(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.
(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.
(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 .
(Continued from Part I and Part II.)
The paper excerpted below supplies some badly needed technical
background for the Wikipedia article on functional decomposition.
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.
Compare and contrast
1. The following excerpt from Wikipedia—
2. A webpage subtitled "Function Decomposition Over a Finite Field."
Related material—
A search tonight for material related to the fourcolor
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 midAugust 2007.)
Fass's interest in function decomposition may or may not
be related to the abovementioned theorem, which
originated in the investigation of functions into the
fourelement 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—
"Design is how it works." — Steve Jobs
An assignment for Jobs in the afterlife—
Discuss the FassFeldman approach to "categorization under
complexity" in the context of the Wikipedia article's
philosophical remarks on "reductionist tradition."
The FassFeldman paper was assigned in an MIT course
for a class on Walpurgisnacht 2003.
"It's the system that matters.
How the data arrange themselves inside it."
— Gravity's Rainbow
"Examples are the stainedglass windows of knowledge."
— Vladimir Nabokov
News item from this afternoon —
The above phrase "mapping systems" suggests a review
of my own very different "map systems." From a search
for that phrase in this journal —
See also "A FourColor Theorem: Function Decomposition
Over a Finite Field."
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 .
Twopart decomposition of base4 array
as two (nonLatin) 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 .
Fourpart decomposition of base2 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.
Pinterest boards uploaded to the new m759.net/piwigo —
Update of May 2 —
Update of May 3 —
Update of May 8 —
Art Space board created at Pinterest
"Now a little trivial heuresis is in order."
— The late Waclaw Szymanski on p. 279 of
"Decompositions of operatorvalued functions
in Hilbert spaces" (Studia Mathematica 50.3
(1974): 265280.)
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.
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."
(A review)
Structured gray matter:
Graphic symmetries of Galois space:
The reason for these graphic symmetries in affine Galois space —
symmetries of the underlying projective Galois space:
* For related remarks, see posts of May 2628, 2012.
My webpage "The Order4 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 fourcolor 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.
The Philosopher's Gaze , by David Michael Levin, The postmetaphysical question—question for a postmetaphysical 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 comingtopass of the ontological difference that is constitutive of all the local figureground 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 responseability, our capacity as beings gifted with vision, to measure up to the responsibility for perceptual responsiveness laid down for us in the "primordial decision" (Entscheid ) of the ontological difference (ibid.). To recognize the operation of the ontological difference taking place in the figureground difference of the perceptual Gestalt is to recognize the ontological difference as the primordial Riß , the primordial Urteil 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 socalled 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.
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.
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 PortRoyal 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 viceversa)
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:
"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 whowouldwanttokissthat aversion, when they were noticed at all— into fascinating royalty, portrayed on stage and screen….
Who bestowed the magic kiss on the mathematical frog?"
Above: Amy Adams in "Sunshine Cleaning"
Related material:
(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
coffinshaped '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.
(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—
2. The six partitions of a tesseract's 16 vertices
into four parallel faces in Diamond Theory in 1937—
"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 science, satisfiability (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
On the Complexity of Combat—
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 Walshfunction symmetry in 1982—
(Click image to enlarge. See also original pdf.)
Note the somewhat confusing resemblance to
a fourcolor decomposition theorem
used in the proof of the diamond theorem.
Coming across John H. Conway's 1991*
pinwheel triangle decomposition this morning—
— suggested a review of a triangle decomposition result from 1984:
Figure A
(Click the below image to enlarge.)
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 16part square map in any natural way to transformations
of the 16part 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.
(Continued from Epiphany and from yesterday.)
Detail from the current American Mathematical Society homepage—
Further detail, with a comparison to Dürer's magic square—
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 4sets corresponding to the
four rows of Dürer's square, and apply the 4color decomposition theorem.
Note the symmetry of the set of 3 line diagrams that result.
Now consider the 4sets 14, 58, 912, and 1316, and note that these
occupy the same positions in the Donmoyer square that 4sets of
like elements occupy in the diamondpuzzle figure below—
Thus the Donmoyer array also enjoys the structural symmetry,
invariant under 322,560 transformations, of the diamondpuzzle 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
— m759 @ 12:00 PM
"… the formula 'Three Hypostases in one Ousia '
Ousia

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 orderfour Latin squares are represented.
For a deeper look at the structure of such squares, let the highschool
chart above be labeled with the letters A through X, and apply the
fourcolor decomposition theorem. The result is 24 structural diagrams—
Some of the squares are structurally congruent under the group of 8 symmetries of the square.
This can be seen in the following regrouping—
(Image corrected on Jan. 25, 2011– "seven" replaced "eight.")
* Retitled "The Order4 (i.e., 4×4) Latin Squares" in the copy at finitegeometry.org/sc.
"Human perception is a saga of created reality. But we were devising entities beyond the agreedupon 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 lowercase omega is found in the standard symbolic representation of the Galois field GF(4)—
A representation of GF(4) that goes beyond the standard representation—
Here the four diagonallydivided twocolor 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.
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 FourColor 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— wellknown as discrete analogues of the trigonometric functions of traditional harmonic analysis.
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.
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 avantgarde 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…."
"… 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) —
See also, from August 1, 2008
(anniversary of Hitler's
opening the 1936 Olympics) —
For Sarah Silverman —
and the 9/9/03 entry
Doonesbury,
August 2122, 2008:
— Susan Sontag,
"Against Interpretation"
"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 openmindedness' 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 […].
"Examples are the stained
glass windows of knowledge."
Click on image to enlarge.
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:
Compare and contrast:
The harmonicanalysis analogy suggests a review of an earlier entry's
link today to 4/30– Structure and Logic— as well as
reexamination of Symmetry and a Trinity
(Dec. 4, 2002).
See also —
A FourColor Theorem,
The Diamond Theorem, and
The Most Violent Poem,
In memory of C. P. Snow,
whose birthday is today
“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***
** Foreword to
A Mathematician’s Apology
Oct. 15, 2004, 7:11:37 PM


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