Log24

Thursday, August 6, 2020

Structure and Mutability . . .

Filed under: General — Tags: , — m759 @ 8:33 PM

Continues.

See a Log24 search for Beadgame Space.

This  post might be regarded as a sort of “checked cell”
for the above concepts listed as tags . . .

Related material from a Log24 search for Structuralism

IMAGE- Stella Octangula and Claude Levi-Strauss

After Personalities . . . Principles

Filed under: General — m759 @ 1:10 PM

In memory of New York personality Pete Hamill ,
who reportedly died yesterday —

Seven years ago yesterday —

The Diamond Theorem, arXiv, 5 August 2013

In memory of another New York  personality, a parking-garage mogul
who reportedly died on August 9, 2005 —

Icon Parking  posts and . . .

Beadgame Space

Monday, May 11, 2020

Finite Geometry at Harvard

Filed under: General — m759 @ 2:34 PM

Now at https://dataverse.harvard.edu

Archived copy of finitegeometry.org/sc .

For some related remarks in a more literary vein,
see posts now tagged Beadgame Space.

Saturday, April 7, 2018

Sides

Filed under: General,Geometry — Tags: , — m759 @ 11:47 AM

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."

Monday, September 5, 2016

Structural Study

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

The Lévi-Strauss “canonic formula” of myth in its original 1955 context,
described as that of permutation groups 

The 1955 Levi-Strauss 'canonic formula' in its original context of permutation groups

Related material in this  journal —

Dueling Formulas and Symmetry.

Tuesday, May 3, 2016

Symmetry

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

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.

Saturday, February 22, 2014

Grundlagenkrise*

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

The title was suggested by a 1921 article
by Hermann Weyl and by a review* of
a more recent publication —

The above Harvard Gazette  piece on Davos is
from St. Ursula’s Day, 2010. See also this  journal
on that date.

See as well a Log24 search for Davos.

A more interesting piece by Peter E. Gordon
(author of the above Davos book) is his review
of Charles Taylor’s A Secular Age .
The review is titled

The Place of the Sacred
in the Absence of God
.”

(The place of the sacred is not, perhaps, Davos,
but a more abstract location.)

* Grundlagenkrise  was a tag for a Jan. 13, 2011,
review in The New Republic  of Gordon’s
book on Cassirer and Heidegger at Davos.

Friday, March 4, 2011

Ageometretos Medeis Eisito*

Filed under: General,Geometry — Tags: — m759 @ 7:11 PM

Your mission, should you choose to accept it…

IMAGE- Future Bead Game Master Joseph Knecht's mission to a Benedictine monastery

See also “Mapping Music” from Harvard Magazine , Jan.-Feb. 2007—

“Life inside an orbifold is a non-Euclidean world”

— as well as the cover story “The Shape of Music” from Princeton Alumni Weekly ,
Feb. 9, 2011, and “Bead Game” + music in this  journal (click, then scroll down).
Those impressed by the phrase “non-Euclidean” may also enjoy
Non-Euclidean Blocks and Pilate Goes to Kindergarten.

The “Bead Game” + music search above includes, notably, a passage describing a
sort of non-Euclidean abacus in the classic 1943 story “Mimsy Were the Borogoves.”
For a visually related experience, see the video “Chord Geometries Demo: Chopin
on a Mobius Strip” at a music.princeton.edu web page.

* Motto of the American Mathematical Society, said to be also the motto of Plato’s Academy.

Monday, February 21, 2011

The Abacus Conundrum*

Filed under: General,Geometry — Tags: , , , — m759 @ 2:02 PM

From Das Glasperlenspiel  (Hermann Hesse, 1943) —

“Bastian Perrot… 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.… …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.”

From “Mimsy Were the Borogoves” (Lewis Padgett, 1943)—

…”Paradine looked up. He frowned, staring. What in—
…”Is that an abacus?” he asked. “Let’s see it, please.”
…Somewhat unwillingly Scott brought the gadget across to his father’s chair. Paradine blinked. The “abacus,” unfolded, was more than a foot square, composed of thin,  rigid wires that interlocked here and there. On the wires the colored beads were strung. They could be slid back and forth, and from one support to another, even at the points of jointure. But— a pierced bead couldn’t cross interlocking  wires—
…So, apparently, they weren’t pierced. Paradine looked closer. Each small sphere had a deep groove running around it, so that it could be revolved and slid along the wire at the same time. Paradine tried to pull one free. It clung as though magnetically. Iron? It looked more like plastic.
…The framework itself— Paradine wasn’t a mathematician. But the angles formed by the wires were vaguely shocking, in their ridiculous lack of Euclidean logic. They were a maze. Perhaps that’s what the gadget was— a puzzle.
…”Where’d you get this?”
…”Uncle Harry gave it to me,” Scott said on the spur of the moment. “Last Sunday, when he came over.” Uncle Harry was out of town, a circumstance Scott well knew. At the age of seven, a boy soon learns that the vagaries of adults follow a certain definite pattern, and that they are fussy about the donors of gifts. Moreover, Uncle Harry would not return for several weeks; the expiration of that period was unimaginable to Scott, or, at least, the fact that his lie would ultimately be discovered meant less to him than the advantages of being allowed to keep the toy.
…Paradine found himself growing slightly confused as he attempted to manipulate the beads. The angles were vaguely illogical. It was like a puzzle. This red bead, if slid along this  wire to that  junction, should reach there— but it didn’t. A maze, odd, but no doubt instructive. Paradine had a well-founded feeling that he’d have no patience with the thing himself.
…Scott did, however, retiring to a corner and sliding beads around with much fumbling and grunting. The beads did  sting, when Scott chose the wrong ones or tried to slide them in the wrong direction. At last he crowed exultantly.
…”I did it, dad!”
…””Eh? What? Let’s see.” The device looked exactly the same to Paradine, but Scott pointed and beamed.
…”I made it disappear.”
…”It’s still there.”
…”That blue bead. It’s gone now.”
…Paradine didn’t believe that, so he merely snorted. Scott puzzled over the framework again. He experimented. This time there were no shocks, even slight. The abacus had showed him the correct method. Now it was up to him to do it on his own. The bizarre angles of the wires seemed a little less confusing now, somehow.
…It was a most instructive toy—
…It worked, Scott thought, rather like the crystal cube.

* Title thanks to Saturday Night Live  (Dec. 4-5, 2010).

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

Wednesday, June 30, 2010

Field Dream

Filed under: General,Geometry — Tags: , — m759 @ 10:23 AM

In memory of Wu Guanzhong, Chinese artist who died in Beijing on Friday

Image-- The Dream of the Expanded Field

“Once Knecht confessed to his teacher that he wished to learn enough to be able to incorporate the system of the I Ching into the Glass Bead Game.  Elder Brother laughed.  ‘Go ahead and try,’ he exclaimed.  ‘You’ll see how it turns out.  Anyone can create a pretty little bamboo garden in the world.  But I doubt that the gardener would succeed in incorporating the world in his bamboo grove.'”

— Hermann Hesse, The Glass Bead Game, translated by Richard and Clara Winston

“The Chinese painter Wu Tao-tzu was famous because he could paint nature in a unique realistic way that was able to deceive all who viewed the picture. At the end of his life he painted his last work and invited all his friends and admirers to its presentation. They saw a wonderful landscape with a romantic path, starting in the foreground between flowers and moving through meadows to high mountains in the background, where it disappeared in an evening fog. He explained that this picture summed up all his life’s work and at the end of his short talk he jumped into the painting and onto the path, walked to the background and disappeared forever.”

Jürgen Teichmann. Teichmann notes that “the German poet Hermann Hesse tells a variation of this anecdote, according to his own personal view, as found in his ‘Kurzgefasster Lebenslauf,’ 1925.”

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

Saturday, January 19, 2008

Saturday January 19, 2008

Filed under: General — Tags: — m759 @ 7:00 AM
In Memory of
Bobby Fischer

Edward Rothstein has a piece on Bobby Fischer in today’s New York Times.  The Rothstein opening:

“There may be only three human activities in which miraculous accomplishment is possible before adulthood: mathematics, music and chess.”

This echoes the opening of a classic George Steiner essay (The New Yorker, Sept. 7, 1968):

“There are three intellectual pursuits, and, so far as I am aware, only three, in which human beings have performed major feats before the age of puberty. They are music, mathematics, and chess.”

— “A Death of Kings,” reprinted in George Steiner: A Reader, Oxford University Press, 1984, pp. 171-178.

Despite its promising (if unoriginal) opening, the New York Times piece is mainly an attack on Fischer’s anti-Jewish stance.  Rothstein actually has little of interest to say about what he calls the “glass-bead games” of music, mathematics, and chess. For a better-written piece on chess and madness, see Charles Krauthammer’s 2005 essay in TIME. The feuilletons of Rothstein and Krauthammer do not, of course, come close to the genuinely bead-game-like writing of Steiner.

Related material on
chess and religion:
Magical Thinking
(December 7th, 2005)

Sunday, November 4, 2007

Sunday November 4, 2007

Filed under: General,Geometry — Tags: — m759 @ 3:00 AM

Talking of Michelangelo:

High Concept

On this date in 1948, T. S. Eliot
won the Nobel Prize in Literature.

The 4x4 square

 Non ha l’ottimo artista in se alcun concetto,
Ch’un marmo solo in se non circoscriva
Col suo soverchio; e solo a quello arriva
La man che ubbidisce all’intelletto.
(The best artist has in himself no concept
in a single block of marble not contained;
only the hand obeying mind will find it.)
— Michelangelo, as quoted
by Erwin Panofsky in

Idea: A Concept in Art Theory

Monday, May 28, 2007

Monday May 28, 2007

Filed under: General,Geometry — Tags: , — m759 @ 5:00 PM
Space-Time

and a Finite Model

Notes by Steven H. Cullinane
May 28, 2007

Part I: A Model of Space-Time

The following paper includes a figure illustrating Penrose’s model of  “complexified, compactified Minkowski space-time as the Klein quadric in complex projective 5-space.”
 
The image “http://www.log24.com/log/pix07/070528-Twistor.jpg” cannot be displayed, because it contains errors.

Click on picture to enlarge.

For some background on the Klein quadric and space-time, see Roger Penrose, “On the Origins of Twistor Theory,” from Gravitation and Geometry: A Volume in Honor of Ivor Robinson, Bibliopolis, 1987.


Part II: A Corresponding Finite Model

 

The Klein quadric also occurs in a finite model of projective 5-space.  See a 1910 paper:

G. M. Conwell, The 3-space PG(3,2) and its group, Ann. of Math. 11, 60-76.

Conwell discusses the quadric, and the related Klein correspondence, in detail.  This is noted in a more recent paper by Philippe Cara:

The image “http://www.log24.com/log/pix07/070528-Quadric.jpg” cannot be displayed, because it contains errors.

 

As Cara goes on to explain, the Klein correspondence underlies Conwell’s discussion of eight heptads.  These play an important role in another correspondence, illustrated in the Miracle Octad Generator of R. T. Curtis, that may be used to picture actions of the large Mathieu group M24.


Related material:

The projective space PG(5,2), home of the Klein quadric in the finite model, may be viewed as the set of 64 points of the affine space AG(6,2), minus the origin.

The 64 points of this affine space may in turn be viewed as the 64 hexagrams of the Classic of Transformation, China’s I Ching.

There is a natural correspondence between the 64 hexagrams and the 64 subcubes of a 4x4x4 cube.  This correspondence leads to a natural way to generate the affine group AGL(6,2).  This may in turn be viewed as a group of over a trillion natural transformations of the 64 hexagrams.

Geometry of the I Ching.
“Once Knecht confessed to his teacher that he wished to learn enough to be able to incorporate the system of the I Ching into the Glass Bead Game.  Elder Brother laughed.  ‘Go ahead and try,’ he exclaimed.  ‘You’ll see how it turns out.  Anyone can create a pretty little bamboo garden in the world.  But I doubt that the gardener would succeed in incorporating the world in his bamboo grove.'”
— Hermann Hesse, The Glass Bead Game,
  translated by Richard and Clara Winston

Thursday, July 20, 2006

Thursday July 20, 2006

Filed under: General — Tags: , , — m759 @ 2:00 AM

Bead Game

Those who clicked on Rieff’s concept in the previous entry will know about the book that Rieff titled Sacred Order/Social Order: My Life among the Deathworks.

That entry, from Tuesday, July 18, was titled “Sacred Order,” and gave as an example the following figure:

The image “http://www.log24.com/log/pix06A/060604-Roots.jpg” cannot be displayed, because it contains errors.
(Based on Weyl’s Symmetry)

For the use of this same figure to represent a theatrical concept–

“It’s like stringing beads on a necklace. By the time the play ends, you have the whole necklace.”

— see Ursprache Revisited (June 9, 2006).

Of course, the figure also includes a cross– or “deathwork”– of sorts.  These incidental social properties of the figure (which is purely mathematical in origin) make it a suitable memorial for a theatre critic who died on the date of the previous entry– July 18– and for whom the American Theatre Wing’s design awards, the Henry Hewes Awards, are named.

“The annual awards honor designers… recognizing not only the traditional design categories of sets, costumes and lighting, but also ‘Notable Effects,’ which encompasses sound, music, video, puppets and other creative elements.” —BroadwayWorld.com

For more on life among the deathworks, see an excellent review of the Rieff book mentioned above.

 

Tuesday, July 18, 2006

Tuesday July 18, 2006

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

Sacred Order

In memory of Philip Rieff, who died on July 1, 2006:

The image “http://www.log24.com/log/pix06A/060604-Roots.jpg” cannot be displayed, because it contains errors.

Related material:

The image ?http://www.log24.com/theory/images/GF64-63cycleA495.gif? cannot be displayed, because it contains errors.

and

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

For details, see the
five Log24 entries ending
on the morning of
Midsummer Day, 2006.

Thanks to University Diaries for pointing out the essay on Rieff.
 
That essay says Rieff had "a dense, knotty, ironic style designed to warn off impatient readers. You had to unpack his aphorisms carefully. And this took a while. As a result, his thinking had a time-release effect." Good for him.  For a related essay (time-release effect unknown), see Hitler's Still Point: A Hate Speech for Harvard.

Tuesday, August 9, 2005

Tuesday August 9, 2005

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

Kaleidoscope

A new web page simplifies the Diamond 16 Puzzle and relates the resulting “kaleidoscope” to Hesse’s Bead Game.

Sunday, September 7, 2003

Sunday September 7, 2003

Filed under: General — Tags: , — m759 @ 11:11 PM

Horse Sense

Mathematicians are familiar with the emblem of Springer Verlag, the principal publisher of higher mathematics.

Ferdinand Springer, son of Julius Springer, founder of Springer Verlag, “was a passionate chess player and published a number of books on the subject. In 1881 this personal hobby and the name Springer led the company to adopt the knight in chess (in German, Springer) as its colophon.”

Hermann Hesse on a certain sort of serenity:

“I would like to say something more to you about cheerful serenity, the serenity of the stars and of the mind…. neither frivolity nor complacency; it is supreme insight and love, affirmation of all reality, alertness on the brink of all depths and abysses; it is a virtue of saints and of knights; it is indestructible and only increases with age and nearness to death. It is the secret of beauty and the real substance of all art.”

— From The Glass Bead Game

A saint and a knight, Jeanne d’Arc, was memorably portrayed by Milla Jovovich in The Messenger.

(Jovovich seems fated to play more-than-human characters in religious epics; see The Fifth Element.)

Another Springer, related to horses and to the accusation of witchcraft faced by Jeanne d’Arc, is Nancy Springer, the author of

The Hex Witch of Seldom.

Springer has written a number of books about horses, as well as other topics.

All of the above…. especially the parts having to do with mathematics and horses… was prompted by my redrawing today of a horse-shape within mathematics.  See my entry The Eight of April 4, 2003, and the horse-figure redrawn at right below.

 



Springer
Verlag



The
Messenger



A
7-Cycle

Believers in the story theory of truth may wish to relate the gifts of Jeanne d’Arc and of the girl in The Hex Witch of Seldom to the legend of Pegasus.  See, for instance,

Plato, Pegasus, and the Evening Star.

For another connection between mathematics and horses, see Sangaku.

Friday, August 1, 2003

Friday August 1, 2003

Filed under: General — Tags: — m759 @ 1:40 PM

Fearful Meditation

Ray Price - Time TIME, Aug. 4, 2003

Ray Price — Time

“The Max D. Barnes-penned title track, with its stark-reality lyrics, is nothing short of haunting: ‘Time is a weapon, it’s cold and it’s cruel; It knows no religion and plays by no rules; Time has no conscience when it’s all said and done; Like a beast in the jungle that devours its young.’ That’s so good, it hurts! Price’s still-amazing vocals are simply the chilling icing on the cake.”

— Lisa Berg, NashvilleCountry.com

O fearful meditation! Where, alack,
Shall time’s best jewel from time’s chest lie hid?

— Shakespeare, Sonnet 65

Clue: click here.  This in turn leads to my March 4 entry Fearful Symmetry, which contains the following:

“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….”

— Hermann Hesse, The Glass Bead Game

“How strange the change from major to minor….”

— Cole Porter, “Every Time We Say Goodbye

Wednesday, July 23, 2003

Wednesday July 23, 2003

Filed under: General,Geometry — Tags: , — m759 @ 4:17 PM

Being Pascal Sauvage

Pascal

“Voilà ce que je sais par une longue expérience de toutes sortes de livres et de personnes. Et sur cela je fais le même jugement de ceux qui disent que les géomètres ne leur donnent rien de nouveau par ces règles, parce qu’ ils les avaient en effet, mais confondues parmi une multitude d’ autres inutiles ou fausses dont ils ne pouvaient pas les discerner, que de ceux qui cherchant un diamant de grand prix

Diamant

parmi un grand nombre de faux, mais qu’ ils n’ en sauraient pas distinguer, se vanteraient, en les tenant tous ensemble, de posséder le véritable aussi bien que celui qui, sans s’ arrêter à ce vil amas, porte la main sur la pierre choisie que l’ on recherche, et pour laquelle on ne jetait pas tout le reste.”

— Blaise Pascal, De l’Esprit Géométrique

La Pensée Sauvage

“….the crowning image of the kaleido­scope, lavishly analogized to the mythwork in a three-hundred-word iconic apotheosis that served to put the wraps on the sustained personification of “la pensée sauvage” in the figure of the bricoleur, in an argument developed across two chapters and some twenty pages in his [Claude Lévi-Strauss’s] most famous book….”

— Robert de Marrais in
Catastrophes, Kaleidoscopes,
String Quartets:
Deploying the Glass Bead Game


Pascal Sauvage

Chiasmus

For more on pensée sauvage, see

“Claude Lévi-Strauss,

Chiasmus

and the Ethnographic Journey.”

Friday, March 21, 2003

Friday March 21, 2003

Filed under: General — Tags: — m759 @ 9:29 AM

ART WARS:

Readings for Bach’s Birthday

Larry J. Solomon:

 

Symmetry as a Compositional Determinant,
Chapter VIII: New Transformations

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

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

Steven H. Cullinane:

Orthogonal Latin Squares as Skew Lines, and

Map Systems

Dorothy Sayers:

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

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

Edward Sapir:

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

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

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