freeman Dyson’s 2008 Einstein Lecture, which was to have been given in October but had to be canceled. Dyson paraphrases a mathematician on Carl Jung’s theory of archetypes:

“… we do not need to accept Jung’s theory as true in order to find it illuminating.”

The same is true of Jung’s remarks on synchronicity.

For example —

Yesterday’s entry, “A Wealth of Algebraic Structure,” lists two articles– each, as it happens, related to Jung’s four-diamond figure from *Aion* as well as to my own Notes on Finite Geometry. The articles were placed online recently by Cambridge University Press on the following dates:

R. T. Curtis’s 1974 article defining his Miracle Octad Generator (MOG) was published online on Oct. 24, 2008.

Curtis’s 1987 article on geometry and algebraic structure in the MOG was published online on Dec. 19, 2008.

On these dates, the entries in this journal discussed…

**Oct. 24:**Cube Space, 1984-2003

Material related to that entry:

**Dec. 19:**

Art and Religion: Inside the White Cube

That entry discusses a book by Mark C. Taylor:

*The Picture in Question: Mark Tansey and the Ends of Representation* (U. of Chicago Press, 1999).

In Chapter 3, “Sutures of Structures,” Taylor asks —

“What, then, is a frame, and what is frame work?”

One possible answer —

Hermann Weyl on the relativity problem in the context of the 4×4 “frame of reference” found in the above Cambridge University Press articles.

“Examples are the stained-glass

windows of knowledge.”

— Vladimir Nabokov

windows of knowledge.”

— Vladimir Nabokov

Comments Off on Tuesday January 6, 2009

**An Indiana Jones Xmas
continues…**

**Chalice, Grail,
Whatever**

Last night on TNT:

The Librarian Part 3:

*Curse of the Judas Chalice*,

in which The Librarian

encounters the mysterious

Professor Lazlo

Related material:

An Arthur Waite quotation

from the Feast of St. Nicholas:

“It is like the *lapis exilis* of

the German Graal legend”

as well as

yesterday’s entry

relating Margaret Wertheim’s

*“Pearly Gates of Cyberspace:
A History of Space from
Dante to the Internet”
* to a different sort of space–

that of the

Professor Laszlo Lovasz’s

“cube space”

*“Click on the Yellow Book.”*

Happy birthday, David Carradine.

Comments Off on Monday December 8, 2008

**Readings for
Devil’s Night**

1. Today’s New York Times reviewof Peter Brook’s production of “The Grand Inquisitor” 2. Mathematics and Theology 3. Christmas, 2005 4. Cube Space, 1984-2003 |

Comments Off on Thursday October 30, 2008

*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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