Log24

Thursday, November 29, 2012

Lines of Symbols

Filed under: General,Geometry — m759 @ 3:00 pm

C. P. Snow on G. H. Hardy, in Snow's foreword to A Mathematician's Apology

"One morning early in 1913, he found, among the letters on his breakfast table, a large untidy envelope decorated with Indian stamps. When he opened it, he found sheets of paper by no means fresh, on which, in a non-English holograph, were line after line of symbols. Hardy glanced at them without enthusiasm. He was by this time, at the age of thirty-six, a world famous mathematician: and world famous mathematicians, he had already discovered, are unusually exposed to cranks. He was accustomed to receiving manuscripts from strangers, proving the prophetic wisdom of the Great Pyramid, the revelations of the Elders of Zion, or the cryptograms that Bacon has inserted in the plays of the so-called Shakespeare."

Some related material (click to enlarge)—

The author links to, but does not name, the source of the above
"line after line of symbols." It is "Visualizing GL(2,p)." See that webpage
for some less esoteric background.

See also the two Wikipedia articles Finite geometry and Hesse configuration
and an image they share—

IMAGE- Image from Wikipedia articles 'Finite geometry' and 'Hesse configuration.'

The Hesse here is not Hermann, but Otto.

Tuesday, March 26, 2013

Raiders of the Lost Symbols

Filed under: General — m759 @ 10:09 am

See Lines of Symbols in this journal.

Friday, May 6, 2022

Interality and the Bead Game

Filed under: General — Tags: , , , — m759 @ 3:00 pm

WIkipedia on the URL suffix ".io" —

"In computer science, "IO" or "I/O" is commonly used
as an abbreviation for input/output, which makes the
.io domain desirable for services that want to be
associated with technology. .io domains are often used
for open source projects, application programming
interfaces ("APIs"), startup companiesbrowser games,
and other online services."

An association with the Bead Game from a post of April 7, 2018

IMAGE- 'Solomon's Cube'

Glasperlenspiel  passage quoted here in Summa Mythologica 

“"I suddenly realized that in the language, or at any rate
in the spirit of the Glass Bead Game, everything actually
was all-meaningful, that every symbol and combination of
symbols led not hither and yon, not to single examples,
experiments, and proofs, but into the center, the mystery
and innermost heart of the world, into primal knowledge.
Every transition from major to minor in a sonata, every
transformation of a myth or a religious cult, every classical
or artistic formulation was, I realized in that flashing moment,
if seen with a truly meditative mind, nothing but a direct route
into the interior of the cosmic mystery, where in the alternation
between inhaling and exhaling, between heaven and earth,
between Yin and Yang, holiness is forever being created.”

A less poetic meditation on the above 4x4x4 design cube —

"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."

See also a related remark by Lévi-Strauss in 1955

"…three different readings become possible:
left to right, top to bottom, front to back."

The recent use by a startup company of the URL "interality.io" suggests
a fourth  reading for the 1955 list of Lévi-Strauss — in and out
i.e., inner and outer group automorphisms —  from a 2011 post
on the birthday of T. S. Eliot :

A transformation:

Inner and outer group automorphisms

Click on the picture for details.

Sunday, April 24, 2022

Structuralism: Three Betweens

Filed under: General — Tags: , , , — m759 @ 10:44 am
 

Tuesday, November 3, 2009

Summa Mythologica

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

Book review by Jadran Mimica in Oceania, Vol. 74, 2003:

"In his classic essay of 1955 'The Structural Study of Myth' Levi-Strauss came up with a universal formula of mythopoeic dynamics

[fx(a) : fy(b) :: fx(b) : fa-1(y)]

that he called canonical 'for it can represent any mythic transformation'. This formulation received its consummation in the four massive Mythologiques volumes, the last of which crystallises the fundamental dialectics of mythopoeic thought: that there is 'one myth only' and the primal ground of this 'one' is 'nothing'. The elucidation of the generative matrix of the myth-work is thus completed as is the self-totalisation of both the thinker and his object."

So there.

At least one mathematician has claimed that the Levi-Strauss formula makes sense. (Jack Morava, arXiv pdf, 2003.)

I prefer the earlier (1943) remarks of Hermann Hesse on transformations of myth:

"…in the spirit of the Glass Bead Game, everything actually was all-meaningful, that every symbol and combination of symbols led not hither and yon, not to single examples, experiments, and proofs, but into the center, the mystery and innermost heart of the world, into primal knowledge. Every transition from major to minor in a sonata, every transformation of a myth or a religious cult, every classical or artistic formulation was, I realized in that flashing moment, if seen with a truly meditative mind, nothing but a direct route into the interior of the cosmic mystery, where in the alternation between inhaling and exhaling, between heaven and earth, between Yin and Yang, holiness is forever being created."

Wednesday, October 9, 2019

The Joy of Six

Note that in the pictures below of the 15 two-subsets of a six-set,
the symbols 1 through 6 in Hudson's square array of 1905 occupy the
same positions as the anticommuting Dirac matrices in Arfken's 1985
square array. Similarly occupying these positions are the skew lines
within a generalized quadrangle (a line complex) inside PG(3,2).

Anticommuting Dirac matrices as spreads of projective lines

Related narrative The "Quantum Tesseract Theorem."

Saturday, April 7, 2018

Sides

The FBI holding cube in "The Blacklist" —

" 'The Front' is not the whole story . . . ."

— Vincent Canby, New York Times  film review, 1976,
     as quoted in Wikipedia.

See also Solomon's Cube in this  journal.

IMAGE- 'Solomon's Cube'

Webpage demonstrating symmetries of 'Solomon's Cube'

Some may view the above web page as illustrating the
Glasperlenspiel  passage quoted here in Summa Mythologica 

“"I suddenly realized that in the language, or at any rate
in the spirit of the Glass Bead Game, everything actually
was all-meaningful, that every symbol and combination of
symbols led not hither and yon, not to single examples,
experiments, and proofs, but into the center, the mystery
and innermost heart of the world, into primal knowledge.
Every transition from major to minor in a sonata, every
transformation of a myth or a religious cult, every classical
or artistic formulation was, I realized in that flashing moment,
if seen with a truly meditative mind, nothing but a direct route
into the interior of the cosmic mystery, where in the alternation
between inhaling and exhaling, between heaven and earth,
between Yin and Yang, holiness is forever being created.”

A less poetic meditation on the above 4x4x4 design cube —

"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."

See also a related remark by Lévi-Strauss in 1955

"…three different readings become possible:
left to right, top to bottom, front to back."

Tuesday, May 3, 2016

Symmetry

A note related to the diamond theorem and to the site
Finite Geometry of the Square and Cube —

The last link in the previous post leads to a post of last October whose
final link leads, in turn, to a 2009 post titled Summa Mythologica .

Webpage demonstrating symmetries of 'Solomon's Cube'

Some may view the above web page as illustrating the
Glasperlenspiel  passage quoted here in Summa Mythologica 

“"I suddenly realized that in the language, or at any rate
in the spirit of the Glass Bead Game, everything actually
was all-meaningful, that every symbol and combination of
symbols led not hither and yon, not to single examples,
experiments, and proofs, but into the center, the mystery
and innermost heart of the world, into primal knowledge.
Every transition from major to minor in a sonata, every
transformation of a myth or a religious cult, every classical
or artistic formulation was, I realized in that flashing moment,
if seen with a truly meditative mind, nothing but a direct route
into the interior of the cosmic mystery, where in the alternation
between inhaling and exhaling, between heaven and earth,
between Yin and Yang, holiness is forever being created.”

A less poetic meditation on the above web page* —

"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."

Update of Sept. 5, 2016 — See also a related remark
by Lévi-Strauss in 1955:  "…three different readings
become possible: left to right, top to bottom, front
to back."

* For the underlying mathematics, see a June 21, 1983, research note.

Friday, March 28, 2014

Chinese Rune

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

"The Geometry of the I Ching introduces something called the Cullinane sequence
for the hexagrams, and uses a notation based on the four sides and two diagonals
in a square to indicate the yin and yang lines. The resulting rune-like symbols
are intriguing…."

— Andreas Schöter's  I Ching  home page

Actually, the geometry is a bit deeper than the rune-like symbols.

" 'Harriet Burden has been really great to me,'
Rune says in an interview, 'not only as a collector
of my work but as a true supporter. And I think of her
as a muse for the project … ' "

— In The Blazing World , the artist known as Rune

Sunday, November 18, 2012

Kernel

Filed under: General — m759 @ 5:24 am

(Continued)

Rachel Dodes in The Wall Street Journal
on All Souls' Day, 2012

"In one of the first lines uttered by Daniel Day-Lewis, playing Abraham Lincoln in the new Steven Spielberg film opening Nov. 9, he says, 'I could be bounded in a nutshell, and count myself a king of infinite space— were it not that I have bad dreams.'

The line was ripped straight from 'Hamlet,' by Lincoln's favorite writer, William Shakespeare. Tony Kushner, the Pulitzer Prize-winning playwright ('Angels in America') who wrote the script for the film, says that Shakespeare, much like Lincoln, 'had extraordinary mastery over the darkest parts of the human spirit.'"

The above quotation omits Shakespeare's words prefacing the nutshell part— "O God."

These same words in a different tongue—  "Hey Ram"— have often been quoted as the last words of Gandhi. (See yesterday's noon post.)

"… for the Highest Essence (brahman ),
which is the core of the world, is identical
with the Highest Self (ātman ), the kernel
of man's existence."

— Heinrich Zimmer, Myths and Symbols
in Indian Art and Civilization
, Pantheon
Books, 1946, page 142 

Related material: A post linked to here on Friday night
that itself links to a different Shakespeare speech.

Thursday, March 1, 2012

Block That Metaphor:

Filed under: General,Geometry — Tags: , , — m759 @ 11:09 pm

The Cube Model and Peano Arithmetic

The eightfold cube  model of the Fano plane may or may not have influenced a new paper (with the date Feb. 10, 2011, in its URL) on an attempted consistency proof of Peano arithmetic—

The Consistency of Arithmetic, by Storrs McCall

"Is Peano arithmetic (PA) consistent?  This paper contains a proof that it is. …

Axiomatic proofs we may categorize as 'syntactic', meaning that they concern only symbols and the derivation of one string of symbols from another, according to set rules.  'Semantic' proofs, on the other hand, differ from syntactic proofs in being based not only on symbols but on a non-symbolic, non-linguistic component, a domain of objects.    If the sole paradigm of 'proof ' in mathematics is 'axiomatic proof ', in which to prove a formula means to deduce it from axioms using specified rules of inference, then Gödel indeed appears to have had the last word on the question of PA-consistency.  But in addition to axiomatic proofs there is another kind of proof.   In this paper I give a proof of PA's consistency based on a formal semantics for PA.   To my knowledge, no semantic consistency proof of Peano arithmetic has yet been constructed.

The difference between 'semantic' and 'syntactic' theories is described by van Fraassen in his book The Scientific Image :

"The syntactic picture of a theory identifies it with a body of theorems, stated in one particular language chosen for the expression of that theory.  This should be contrasted with the alternative of presenting a theory in the first instance by identifying a class of structures as its models.  In this second, semantic, approach the language used to express the theory is neither basic nor unique; the same class of structures could well be described in radically different ways, each with its own limitations.  The models occupy centre stage." (1980, p. 44)

Van Fraassen gives the example on p. 42 of a consistency proof in formal geometry that is based on a non-linguistic model.  Suppose we wish to prove the consistency of the following geometric axioms:

A1.  For any two lines, there is at most one point that lies on both.
A2.  For any two points, there is exactly one line that lies on both.
A3.  On every line there lie at least two points.

The following diagram shows the axioms to be consistent:

Figure 1
 

The consistency proof is not a 'syntactic' one, in which the consistency of A1-A3 is derived as a theorem of a deductive system, but is based on a non-linguistic structure.  It is a semantic as opposed to a syntactic proof.  The proof constructed in this paper, like van Fraassen's, is based on a non-linguistic component, not a diagram in this case but a physical domain of three-dimensional cube-shaped blocks. ….

… The semantics presented in this paper I call 'block semantics', for reasons that will become clear….  Block semantics is based on domains consisting of cube-shaped objects of the same size, e.g. children's wooden building blocks.  These can be arranged either in a linear array or in a rectangular array, i.e. either in a row with no space between the blocks, or in a rectangle composed of rows and columns.  A linear array can consist of a single block, and the order of individual blocks in a linear or rectangular array is irrelevant. Given three blocks A, B and C, the linear arrays ABC and BCA are indistinguishable.  Two linear arrays can be joined together or concatenated into a single linear array, and a rectangle can be re-arranged or transformed into a linear array by successive concatenation of its rows.  The result is called the 'linear transformation' of the rectangle.  An essential characteristic of block semantics is that every domain of every block model is finite.  In this respect it differs from Tarski’s semantics for first-order logic, which permits infinite domains.  But although every block model is finite, there is no upper limit to the number of such models, nor to the size of their domains.

It should be emphasized that block models are physical models, the elements of which can be physically manipulated.  Their manipulation differs in obvious and fundamental ways from the manipulation of symbols in formal axiomatic systems and in mathematics.  For example the transformations described above, in which two linear arrays are joined together to form one array, or a rectangle of blocks is re-assembled into a linear array, are physical transformations not symbolic transformations. …" 

Storrs McCall, Department of Philosophy, McGill University

See also…

Friday, March 18, 2011

Defining Configurations*

Filed under: General,Geometry — Tags: , — m759 @ 7:00 pm

The On-Line Encyclopedia of Integer Sequences has an article titled "Number of combinatorial configurations of type (n_3)," by N.J.A. Sloane and D. Glynn.

From that article:

  • DEFINITION: A combinatorial configuration of type (n_3) consists of an (abstract) set of n points together with a set of n triples of points, called lines, such that each point belongs to 3 lines and each line contains 3 points.
  • EXAMPLE: The unique (8_3) configuration consists of the triples 125, 148, 167, 236, 278, 347, 358, 456.

The following corrects the word "unique" in the example.

http://www.log24.com/log/pix11/110320-MoebiusKantorConfig500w.jpg

* This post corrects an earlier post, also numbered 14660 and dated 7 PM March 18, 2011, that was in error.
   The correction was made at about 11:50 AM on March 20, 2011.

_____________________________________________________________

Update of March 21

The problem here is of course with the definition. Sloane and Glynn failed to include in their definition a condition that is common in other definitions of configurations, even abstract or purely "combinatorial" configurations. See, for instance, Configurations of Points and Lines , by Branko Grunbaum (American Mathematical Society, 2009), p. 17—

In the most general sense we shall consider combinatorial (or abstract) configurations; we shall use the term set-configurations as well. In this setting "points" are interpreted as any symbols (usually letters or integers), and "lines" are families of such symbols; "incidence" means that a "point" is an element of a "line". It follows that combinatorial configurations are special kinds of general incidence structures. Occasionally, in order to simplify and clarify the language, for "points" we shall use the term marks, and for "lines" we shall use blocks. The main property of geometric configurations that is preserved in the generalization to set-configurations (and that characterizes such configurations) is that two marks are incident with at most one block, and two blocks with at most one mark.

Whether or not omitting this "at most one" condition from the definition is aesthetically the best choice, it dramatically changes the number  of configurations in the resulting theory, as the above (8_3) examples show.

Update of March 22 (itself updated on March 25)

For further background on configurations, see Dolgachev—

http://www.log24.com/log/pix11/110322-DolgachevIntro.gif

Note that the two examples Dolgachev mentions here, with 16 points and 9 points, are not unrelated to the geometry of 4×4 and 3×3 square arrays. For the Kummer and related 16-point configurations, see section 10.3, "The Three Biplanes of Order 4," in Burkard Polster's A Geometrical Picture Book  (Springer, 1998). See also the 4×4 array described by Gordon Royle in an undated web page and in 1980 by Assmus and Sardi. For the Hesse configuration, see (for instance) the passage from Coxeter quoted in Quaternions in an Affine Galois Plane.

Update of March 27

See the above link to the (16,6) 4×4 array and the (16,6) exercises using this array in R.D. Carmichael's classic Introduction to the Theory of Groups of Finite Order  (1937), pp. 42-43. For a connection of this sort of 4×4 geometry to the geometry of the diamond theorem, read "The 2-subsets of a 6-set are the points of a PG(3,2)" (a note from 1986) in light of R.W.H.T. Hudson's 1905 classic Kummer's Quartic Surface , pages 8-9, 16-17, 44-45, 76-77, 78-79, and 80.

Sunday, August 15, 2010

The Game

Filed under: General — Tags: , , — m759 @ 11:07 pm
'Magister Ludi,' or 'The Glass Bead Game,' by Hermann Hesse

We shall now give a brief summary of the beginnings of the Glass Bead Game. It appears to have arisen simultaneously in Germany and in England. In both countries, moreover, it was originally a kind of exercise employed by those small groups of musicologists and musicians who worked and studied in the new seminaries of musical theory. If we compare the original state of the Game with its subsequent developments and its present form, it is much like comparing a musical score of the period before 1500, with its primitive notes and absence of bar lines, with an eighteenth-century score, let alone with one from the nineteenth with its confusing excess of symbols for dynamics, tempi, phrasing, and so on, which often made the printing of such scores a complex technical problem.

The Game was at first nothing more than a witty method for developing memory and ingenuity among students and musicians. And as we have said, it was played both in England and Germany before it was ‘invented’ here in the Musical Academy of Cologne, and was given the name it bears to this day, after so many generations, although it has long ceased to have anything to do with glass beads.

The inventor, Bastian Perrot of Calw, a rather eccentric but clever, sociable, and humane musicologist, used glass beads instead of letters, numerals, notes, or other graphic symbols. Perrot, who incidentally has also bequeathed to us a treatise on the Apogee and Decline of Counterpoint, found that the pupils at the Cologne Seminary had a rather elaborate game they used to play. One would call out, in the standardized abbreviations of their science, motifs or initial bars of classical compositions, whereupon the other had to respond with the continuation of the piece, or better still with a higher or lower voice, a contrasting theme, and so forth. It was an exercise in memory and improvisation quite similar to the sort of thing probably in vogue among ardent pupils of counterpoint in the days of Schütz, Pachelbel, and Bach — although it would then not have been done in theoretical formulas, but in practice on the cembalo, lute, or flute, or with the voice.

Bastian Perrot in all probability was a member of the Journeyers to the East. He was partial to handicrafts and had himself built several pianos and clavichords in the ancient style. Legend has it that he was adept at playing the violin in the old way, forgotten since 1800, with a high-arched bow and hand-regulated tension of the bow hairs. Given these interests, it was perhaps only natural that he should have constructed a frame, modeled on a child’s abacus, a frame with several dozen wires on which could be strung glass beads of various sizes, shapes, and colors. The wires corresponded to the lines of the musical staff, the beads to the time-values of the notes, and so on. In this way he could represent with beads musical quotations or invented themes, could alter, transpose, and develop them, change them and set them in counterpoint to one another. In technical terms this was a mere plaything, but the pupils liked it; it was imitated and became fashionable in England too. For a time the game of musical exercises was played in this charmingly primitive manner. And as is so often the case, an enduring and significant institution received its name from a passing and incidental circumstance. For what later evolved out of that students’ sport and Perrot’s bead-strung wires bears to this day the name by which it became popularly known, the Glass Bead Game.

Hermann Hesse

“For although in a certain sense and for light-minded persons non-existent things can be more easily and irresponsibly represented in words than existing things, for the serious and conscientious historian it is just the reverse. Nothing is harder, yet nothing is more necessary, than to speak of certain things whose existence is neither demonstrable nor probable. The very fact that serious and conscientious men treat them as existing things brings them a step closer to existence and to the possibility of being born.”

— “Albertus Secundus,” epigraph to The Glass Bead Game

From DownloadThat.com

(Click to enlarge.)

http://www.log24.com/log/pix10B/100815-ThePaletteSm.jpg

Tuesday, November 3, 2009

Summa Mythologica

Filed under: General,Geometry — Tags: , , — m759 @ 10:10 pm

Book review by Jadran Mimica in Oceania, Vol. 74, 2003:

"In his classic essay of 1955 'The Structural Study of Myth' Levi-Strauss came up with a universal formula of mythopeic dynamics

[fx(a) : fy(b) :: fx(b) : fa-1(y)]

that he called canonical 'for it can represent any mythic transformation'. This formulation received its consummation in the four massive Mythologiques volumes, the last of which crystallises the fundamental dialectics of mythopoeic thought: that there is 'one myth only' and the primal ground of this 'one' is 'nothing'. The elucidation of the generative matrix of the myth-work is thus completed as is the self-totalisation of both the thinker and his object."

So there.

At least one mathematician has claimed that the Levi-Strauss formula makes sense. (Jack Morava, arXiv pdf, 2003.)

I prefer the earlier (1943) remarks of Hermann Hesse on transformations of myth:

"…in the spirit of the Glass Bead Game, everything actually was all-meaningful, that every symbol and combination of symbols led not hither and yon, not to single examples, experiments, and proofs, but into the center, the mystery and innermost heart of the world, into primal knowledge. Every transition from major to minor in a sonata, every transformation of a myth or a religious cult, every classical or artistic formulation was, I realized in that flashing moment, if seen with a truly meditative mind, nothing but a direct route into the interior of the cosmic mystery, where in the alternation between inhaling and exhaling, between heaven and earth, between Yin and Yang, holiness is forever being created."

Friday, August 28, 2009

Friday August 28, 2009

Filed under: General — m759 @ 3:09 am
Rites of Passage

“Things fall apart;
   the centre cannot hold….

Part I:

“Inside the church, the grief was real. Sen. Edward Kennedy’s voice caught as he read his lovely eulogy, and when he was done, Caroline Kennedy Schlossberg stood up and hugged him. She bravely read from Shakespeare’s ‘The Tempest‘ (‘Our revels now are ended. We are such stuff as dreams are made on‘). Many of the 315 mourners, family and friends of the Kennedys and Bessettes, swallowed hard through a gospel choir’s rendition of ‘Amazing Grace,’ and afterward, they sang lustily as Uncle Teddy led the old Irish songs at the wake.”

Newsweek magazine, issue dated August 2, 1999

Part II:

The Ba gua (Chinese….) are eight diagrams used in Taoist cosmology to represent a range of interrelated concepts. Each consists of three lines, each either ‘broken’ or ‘unbroken,’ representing a yin line or a yang line, respectively. Due to their tripartite structure, they are often referred to as ‘trigrams’ in English. —Wikipedia

Part III:

3x3 array of symbols, cover of 'Dorm Room Feng Shui'

Above: detail from the cover of…

Bagua in Brief, from 'Dorm Room Feng Shui'
Figures explaining 'Dorm Room Feng Shui'

Sunday, May 25, 2008

Sunday May 25, 2008

Filed under: General,Geometry — Tags: , , , — m759 @ 9:00 am
 
Wechsler Cubes
"Confusion is nothing new."
— Song lyric, Cyndi Lauper
 
Part I:
Magister Ludi

Hermann Hesse's 1943 The Glass Bead Game (Picador paperback, Dec. 6, 2002, pp. 139-140)–

"For the present, the Master showed him a bulky memorandum, a proposal he had received from an organist– one of the innumerable proposals which the directorate of the Game regularly had to examine. Usually these were suggestions for the admission of new material to the Archives. One man, for example, had made a meticulous study of the history of the madrigal and discovered in the development of the style a curved that he had expressed both musically and mathematically, so that it could be included in the vocabulary of the Game. Another had examined the rhythmic structure of Julius Caesar's Latin and discovered the most striking congruences with the results of well-known studies of the intervals in Byzantine hymns. Or again some fanatic had once more unearthed some new cabala hidden in the musical notation of the fifteenth century. Then there were the tempestuous letters from abstruse experimenters who could arrive at the most astounding conclusions from, say, a comparison of the horoscopes of Goethe and Spinoza; such letters often included pretty and seemingly enlightening geometric drawings in several colors."

Part II:
A Bulky Memorandum

From Siri Hustvedt, author of Mysteries of the Rectangle: Essays on Painting (Princeton Architectural Press, 2005)– What I Loved: A Novel (Picador paperback, March 1, 2004, page 168)–

A description of the work of Bill Wechsler, a fictional artist:

"Bill worked long hours on a series of autonomous pieces about numbers. Like O's Journey, the works took place inside glass cubes, but these were twice as large– about two feet square. He drew his inspiration from sources as varied as the Cabbala, physics, baseball box scores, and stock market reports. He painted, cut, sculpted, distorted, and broke the numerical signs in each work until they became unrecognizable. He included figures, objects, books, windows, and always the written word for the number. It was rambunctious art, thick with allusion– to voids, blanks, holes, to monotheism and the individual, the the dialectic and yin-yang, to the Trinity, the three fates, and three wishes, to the golden rectangle, to seven heavens, the seven lower orders of the sephiroth, the nine Muses, the nine circles of Hell, the nine worlds of Norse mythology, but also to popular references like A Better Marriage in Five Easy Lessons and Thinner Thighs in Seven Days. Twelve-step programs were referred to in both cube one and cube two. A miniature copy of a book called The Six Mistakes Parents Make Most Often lay at the bottom of cube six. Puns appeared, usually well disguised– one, won; two, too, and Tuesday; four, for, forth; ate, eight. Bill was partial to rhymes as well, both in images and words. In cube nine, the geometric figure for a line had been painted on one glass wall. In cube three, a tiny man wearing the black-and-white prison garb of cartoons and dragging a leg iron has

— End of page 168 —

opened the door to his cell. The hidden rhyme is "free." Looking closely through the walls of the cube, one can see the parallel rhyme in another language: the German word drei is scratched into one glass wall. Lying at the bottom of the same box is a tiny black-and-white photograph cut from a book that shows the entrance to Auschwitz: ARBEIT MACHT FREI. With every number, the arbitrary dance of associations worked togethere to create a tiny mental landscape that ranged in tone from wish-fulfillment dream to nightmare. Although dense, the effect of the cubes wasn't visually disorienting. Each object, painting, drawing, bit of text, or sculpted figure found its rightful place under the glass according to the necessary, if mad, logic of numerical, pictorial, and verbal connection– and the colors of each were startling. Every number had been given a thematic hue. Bill had been interested in Goethe's color wheel and in Alfred Jensen's use of it in his thick, hallucinatory paintings of numbers. He had assigned each number a color. Like Goethe, he included black and white, although he didn't bother with the poet's meanings. Zero and one were white. Two was blue. Three was red, four was yellow, and he mixed colors: pale blue for five, purples in six, oranges in seven, greens in eight, and blacks and grays in nine. Although other colors and omnipresent newsprint always intruded on the basic scheme, the myriad shades of a single color dominated each cube.

The number pieces were the work of a man at the top of his form. An organic extension of everything Bill had done before, these knots of symbols had an explosive effect. The longer I looked at them, the more the miniature constructions seemed on the brink of bursting from internal pressure. They were tightly orchestrated semantic bombs through which Bill laid bare the arbitrary roots of meaning itself– that peculiar social contract generated by little squiggles, dashes, lines, and loops on a page."

Part III:
Wechsler Cubes
(named not for
Bill Wechsler, the
fictional artist above,
but for the non-fictional
David Wechsler) –

 

From 2002:

 

Above: Dr. Harrison Pope, Harvard professor of psychiatry, demonstrates the use of the Wechsler Adult Intelligence Scale "block design" subtest.

 
Part IV:
A Magic Gallery
Log24, March 4, 2004
 

ZZ
WW

Figures from the
Kaleidoscope Puzzle
of Steven H. Cullinane:


Poem by Eugen Jost:
Zahlen und Zeichen,
Wörter und Worte

Mit Zeichen und Zahlen
vermessen wir Himmel und Erde
schwarz
auf weiss
schaffen wir neue Welten
oder gar Universen

 Numbers and Names,
Wording and Words

With numbers and names
we measure heaven and earth
black
on white
we create new worlds
and universes

English translation
by Catherine Schelbert

A related poem:

Alphabets
by Hermann Hesse

From time to time
we take our pen in hand
and scribble symbols
on a blank white sheet
Their meaning is
at everyone's command;
it is a game whose rules
are nice and neat.

But if a savage
or a moon-man came
and found a page,
a furrowed runic field,
and curiously studied
lines and frame:
How strange would be
the world that they revealed.
a magic gallery of oddities.
He would see A and B
as man and beast,
as moving tongues or
arms or legs or eyes,
now slow, now rushing,
all constraint released,
like prints of ravens'
feet upon the snow.
He'd hop about with them,
fly to and fro,
and see a thousand worlds
of might-have-been
hidden within the black
and frozen symbols,
beneath the ornate strokes,
the thick and thin.
He'd see the way love burns
and anguish trembles,
He'd wonder, laugh,
shake with fear and weep
because beyond this cipher's
cross-barred keep
he'd see the world
in all its aimless passion,
diminished, dwarfed, and
spellbound in the symbols,
and rigorously marching
prisoner-fashion.
He'd think: each sign
all others so resembles
that love of life and death,
or lust and anguish,
are simply twins whom
no one can distinguish…
until at last the savage
with a sound
of mortal terror
lights and stirs a fire,
chants and beats his brow
against the ground
and consecrates the writing
to his pyre.
Perhaps before his
consciousness is drowned
in slumber there will come
to him some sense
of how this world
of magic fraudulence,
this horror utterly
behind endurance,
has vanished as if
it had never been.
He'll sigh, and smile,
and feel all right again.

— Hermann Hesse (1943),
"Buchstaben," from
Das Glasperlenspiel,
translated by
Richard and Clara Winston

Tuesday, June 27, 2006

Tuesday June 27, 2006

Filed under: General,Geometry — m759 @ 10:31 am
Chinese Jar
Revisited

In memory of
Irving Kaplansky,
who died on
Sunday, June 25, 2006

“Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.”

T. S. Eliot


Kaplansky received his doctorate in mathematics at Harvard in 1941 as the first Ph.D. student of Saunders Mac Lane.

From the April 25, 2005, Harvard Crimson:

Ex-Math Prof Mac Lane, 95, Dies

Gade University Professor of Mathematics Barry Mazur, a friend of the late Mac Lane, recalled that [a Mac Lane paper of 1945] had at first been rejected from a lower-caliber mathematical journal because the editor thought that it was “more devoid of content” than any other he had read.

“Saunders wrote back and said, ‘That’s the point,'” Mazur said. “And in some ways that’s the genius of it. It’s the barest, most Beckett-like vocabulary that incorporates the theory and nothing else.”

He likened it to a sparse grammar of nouns and verbs and a limited vocabulary that is presented “in such a deft way that it will help you understand any language you wish to understand and any language will fit into it.”

A sparse grammar of lines from Charles Sanders Peirce (Harvard College, class of 1859):

The image “http://www.log24.com/theory/images/PeirceBox.bmp” cannot be displayed, because it contains errors.

The image “http://www.log24.com/theory/images/PeirceSymbols1.jpg” cannot be displayed, because it contains errors.

It is true of this set of binary connectives, as it is true of logic generally, that (as alleged above of Mac Lane’s category theory) “it will help you understand any language you wish to understand and any language will fit into it.” Of course, a great deal of questionable material has been written about these connectives. (See, for instance, Piaget and De Giacomo.) For remarks on the connectives that are not questionable, see Wittgenstein’s Tractatus Logico-Philosophicus (English version, 1922), section 5.101, and Knuth’s “Boolean Basics” (draft, 2006).

Related entry: Binary Geometry.

Friday, April 15, 2005

Friday April 15, 2005

Filed under: General — Tags: — m759 @ 7:11 am
Leonardo Day

The image “http://www.log24.com/log/pix05/050415-Google.gif” cannot be displayed, because it contains errors.

In memory of Leonardo and of Chen Yifei (previous entry), a link to the Sino-Judaic Institute’s review of Chen’s film “Escape to Shanghai” —

The image “http://www.log24.com/log/pix05/050415-PointsEast.gif” cannot be displayed, because it contains errors.
Click on the above for details.

Related material
from Log24.net:


Saturday, December 27, 2003  10:21 PM

Toy

“If little else, the brain is an educational toy.  While it may be a frustrating plaything — one whose finer points recede just when you think you are mastering them — it is nonetheless perpetually fascinating, frequently surprising, occasionally rewarding, and it comes already assembled; you don’t have to put it together on Christmas morning.

The problem with possessing such an engaging toy is that other people want to play with it, too.  Sometimes they’d rather play with yours than theirs.  Or they object if you play with yours in a different manner from the way they play with theirs.  The result is, a few games out of a toy department of possibilities are universally and endlessly repeated.  If you don’t play some people’s game, they say that you have ‘lost your marbles,’ not recognizing that,

while Chinese checkers is indeed a fine pastime, a person may also play dominoes, chess, strip poker, tiddlywinks, drop-the-soap or Russian roulette with his brain.

One brain game that is widely, if poorly, played is a gimmick called ‘rational thought.’ “

— Tom Robbins, Even Cowgirls Get the Blues

Sol LeWitt
June 12, 1969
:

“I took the number twenty-four and there’s twenty-four ways of expressing the numbers one, two, three, four.  And I assigned one kind of line to one, one to two, one to three, and one to four.  One was a vertical line, two was a horizontal line, three was diagonal left to right, and four was diagonal right to left.  These are the basic kind of directions that lines can take…. the absolute ways that lines can be drawn.   And I drew these things as parallel lines very close to one another in boxes.  And then there was a system of changing them so that within twenty-four pages there were different arrangements of actually sixteen squares, four sets of four.  Everything was based on four.  So this was kind of a… more of a… less of a rational… I mean, it gets into the whole idea of methodology.”

Yes, it does.
See Art Wars, Poetry’s Bones, and Time Fold.


Friday, December 26, 2003  7:59 PM

ART WARS, St. Stephen’s Day:

The Magdalene Code

Got The Da Vinci Code for Xmas.

From page 262:

When Langdon had first seen The Little Mermaid, he had actually gasped aloud when he noticed that the painting in Ariel’s underwater home was none other than seventeenth-century artist Georges de la Tour’s The Penitent Magdalene — a famous homage to the banished Mary Magdalene — fitting decor considering the movie turned out to be a ninety-minute collage of blatant symbolic references to the lost sanctity of Isis, Eve, Pisces the fish goddess, and, repeatedly, Mary Magdalene.

Related Log24 material —

December 21, 2002:

A Maiden’s Prayer

The Da Vinci Code, pages 445-446:

“The blade and chalice?” Marie asked.  “What exactly do they look like?”

Langdon sensed she was toying with him, but he played along, quickly describing the symbols.

A look of vague recollection crossed her face.  “Ah, yes, of course.  The blade represents all that is masculine.  I believe it is drawn like this, no?”  Using her index finger, she traced a shape on her palm.

“Yes,” Langdon said.  Marie had drawn the less common “closed” form of the blade, although Langdon had seen the symbol portrayed both ways.

“And the inverse,” she said, drawing again upon her palm, “is the chalice, which represents the feminine.”

“Correct,” Langdon said….

… Marie turned on the lights and pointed….

“There you are, Mr. Langdon.  The blade and chalice.”….

“But that’s the Star of Dav–“

Langdon stopped short, mute with amazement as it dawned on him.

The blade and chalice.

Fused as one.

The Star of David… the perfect union of male and female… Solomon’s Seal… marking the Holy of Holies, where the male and female deities — Yahweh and Shekinah — were thought to dwell.

Related Log24 material —

May 25, 2003:
Star Wars.
 


Concluding remark of April 15, 2005:
For a more serious approach to portraits of
redheads, see Chen Yifei’s work.

The image “http://www.log24.com/log/pix05/050415-TheDuet-ChenYifei.jpg” cannot be displayed, because it contains errors.

Thursday, March 4, 2004

Thursday March 4, 2004

Filed under: General — Tags: , — m759 @ 1:44 pm

ZZ

Mit Zeichen und Zahlen
vermessen wir Himmel und Erde
schwarz
auf weiss
schaffen wir neue Welten
oder gar Universen
 
With numbers and names
we measure heaven and earth
black
on white
we create new worlds
and universes
 
— from “Zahlen und Zeichen,
  Wörter und Worte”
 
 

“Numbers and Names,
Wording and Words”

by Eugen Jost

English translation by Catherine Schelbert

Alphabets

From time to time we take our pen in hand
And scribble symbols on a blank white sheet
Their meaning is at everyone’s command;
It is a game whose rules are nice and neat.

But if a savage or a moon-man came
And found a page, a furrowed runic field,
And curiously studied lines and frame:
How strange would be the world that they revealed.
A magic gallery of oddities.
He would see A and B as man and beast,
As moving tongues or arms or legs or eyes,
Now slow, now rushing, all constraint released,
Like prints of ravens’ feet upon the snow.
He’d hop about with them, fly to and fro,
And see a thousand worlds of might-have-been
Hidden within the black and frozen symbols,
Beneath the ornate strokes, the thick and thin.
He’d see the way love burns and anguish trembles,
He’d wonder, laugh, shake with fear and weep
Because beyond this cipher’s cross-barred keep
He’d see the world in all its aimless passion,
Diminished, dwarfed, and spellbound in the symbols,
And rigorously marching prisoner-fashion.
He’d think: each sign all others so resembles
That love of life and death, or lust and anguish,
Are simply twins whom no one can distinguish …
Until at last the savage with a sound
Of mortal terror lights and stirs a fire,
Chants and beats his brow against the ground
And consecrates the writing to his pyre.
Perhaps before his consciousness is drowned
In slumber there will come to him some sense
Of how this world of magic fraudulence,
This horror utterly behind endurance,
Has vanished as if it had never been.
He’ll sigh, and smile, and feel all right again.

— Hermann Hesse (1943),
Buchstaben” from Das Glasperlenspiel,
translated by Richard and Clara Winston

See also the previous entry,
on the dream
of El Pato-lógico.

Friday, February 20, 2004

Friday February 20, 2004

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

The Da Vinci Code
and
Symbology at Harvard

The protagonist of the recent bestseller The Da Vinci Code is Robert Langdon, "a professor of Religious Symbology at Harvard University."  A prominent part in the novel is played by the well-known Catholic organization Opus Dei.  Less well known (indeed, like Langdon, nonexistent) is the academic discipline of "symbology."  (For related disciplines that do exist, click here.) What might a course in this subject at Harvard be like?

Harvard Crimson, April 10, 2003:

While Opus Dei members said that they do not refer to their practices of recruitment as "fishing," the Work’s founder does describe the process of what he calls "winning new apostles" with an aquatic metaphor.

Point #978 of The Way invokes a passage in the New Testament in which Jesus tells Peter that he will make him a "fisher of men." The point reads:

" ‘Follow me, and I will make you into fishers of men.’ Not without reason does our Lord use these words: men—like fish—have to be caught by the head. What evangelical depth there is in the ‘intellectual apostolate!’ ”

IMAGE- Escher, 'Fishes and Scales'

IMAGE- Cullinane, 'Invariance'

Exercise for Symbology 101:

Describe the symmetry
in each of the pictures above.
Show that the second picture
retains its underlying structural
symmetry under a group of
322,560 transformations.

Having reviewed yesterday's notes
on Gombrich, Gadamer, and Panofsky,
discuss the astrological meaning of
the above symbols in light of
today's date, February 20.

Extra credit:

Relate the above astrological
symbolism to the four-diamond
symbol in Jung's Aion.

Happy metaphors!

Robert Langdon

Tuesday, March 4, 2003

Tuesday March 4, 2003

Filed under: General — Tags: — m759 @ 9:25 pm

Fearful Symmetry

I just Googled this phrase and found the following site, which turns out to be related to my previous entry on the Bead Game and the death of John P. Thompson.

Fearful Symmetry:
The Music Master’s Lecture
,

by Daniel d’Quincy.

This in turn links to an excerpt from The Glass Bead Game that includes this passage: 

“I suddenly realized that in the language, or at any rate in the spirit of the Glass Bead Game, everything actually was all-meaningful, that every symbol and combination of symbols led not hither and yon, not to single examples, experiments, and proofs, but into the center, the mystery and innermost heart of the world, into primal knowledge. Every transition from major to minor in a sonata, every transformation of a myth or a religious cult, every classical or artistic formulation was, I realized in that flashing moment, if seen with a truly meditative mind, nothing but a direct route into the interior of the cosmic mystery, where in the alternation between inhaling and exhaling, between heaven and earth, between Yin and Yang, holiness is forever being created.”

It is very easy to get dangerously confused about holiness, but here are some relevant quotes:

“You will have to allow me to digress a bit in order to bring ourselves to a sufficiently elevated perspective… I warn you, it will require an attitude of playfulness on your part. Our approach will aim more at sincerity than seriousness. The attitude I’m aiming at is best expressed, I suppose, in the playing of a unique game, known by its German name as Das Glasperlenspiel, and which we may translate as the Glass Bead Game.”

— Daniel d’Quincy, Fearful Symmetry 

“7:11”

— God himself said this, at least according to the previous entry and to my Jan. 28 entry, State of the Communion.

“Seven is heaven.”

— See my web page Eight is a Gate.

“An excellent example of a ‘universal’ in the sense of Charles Williams, Jung, or Plato is Hexagram 11 in China’s 3,000-year-old classic, the I Ching:

Hexagram 11

‘Heaven and earth unite:
 the image of PEACE.’ 
 (Wilhelm/Baynes translation,
 Princeton University Press, 1967)” 

— S. H. Cullinane, Plato, Pegasus, and the Evening Star

Thus we may associate the numbers 7 and 11 with the notions of heaven and peace; for a somewhat darker association of the time 7:11 with Kali as Time the Destroyer, see my last entry and also my previous entries

Fat Man and Dancing Girl (Feb. 18, 2003), and 

Time and Eternity (Feb. 1, 2003).

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