Tuesday, February 20, 2018

The System

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

"It's the system  that matters. 
How the data arrange themselves inside it."

— Gravity's Rainbow  

"Examples are the stained-glass windows of knowledge."

— Vladimir Nabokov   

Map Systems (decomposition of functions over a finite field)

Wednesday, November 22, 2017

“Design is how it works” — Steve Jobs

Filed under: General,Geometry — m759 @ 1:00 PM

News item from this afternoon —

Apple AI research on 'mapping systems'

The above phrase "mapping systems" suggests a review
of my own very different  "map systems." From a search
for that phrase in this journal —

Map Systems (decomposition of functions over a finite field)

See also "A Four-Color Theorem: Function Decomposition
Over a Finite Field.

Saturday, September 3, 2016

Resplendent Triviality

Filed under: General,Geometry — m759 @ 11:30 AM

See The Echo in Plato's Cave and
a four-color decomposition theorem.

An illustration —

A four-color decomposition theorem, illustrated

Wednesday, December 30, 2015

Inverse Image

Filed under: General,Geometry — m759 @ 7:00 AM

The previous post discussed some art related to the
deceptively simple concept of "four colors."

For other related material, see posts that contain a link 
to "mapsys.html."

Friday, August 14, 2015

Discrete Space

Filed under: General,Geometry — Tags: — m759 @ 7:24 AM

(A review)

Galois space:

Image-- examples from Galois affine geometry

Counting symmetries of  Galois space:
IMAGE - The Diamond Theorem

The reason for these graphic symmetries in affine Galois space —

symmetries of the underlying projective Galois space:

Tuesday, June 9, 2015

Colorful Song

Filed under: General,Geometry — Tags: — m759 @ 8:40 PM

For geeks* —

Domain, Domain on the Range , "

where Domain = the Galois tesseract  and
Range = the four-element Galois field.

This post was suggested by the previous post,
by a Log24 search for Knight + Move, and by
the phrase "discouraging words" found in that search.

* A term from the 1947 film "Nightmare Alley."

Sunday, September 14, 2014


Filed under: General,Geometry — Tags: , — m759 @ 9:26 AM

Structured gray matter:

Graphic symmetries of Galois space:
IMAGE - The Diamond Theorem

The reason for these graphic symmetries in affine  Galois space —

symmetries of the underlying projective  Galois space:

Thursday, July 17, 2014

Paradigm Shift:

Filed under: General,Geometry — Tags: — m759 @ 11:01 AM

Continuous Euclidean space to discrete Galois space*

Euclidean space:

Point, line, square, cube, tesseract

From a page by Bryan Clair

Counting symmetries in Euclidean space:

Galois space:

Image-- examples from Galois affine geometry

Counting symmetries of  Galois space:
IMAGE - The Diamond Theorem

The reason for these graphic symmetries in affine Galois space —

symmetries of the underlying projective Galois space:

* For related remarks, see posts of May 26-28, 2012.

Saturday, February 1, 2014

The Delft Version

Filed under: General,Geometry — m759 @ 7:00 AM

My webpage "The Order-4 Latin Squares" has a rival—

"Latin squares of order 4: Enumeration of the
 24 different 4×4 Latin squares. Symmetry and
 other features."

The author — Yp de Haan, a professor emeritus of
materials science at Delft University of Technology —

The main difference between de Haan's approach and my own
is my use of the four-color decomposition theorem, a result that
I discovered in 1976.  This would, had de Haan known it, have
added depth to his "symmetry and other features" remarks.

Tuesday, September 3, 2013

“The Stone” Today Suggests…

Filed under: General,Geometry — m759 @ 12:31 PM

A girl's best friend?

The Philosopher's Gaze , by David Michael Levin,
U. of California Press, 1999, in III.5, "The Field of Vision," pp. 174-175—

The post-metaphysical question—question for a post-metaphysical phenomenology—is therefore: Can the perceptual field, the ground of perception, be released  from our historical compulsion to represent it in a way that accommodates our will to power and its need to totalize and reify the presencing of being? In other words: Can the ground be experienced as  ground? Can its hermeneutical way of presencing, i.e., as a dynamic interplay of concealment and unconcealment, be given appropriate  respect in the receptivity of a perception that lets itself  be appropriated by  the ground and accordingly lets  the phenomenon of the ground be  what and how it is? Can the coming-to-pass of the ontological difference that is constitutive of all the local figure-ground differences taking place in our perceptual field be made visible hermeneutically, and thus without violence to its withdrawal into concealment? But the question concerning the constellation of figure and ground cannot be separated from the question concerning the structure of subject and object. Hence the possibility of a movement beyond metaphysics must also think the historical possibility of breaking out of this structure into the spacing of the ontological difference: différance , the primordial, sensuous, ekstatic écart . As Heidegger states it in his Parmenides lectures, it is a question of "the way historical man belongs within the bestowal of being (Zufügung des Seins ), i.e., the way this order entitles him to acknowledge being and to be the only being among all beings to see  the open" (PE* 150, PG** 223. Italics added). We might also say that it is a question of our response-ability, our capacity as beings gifted with vision, to measure up to the responsibility for perceptual responsiveness laid down for us in the "primordial de-cision" (Entscheid ) of the ontological difference (ibid.). To recognize the operation of the ontological difference taking place in the figure-ground difference of the perceptual Gestalt  is to recognize the ontological difference as the primordial Riß , the primordial Ur-teil  underlying all our perceptual syntheses and judgments—and recognize, moreover, that this rift, this  division, decision, and scission, an ekstatic écart  underlying and gathering all our so-called acts of perception, is also the only "norm" (ἀρχή ) by which our condition, our essential deciding and becoming as the ones who are gifted with sight, can ultimately be judged.

* PE: Parmenides  of Heidegger in English— Bloomington: Indiana University Press, 1992

** PG: Parmenides  of Heidegger in German— Gesamtausgabe , vol. 54— Frankfurt am Main: Vittorio Klostermann, 1992

Examples of "the primordial Riß " as ἀρχή  —

For an explanation in terms of mathematics rather than philosophy,
see the diamond theorem. For more on the Riß  as ἀρχή , see
Function Decomposition Over a Finite Field.

Sunday, December 9, 2012

Deep Structure

Filed under: General,Geometry — Tags: , — m759 @ 10:18 AM

The concept of "deep structure," once a popular meme,
has long been abandoned by Chomskians.

It still applies, however, to the 1976 mathematics, diamond theory  ,
underlying the formal patterns discussed in a Royal Society paper
this year.

A review of deep structure, from the Wikipedia article Cartesian linguistics

[Numbers in parentheses refer to pages in the original 1966 Harper edition of Chomsky's book Cartesian Linguistics .]

Deep structure vs. surface structure

"Pursuing the fundamental distinction between body and mind, Cartesian linguistics characteristically assumes that language has two aspects" (32). These are namely the sound/character of a linguistic sign and its significance (32). Semantic interpretation or phonetic interpretation may not be identical in Cartesian linguistics (32). Deep structures are often only represented in the mind (a mirror of thought), as opposed to surface structures, which are not.

Deep structures vary less between languages than surface structures. For instance, the transformational operations to derive surface forms of Latin and French may obscure common features of their deep structures (39). Chomsky proposes, "In many respects, it seems to me quite accurate, then, to regard the theory of transformational generative grammar, as it is developing in current work, as essentially a modern and more explicit version of the Port-Royal theory" (39).

Summary of Port Royal Grammar

The Port Royal Grammar is an often cited reference in Cartesian Linguistics  and is considered by Chomsky to be a more than suitable example of Cartesian linguistic philosophy. "A sentence has an inner mental aspect (a deep structure that conveys its meaning) and an outer, physical aspect as a sound sequence"***** This theory of deep and surface structures, developed in Port Royal linguistics, meets the formal requirements of language theory. Chomsky describes it in modern terms as "a base system that generates deep structures and a transformational system that maps these into surface structures", essentially a form of transformational grammar akin to modern studies (42).

The corresponding concepts from diamond theory are

"Deep structure"— The line diagrams indicating the underlying
structure of varying patterns

"A base system that generates deep structures"—
Group actions on square arrays for instance, on the 4×4 square

"A transformational system"— The decomposition theorem 
that maps deep structure into surface structure (and vice-versa)

Monday, September 17, 2012

Pattern Conception

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

( Continued from yesterday's post FLT )

Context Part I —

"In 1957, George Miller initiated a research programme at Harvard University to investigate rule-learning, in situations where participants are exposed to stimuli generated by rules, but are not told about those rules. The research program was designed to understand how, given exposure to some finite subset of stimuli, a participant could 'induce' a set of rules that would allow them to recognize novel members of the broader set. The stimuli in question could be meaningless strings of letters, spoken syllables or other sounds, or structured images. Conceived broadly, the project was a seminal first attempt to understand how observers, exposed to a set of stimuli, could come up with a set of principles, patterns, rules or hypotheses that generalized over their observations. Such abstract principles, patterns, rules or hypotheses then allow the observer to recognize not just the previously seen stimuli, but a wide range of other stimuli consistent with them. Miller termed this approach 'pattern conception ' (as opposed to 'pattern perception'), because the abstract patterns in question were too abstract to be 'truly perceptual.'….

…. the 'grammatical rules' in such a system are drawn from the discipline of formal language theory  (FLT)…."

— W. Tecumseh Fitch, Angela D. Friederici, and Peter Hagoort, "Pattern Perception and Computational Complexity: Introduction to the Special Issue," Phil. Trans. R. Soc. B  (2012) 367, 1925-1932 

Context Part II —

IMAGE- Wikipedia article 'Formal language'

Context Part III —

A four-color theorem describes the mathematics of
general  structures, not just symbol-strings, formed from
four kinds of things— for instance, from the four elements
of the finite Galois field GF(4), or the four bases of DNA.

Context Part IV —

A quotation from William P. Thurston, a mathematician
who died on Aug. 21, 2012—

"It may sound almost circular to say that
what mathematicians are accomplishing
is to advance human understanding of mathematics.
I will not try to resolve this
by discussing what mathematics is,
because it would take us far afield.
Mathematicians generally feel that they know
what mathematics is, but find it difficult
to give a good direct definition.
It is interesting to try. For me,
'the theory of formal patterns'
has come the closest, but to discuss this
would be a whole essay in itself."

Related material from a literate source—

"So we moved, and they, in a formal pattern"

Formal Patterns—

Not formal language theory  but rather
finite projective geometry  provides a graphic grammar
of abstract design

IMAGE- Harvard Crimson ad, Easter Sunday, 2008: 'Finite projective geometry as a graphic grammar of abstract design'

See also, elsewhere in this journal,
Crimson Easter Egg and Formal Pattern.

Sunday, September 9, 2012

Decomposition Sermon

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

(Continued from Walpurgisnacht 2012)

Wikipedia article on functional decomposition

"Outside of purely mathematical considerations,
perhaps the greatest value of functional decomposition
is the insight it provides into the structure of the world."

Certainly this is true for the sort of decomposition
known as harmonic analysis .

It is not, however, true of my own decomposition theorem,
which deals only with structures made up of at most four
different sorts of elementary parts.

But my own approach has at least some poetic value.

See the four elements of the Greeks in (for instance)
Eliot's Four Quartets  and in Auden's For the Time Being .

Friday, August 24, 2012

A Colorful Tale

Filed under: General,Geometry — m759 @ 6:06 AM

(Continued from July 19, 2008)

From the Diamond 16 Puzzle

IMAGE- The Diamond 16 Puzzle

The resemblance between the "quadrants" part of
the above picture and the new Microsoft symbol

IMAGE- New Microsoft symbol, August 2012

— is of course purely coincidental, as is the fact
that the new symbol illustrates four colors.

Monday, April 30, 2012

Decomposition (continued)

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

Compare and contrast

1. The following excerpt from Wikipedia

IMAGE- Excerpt from 'Functional decomposition' article at Wikipedia

2. A webpage subtitled "Function Decomposition Over a Finite Field."

Related material—

Decomposition and Jews Telling Stories.

Saturday, March 3, 2012


Filed under: General,Geometry — m759 @ 3:33 AM

A search tonight for material related to the four-color
decomposition theorem yielded the Wikipedia article
Functional decomposition.

The article, of more philosophical than mathematical
interest, is largely due to one David Fass at Rutgers.

(See the article's revision history for mid-August 2007.)

Fass's interest in function decomposition may or may not
be related to the above-mentioned theorem, which 
originated in the investigation of functions into the
four-element Galois field from a 4×4 square domain.

Some related material involving Fass and 4×4 squares—

A 2003 paper he wrote with Jacob Feldman—

(Click to enlarge.)

"Design is how it works." — Steve Jobs

An assignment for Jobs in the afterlife—

Discuss the Fass-Feldman approach to "categorization under
complexity" in the context of the Wikipedia article's
philosophical remarks on "reductionist tradition."

The Fass-Feldman paper was assigned in an MIT course
for a class on Walpurgisnacht 2003.

Saturday, February 18, 2012


Filed under: General,Geometry — Tags: — m759 @ 8:48 AM

Pentagram design agency on the new Windows 8 logo

"… the logo re-imagines the familiar four-color symbol
as a modern geometric shape"—


Sam Moreau, Principal Director of User Experience for Windows,

On Redesigning the Windows Logo

"To see what is in front of one's nose
needs a constant struggle."
George Orwell

That is the feeling we had when Paula Scher
(from the renowned Pentagram design agency)
showed us her sketches for the new Windows logo.

Related material:


Thursday, February 9, 2012

ART WARS continued

Filed under: General,Geometry — m759 @ 1:06 PM

On the Complexity of Combat—

(Click to enlarge.)

The above article (see original pdf), clearly of more 
theoretical than practical interest, uses the concept
of "symmetropy" developed by some Japanese

For some background from finite geometry, see
Symmetry of Walsh Functions. For related posts
in this journal, see Smallest Perfect Universe.

Update of 7:00 PM EST Feb. 9, 2012—

Background on Walsh-function symmetry in 1982—

(Click image to enlarge. See also original pdf.)

Note the somewhat confusing resemblance to
a four-color decomposition theorem
used in the proof of the diamond theorem

Tuesday, January 10, 2012

Defining Form

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

(Continued from Epiphany and from yesterday.)

Detail from the current American Mathematical Society homepage


Further detail, with a comparison to Dürer's magic square—

http://www.log24.com/log/pix12/120110-Donmoyer-Still-Life-Detail.jpg http://www.log24.com/log/pix12/120110-DurerSquare.jpg

The three interpenetrating planes in the foreground of Donmoyer's picture
provide a clue to the structure of the the magic square array behind them.

Group the 16 elements of Donmoyer's array into four 4-sets corresponding to the
four rows of Dürer's square, and apply the 4-color decomposition theorem.
Note the symmetry of the set of 3 line diagrams that result.

Now consider the 4-sets 1-4, 5-8, 9-12, and 13-16, and note that these
occupy the same positions in the Donmoyer square that 4-sets of
like elements occupy in the diamond-puzzle figure below—


Thus the Donmoyer array also enjoys the structural  symmetry,
invariant under 322,560 transformations, of the diamond-puzzle figure.

Just as the decomposition theorem's interpenetrating lines  explain the structure
of a 4×4 square , the foreground's interpenetrating planes  explain the structure
of a 2x2x2 cube .

For an application to theology, recall that interpenetration  is a technical term
in that field, and see the following post from last year—

Saturday, June 25, 2011


Theology for Antichristmas

— m759 @ 12:00 PM

Hypostasis (philosophy)

"… the formula 'Three Hypostases  in one Ousia '
came to be everywhere accepted as an epitome
of the orthodox doctrine of the Holy Trinity.
This consensus, however, was not achieved
without some confusion…." —Wikipedia



Click for further details:



Monday, August 8, 2011

Diamond Theory vs. Story Theory (continued)

Filed under: General,Geometry — m759 @ 5:01 PM

Some background

Richard J. Trudeau, a mathematics professor and Unitarian minister, published in 1987 a book, The Non-Euclidean Revolution , that opposes what he calls the Story Theory of truth [i.e., Quine, nominalism, postmodernism] to what he calls the traditional Diamond Theory of truth [i.e., Plato, realism, the Roman Catholic Church]. This opposition goes back to the medieval "problem of universals" debated by scholastic philosophers.

(Trudeau may never have heard of, and at any rate did not mention, an earlier 1976 monograph on geometry, "Diamond Theory," whose subject and title are relevant.)

From yesterday's Sunday morning New York Times

"Stories were the primary way our ancestors transmitted knowledge and values. Today we seek movies, novels and 'news stories' that put the events of the day in a form that our brains evolved to find compelling and memorable. Children crave bedtime stories…."

Drew Westen, professor at Emory University

From May 22, 2009

Poster for 'Diamonds' miniseries on ABC starting May 24, 2009

The above ad is by
  Diane Robertson Design—

Credit for 'Diamonds' miniseries poster: Diane Robertson Design, London

Diamond from last night’s
Log24 entry, with
four colored pencils from
Diane Robertson Design:

Diamond-shaped face of Durer's 'Melencolia I' solid, with  four colored pencils from Diane Robertson Design
See also
A Four-Color Theorem.

For further details, see Saturday's correspondences
and a diamond-related story from this afternoon's
online New York Times.

Saturday, August 6, 2011


Filed under: General,Geometry — Tags: , — m759 @ 2:00 PM

Comme de longs échos qui de loin se confondent
Dans une ténébreuse et profonde unité….

— Baudelaire, "Correspondances "

From "A Four-Color Theorem"


Figure 1

Note that this illustrates a natural correspondence

(A) the seven highly symmetrical four-colorings
      of the 4×2 array at the left of Fig. 1, and

(B) the seven points of the smallest
      projective plane at the right of Fig. 1.

To see the correspondence, add, in binary
fashion, the pairs of projective points from the
"points" section that correspond to like-colored
squares in a four-coloring from the left of Fig. 1.
(The correspondence can, of course, be described
in terms of cosets rather than of colorings.)

A different correspondence between these 7 four-coloring
structures and these 7 projective-line structures appears in
a structural analysis of the Miracle Octad Generator
(MOG) of R.T. Curtis—


Figure 2

Here the correspondence between the 7 four-coloring structures (left section) and the 7 projective-line structures (center section) is less obvious, but more fruitful.  It yields, as shown, all of the 35 partitions of an 8-element set  (an 8-set ) into two 4-sets. The 7 four-colorings in Fig. 2 also appear in the 35 4×4 parts of the MOG that correspond, in a way indicated by Fig. 2, to the 35 8-set paritions. This larger correspondence— of 35 4×2 arrays with 35 4×4 arrays— is  the MOG, at least as it was originally defined. See The MOG, Generating the Octad Generator, and Eightfold Geometry.


For some applications of the Curtis MOG, see
(for instance) Griess's Twelve Sporadic Groups .

Saturday, January 22, 2011

High School Squares*

Filed under: General,Geometry — Tags: — m759 @ 1:20 AM

The following is from the weblog of a high school mathematics teacher—


This is related to the structure of the figure on the cover of the 1976 monograph Diamond Theory


Each small square pattern on the cover is a Latin square,
with elements that are geometric figures rather than letters or numerals.
All order-four Latin squares are represented.

For a deeper look at the structure of such squares, let the high-school
chart above be labeled with the letters A through X, and apply the
four-color decomposition theorem.  The result is 24 structural diagrams—

    Click to enlarge

IMAGE- The Order-4 (4x4) Latin Squares

Some of the squares are structurally congruent under the group of 8 symmetries of the square.

This can be seen in the following regrouping—

   Click to enlarge

IMAGE- The Order-4 (4x4) Latin Squares, with Congruent Squares Adjacent

      (Image corrected on Jan. 25, 2011– "seven" replaced "eight.")

* Retitled "The Order-4 (i.e., 4×4) Latin Squares" in the copy at finitegeometry.org/sc.

Saturday, July 3, 2010

Beyond the Limits

Filed under: General,Geometry — Tags: — m759 @ 7:29 PM

"Human perception is a saga of created reality. But we were devising entities beyond the agreed-upon limits of recognition or interpretation…."

– Don DeLillo, Point Omega

Capitalized, the letter omega figures in the theology of two Jesuits, Teilhard de Chardin and Gerard Manley Hopkins. For the former, see a review of DeLillo. For the latter, see James Finn Cotter's Inscape  and "Hopkins and Augustine."

The lower-case omega is found in the standard symbolic representation of the Galois field GF(4)—

GF(4) = {0, 1, ω, ω2}

A representation of GF(4) that goes beyond the standard representation—


Here the four diagonally-divided two-color squares represent the four elements of GF(4).

The graphic properties of these design elements are closely related to the algebraic properties of GF(4).

This is demonstrated by a decomposition theorem used in the proof of the diamond theorem.

To what extent these theorems are part of "a saga of created reality" may be debated.

I prefer the Platonist's "discovered, not created" side of the debate.

Thursday, May 27, 2010

A Gathering for Gardner

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

"You ain't been blue; no, no, no.
 You ain't been blue,
 Till you've had that mood indigo."
 — Song lyrics, authorship disputed


Indigo (web color) (#4B0082)

"Pigment indigo (web color indigo) represents
 the way the color indigo was always reproduced
 in pigments, paints, or colored pencils in the 1950s."

Related mythology:

Indigo Children and the classic
1964 film Children of the Damned

Image-- Children of the Damned take sanctuary in St. Dunstan's Church.

Related non-mythology:

Colored pencils

Image-- Diamond-shaped face of Durer's 'Melencolia I' solid, with four colored pencils from Diane Robertson Design

Saturday, December 26, 2009

Annals of Philosophy

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

Towards a Philosophy of Real Mathematics, by David Corfield, Cambridge U. Press, 2003, p. 206:

"Now, it is no easy business defining what one means by the term conceptual…. I think we can say that the conceptual is usually expressible in terms of broad principles. A nice example of this comes in the form of harmonic analysis, which is based on the idea, whose scope has been shown by George Mackey (1992) to be immense, that many kinds of entity become easier to handle by decomposing them into components belonging to spaces invariant under specified symmetries."

For a simpler example of this idea, see the entities in The Diamond Theorem, the decomposition in A Four-Color Theorem, and the space in Geometry of the 4×4 Square.  The decomposition differs from that of harmonic analysis, although the subspaces involved in the diamond theorem are isomorphic to Walsh functions— well-known as discrete analogues of the trigonometric functions of traditional harmonic analysis.

Saturday, September 19, 2009

Saturday September 19, 2009

Filed under: General,Geometry — m759 @ 4:23 PM

Old Year, Raus!

Also in today’s New York Times obituaries index:

 John T. Elson, Editor Who Asked
“Is God Dead?” at Time, Dies at 78

John T. Elson and Budd Schulberg

Wikipedia article on George Polya:

  • Look for a pattern
  • Draw a picture
  • Solve a simpler problem
  • Use a model
  • Work backward

From the date of Elson’s death:

Cube, 4x4x4

Four coloring pencils, of four different colors

Related material:
A Four-Color Theorem.”

Monday, September 7, 2009

Monday September 7, 2009

Filed under: General,Geometry — Tags: — m759 @ 12:00 AM

Magic Boxes

"Somehow it seems to fill my head with ideas– only I don't exactly know what they are!…. Let's have a look at the garden first!"

— A passage from Lewis Carroll's Through the Looking-Glass. The "garden" part– but not the "ideas" part– was quoted by Jacques Derrida in Dissemination in the epigraph to Chapter 7, "The Time before First."

 on the passage:

Part I    "The Magic Box,"  shown on Turner Classic Movies earlier tonight

Part II: "Mimsy Were the Borogoves," a classic science fiction story:

"… he lifted a square, transparent crystal block, small enough to cup in his palm– much too small to contain the maze of apparatus within it. In a moment Scott had solved that problem. The crystal was a sort of magnifying glass, vastly enlarging the things inside the block. Strange things they were, too. Miniature people, for example– They moved. Like clockwork automatons, though much more smoothly. It was rather like watching a play."

Part III:  A Crystal Block

Cube, 4x4x4

Four coloring pencils, of four different colors

Image of pencils is by
Diane Robertson Design.

Related material:
"A Four-Color Theorem."

Part IV:

David Carradine displays a yellow book-- the Princeton I Ching.

"Click on the Yellow Book."

Sunday, September 6, 2009

Sunday September 6, 2009

Filed under: General,Geometry — m759 @ 11:18 PM
Magic Boxes

Part I: “The Magic Box,” shown on Turner Classic Movies tonight

Part II: “Mimsy Were the Borogoves,” a classic science fiction story:

“… he lifted a square, transparent crystal block, small enough to cup in his palm– much too small to contain the maze of apparatus within it. In a moment Scott had solved that problem. The crystal was a sort of magnifying glass, vastly enlarging the things inside the block. Strange things they were, too. Miniature people, for example–

They moved. Like clockwork automatons, though much more smoothly. It was rather like watching a play.”



Image of pencils is by
Diane Robertson Design.

Related material:
A Four-Color Theorem.”

Friday, May 22, 2009

Friday May 22, 2009

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

New York Times
banner this morning:

NYT banner, 9:21 AM Friday, May 22, 2009-- Ears are ads for HSBC.

Click to enlarge.

Related material from
July 11, 2008:

HSBC logo with framed version

The HSBC Logo Designer —

Henry Steiner

Henry Steiner, designerHe is an internationally recognized corporate identity consultant. Based in Hong Kong, his work for clients such as HongkongBank, IBM and Unilever is a major influence in Pacific Rim design.

Born in Austria and raised in New York, Steiner was educated at Yale under Paul Rand and attended the Sorbonne as a Fulbright Fellow. He is a past President of Alliance Graphique Internationale. Other professional affiliations include the American Institute of Graphic Arts, Chartered Society of Designers, Design Austria, and the New York Art Directors' Club.

His Cross-Cultural Design: Communicating in the Global Marketplace was published by Thames and Hudson (1995).



Charles Taylor,
"Epiphanies of Modernism,"
Chapter 24 of Sources of the Self
  (Cambridge U. Press, 1989, p. 477):


"… the object sets up
 a kind of frame or space or field
   within which there can be epiphany."


Related material suggested by
an ad last night on
ABC's Ugly Betty season finale:

Poster for 'Diamonds' miniseries on ABC starting May 24, 2009

Credit for 'Diamonds' miniseries poster: Diane Robertson Design, London

Diamond from last night's
Log24 entry, with
four colored pencils from
Diane Robertson Design:

Diamond-shaped face of Durer's 'Melencolia I' solid, with  four colored pencils from Diane Robertson Design
See also
A Four-Color Theorem.

Friday, August 22, 2008

Friday August 22, 2008

Filed under: General,Geometry — m759 @ 5:01 AM

Tentative movie title:

Kohs Block Design Test

The Kohs Block Design
Intelligence Test

Samuel Calmin Kohs, the designer (but not the originator) of the above intelligence test, would likely disapprove of the "Aryan Youth types" mentioned in passing by a film reviewer in today's New York Times. (See below.) The Aryan Youth would also likely disapprove of Dr. Kohs.

Related material from
Notes on Finite Geometry:

Kohs Block Design figure illustrating the four-color decomposition theorem

Other related material:

1.  Wechsler Cubes (intelligence testing cubes derived from the Kohs cubes shown above). See…

Harvard psychiatry and…
The Montessori Method;
The Crimson Passion;
The Lottery Covenant.

2.  Wechsler Cubes of a different sort (Log24, May 25, 2008)

3.  Manohla Dargis in today's New York Times:

"… 'Momma’s Man' is a touchingly true film, part weepie, part comedy, about the agonies of navigating that slippery slope called adulthood. It was written and directed by Azazel Jacobs, a native New Yorker who has set his modestly scaled movie with a heart the size of the Ritz in the same downtown warren where he was raised. Being a child of the avant-garde as well as an A student, he cast his parents, the filmmaker Ken Jacobs and the artist Flo Jacobs, as the puzzled progenitors of his centerpiece, a wayward son of bohemia….

In American movies, growing up tends to be a job for either Aryan Youth types or the oddballs and outsiders…."

4.  The bohemian who named his son Azazel:

"… I think that the deeper opportunity, the greater opportunity film can offer us is as an exercise of the mind. But an exercise, I hate to use the word, I won't say 'soul,' I won't say 'soul' and I won't say 'spirit,' but that it can really put our deepest psychological existence through stuff. It can be a powerful exercise. It can make us think, but I don't mean think about this and think about that. The very, very process of powerful thinking, in a way that it can afford, is I think very, very valuable. I basically think that the mind is not complete yet, that we are working on creating the mind. Okay. And that the higher function of art for me is its contribution to the making of mind."

Interview with Ken Jacobs, UC Berkeley, October 1999

5.  For Dargis's "Aryan Youth types"–

From a Manohla Dargis
New York Times film review
of April 4, 2007
   (Spy Wednesday) —

Scene from Paul Verhoeven's film 'Black Book'

See also, from August 1, 2008
(anniversary of Hitler's
opening the 1936 Olympics) —

For Sarah Silverman

and the 9/9/03 entry 

Olympic Style.

August 21-22, 2008:


Monday, August 11, 2008

Monday August 11, 2008

Filed under: General,Geometry — m759 @ 9:00 PM
 New Illustration
for the four-color
decomposition theorem:

Four-color decompostion applied to the 8-point binary affine space

Tuesday, June 24, 2008

Tuesday June 24, 2008

Filed under: General,Geometry — m759 @ 5:01 AM
Plato’s Cave, continued:

                     … we know that we use
Only the eye as faculty, that the mind
Is the eye, and that this landscape of the mind

Is a landscape only of the eye; and that
We are ignorant men incapable
Of the least, minor, vital metaphor….

— Wallace Stevens, “Crude Foyer”

                                               … So, so,
O son of man, the ignorant night, the travail
Of early morning, the mystery of the beginning
Again and again,
                         while history is unforgiven.

— Delmore Schwartz,
  “In the Naked Bed, in Plato’s Cave

The Echo in Plato’s Cave:

Somewhere between
a flagrant triviality and
a resplendent Trinity we
have what might be called
“a resplendent triviality.”

For further details, see
A Four-Color Theorem.”

Monday, June 9, 2008

Monday June 9, 2008

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

"With respect, you only interpret."
"Countries have gone to war
after misinterpreting one another."

The Interpreter

"Once upon a time (say, for Dante),
it must have been a revolutionary
and creative move to design works
of art so that they might be
experienced on several levels."

— Susan Sontag,
"Against Interpretation"

Edward Rothstein in today's New York Times review of San Francisco's new Contemporary Jewish Museum:

"An introductory wall panel tells us that in the Jewish mystical tradition the four letters [in Hebrew] of pardes each stand for a level of biblical interpretation: very roughly, the literal, the allusive, the allegorical and the hidden. Pardes, we are told, became the museum’s symbol because it reflected the museum’s intention to cultivate different levels of interpretation: 'to create an environment for exploring multiple perspectives, encouraging open-mindedness' and 'acknowledging diverse backgrounds.' Pardes is treated as a form of mystical multiculturalism.

But even the most elaborate interpretations of a text or tradition require more rigor and must begin with the literal. What is being said? What does it mean? Where does it come from and where else is it used? Yet those are the types of questions– fundamental ones– that are not being asked or examined […].

How can multiple perspectives and open-mindedness and diverse backgrounds be celebrated without a grounding in knowledge, without history, detail, object and belief?"

"It's the system that matters.
How the data arrange
themselves inside it."

Gravity's Rainbow  

"Examples are the stained-
glass windows of knowledge."

Vladimir Nabokov  

Map Systems (decomposition of functions over a finite field)

Click on image to enlarge.   

Sunday, April 13, 2008

Sunday April 13, 2008

Filed under: General,Geometry — m759 @ 7:59 AM
The Echo
in Plato’s Cave

“It is said that the students of medieval Paris came to blows in the streets over the question of universals. The stakes are high, for at issue is our whole conception of our ability to describe the world truly or falsely, and the objectivity of any opinions we frame to ourselves. It is arguable that this is always the deepest, most profound problem of philosophy.”

— Simon Blackburn, Think (Oxford, 1999)

Michael Harris, mathematician at the University of Paris:

“… three ‘parts’ of tragedy identified by Aristotle that transpose to fiction of all types– plot (mythos), character (ethos), and ‘thought’ (dianoia)….”

— paper (pdf) to appear in Mathematics and Narrative, A. Doxiadis and B. Mazur, eds.

Mythos —

A visitor from France this morning viewed the entry of Jan. 23, 2006: “In Defense of Hilbert (On His Birthday).” That entry concerns a remark of Michael Harris.

A check of Harris’s website reveals a new article:

“Do Androids Prove Theorems in Their Sleep?” (slighly longer version of article to appear in Mathematics and Narrative, A. Doxiadis and B. Mazur, eds.) (pdf).

From that article:

“The word ‘key’ functions here to structure the reading of the article, to draw the reader’s attention initially to the element of the proof the author considers most important. Compare E.M. Forster in Aspects of the Novel:

[plot is] something which is measured not be minutes or hours, but by intensity, so that when we look at our past it does not stretch back evenly but piles up into a few notable pinnacles.”

Ethos —

“Forster took pains to widen and deepen the enigmatic character of his novel, to make it a puzzle insoluble within its own terms, or without. Early drafts of A Passage to India reveal a number of false starts. Forster repeatedly revised drafts of chapters thirteen through sixteen, which comprise the crux of the novel, the visit to the Marabar Caves. When he began writing the novel, his intention was to make the cave scene central and significant, but he did not yet know how:

When I began a A Passage to India, I knew something important happened in the Malabar (sic) Caves, and that it would have a central place in the novel– but I didn’t know what it would be… The Malabar Caves represented an area in which concentration can take place. They were to engender an event like an egg.”

E. M. Forster: A Passage to India, by Betty Jay

Dianoia —

Flagrant Triviality
or Resplendent Trinity?

“Despite the flagrant triviality of the proof… this result is the key point in the paper.”

— Michael Harris, op. cit., quoting a mathematical paper

Online Etymology Dictionary

c.1500, “resplendent,” from L. flagrantem (nom. flagrans) “burning,” prp. of flagrare “to burn,” from L. root *flag-, corresponding to PIE *bhleg (cf. Gk. phlegein “to burn, scorch,” O.E. blæc “black”). Sense of “glaringly offensive” first recorded 1706, probably from common legalese phrase in flagrante delicto “red-handed,” lit. “with the crime still blazing.”

A related use of “resplendent”– applied to a Trinity, not a triviality– appears in the Liturgy of Malabar:


The Liturgies of SS. Mark, James, Clement, Chrysostom, and Basil, and the Church of Malabar, by the Rev. J.M. Neale and the Rev. R.F. Littledale, reprinted by Gorgias Press, 2002

On Universals and
A Passage to India:

“”The universe, then, is less intimation than cipher: a mask rather than a revelation in the romantic sense. Does love meet with love? Do we receive but what we give? The answer is surely a paradox, the paradox that there are Platonic universals beyond, but that the glass is too dark to see them. Is there a light beyond the glass, or is it a mirror only to the self? The Platonic cave is even darker than Plato made it, for it introduces the echo, and so leaves us back in the world of men, which does not carry total meaning, is just a story of events.”

— Betty Jay,  op. cit.


Judy Davis in the Marabar Caves

In mathematics
(as opposed to narrative),
somewhere between
 a flagrant triviality and
a resplendent Trinity we
have what might be called
“a resplendent triviality.”

For further details, see
A Four-Color Theorem.”

Wednesday, October 24, 2007

Wednesday October 24, 2007

Filed under: General,Geometry — Tags: , — m759 @ 11:11 PM
Descartes's Twelfth Step

An earlier entry today ("Hollywood Midrash continued") on a father and son suggests we might look for an appropriate holy ghost. In that context…


A search for further background on Emmanuel Levinas, a favorite philosopher of the late R. B. Kitaj (previous two entries), led (somewhat indirectly) to the following figures of Descartes:

The color-analogy figures of Descartes
This trinity of figures is taken from Descartes' Rule Twelve in Rules for the Direction of the Mind. It seems to be meant to suggest an analogy between superposition of colors and superposition of shapes.Note that the first figure is made up of vertical lines, the second of vertical and horizontal lines, and the third of vertical, horizontal, and diagonal lines. Leon R. Kass recently suggested that the Descartes figures might be replaced by a more modern concept– colors as wavelengths. (Commentary, April 2007). This in turn suggests an analogy to Fourier series decomposition of a waveform in harmonic analysis. See the Kass essay for a discussion of the Descartes figures in the context of (pdf) Science, Religion, and the Human Future (not to be confused with Life, the Universe, and Everything).

Compare and contrast:

The harmonic-analysis analogy suggests a review of an earlier entry's
link today to 4/30–  Structure and Logic— as well as
re-examination of Symmetry and a Trinity

(Dec. 4, 2002).

See also —

A Four-Color Theorem,
The Diamond Theorem, and
The Most Violent Poem,

Emma Thompson in 'Wit'

from Mike Nichols's birthday, 2003.

Friday, March 21, 2003

Friday March 21, 2003

Filed under: General — m759 @ 9:29 AM


Readings for Bach’s Birthday

Larry J. Solomon:

Symmetry as a Compositional Determinant,
Chapter VIII: New Transformations

In Solomon’s work, a sequence of notes is represented as a set of positions within a Latin square:

Transformations of the Latin square correspond to transformations of the musical notes.  For related material, see The Glass Bead Game, by Hermann Hesse, and Charles Cameron’s sites on the Game.

Steven H. Cullinane:

Orthogonal Latin Squares as Skew Lines, and

Map Systems

Dorothy Sayers:

“The function of imaginative speech is not to prove, but to create–to discover new similarities, and to arrange them to form new entities, to build new self-consistent worlds out of the universe of undifferentiated mind-stuff.” (Christian Letters to a Post-Christian World, Grand Rapids: Eerdmans, 1969, p. xiii)

— Quoted by Timothy A. Smith, “Intentionality and Meaningfulness in Bach’s Cyclical Works

Edward Sapir:

“…linguistics has also that profoundly serene and satisfying quality which inheres in mathematics and in music and which may be described as the creation out of simple elements of a self-contained universe of forms.  Linguistics has neither the sweep nor the instrumental power of mathematics, nor has it the universal aesthetic appeal of music.  But under its crabbed, technical, appearance there lies hidden the same classical spirit, the same freedom in restraint, which animates mathematics and music at their purest.”

 “The Grammarian and his Language,”
    American Mercury 1:149-155, 1924

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)

Creative Commons License
This work is licensed under a
Creative Commons Attribution-NonCommercial-NoDerivs 2.5 License

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


Initial Xanga entry.  Updated Nov. 18, 2006.

Powered by WordPress