Tuesday, October 20, 2020

The Browning Methods

Filed under: General — Tags: , — m759 @ 6:00 AM

The Ballad of Goo Ballou —
the Sequel to . . .

Let me count  the ways” is an appropriate request
for students of the discrete ,  as opposed to the
continuous , which instead requires measurement .

Related academic material —

Raymond Cattell on crystallized  vs. fluid  intelligence.
For a more literary approach, see Crystal and Dragon
and For Trevanian.

This post was inspired in part by
the American Sequel Society and . . .

Wednesday, July 29, 2020

Duren, Not Durin

Filed under: General — m759 @ 9:54 AM

A flashback from Log24 posts of July 9-11, 2020,
now tagged Structure and Mutability

Quote related to the 'Crystal and Dragon' concept.

For such temptation, see
Dwarves named “Durin.”

Friday, July 17, 2020

Poetic as Well as Prosaic

Filed under: General — Tags: , — m759 @ 9:51 AM

Prosaic —

Structure and Mutability

Poetic —

Crystal and Dragon


Prosaic —

These devices may have some
theoretical as well as practical value.

Poetic —

Counting symmetries with the orbit-stabilizer theorem

Saturday, July 11, 2020

Philosophy for Murdoch Fans

Filed under: General — Tags: — m759 @ 5:03 AM

The previous post contained a passage from Iris Murdoch’s
1961 essay “Against Dryness.”  Some related philosophy —

'Crystal and Dragon' by David Wade, publisher's description

For those who prefer pure mathematics to philosophical ruminations
there are some relevant remarks in my webpage of August 27, 2003.

Thursday, July 9, 2020

Fashion Space

Filed under: General — Tags: — m759 @ 7:59 PM

In memoriam

Click the quotation below for “Foster’s Space” posts.

Quote related to the 'Crystal and Dragon' concept.

Sunday, October 2, 2016

Happy Birthday, Wallace Stevens

Filed under: General — Tags: — m759 @ 8:28 AM

Log24 in review — Logos and Logic,  Crystal and Dragon .

Sunday, December 6, 2009

Holiday Book, continued

Filed under: General — m759 @ 2:02 PM

From the Red Book of Jung:

The Red Book of Jung

Related material:

Smaug and the Arkenstone
in the Red Book of Bilbo
and Crystal and Dragon.

Thursday, November 6, 2008

Thursday November 6, 2008

Filed under: General,Geometry — Tags: — m759 @ 10:07 AM

Death of a Classmate

Michael Crichton,
Harvard College, 1964

Authors Michael Crichton and David Foster Wallace in NY Times obituaries, Thursday, Nov.  6, 2008

Authors Michael Crichton and
David Foster Wallace in today’s
New York Times obituaries

The Times’s remarks above
on the prose styles of
Crichton and Wallace–
“compelling formula” vs.
“intricate complexity”–
suggest the following works
of visual art in memory
of Crichton.


Crystal from 'Diamond Theory'


(from Crichton’s
Jurassic Park)–

Dragon Curve from 'Jurassic Park'

For the mathematics
(dyadic harmonic analysis)
relating these two figures,
see Crystal and Dragon.

Some philosophical
remarks related to
the Harvard background
that Crichton and I share–

Hitler’s Still Point

The Crimson Passion.

Wednesday, August 20, 2008

Wednesday August 20, 2008

Filed under: General — m759 @ 11:29 PM
For Madeleine L’Engle,
wherever she may be

The entries of yesterday (updated today) and the day before suggest a flashback to the five “Dungeons & Dragons” entries ending on March 6, 2008.  For more about dungeons, see Jan. 7, 2007. For more about dragons, see Crystal and Dragon: The Cosmic Dance of Symmetry and Chaos in Nature, Art and Consciousness, by David Wade.

Sunday, July 2, 2006

Sunday July 2, 2006

Filed under: General,Geometry — m759 @ 9:29 AM

The Rock and the Serpent

In a search for a title to express
the contrast between truth and lies,
an analogy between the phrases

Crystal and Dragon” and
Mathematics and Narrative

suggests a similar phrase,

“The Rock and the Serpent.”

A web search for related titles leads to a book by Alice Thomas Ellis:

Serpent on the Rock: A Personal View of Christianity. (See a review.)

(This in turn leads to an article on Ellis’s husband, the late Colin Haycraft, publisher.)

For an earlier discussion of Ellis in this weblog, see Three Eleanors (March 12, 2005).

That entry brings us back to the theme of truth and lies with its link to an article from the Catholic publication Commonweal:

Getting to Truth by Lying.

Christians who wish to lie more effectively may consult a book by the author of the Commonweal article:

The image “http://www.log24.com/log/pix05/050312-Form.jpg” cannot be displayed, because it contains errors.

For a more sympathetic view of
suffering stemming from
Christian narrative,

The image “http://www.log24.com/log/pix06A/LecturesOnDonQuixote.gif” cannot be displayed, because it contains errors.

(Click on cover
for details. See also Log24
entries on Guy Davenport,
who wrote the foreword.)

Wednesday, June 21, 2006

Wednesday June 21, 2006

Filed under: General,Geometry — m759 @ 10:00 AM

Go with the Flow

The previous entry links to a document that discusses the mathematical concept of “Ricci flow (pdf).”

Though the concept was not named for him, this seems as good a time as any to recall the virtues of St. Matteo Ricci, a Jesuit who died in Beijing on May 11, 1610. (The Church does not yet recognize him as a saint; so much the worse for the Church.)

There was no Log24 entry on Ricci’s saint’s day, May 11, this year, but an entry for 4:29 PM May 10, 2006, seems relevant, since Beijing is 12 hours ahead of my local (Eastern US) time.

Ricci is famous for constructing
a “memory palace.”
Here is my equivalent,
from the May 10 entry:

The image ?http://www.log24.com/theory/images/MySpace.jpg? cannot be displayed, because it contains errors.

The relevance of this structure
to memory and to Chinese culture
is given in Dragon School and in
Geometry of the 4x4x4 Cube.

For some related remarks on
the colloquial, rather than the
mathematical, concept of flow,
Philosophy, Religion, and Science
as well as Crystal and Dragon.

Yesterday’s entry on the 1865
remarks on aesthetics of
Gerard Manley Hopkins,
who later became a Jesuit,
may also have some relevance.

Wednesday, August 27, 2003

Wednesday August 27, 2003

Filed under: General — m759 @ 3:40 AM

Crystal and Dragon

David Wade published a book called Crystal and Dragon in 1993 about the apparent opposites of structure and fluidity, order and chaos, law and freedom, and so on.

Here is a page on these concepts as they relate to my mathematical work.

Saturday, July 20, 2002

Saturday July 20, 2002

Filed under: General,Geometry — Tags: , — m759 @ 10:13 PM

ABSTRACT: Finite projective geometry explains the surprising symmetry properties of some simple graphic designs– found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the "Miracle Octad Generator" of R. T. Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.
We regard the four-diamond figure D above as a 4×4 array of two-color diagonally-divided square tiles.

Let G be the group of 322,560 permutations of these 16 tiles generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2×2 quadrants.

THEOREM: Every G-image of D (as at right, below) has some ordinary or color-interchange symmetry.


For an animated version, click here.


Some of the patterns resulting from the action of G on D have been known for thousands of years. (See Jablan, Symmetry and Ornament, Ch. 2.6.) It is perhaps surprising that the patterns' interrelationships and symmetries can be explained fully only by using mathematics discovered just recently (relative to the patterns' age)– in particular, the theory of automorphism groups of finite geometries.

Using this theory, we can summarize the patterns' properties by saying that G is isomorphic to the affine group A on the linear 4-space over GF(2) and that the 35 structures of the 840 = 35 x 24 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2).

This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets, and indicating the locations of these two-sets of tiles within the 4×4 patterns. The lines of the line diagrams may be added in a binary fashion (i.e., 1+1=0). Each three-set of line diagrams sums to zero– i.e., each diagram in a three-set is the binary sum of the other two diagrams in the set. Thus, the 35 three-sets of line diagrams correspond to the 35 three-point lines of the finite projective 3-space PG(3,2).

For example, here are the line diagrams for the figures above:

Shown below are the 15 possible line diagrams resulting from row/column/quadrant permutations. These 15 diagrams may, as noted above, be regarded as the 15 points of the projective 3-space PG(3,2).

The symmetry of the line diagrams accounts for the symmetry of the two-color patterns. (A proof shows that a 2nx2n two-color triangular half-squares pattern with such line diagrams must have a 2×2 center with a symmetry, and that this symmetry must be shared by the entire pattern.)

Among the 35 structures of the 840 4×4 arrays of tiles, orthogonality (in the sense of Latin-square orthogonality) corresponds to skewness of lines in the finite projective space PG(3,2). This was stated by the author in a 1978 note. (The note apparently had little effect. A quarter-century later, P. Govaerts, D. Jungnickel, L. Storme, and J. A. Thas wrote that skew (i.e., nonintersecting) lines in a projective space seem "at first sight not at all related" to orthogonal Latin squares.)

We can define sums and products so that the G-images of D generate an ideal (1024 patterns characterized by all horizontal or vertical "cuts" being uninterrupted) of a ring of 4096 symmetric patterns. There is an infinite family of such "diamond" rings, isomorphic to rings of matrices over GF(4).

The proof uses a decomposition technique for functions into a finite field that might be of more general use.

The underlying geometry of the 4×4 patterns is closely related to the Miracle Octad Generator of R. T. Curtis– used in the construction of the Steiner system S(5,8,24)– and hence is also related to the Leech lattice, which, as Walter Feit has remarked, "is a blown up version of S(5,8,24)."

For a movable JavaScript version of these 4×4 patterns, see The Diamond 16 Puzzle.

The above is an expanded version of Abstract 79T-A37, "Symmetry invariance in a diamond ring," by Steven H. Cullinane, Notices of the American Mathematical Society, February 1979, pages A-193, 194.

For a discussion of other cases of the theorem, click here.

Related pages:

The Diamond 16 Puzzle

Diamond Theory in 1937:
A Brief Historical Note

Notes on Finite Geometry

Geometry of the 4×4 Square

Binary Coordinate Systems

The 35 Lines of PG(3,2)

Map Systems:
Function Decomposition over a Finite Field

The Diamond Theorem–
The 2×2, the 2x2x2, the 4×4, and the 4x4x4 Cases

Diamond Theory

Latin-Square Geometry

Walsh Functions


The Diamond Theory of Truth

Geometry of the I Ching

Solomon's Cube and The Eightfold Way

Crystal and Dragon in Diamond Theory

The Form, the Pattern

The Grid of Time

Block Designs

Finite Relativity

Theme and Variations

Models of Finite Geometries

Quilt Geometry

Pattern Groups

The Fano Plane Revisualized,
or the Eightfold Cube

The Miracle Octad Generator


Visualizing GL(2,p)

Jung's Imago

Author's home page

AMS Mathematics Subject Classification:

20B25 (Group theory and generalizations :: Permutation groups :: Finite automorphism groups of algebraic, geometric, or combinatorial structures)

05B25 (Combinatorics :: Designs and configurations :: Finite geometries)

51E20 (Geometry :: Finite geometry and special incidence structures :: Combinatorial structures in finite projective spaces)

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Page created Jan. 6, 2006, by Steven H. Cullinane      diamondtheorem.com


Initial Xanga entry.  Updated Nov. 18, 2006.

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