Log24

Saturday, September 3, 2016

Decomposition Song

Filed under: General — m759 @ 1:00 AM

In Dead Earnest

(Poem by Lee Hays, performed by Pete Seeger)

If I should die before I wake,
All my bone and sinew take
Put me in the compost pile
To decompose me for a while . . . .

For a different sort of decomposition, see the previous post.

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 .

Monday, April 30, 2012

Decomposition– Part III

Filed under: General,Geometry — m759 @ 5:27 PM

(Continued from Part I and Part II.)

The paper excerpted below supplies some badly needed technical
background for the Wikipedia article on functional decomposition.

IMAGE- Excerpt from 'Unified Approach to Functional Decompositions of Switching Functions,' by Marek A. Perkowski et al., 1995

The preprint above gives the precise definitions and technical references
that are completely absent from Wikipedia's Functional decomposition.

For some related material on 4×4 arrays like those in the above figure
see Decomposition Part I and Geometry of the 4×4 Square.

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

Decomposition

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.

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.
"

Wednesday, October 18, 2017

Dürer for St. Luke’s Day

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 1:00 PM

Structure of the Dürer magic square 

16   3   2  13
 5  10  11   8   decreased by 1 is …
 9   6   7  12
 4  15  14   1

15   2   1  12
 4   9  10   7
 8   5   6  11
 3  14  13   0 .

Base 4 —

33  02  01  30
10  21  22  13
20  11  12  23 
03  32  31  00 .

Two-part decomposition of base-4 array
as two (non-Latin) orthogonal arrays

3 0 0 3     3 2 1 0
1 2 2 1     0 1 2 3
2 1 1 2     0 1 2 3
0 3 3 0     3 2 1 0 .

Base 2 –

1111  0010  0001  1100
0100  1001  1010  0111
1000  0101  0110  1011
0011  1110  1101  0000 .

Four-part decomposition of base-2 array
as four affine hyperplanes over GF(2) —

1001  1001  1100  1010
0110  1001  0011  0101
1001  0110  0011  0101
0110  0110  1100  1010 .

— Steven H. Cullinane,
  October 18, 2017

See also recent related analyses of
noted 3×3 and 5×5 magic squares.

Tuesday, May 2, 2017

Image Albums

Filed under: General,Geometry — Tags: — m759 @ 1:05 PM

Pinterest boards uploaded to the new m759.net/piwigo

Diamond Theorem 

Diamond Theorem Correlation

Miracle Octad Generator

The Eightfold Cube

Six-Set Geometry

Diamond Theory Cover

Update of May 2 —

Four-Color Decomposition

Binary Galois Spaces

The Galois Tesseract

Update of May 3 —

Desargues via Galois

The Tetrahedral Model

Solomon's Cube

Update of May 8 —

Art Space board created at Pinterest

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

Friday, September 2, 2016

Heuresis

Filed under: General — Tags: — m759 @ 7:22 PM

"Now a little trivial heuresis is in order."

The late Waclaw Szymanski on p. 279 of
"Decompositions of operator-valued functions 
in Hilbert spaces
" (Studia Mathematica  50.3
(1974): 265-280.)

See "A Talisman for Finkelstein," from midnight
on the reported date of Szymanski's death.  That post
refers to "the correspondence in the previous post
between Figures A and B" as does this  post

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:

Sunday, September 14, 2014

Sensibility

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.

Sunday, November 24, 2013

Logic for Jews*

Filed under: General,Geometry — Tags: , — m759 @ 7:20 AM

The search for 1984 at the end of last evening's post
suggests the following Sunday meditation.

My own contribution to this genre—

A triangle-decomposition result from 1984:

American Mathematical Monthly ,  June-July 1984, p. 382

MISCELLANEA, 129

Triangles are square

"Every triangle consists of n  congruent copies of itself"
is true if and only if  is a square. (The proof is trivial.) 
— Steven H. Cullinane

The Orwell slogans are false. My own is not.

* The "for Jews" of the title applies to some readers of Edward Frenkel.

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.

Saturday, August 3, 2013

In the Details

Filed under: General — m759 @ 11:01 AM

By chance, the latest* remarks in philosopher Colin McGinn's
weblog were posted (yesterday) at 10:04 AM.

Checking, in my usual mad way, for synchronicity, I find
the following from this  weblog on the date  10/04 (2012)—

Note too the time of this morning's previous post here
(on McGinn)— 9:09 AM.  Another synchronistic check
yields Log24 posts from 9/09 (2012):

Related to this last post:

Detail from a stock image suggested by the web page of
a sociologist (Harvard '64) at the University of Washington in Seattle—

Note, on the map of  Wyoming, Devil's Gate.

There are, of course, many such gates.

* Correction (of about 11:20 AM Aug. 3):
  Later  remarks by McGinn were  posted at 10:06 AM today.  
  They included the phrase "The devil is in the details."
  Yet another check for synchronicity leads to
  10/06 (2012) in this  journal with its post related to McGinn's
  weblog remarks yesterday on philosophy and art.
  That 10/06 Log24  post is somewhat in the spirit of other
  remarks by McGinn discussed in a 2009 Harvard Crimson  review.

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)

Thursday, September 6, 2012

Decomp Revisited

Filed under: General — m759 @ 1:11 PM

Frogs:

"Some mathematicians are birds, others are frogs. Birds fly high in the air and survey broad vistas of mathematics out to the far horizon. They delight in concepts that unify our thinking and bring together diverse problems from different parts of the landscape. Frogs live in the mud below and see only the flowers that grow nearby. They delight in the details of particular objects, and they solve problems one at a time."

— Freeman Dyson (See July 22, 2011)

A Rhetorical Question:

Robert Osserman in 2004

"The past decade has been an exciting one in the world of mathematics and a fabulous one (in the literal sense) for mathematicians, who saw themselves transformed from the frogs of fairy tales— regarded with a who-would-want-to-kiss-that aversion, when they were noticed at all— into fascinating royalty, portrayed on stage and screen….

Who bestowed the magic kiss on the mathematical frog?"

A Rhetorical Answer:

http://www.log24.com/log/pix11C/111130-SunshineCleaning.jpg

Above: Amy Adams in "Sunshine Cleaning"

Related material:

Sunday, August 19, 2012

O Marks the Spot

Filed under: General — m759 @ 9:00 PM

(Continued :  See Identity,  decomposition, and Sunshine Cleaning . )

"What, one might ask, does the suave, debonaire
Roger Thornhill have to do with the notion of
decomposition (emphasized by the unusual
coffin-shaped 'O') implied in the acronym
formed by his initials?" 

— Paul Gordon, Dial "M" for Mother ,
     Fairleigh Dickinson University Press, 2008, page 97

"To stay with the context of Cavell's brilliant reading 
of the film's relation to Hamlet, 'there is something rot -ten
in North by Northwest ' that also needs to be explained."

— Paul Gordon, op. cit., page 98

Related remarks— Sunday morning, May 20, 2007.

Monday, August 13, 2012

Raiders of the Lost Tesseract

Filed under: General,Geometry — Tags: — m759 @ 3:33 PM

(An episode of Mathematics and Narrative )

A report on the August 9th opening of Sondheim's Into the Woods

Amy Adams… explained why she decided to take on the role of the Baker’s Wife.

“It’s the ‘Be careful what you wish’ part,” she said. “Since having a child, I’m really aware that we’re all under a social responsibility to understand the consequences of our actions.” —Amanda Gordon at businessweek.com

Related material—

Amy Adams in Sunshine Cleaning  "quickly learns the rules and ropes of her unlikely new market. (For instance, there are products out there specially formulated for cleaning up a 'decomp.')" —David Savage at Cinema Retro

Compare and contrast…

1.  The following item from Walpurgisnacht 2012

IMAGE- Excerpt from 'Unified Approach to Functional Decompositions of Switching Functions,' by Marek A. Perkowski et al., 1995

2.  The six partitions of a tesseract's 16 vertices 
       into four parallel faces in Diamond Theory in 1937

Tuesday, May 1, 2012

What is Truth? (continued)

Filed under: General — m759 @ 11:01 PM

"There is a pleasantly discursive treatment of
 Pontius Pilate's unanswered question 'What is truth?'"
— H. S. M. Coxeter, 1987

Returning to the Walpurgisnacht posts
Decomposition (continued) and
Decomposition– Part III —

Some further background…

SAT

(Not  a Scholastic Aptitude Test)

"In computer sciencesatisfiability (often written
in all capitals or abbreviated 
SAT) is the problem
of determining if the variables of a given 
Boolean
 formula can be assigned in such a way as to
make the formula evaluate to TRUE."

— Wikipedia article Boolean satisfiability problem

For the relationship of logic decomposition to SAT,
see (for instance) these topics in the introduction to—

Advanced Techniques in Logic Synthesis,
Optimizations and Applications* 

Click image for a synopsis.

* Edited by Sunil P. Khatri and Kanupriya Gulati

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
researchers.

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

Thursday, January 12, 2012

Triangles Are Square

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

Coming across John H. Conway's 1991*
pinwheel  triangle decomposition this morning—

http://www.log24.com/log/pix12/120112-ConwayTriangleDecomposition.jpg

— suggested a review of a triangle decomposition result from 1984:

IMAGE- Triangle and square, each with 16 parts

Figure A

(Click the below image to enlarge.)

IMAGE- 'Triangles Are Square,' by Steven H. Cullinane (American Mathematical Monthly, 1985)

The above 1985 note immediately suggests a problem—

What mappings of a square  with c 2 congruent parts
to a triangle  with c 2 congruent parts are "natural"?**

(In Figure A above, whether the 322,560 natural transformations
of the 16-part square map in any natural way to transformations
of the 16-part triangle is not immediately apparent.)

* Communicated to Charles Radin in January 1991. The Conway
  decomposition may, of course, have been discovered much earlier.

** Update of Jan. 18, 2012— For a trial solution to the inverse
    problem, see the "Triangles are Square" page at finitegeometry.org.

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

http://www.log24.com/log/pix12/120110-AMS_page-Detail.jpg

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—

http://www.log24.com/log/pix12/120110-DiamondPuzzleFigure.jpg

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

http://www.log24.com/log/pix11A/110625-CubeHypostases.gif

Ousia

Click for further details:

http://www.log24.com/log/pix11A/110625-ProjectiveTrinitySm.jpg

 

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—

http://www.log24.com/log/pix11/110121-LatinSquares4x4.jpg

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

http://www.log24.com/log/pix11/110122-DiamondTheoryCover.jpg

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—

http://www.log24.com/log/pix10A/100703-Elements.gif

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.

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.

Friday, August 22, 2008

Friday August 22, 2008

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

Tentative movie title:
Blockheads

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.

Doonesbury,
August 21-22, 2008:

http://www.log24.com/log/pix08A/080821-22-db16color.gif
 

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

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.   
 

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…

Descartes

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, October 15, 2004

Friday October 15, 2004

Filed under: General — m759 @ 7:11 PM
Snow Jobs

In memory of C. P. Snow,
whose birthday is today

“Without the narrative prop of
High Table dinner conversation
at Cambridge, Snow would be lost.”
— Roger Kimball*

The image “http://www.log24.com/log/pix04A/041015-Sup.jpg” cannot be displayed, because it contains errors.

“It was a perfectly ordinary night
at Christ’s high table, except that
Hardy was dining as a guest.”
— C. P. Snow**

“666=2.3.3.37, and there is
no other decomposition.”
— G. H. Hardy***

* The Two Cultures Today

** Foreword to
A Mathematician’s Apology

*** A Mathematician’s Apology

Oct. 15, 2004, 7:11:37 PM

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.

Example:


For an animated version, click here.

Remarks:

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

Inscapes

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

Kaleidoscope

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

 

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