Wednesday, April 8, 2009
“Since 1963, when Pynchon’s first novel, V., came out, the writer– widely considered America’s most important novelist since World War II– has become an almost mythical figure, a kind of cross between the Nutty Professor (Jerry Lewis’s) and Caine in Kung Fu.”
— Nancy Jo Sales in the November 11, 1996, issue of New York Magazine
A Cross Between
(Click on images for their
source in past entries.)
Comments Off on Wednesday April 8, 2009
Saturday, April 4, 2009
Steiner Systems
"Music, mathematics, and chess are in vital respects dynamic acts of location. Symbolic counters are arranged in significant rows. Solutions, be they of a discord, of an algebraic equation, or of a positional impasse, are achieved by a regrouping, by a sequential reordering of individual units and unit-clusters (notes, integers, rooks or pawns). The child-master, like his adult counterpart, is able to visualize in an instantaneous yet preternaturally confident way how the thing should look several moves hence. He sees the logical, the necessary harmonic and melodic argument as it arises out of an initial key relation or the preliminary fragments of a theme. He knows the order, the appropriate dimension, of the sum or geometric figure before he has performed the intervening steps. He announces mate in six because the victorious end position, the maximally efficient configuration of his pieces on the board, lies somehow 'out there' in graphic, inexplicably clear sight of his mind…."
"… in some autistic enchantment,
pure as one of Bach's inverted canons or Euler's formula for polyhedra."
— George Steiner, "A Death of Kings," in The New Yorker, issue dated Sept. 7, 1968
Comments Off on Saturday April 4, 2009
Annual Tribute to
The Eight
Other knight figures:
Click on the SpringerLink
knight for a free copy
(pdf, 1.2 mb) of
the following paper
dealing with the geometry
underlying the R.T. Curtis
knight figures above:
Context:
Literature and Chess and
Sporadic Group References
Details:
| Adapted (for HTML) from the opening paragraphs of the above paper, W. Jonsson's 1970 "On the Mathieu Groups M22, M23, M24…"–
"[A]… uniqueness proof is offered here based upon a detailed knowledge of the geometric aspects of the elementary abelian group of order 16 together with a knowledge of the geometries associated with certain subgroups of its automorphism group. This construction was motivated by a question posed by D.R. Hughes and by the discussion Edge [5] (see also Conwell [4]) gives of certain isomorphisms between classical groups, namely
PGL(4,2)~PSL(4,2)~SL(4,2)~A8,
PSp(4,2)~Sp(4,2)~S6,
where A8 is the alternating group on eight symbols, S6 the symmetric group on six symbols, Sp(4,2) and PSp(4,2) the symplectic and projective symplectic groups in four variables over the field GF(2) of two elements, [and] PGL, PSL and SL are the projective linear, projective special linear and special linear groups (see for example [7], Kapitel II).
The symplectic group PSp(4,2) is the group of collineations of the three dimensional projective space PG(3,2) over GF(2) which commute with a fixed null polarity tau…."
References
4. Conwell, George M.: The three space PG(3,2) and its group. Ann. of Math. (2) 11, 60-76 (1910).
5. Edge, W.L.: The geometry of the linear fractional group LF(4,2). Proc. London Math. Soc. (3) 4, 317-342 (1954).
7. Huppert, B.: Endliche Gruppen I. Berlin-Heidelberg-New York: Springer 1967. |
Comments Off on Saturday April 4, 2009
Friday, April 3, 2009
“Philosophers ponder the idea of identity: what it is to give something a name on Monday and have it respond to that name on Friday….”
— Bernard Holland in The New York Times of Monday, May 20, 1996
Yesterday’s afternoon entry cited philosopher John Holbo on chess. This, together with Holland’s remark above and Monday’s entries on Zizek, suggests…
Holbo on Zizek(
pdf, 11 pages)
In this excellent analysis,
Holbo quotes Kierkegaard:
“… the knight of faith
‘has the pain of being unable to
make himself intelligible to others'”
(Kierkegaard, Fear and Trembling)
For some material that may serve to illustrate Kierkegaard’s remark, see Log24 on
Twelfth Night and
Epiphany this year.
“… There was a problem laid out on the board, a six-mover. I couldn’t solve it, like a lot of my problems. I reached down and moved a knight…. I looked down at the chessboard. The move with the knight was wrong. I put it back where I had moved it from. Knights had no meaning in this game. It wasn’t a game for knights.”
— Raymond Chandler, The Big Sleep
Context: the five entries
ending at 9:26 AM
on March 10, 2009…
and, for Kierkegaard,
Diamonds Are Forever.
Comments Off on Friday April 3, 2009
Thursday, April 2, 2009
Transformative
Hermeneutics
In memory of
physics historian
Martin J. Klein,
(June 25, 1924-
March 28, 2009)
"… in physics itself, there was what appeared, briefly, to be an ending, which then very quickly gave way to a new beginning: The quest for the ultimate building-blocks of the universe had been taken down to the molecular level in nineteenth-century kinetic theory… and finally to the nuclear level in the second and third decades of the twentieth century. For a moment in the 1920s the quest appeared to have ended…. However… this paradise turned out to be, if not exactly a fool's paradise, then perhaps an Eden lost."
— No Truth Except in the Details: Essays in Honor of Martin J. Klein, introduction by A.J. Kox and Daniel Siegel, June 25, 1994
New York Times obituary dated April 1, 2009:
"Martin J. Klein, a historian of modern physics…. died Saturday, [March 28, 2009] in Chapel Hill, N.C. He was 84 and lived in Chapel Hill."
Klein edited, among other things, Paul Ehrenfest: Collected Scientific Papers (publ. by North-Holland, Amsterdam, 1959).
Related material:
"Almost every famous chess game
is a well-wrought urn
in Cleanth Brooks’ sense."
— John Holbo,
Now We See
Wherein Lies the Pleasure
"The entire sequence of moves in these… chapters reminds one– or should remind one– of a certain type of chess problem where the point is not merely the finding of a mate in so many moves, but what is termed 'retrograde analysis'…."
— Vladimir Nabokov, foreword to The Defense
Comments Off on Thursday April 2, 2009
Sunday, March 29, 2009
Getting All
the Meaning In
Webpage heading for the
2009 meeting of the
American Comparative
Literature Association:
The mysterious symbols on
the above map suggest the
following reflections:
From A Cure of the Mind: The Poetics of Wallace Stevens, by Theodore Sampson, published by Black Rose Books Ltd., 2000–
Page x:
"… if what he calls 'the spirit's alchemicana' (CP [Collected Poems] 471) addresses itself to the irrational element in poetry, to what extent is such an element dominant in his theory and practice of poetry, and therefore in what way is Stevens' intricate verbal music dependent on his irrational use of language– a 'pure rhetoric of a language without words?' (CP 374)?"
Related material:
| From "'When Novelists Become Cubists:' The Prose Ideograms of Guy Davenport," by Andre Furlani:
Laurence Zachar argues that Davenport's writing is situated "aux frontieres intergeneriques" where manifold modes are brought into concord: "L'etonnant chez Davenport est la facon don't ce materiau qui parait l'incarnation meme du chaos– hermetique, enigmatique, obscur, avec son tropplein de references– se revele en fait etre construit, ordonne, structure. Plus l'on s'y plonge, et plus l'on distingue de cohesion dans le texte." 'What astonishes in Davenport is the way in which material that seems the very incarnation of chaos– hermetic, enigmatic, obscure, with its proliferation of allusions– in fact reveals itself to be constructed, organized, structured. The more one immerses oneself in them the more one discerns the texts' cohesion.' (62).
Davenport also works along the intergeneric border between text and graphic, for he illustrates many of his texts. (1) "The prime use of words is for imagery: my writing is drawing," he states in an interview (Hoeppfner 123). Visual imagery is not subordinated to writing in Davenport, who draws on the assemblage practice of superimposing image and writing. "I trust the image; my business is to get it onto the page," he writes in the essay "Ernst Machs Max Ernst." "A page, which I think of as a picture, is essentially a texture of images. […] The text of a story is therefore a continuous graph, kin to the imagist poem, to a collage (Ernst, Willi Baumeister, El Lissitzky), a page of Pound, a Brakhage film" (Geography 374-75).
Note:
(1.) Davenport is an illustrator of books (such as Hugh Kenner's The Stoic Comedians and The Counterfeiters) and journals (such as The Kenyon Review, Parnassus, and Paideuma). His art is the subject of Erik Anderson Reece's monograph, A Balance of Quinces, which reveals the inseparable relationship between Davenport's literary and pictorial work.
References:
Davenport, Guy. The Geography of the Imagination. San Francisco: North Point Press, 1981. Rpt. New York: Pantheon, 1992.
Hoepffner, Bernard. "Pleasant Hill: An Interview with Guy Davenport." Conjunctions 24 (1995): 118-24.
Reece, Erik Anderson. A Balance of Quinces: The Paintings and Drawings of Guy Davenport. New York: New Directions, 1996.
Zachar, Laurence. "Guy Davenport: Une Mosaique du genres." Recherches Anglaises et Nord-Americaines 21 (1994): 51-63.
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"… when novelists become Cubists; that is, when they see the possibilities of making a hieroglyph, a coherent symbol, an ideogram of the total work. A symbol comes into being when an artist sees that it is the only way to get all the meaning in."
— Guy Davenport, The Geography of the Imagination
See also last night's
commentary on the
following symbols:
Comments Off on Sunday March 29, 2009
Saturday, March 28, 2009
The Rest
of the Story
Today's previous entry discussed the hermeneutics of the midday NY and PA lottery numbers.
The rest of the story:
Lotteries
on Reba's
birthday,
2009 |
Pennsylvania
(No revelation) |
New York
(Revelation) |
Mid-day
(No belief) |
No belief,
no revelation
726 |
Revelation
without belief
378 |
Evening
(Belief) |
Belief without
revelation
006 |
Belief and
revelation
091 |
Interpretations of the evening numbers–
The PA evening number, 006, may be viewed as a followup to the PA midday 726 (or 7/26, the birthday of Kate Beckinsale and Carl Jung). Here 006 is the prestigious "00" number assigned to Beckinsale.
Will: Do you like apples?
Clark: Yeah.
Will: Well, I got her number.
How do you like them apples?
— "Good Will Hunting"
The NY evening number, 091, may be viewed as a followup to the NY midday 378 (the number of pages in The Innermost Kernel by Suzanne Gieser, published by Springer, 2005)–
Page 91: The entire page is devoted to the title of the book's Part 3– "The Copenhagen School and Psychology"–
The next page begins: "With the crisis of physics, interest in epistemological and psychological questions grew among many theoretical physicists. This interest was particularly marked in the circle around Niels Bohr."
A particularly
marked circle
from March 15:
The circle above is
marked with a version of
the classic Chinese symbol
adopted as a personal emblem
by Danish physicist Niels Bohr,
leader of the Copenhagen School.
"Two things of opposite natures seem to depend
On one another, as a man depends
On a woman, day on night, the imagined
On the real. This is the origin of change.
Winter and spring, cold copulars, embrace
And forth the particulars of rapture come."
-- Wallace Stevens,
"Notes Toward a Supreme Fiction,"
Canto IV of "It Must Change"
The square above is marked
with a graphic design
related to the four-diamond
figure of Jung's Aion.
Comments Off on Saturday March 28, 2009
Saturday, March 21, 2009
Counters in Rows
"Music, mathematics, and chess are in vital respects dynamic acts of location. Symbolic counters are arranged in significant rows. Solutions, be they of a discord, of an algebraic equation, or of a positional impasse, are achieved by a regrouping, by a sequential reordering of individual units and unit-clusters (notes, integers, rooks or pawns)."
— George Steiner
(See March 10, "Language Game.")
Comments Off on Saturday March 21, 2009
Thursday, March 19, 2009
Two-Face
[Note: Janus is Roman, not Greek, and
the photo is from one “Fubar Obfusco”] 
Click on image for details.
From January 8:
Religion and Narrative, continued:
| A Public Square
In memory of
Richard John Neuhaus,
who died today at 72:
“It seems, as one becomes older,
That the past has another pattern,
and ceases to be a mere sequence….”
— T. S. Eliot, Four Quartets
Click on image for details.
See also The Folding.
Posted 1/8/2009 7:00 PM |
|
Context:
Notes on Mathematics and Narrative
(entries in chronological order,
March 13 through 19)
Comments Off on Thursday March 19, 2009
Tuesday, March 17, 2009
"The diagram above is from a ninth century manuscript of Apuleius' commentary on Aristotle's Perihermaneias, probably one of the oldest surviving pictures of the square."
— Edward Buckner at The Logic Museum
From the webpage "Semiotics for Beginners: Paradigmatic Analysis," by Daniel Chandler:
The Semiotic Square
"The structuralist semiotician Algirdas Greimas introduced the semiotic square (which he adapted from the 'logical square' of scholastic philosophy) as a means of analysing paired concepts more fully (Greimas 1987,* xiv, 49). The semiotic square is intended to map the logical conjunctions and disjunctions relating key semantic features in a text. Fredric Jameson notes that 'the entire mechanism… is capable of generating at least ten conceivable positions out of a rudimentary binary opposition' (in Greimas 1987,* xiv). Whilst this suggests that the possibilities for signification in a semiotic system are richer than the either/or of binary logic, but that [sic] they are nevertheless subject to 'semiotic constraints' – 'deep structures' providing basic axes of signification."
* Greimas, Algirdas (1987): On Meaning: Selected Writings in Semiotic Theory (trans. Paul J Perron & Frank H Collins). London: Frances Pinter
Another version of the semiotic square:
Krauss says that her figure "is, of course, a Klein Group."
Here is a more explicit figure representing the Klein group:
There is also the logical
diamond of opposition —
A semiotic (as opposed to logical)
diamond has been used to illustrate
remarks by Fredric Jameson,
a Marxist literary theorist:
|
"Introduction to Algirdas Greimas, Module on the Semiotic Square," by Dino Felluga at Purdue University–
The semiotic square has proven to be an influential concept not only in narrative theory but in the ideological criticism of Fredric Jameson, who uses the square as "a virtual map of conceptual closure, or better still, of the closure of ideology itself" ("Foreword"* xv). (For more on Jameson, see the [Purdue University] Jameson module on ideology.)
Greimas' schema is useful since it illustrates the full complexity of any given semantic term (seme). Greimas points out that any given seme entails its opposite or "contrary." "Life" (s1) for example is understood in relation to its contrary, "death" (s2). Rather than rest at this simple binary opposition (S), however, Greimas points out that the opposition, "life" and "death," suggests what Greimas terms a contradictory pair (-S), i.e., "not-life" (-s1) and "not-death" (-s2). We would therefore be left with the following semiotic square (Fig. 1):
As Jameson explains in the Foreword to Greimas' On Meaning, "-s 1 and -s 2"—which in this example are taken up by "not-death" and "not-life"—"are the simple negatives of the two dominant terms, but include far more than either: thus 'nonwhite' includes more than 'black,' 'nonmale' more than 'female'" (xiv); in our example, not-life would include more than merely death and not-death more than life.
* Jameson, Fredric. "Foreword." On Meaning: Selected Writings in Semiotic Theory. By Algirdas Greimas. Trans. Paul J. Perron and Frank H. Collins. Minneapolis: U of Minnesota P, 1976.
|
"The Game in the Ship cannot be approached as a job, a vocation, a career, or a recreation. To the contrary, it is Life and Death itself at work there. In the Inner Game, we call the Game Dhum Welur, the Mind of God."
— The Gameplayers of Zan, by M.A. Foster
"For every kind of vampire,
there is a kind of cross."
— Thomas Pynchon,
Gravity's Rainbow
Crosses used by semioticians
to baffle their opponents
are illustrated above.
Some other kinds of crosses,
and another kind of opponent:
Monday, July 11, 2005
Logos
for St. Benedict's Day
Click on either of the logos below for religious meditations– on the left, a Jewish meditation from the Conference of Catholic Bishops; on the right, an Aryan meditation from Stormfront.org.
 
Both logos represent different embodiments of the "story theory" of truth, as opposed to the "diamond theory" of truth. Both logos claim, in their own ways, to represent the eternal Logos of the Christian religion. I personally prefer the "diamond theory" of truth, represented by the logo below.
See also the previous entry
(below) and the entries
of 7/11, 2003.
Sunday, July 10, 2005
|
From Artemiadis's website:
|
|
1986:
|
Elected Regular Member
of the Academy of Athens
|
|
1999:
|
Vice President
of the Academy of Athens
|
|
2000:
|
President
of the Academy of Athens
|
"First of all, I'd like to
thank the Academy…"
— Remark attributed to Plato
|
Comments Off on Tuesday March 17, 2009
Sunday, March 15, 2009
Angels, Demons,
"Symbology"
"On Monday morning, 9 March, after visiting the Mayor of Rome and the Municipal Council on the Capitoline Hill, the Holy Father spoke to the Romans who gathered in the square outside the Senatorial Palace…
'… a verse by Ovid, the great Latin poet, springs to mind. In one of his elegies he encouraged the Romans of his time with these words:
"Perfer et obdura: multo graviora tulisti."
"Hold out and persist:
you have got through
far more difficult situations."
(Tristia, Liber V, Elegia XI, verse 7).'"
Note the color-interchange
symmetry of each symbol
under 180-degree rotation.
Related material:
The Illuminati Diamond:
Dan Brown's novel Angels & Demons introduced in the year 2000 the fictional academic discipline of "symbology" and a fictional Harvard professor of that discipline, Robert Langdon (named after ambigram* artist John Langdon).
Tom Hanks as Robert Langdon
A possible source for Brown's term "symbology" is a 1995 web page, "The Rotation of the Elements," by one "John Opsopaus." (Cf. Art History Club.)
"The four qualities are the key to understanding the rotation of the elements and many other applications of the symbology of the four elements." –John Opsopaus
* "…ambigrams were common in symbology…." —Angels & Demons
Comments Off on Sunday March 15, 2009
Saturday, March 14, 2009
Flowers for Barry
|
On Time
(in Mathematics and Literature)
“… I want to spend these twenty minutes savoring, and working up, the real complexity of the metaphorical relationship of time and distance– to defamiliarize it for us. And then I will give a few examples of how imaginative literature makes use of the inherent strangeness in this relationship:
Time ↔ Distance.
And finally I will offer my opinion (which I think must be everyone’s opinion) about why we derive significant– but not total– comfort from this equation.”
— Barry Mazur, March 8, 2009, draft (pdf) of talk for conference on comparative literature* |
Another version of
Mazur’s metaphor
Time ↔ Distance:
— Steven H. Cullinane,
October 8, 2003
For some context in
comparative literature,
see Time Fold
(Oct. 10, 2003)
and A Hanukkah Tale
(Dec. 22, 2008).
Related material:
Rat Psychology
yesterday.
Comments Off on Saturday March 14, 2009
Wednesday, March 11, 2009
The Times describes one of the empty rooms on exhibit as…
"… Yves Klein’s 'La spécialisation de la sensibilité à l’état matière première en sensibilité picturale stabilisée, Le Vide' ('The Specialization of Sensibility in the Raw Material State Into Stabilized Pictorial Sensibility, the Void')"
This is a mistranslation. See "An Aesthetics of Matter" (pdf), by Kiyohiko Kitamura and Tomoyuki Kitamura, pp. 85-101 in International Yearbook of Aesthetics, Volume 6, 2002—
"The exhibition «La spécialisation de la sensibilité à l’état matière-première en sensibilité picturale stabilisée», better known as «Le Vide» (The Void) was held at the Gallery Iris Clert in Paris from April 28th till May 5th, 1955." –p. 94
"… «Sensibility in the state of prime matter»… filled the emptiness." –p. 95
Kitamura and Kitamura translate matière première correctly as "prime matter" (the prima materia of the scholastic philosophers) rather than "raw material." (The phrase in French can mean either.)
Comments Off on Wednesday March 11, 2009
Monday, March 9, 2009
Humorism
"Always with a
little humor."
— Dr. Yen Lo
From Temperament: A Brief Survey
For other interpretations
of the above shape, see
The Illuminati Diamond.
from Jung's Aion:
"From the circle and quaternity motif is derived the symbol of the geometrically formed crystal and the wonder-working stone. From here analogy formation leads on to the city, castle, church, house, room, and vessel. Another variant is the wheel. The former motif emphasizes the ego’s containment in the greater dimension of the self; the latter emphasizes the rotation which also appears as a ritual circumambulation. Psychologically, it denotes concentration on and preoccupation with a centre…." –Jung,
Collected Works, Vol. 9, Part II, paragraph 352
As for rotation, see the ambigrams in Dan Brown's Angels & Demons (to appear as a film May 15) and the following figures:
Comments Off on Monday March 9, 2009
Saturday, March 7, 2009
One or Two Ideas
|
From James Joyce's A Portrait of the Artist as a Young Man:
he hearth and began to stroke his chin.
–When may we expect to have something from you on the esthetic question? he asked.
–From me! said Stephen in astonishment. I stumble on an idea once a fortnight if I am lucky.
–These questions are very profound, Mr Dedalus, said the dean. It is like looking down from the cliffs of Moher into the depths. Many go down into the depths and never come up. Only the trained diver can go down into those depths and explore them and come to the surface again.
–If you mean speculation, sir, said Stephen, I also am sure that there is no such thing as free thinking inasmuch as all thinking must be bound by its own laws.
–Ha!
–For my purpose I can work on at present by the light of one or two ideas of Aristotle and Aquinas.
–I see. I quite see your point.
|
Besides being Mondrian's birthday, today is also the dies natalis (in the birth-into-heaven sense) of St. Thomas Aquinas and, for those who believe worthy pre-Christians also enter heaven, possibly of Aristotle.
Pope Benedict XVI explained the dies natalis concept on Dec. 26, 2006:
"For believers the day of death, and even more the day of martyrdom, is not the end of all; rather, it is the 'transit' towards immortal life. It is the day of definitive birth, in Latin, dies natalis."
The Pope's remarks on that date
were in St. Peter's Square.
Pictorial version
of Hexagram 20,
Contemplation (View)
Comments Off on Saturday March 7, 2009
Friday, March 6, 2009
The Illuminati Stone
TV listing for this evening —
Family Channel, 7:30 PM:
"Harry Potter and
the Sorcerer's Stone"
In other entertainment news —
Scheduled to open May 15:
"Only gradually did I discover
what the mandala really is:
'Formation, Transformation,
Eternal Mind's eternal recreation'"
(Faust, Part Two)
— Carl Gustav Jung
Related material:
|
"For just about half a century, E.J. Holmyard's concisely-titled Alchemy has served as a literate, well-informed, and charming introduction to the history and literature of Western alchemy." —Ian Myles Slater
|
For more about this
"prime matter" (prima materia)
see The Diamond Archetype
and Holy the Firm.
Background:
Holmyard —
— and Aristotle's
On Generation and Corruption.
Comments Off on Friday March 6, 2009
Monday, March 2, 2009
Today in History – March 2
By The Associated Press
Today is Monday, March 2, the 61st day of 2009. There are 304 days left in the year.
Today's Highlight in History:
On March 2, 1939, Roman Catholic Cardinal Eugenio Pacelli was elected Pope on his 63rd birthday; he took the name Pius XII.
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Log24 on June 9, 2008—
From Gravity's Rainbow (Penguin Classics, 1995), page 563:
"He brings out the mandala he found.
'What's it mean?'
[….]
Slothrop gives him the mandala. He hopes it will work like the mantra that Enzian told him once, mba-kayere (I am passed over), mba-kayere… a spell […]. A mezuzah. Safe passage through a bad night…."
In lieu of Slothrop's mandala, here is another…
Related mandalas:
and
For further details,
click on any of the
three mandalas above.
|
Comments Off on Monday March 2, 2009
Sunday, March 1, 2009
Solomon's Cube
continued
"There is a book… called A Fellow of Trinity, one of series dealing with what is supposed to be Cambridge college life…. There are two heroes, a primary hero called Flowers, who is almost wholly good, and a secondary hero, a much weaker vessel, called Brown. Flowers and Brown find many dangers in university life, but the worst is a gambling saloon in Chesterton run by the Misses Bellenden, two fascinating but extremely wicked young ladies. Flowers survives all these troubles, is Second Wrangler and Senior Classic, and succeeds automatically to a Fellowship (as I suppose he would have done then). Brown succumbs, ruins his parents, takes to drink, is saved from delirium tremens during a thunderstorm only by the prayers of the Junior Dean, has much difficulty in obtaining even an Ordinary Degree, and ultimately becomes a missionary. The friendship is not shattered by these unhappy events, and Flowers's thoughts stray to Brown, with affectionate pity, as he drinks port and eats walnuts for the first time in Senior Combination Room."
— G. H. Hardy, A Mathematician's Apology
"The Solomon Key is the working title of an unreleased novel in progress by American author Dan Brown. The Solomon Key will be the third book involving the character of the Harvard professor Robert Langdon, of which the first two were Angels & Demons (2000) and The Da Vinci Code (2003)." — Wikipedia
"One has O+(6) ≅ S8, the symmetric group of order 8! …."
— "Siegel Modular Forms and Finite Symplectic Groups," by Francesco Dalla Piazza and Bert van Geemen, May 5, 2008, preprint.
"The complete projective group of collineations and dualities of the [projective] 3-space is shown to be of order [in modern notation] 8! …. To every transformation of the 3-space there corresponds a transformation of the [projective] 5-space. In the 5-space, there are determined 8 sets of 7 points each, 'heptads' …."
— George M. Conwell, "The 3-space PG(3, 2) and Its Group," The Annals of Mathematics, Second Series, Vol. 11, No. 2 (Jan., 1910), pp. 60-76
"It must be remarked that these 8 heptads are the key to an elegant proof…."
— Philippe Cara, "RWPRI Geometries for the Alternating Group A8," in Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference (July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis, James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97
Comments Off on Sunday March 1, 2009
Friday, February 27, 2009
Time and Chance
continued
Today's Pennsylvania lottery numbers suggest the following meditations…
Midday: Lot 497, Bloomsbury Auctions May 15, 2008– Raum und Zeit (Space and Time), by Minkowski, 1909. Background: Minkowski Space and "100 Years of Space-Time."*
Evening: 5/07, 2008, in this journal– "Forms of the Rock."
Related material:
A current competition at Harvard Graduate School of Design, "The Space of Representation," has a deadline of 8 PM tonight, February 27, 2009.
The announcement of the competition quotes the Marxist Henri Lefebvre on "the social production of space."
A related quotation by Lefebvre (cf. 2/22 2009):
"… an epoch-making event so generally ignored that we have to be reminded of it at every moment. The fact is that around 1910 a certain space was shattered… the space… of classical perspective and geometry…."
— Page 25 of The Production of Space (Blackwell Publishing, 1991)
This suggests, for those who prefer Harvard's past glories to its current state, a different Raum from the Zeit 1910.
In January 1910 Annals of Mathematics, then edited at Harvard, published George M. Conwell's "The 3-space PG(3, 2) and Its Group." This paper, while perhaps neither epoch-making nor shattering, has a certain beauty. For some background, see this journal on February 24, 2009.†
* Ending on Stephen King's birthday, 2008
† Mardi Gras
Comments Off on Friday February 27, 2009
Tuesday, February 24, 2009
Hollywood Nihilism
Meets
Pantheistic Solipsism
Tina Fey to Steve Martin
at the Oscars:
"Oh, Steve, no one wants
to hear about our religion
… that we made up."
|
From Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, p. 117:
… in 'The Pediment of Appearance,' a slight narrative poem in Transport to Summer…
A group of young men enter some woods 'Hunting for the great ornament, The pediment of appearance.' Though moving through the natural world, the young men seek the artificial, or pure form, believing that in discovering this pediment, this distillation of the real, they will also discover the 'savage transparence,' the rude source of human life. In Stevens's world, such a search is futile, since it is only through observing nature that one reaches beyond it to pure form. As if to demonstrate the degree to which the young men's search is misaligned, Stevens says of them that 'they go crying/The world is myself, life is myself,' believing that what surrounds them is immaterial. Such a proclamation is a cardinal violation of Stevens's principles of the imagination.
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Superficially the young men's philosophy seems to resemble what Wikipedia calls "pantheistic solipsism"– noting, however, that "This article has multiple issues."
As, indeed, does pantheistic solipsism– a philosophy (properly called "eschatological pantheistic multiple-ego solipsism") devised, with tongue in cheek, by science-fiction writer Robert A. Heinlein.
Despite their preoccupation with solipsism, Heinlein and Stevens point, each in his own poetic way, to a highly non-solipsistic topic from pure mathematics that is, unlike the religion of Martin and Fey, not made up– namely, the properties of space.
Heinlein:
"Sharpie, we have condensed six dimensions into four, then we either work by analogy into six, or we have to use math that apparently nobody but Jake and my cousin Ed understands. Unless you can think of some way to project six dimensions into three– you seem to be smart at such projections."
I closed my eyes and thought hard. "Zebbie, I don't think it can be done. Maybe Escher could have done it."
Stevens:
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A discussion of Stevens's late poem "The Rock" (1954) in Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, p. 120:
For Stevens, the poem "makes meanings of the rock." In the mind, "its barrenness becomes a thousand things/And so exists no more." In fact, in a peculiar irony that only a poet with Stevens's particular notion of the imagination's function could develop, the rock becomes the mind itself, shattered into such diamond-faceted brilliance that it encompasses all possibilities for human thought:
The rock is the gray particular of man's life,
The stone from which he rises, up—and—ho,
The step to the bleaker depths of his descents ...
The rock is the stern particular of the air,
The mirror of the planets, one by one,
But through man's eye, their silent rhapsodist,
Turquoise the rock, at odious evening bright
With redness that sticks fast to evil dreams;
The difficult rightness of half-risen day.
The rock is the habitation of the whole,
Its strength and measure, that which is near,
point A
In a perspective that begins again
At B: the origin of the mango's rind.
(Collected Poems, 528)
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Stevens's rock is associated with empty space, a concept that suggests "nothingness" to one literary critic:
B. J. Leggett, "Stevens's Late Poetry" in
The Cambridge Companion to Wallace Stevens— On the poem "The Rock":
"… the barren rock of the title is Stevens's symbol for the nothingness that underlies all existence, 'That in which space itself is contained'…. Its subject is its speaker's sense of nothingness and his need to be cured of it."
This interpretation might appeal to Joan Didion, who, as author of the classic novel
Play It As It Lays, is perhaps the world's leading expert on
Hollywood nihilism.
More positively…
Space is, of course, also a topic
in pure mathematics…
For instance, the 6-dimensional
affine space (or the corresponding
5-dimensional projective space)
over the two-element Galois field
can be viewed as an illustration of
Stevens's metaphor in "The Rock."
Heinlein should perhaps have had in mind
the Klein correspondence when he discussed "some way to project six dimensions into three." While such a projection is of course trivial for anyone who has taken an undergraduate course in linear algebra, the following remarks by Philippe Cara present a much more meaningful mapping, using the Klein correspondence, of structures in six (affine) dimensions to structures in three.
Cara:
Here the 6-dimensional affine
space contains the 63 points
of PG(5, 2), plus the origin, and
the 3-dimensional affine
space contains as its 8 points
Conwell's eight "heptads," as in
Generating the Octad Generator.
Comments Off on Tuesday February 24, 2009
Sunday, February 22, 2009
Themes and
Variations
“Designed objects, Brock writes, can be broken down into ‘themes’ and ‘transformations.’ A theme is a motif, such as an S-curve; a transformation might see that curve appear elsewhere in the design, but stretched, rotated 90 degrees, mirrored, or otherwise reworked.
Aesthetic satisfaction comes from an apprehension of how those themes and transformations relate to each other, or of what Brock calls their ‘relative complexity.’ Basically– and this is the nub of it– ‘if the theme is simple, then we are most satisfied when its echoes are complex… and vice versa.'”
Related material:
Theme
and Variations
See also earlier tributes to
Hollywood Game Theory
and Hollywood Religion:
For some variations on the
above checkerboard theme, see
Finite Relativity and
A Wealth of Algebraic Structure.
Comments Off on Sunday February 22, 2009
Friday, February 20, 2009
The Cross
of Constantine
mentioned in
this afternoon's entry
"Emblematizing the Modern"
was the object of a recent
cinematic chase sequence
(successful and inspiring)
starring Mira Sorvino
at the Metropolitan
Museum of Art.
In memory of
Dr. Hunter S. Thompson,
dead by his own hand
on this date
four years ago —
Click for details.
There is
another sort of object
we may associate with a
different museum and with
a modern Constantine …
See "Art Wars for MoMA"
(Dec. 14, 2008).
This object, modern
rather than medieval,
is the ninefold square:
It may suit those who,
like Rosalind Krauss
(see "Emblematizing"),
admire the grids of modern art
but view any sort of Christian
cross with fear and loathing.
For some background that
Dr. Thompson might appreciate,
see notes on Geometry and Death
in this journal, June 1-15, 2007,
and the five Log24 entries
ending at 9 AM Dec. 10. 2006,
which include this astute
observation by J. G. Ballard:
"Modernism's attempt to build a better world with the aid of science and technology now seems almost heroic. Bertolt Brecht, no fan of modernism, remarked that the mud, blood and carnage of the first world war trenches left its survivors longing for a future that resembled a white-tiled bathroom."
Selah.
Comments Off on Friday February 20, 2009
Note that in applications, the vertical axis
of the Cross of Descartes often symbolizes
the timeless (money, temperature, etc.)
while the horizontal axis often symbolizes time.
T.S. Eliot:
"Men’s curiosity searches past and future
And clings to that dimension. But to apprehend
The point of intersection of the timeless
With time, is an occupation for the saint…."
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There is a reason, apart from her
ethnic origins, that Rosalind Krauss (cf.
9/13/06) rejects, with a shudder, the cross as a key to "the Pandora's box of spiritual reference that is opened once one uses it." The rejection occurs in the context of her attempt to establish not the cross, but the
grid, as a religious symbol:
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"In suggesting that the success [1] of the grid
is somehow connected to its structure as myth,
I may of course be accused of stretching a point
beyond the limits of common sense, since myths
are stories, and like all narratives they unravel
through time, whereas grids are not only spatial
to start with, they are visual structures
that explicitly reject a narrative
or sequential reading of any kind.
[1] Success here refers to
three things at once:
a sheerly quantitative success,
involving the number of artists
in this century who have used grids;
a qualitative success through which
the grid has become the medium
for some of the greatest works
of modernism; and an ideological
success, in that the grid is able–
in a work of whatever quality–
to emblematize the Modern."
— Rosalind Krauss, "Grids" (1979)
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Related material:
Time Fold and Weyl on
objectivity and frames of reference.
See also Stambaugh on
The Formless Self
as well as
A Study in Art Education
and
Jung and the Imago Dei.
Comments Off on Friday February 20, 2009
Tuesday, February 17, 2009
Diamond-Faceted:
Transformations
of the Rock
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A discussion of Stevens's late poem "The Rock" (1954) in Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, p. 120:
For Stevens, the poem "makes meanings of the rock." In the mind, "its barrenness becomes a thousand things/And so exists no more." In fact, in a peculiar irony that only a poet with Stevens's particular notion of the imagination's function could develop, the rock becomes the mind itself, shattered into such diamond-faceted brilliance that it encompasses all possibilities for human thought:
The rock is the gray particular of man's life,
The stone from which he rises, up—and—ho,
The step to the bleaker depths of his descents ...
The rock is the stern particular of the air,
The mirror of the planets, one by one,
But through man's eye, their silent rhapsodist,
Turquoise the rock, at odious evening bright
With redness that sticks fast to evil dreams;
The difficult rightness of half-risen day.
The rock is the habitation of the whole,
Its strength and measure, that which is near,
point A
In a perspective that begins again
At B: the origin of the mango's rind.
(Collected Poems, 528)
|
A mathematical version of
this poetic concept appears
in a rather cryptic note
from 1981 written with
Stevens's poem in mind:
For some explanation of the
groups of 8 and 24
motions referred to in the note,
see an earlier note from 1981.
For the Perlis "diamond facets,"
see the Diamond 16 Puzzle.
For a much larger group
of motions, see
Solomon's Cube.
As for "the mind itself"
and "possibilities for
human thought," see
Geometry of the I Ching.
Comments Off on Tuesday February 17, 2009
Monday, February 9, 2009
The Vision Thing
The British Academy Awards last night showed two Paul Newman clips:
"Sometimes nothin' can be a real cool hand."
"Boy, I got vision and the rest of the world wears bifocals."
Related material: This journal, September 2008.
As for bifocals…
Pennsylvania Lottery
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Comments Off on Monday February 9, 2009
Thursday, February 5, 2009
Through the
Looking Glass:
A Sort of Eternity
From the new president’s inaugural address:
“… in the words of Scripture, the time has come to set aside childish things.”
The words of Scripture:
| 9 |
For we know in part, and we prophesy in part. |
| 10 |
But when that which is perfect is come, then that which is in part shall be done away. |
| 11 |
When I was a child, I spake as a child, I understood as a child, I thought as a child: but when I became a man, I put away childish things. |
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For now we see through a glass, darkly, but then face to face: now I know in part; but then shall I know even as also I am known.
— First Corinthians 13 |
“through a glass”—
[di’ esoptrou].
By means of
a mirror [esoptron].
Childish things:
© 2005 The Institute for Figuring
Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)
Not-so-childish:
Three planes through
the center of a cube
that split it into
eight subcubes:
Through a glass, darkly:
A group of 8 transformations is
generated by affine reflections
in the above three planes.
Shown below is a pattern on
the faces of the 2x2x2 cube
that is symmetric under one of
these 8 transformations–
a 180-degree rotation:
(Click on image
for further details.)
But then face to face:
A larger group of 1344,
rather than 8, transformations
of the 2x2x2 cube
is generated by a different
sort of affine reflections– not
in the infinite Euclidean 3-space
over the field of real numbers,
but rather in the finite Galois
3-space over the 2-element field.
Galois age fifteen,
drawn by a classmate.
These transformations
in the Galois space with
finitely many points
produce a set of 168 patterns
like the one above.
For each such pattern,
at least one nontrivial
transformation in the group of 8
described above is a symmetry
in the Euclidean space with
infinitely many points.
For some generalizations,
see Galois Geometry.
Related material:
The central aim of Western religion–
"Each of us has something to offer the Creator...
the bridging of
masculine and feminine,
life and death.
It's redemption.... nothing else matters."
-- Martha Cooley in The Archivist (1998)
The central aim of Western philosophy–
Dualities of Pythagoras
as reconstructed by Aristotle:
Limited Unlimited
Odd Even
Male Female
Light Dark
Straight Curved
... and so on ....
“Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy.”
— Jamie James in The Music of the Spheres (1993)
“In the garden of Adding
live Even and Odd…
And the song of love’s recision
is the music of the spheres.”
— The Midrash Jazz Quartet in City of God, by E. L. Doctorow (2000)
A quotation today at art critic Carol Kino’s website, slightly expanded:
“Art inherited from the old religion
the power of consecrating things
and endowing them with
a sort of eternity;
museums are our temples,
and the objects displayed in them
are beyond history.”
— Octavio Paz,”Seeing and Using: Art and Craftsmanship,” in Convergences: Essays on Art and Literature (New York: Harcourt Brace Jovanovich 1987), 52
From Brian O’Doherty’s 1976 Artforum essays– not on museums, but rather on gallery space:
“Inside the White Cube“
“We have now reached
a point where we see
not the art but the space first….
An image comes to mind
of a white, ideal space
that, more than any single picture,
may be the archetypal image
of 20th-century art.”
“Space: what you
damn well have to see.”
— James Joyce, Ulysses
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Comments Off on Thursday February 5, 2009
Monday, February 2, 2009
The Candlebrow Conference
in Pynchon's Against the Day:
The conferees had gathered here from all around the world…. Their spirits all one way or another invested in, invested by, the siegecraft of Time and its mysteries.
"Fact is, our system of so-called linear time is based on a circular or, if you like, periodic phenomenon– the earth's own spin. Everything spins, up to and including, probably, the whole universe. So we can look to the prairie, the darkening sky, the birthing of a funnel-cloud to see in its vortex the fundamental structure of everything–"
Quaternion by
S. H. Cullinane
"Um, Professor–"….
… Those in attendance, some at quite high speed, had begun to disperse, the briefest of glances at the sky sufficing to explain why. As if the professor had lectured it into being, there now swung from the swollen and light-pulsing clouds to the west a classic prairie "twister"….
… In the storm cellar, over semiliquid coffee and farmhouse crullers left from the last twister, they got back to the topic of periodic functions….
"Eternal Return, just to begin with. If we may construct such functions in the abstract, then so must it be possible to construct more secular, more physical expressions."
"Build a time machine."
"Not the way I would have put it, but if you like, fine."
Vectorists and Quaternionists in attendance reminded everybody of the function they had recently worked up….
"We thus enter the whirlwind. It becomes the very essence of a refashioned life, providing the axes to which everything will be referred. Time no long 'passes,' with a linear velocity, but 'returns,' with an angular one…. We are returned to ourselves eternally, or, if you like, timelessly."
"Born again!" exclaimed a Christer in the gathering, as if suddenly enlightened.
Above, the devastation had begun. |
Comments Off on Monday February 2, 2009
Sunday, February 1, 2009
Comments Off on Sunday February 1, 2009
Friday, January 30, 2009
Two-Part Invention
This journal on
October 8, 2008,
at noon:
“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?'”
— H. S. M. Coxeter, introduction to Richard J. Trudeau’s remarks on the “story theory” of truth as opposed to the “diamond theory” of truth in The Non-Euclidean Revolution
Trudeau’s 1987 book uses the phrase “diamond theory” to denote the philosophical theory, common since Plato and Euclid, that there exist truths (which Trudeau calls “diamonds”) that are certain and eternal– for instance, the truth in Euclidean geometry that the sum of a triangle’s angles is 180 degrees.
Insidehighered.com on
the same day, October 8, 2008,
at 12:45 PM EDT
“Future readers may consider Updike our era’s Mozart; Mozart was once written off as a too-prolific composer of ‘charming nothings,’ and some speak of Updike that way.”
— Comment by BPJ |
“Birthday, death-day–
what day is not both?”
—
John Updike
Updike died on January 27.
On the same date,
Mozart was born.
Requiem
Mr. Best entered,
tall, young, mild, light.
He bore in his hand
with grace a notebook,
new, large, clean, bright.
— James Joyce, Ulysses,
Shakespeare and Company,
Paris, 1922, page 178 |
Related material:
Comments Off on Friday January 30, 2009
Wednesday, January 14, 2009
Eight is a Gate
From the most highly
rated negative review:
“I never did figure out
what ‘The Eight’ was.”
Various approaches
to this concept
(click images for details):
Comments Off on Wednesday January 14, 2009
Thursday, January 8, 2009
Report of Arrival
A PBS broadcast of Cyrano de Bergerac was shown yesterday nationally and this evening, a day late, by WNED TV, Buffalo.
From the translation by Anthony Burgess:
Cyrano speaks of falling leaves–
They fall well. With a sort of panache.
They plume down in their last
Loveliness, disguising their fear
Of being dried and pounded to ash
To mix with the common dust.
They go in grace, making their fall appear
Like flying.
ROXANE You’re melancholy today.
CYRANO Never. I’m not the melancholy sort.
ROXANE Very well, then. We’ll let
The leaves of the fall fall while you
Turn the leaves of my gazette.
What’s new at court?
CYRANO … There have been some scandals
To do with witches. A bishop went to heaven,
Or so it’s believed: there’s been as yet no report
Of his arrival….”
Later….
CYRANO … See it there, a white plume
Over the battle– A diamond in the ash
Of the ultimate combustion–
My panache.”
Comments Off on Thursday January 8, 2009
Tuesday, January 6, 2009
Archetypes, Synchronicity,
and Dyson on Jung
The current (Feb. 2009) Notices of the American Mathematical Society has a written version of Freeman Dyson’s 2008 Einstein Lecture, which was to have been given in October but had to be canceled. Dyson paraphrases a mathematician on Carl Jung’s theory of archetypes:
“… we do not need to accept Jung’s theory as true in order to find it illuminating.”
The same is true of Jung’s remarks on synchronicity.
For example —
Yesterday’s entry, “A Wealth of Algebraic Structure,” lists two articles– each, as it happens, related to Jung’s four-diamond figure from Aion as well as to my own Notes on Finite Geometry. The articles were placed online recently by Cambridge University Press on the following dates:
R. T. Curtis’s 1974 article defining his Miracle Octad Generator (MOG) was published online on Oct. 24, 2008.
Curtis’s 1987 article on geometry and algebraic structure in the MOG was published online on Dec. 19, 2008.
On these dates, the entries in this journal discussed…
Oct. 24:
Cube Space, 1984-2003
Material related to that entry:
Dec. 19:
Art and Religion: Inside the White Cube
That entry discusses a book by Mark C. Taylor:
The Picture in Question: Mark Tansey and the Ends of Representation (U. of Chicago Press, 1999).
In Chapter 3, “Sutures of Structures,” Taylor asks —
“What, then, is a frame, and what is frame work?”
One possible answer —
Hermann Weyl on the relativity problem in the context of the 4×4 “frame of reference” found in the above Cambridge University Press articles.
“Examples are the stained-glass
windows of knowledge.”
— Vladimir Nabokov
Comments Off on Tuesday January 6, 2009
Monday, January 5, 2009
A 1987 article by R. T. Curtis on the geometry of his Miracle Octad Generator (MOG) as it relates to the geometry of the 4×4 square is now available online ($20):
Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:
“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”
(Received July 20 1987)
— Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353, doi:10.1017/S0013091500004600.
(Published online by Cambridge University Press 19 Dec 2008.)
In the above article, Curtis explains how two-thirds of his 4×6 MOG array may be viewed as the 4×4 model of the four-dimensional affine space over GF(2). (His earlier 1974 paper (below) defining the MOG discussed the 4×4 structure in a purely combinatorial, not geometric, way.)
For further details, see The Miracle Octad Generator as well as Geometry of the 4×4 Square and Curtis’s original 1974 article, which is now also available online ($20):
A new combinatorial approach to M24, by R. T. Curtis. Abstract:
“In this paper, we define M24 from scratch as the subgroup of S24 preserving a Steiner system S(5, 8, 24). The Steiner system is produced and proved to be unique and the group emerges naturally with many of its properties apparent.”
(Received June 15 1974)
— Mathematical Proceedings of the Cambridge Philosophical Society (1976), 79: 25-42, doi:10.1017/S0305004100052075.
(Published online by Cambridge University Press 24 Oct 2008.)
* For instance:
Click for details.
Comments Off on Monday January 5, 2009
Monday, December 22, 2008
The Folding
Hamlet, Act 1, Scene 5 —
Ghost:
"I could a tale unfold whose lightest word
Would harrow up thy soul, freeze thy young blood,
Make thy two eyes, like stars, start from their spheres,
Thy knotted and combined locks to part
And each particular hair to stand on end,
Like quills upon the fretful porpentine:
But this eternal blazon must not be
To ears of flesh and blood. List, list, O, list!"
This recalls the title of a piece in this week's New Yorker:"The Book of Lists:
Susan Sontag’s early journals." (See Log24 on Thursday, Dec. 18.)
In the rather grim holiday spirit of that piece, here are some journal notes for Sontag, whom we may imagine as the ghost of Hanukkah past.
There are at least two ways of folding a list (or tale) to fit a rectangular frame.The normal way, used in typesetting English prose and poetry, starts at the top, runs from left to right, jumps down a line, then again runs left to right, and so on until the passage is done or the bottom right corner of the frame is reached.
The boustrophedonic way again goes from top to bottom, with the first line running from left to right, the next from right to left, the next from left to right, and so on, with the lines' directions alternating.
The word "boustrophedon" is from the Greek words describing the turning, at the end of each row, of an ox plowing (or "harrowing") a field.
The Tale of
the Eternal Blazon
by Washington Irving
"Blazon meant originally a shield, and then the heraldic bearings on a shield.
Later it was applied to the art of describing or depicting heraldic bearings
in the proper manner; and finally the term came to signify ostentatious display
and also description or record by words or other means. In Hamlet, Act I. Sc. 5,
the Ghost, while talking with Prince Hamlet, says:
'But this eternal blazon
must not be
To ears of flesh and blood.'
Eternal blazon signifies revelation or description of things pertaining to eternity."
— Irving's Sketch Book, p. 461
By Washington Irving and Mary Elizabeth Litchfield, Ginn & Company, 1901
Related material:
Folding (and harrowing up)
some eternal blazons —
These are the foldings
described above.
They are two of the 322,560
natural ways to fit
the list (or tale)
"1, 2, 3, … 15, 16"
into a 4×4 frame.
For further details, see
The Diamond 16 Puzzle.
Moral of the tale:
Cynthia Zarin in The New Yorker, issue dated April 12, 2004–
"Time, for L'Engle, is accordion-pleated. She elaborated, 'When you bring a sheet off the line, you can't handle it until it's folded, and in a sense, I think, the universe can't exist until it's folded– or it's a story without a book.'"
Comments Off on Monday December 22, 2008
Sunday, December 21, 2008
Interpretive Grids
The 15 grids in the picture at right above may be regarded as
interpreting the structure of the space at left above.
This pair of pictures was suggested by yesterday’s entry at Ars Mathematica containing the phrase “a dramatic extension of the notion of points.”
For other uses of the phrase “interpretive grid,” see today’s previous entry.
Comments Off on Sunday December 21, 2008
Friday, December 19, 2008
Inside the
White Cube
Part I: The White Cube
Part II: Inside
Part III: Outside
Click to enlarge.
Mark Tansey, The Key (1984)
For remarks on religion
related to the above, see
Log24 on the Garden of Eden
and also Mark C. Taylor,
"What Derrida Really Meant"
(New York Times, Oct. 14, 2004).
For some background on Taylor,
see Wikipedia. Taylor, Chairman
of the Department of Religion at
Columbia University, has a
1973 doctorate in religion from
Harvard University. His opinion
of Derrida indicates that his
sympathies lie more with
the serpent than with the angel
in the Tansey picture above.
For some remarks by Taylor on
the art of Tansey relevant to the
structure of the white cube
(Part I above), see Taylor's
The Picture in Question:
Mark Tansey and the
Ends of Representation
(U. of Chicago Press, 1999):
From Chapter 3,
"Sutures* of Structures," p. 58:
"What, then, is a frame, and what is frame work?
This question is deceptive in its simplicity. A frame is, of course, 'a basic skeletal structure designed to give shape or support' (American Heritage Dictionary)…. when the frame is in question, it is difficult to determine what is inside and what is outside. Rather than being on one side or the other, the frame is neither inside nor outside. Where, then, Derrida queries, 'does the frame take place….'"
* P. 61:
"… the frame forms the suture of structure. A suture is 'a seamless [sic**] joint or line of articulation,' which, while joining two surfaces, leaves the trace of their separation."
** A dictionary says "a seamlike joint or line of articulation," with no mention of "trace," a term from Derrida's jargon.
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Comments Off on Friday December 19, 2008
Tuesday, December 16, 2008
The Square Wheel
(continued)
From The n-Category Cafe today:
David Corfield at 2:33 PM UTC quoting a chapter from a projected second volume of a biography:
"Grothendieck’s spontaneous reaction to whatever appeared to be causing a difficulty… was to adopt and embrace the very phenomenon that was problematic, weaving it in as an integral feature of the structure he was studying, and thus transforming it from a difficulty into a clarifying feature of the situation."
John Baez at 7:14 PM UTC on research:
"I just don’t want to reinvent a wheel, or waste my time inventing a square one."
For the adoption and embracing of such a problematic phenomenon, see The Square Wheel (this journal, Sept. 14, 2004).
For a connection of the square wheel with yesterday's entry for Julie Taymor's birthday, see a note from 2002:
Comments Off on Tuesday December 16, 2008
Sunday, December 14, 2008
Epigraphs
The New York Times of Sunday, May 6, 2007, on a writer of pulp fiction:
His early novels, written in two weeks or less, were published in double-decker Ace paperbacks that included two books in one, with a lurid cover for each. “If the Holy Bible was printed as an Ace Double,” an editor once remarked, “it would be cut down to two 20,000-word halves with the Old Testament retitled as ‘Master of Chaos’ and the New Testament as ‘The Thing With Three Souls.'”
Epigraph for Part One:
“Ours is a very gutsy religion, Cullinane.“
— James A. Michener
Lurid cover:
The Pussycat
Epigraph for Part Two:
“Beware lest you believe that you can comprehend the Incomprehensible….“
— Saint Bonaventure
Lurid cover:
The Owl
Click on the image for a
relevant Wallace Stevens poem.
Comments Off on Sunday December 14, 2008
Wednesday, December 10, 2008
Comments Off on Wednesday December 10, 2008
Sunday, December 7, 2008
Space and
the Soul
On a book by Margaret Wertheim:
“She traces the history of space beginning with the cosmology of Dante. Her journey continues through the historical foundations of celestial space, relativistic space, hyperspace, and, finally, cyberspace.” –Joe J. Accardi, Northeastern Illinois Univ. Lib., Chicago, in Library Journal, 1999 (quoted at Amazon.com)
There are also other sorts of space.
© 2005 The Institute for Figuring
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Saturday, December 6, 2008
Another Opening,
Another Show
"While feasts of Saint Nicholas are not observed nationally, cities with strong German influences like Milwaukee, Cincinnati, and St. Louis celebrate St. Nick's Day on a scale similar to the German custom." —
Wikipedia
A footprint from Germany:
The link in the above footprint leads
to an entry of July 5, 2006.
The access method:
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"The Python urllib module implements a fairly high-level abstraction for making any web object with a URL act like a Python file: i.e., you open it, and get back an object…."
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For more pictures and discussion
of the object fetched by Python,
see
AntiChristmas 2007.
For a larger and more sophisticated
relative of that object,
see Solomon's Cube and
the related three presents
from the German link's target:
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Friday, December 5, 2008
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Thursday, December 4, 2008
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Sunday, November 30, 2008
Abstraction and Faith
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From Sol LeWitt: A Retrospective, edited by Gary Garrels, Yale University Press, 2000, p. 376:
THE SQUARE AND THE CUBE
by Sol LeWitt
The best that can be said for either the square or the cube is that they are relatively uninteresting in themselves. Being basic representations of two- and three-dimensional form, they lack the expressive force of other more interesting forms and shapes. They are standard and universally recognized, no initiation being required of the viewer; it is immediately evident that a square is a square and a cube a cube. Released from the necessity of being significant in themselves, they can be better used as grammatical devices from which the work may proceed.
Reprinted from Lucy R. Lippard et al., "Homage to the Square," Art in America 55, No. 4 (July-August 1967): 54. (LeWitt's contribution was originally untitled.)
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A vulgarized version
of LeWitt's remarks
appears on a webpage of
the National Gallery of Art.
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Today's Sermon
"Closing the Circle on Abstract Art"
On Kirk Varnedoe's National Gallery lectures in 2003 (Philip Kennicott, Washington Post, Sunday, May 18, 2003):
"Varnedoe's lectures were ultimately about faith, about his faith in the power of abstraction, and abstraction as a kind of anti-religious faith in itself."
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Monday, November 24, 2008
Frame Tale
Click on image for details.
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Sunday, November 23, 2008
The Idea of Identity
“The first credential
we should demand of a critic
is his ideograph of the good.”
— Ezra Pound,
How to Read
Music critic Bernard Holland in The New York Times on Monday, May 20, 1996:
The Juilliard’s
Half-Century Ripening
Philosophers ponder the idea of identity: what it is to give something a name on Monday and have it respond to that name on Friday regardless of what might have changed in the interim. Medical science tells us that the body’s cells replace themselves wholesale within every seven years, yet we tell ourselves that we are what we were….
Schubert at the end of his life had already passed on to another level of spirit. Beethoven went back and forth between the temporal world and the world beyond right up to his dying day.
Exercise
Part I:
Apply Holland’s Monday-to-Friday “idea of identity” to the lives and deaths during the week of Monday, Nov. 10 (“Frame Tales“), through Friday, Nov. 14, of a musician and a maker of music documentaries– Mitch Mitchell (d. Nov. 12) and Baird Bryant (d. Nov. 13).
Part II:
Apply Holland’s “idea of identity” to last week (Monday, Nov. 17, through Friday, Nov. 21), combining it with Wigner’s remarks on invariance (discussed here on Monday) and with the “red dragon” (Log24, Nov. 15) concept of flight over “the Hump”– the Himalayas– and the 1991 documentary filmed by Bryant, “Heart of Tibet.”
Part III:
Discuss Parts I and II in the context of Eliot’s Four Quartets. (See Time Fold, The Field of Reason, and Balance.)
Comments Off on Sunday November 23, 2008
Wednesday, November 19, 2008
"Through the unknown,
remembered gate…."
— Four Quartets
(Epigraph to the introduction,
Parallelisms of Complete Designs
by Peter J. Cameron,
Merton College, Oxford)
"It's still the same old story…."
— Song lyric
The Great Gatsby, Chapter 6:
"An instinct toward his future glory had led him, some months before, to the small Lutheran college of St. Olaf in southern Minnesota. He stayed there two weeks, dismayed at its ferocious indifference to the drums of his destiny, to destiny itself, and despising the janitor’s work with which he was to pay his way through."
There is a link to an article on St. Olaf College in Arts & Letters Daily today:
"John Milton, boring? Paradise Lost has a little bit of something for everybody. Hot sex! Hellfire! Some damned good poetry, too…" more»
The "more" link is to The Chronicle of Higher Education.
For related material on Paradise Lost and higher education, see Mathematics and Narrative.
Comments Off on Wednesday November 19, 2008
Sunday, November 16, 2008
Art and Lies
Observations suggested by an article on author Lewis Hyde– "What is Art For?"– in today's New York Times Magazine:
Margaret Atwood (pdf) on Lewis Hyde's
Trickster Makes This World: Mischief, Myth, and Art —
"Trickster," says Hyde, "feels no anxiety when he deceives…. He… can tell his lies with creative abandon, charm, playfulness, and by that affirm the pleasures of fabulation." (71) As Hyde says, "… almost everything that can be said about psychopaths can also be said about tricksters," (158), although the reverse is not the case. "Trickster is among other things the gatekeeper who opens the door into the next world; those who mistake him for a psychopath never even know such a door exists." (159)
What is "the next world"? It might be the Underworld….
The pleasures of fabulation, the charming and playful lie– this line of thought leads Hyde to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the con-artist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation and art all come from the same ancient root, a word meaning to join, to fit, and to make. (254) If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist. Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart.
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"What happened to that… cube?"
Apollinax laughed until his eyes teared. "I'll give you a hint, my dear. Perhaps it slid off into a higher dimension."
"Are you pulling my leg?"
"I wish I were," he sighed. "The fourth dimension, as you know, is an extension along a fourth coordinate perpendicular to the three coordinates of three-dimensional space. Now consider a cube. It has four main diagonals, each running from one corner through the cube's center to the opposite corner. Because of the cube's symmetry, each diagonal is clearly at right angles to the other three. So why shouldn't a cube, if it feels like it, slide along a fourth coordinate?"
— "Mr. Apollinax Visits New York," by Martin Gardner, Scientific American, May 1961, reprinted in The Night is Large
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Comments Off on Sunday November 16, 2008
From Koestler’s Darkness at Noon, a fictional Communist on propaganda:
“It is necessary to hammer every sentence into the masses by repetition and simplification. What is presented as right must shine like gold; what is presented as wrong must be black as pitch.”
Thanks for this quotation to Kati Marton, author of The Great Escape: Nine Jews Who Fled Hitler and Changed the World (Simon & Schuster, paperback edition Nov. 6, 2007). One of Marton’s nine was Koestler.
From another book related to this exodus:
“Riesz was one of the most elegant mathematical writers in the world, known for his precise, concise, and clear expositions. He was one of the originators of the theory of function spaces– an analysis which is geometrical in nature.”
— Stanislaw Ulam, Adventures of a Mathematician
And from Gian-Carlo Rota, a friend of Ulam:
“Riesz’s example is well worth following today.”
Related material: Misunderstanding in the Theory of Design and Geometry for Jews.
For a different approach to ethnicity and the number nine that is also “geometrical in nature,” see The Pope in Plato’s Cave and the four entries preceding it, as well as A Study in Art Education.
Comments Off on Sunday November 16, 2008
Monday, November 10, 2008
Frame Tales
From June 30 —
("Will this be on the test?")
Frame Tale One:
Frame Tale Two:
Barry Sharples
on his version of the
Kaleidoscope Puzzle —
Background:
"A possible origin of this puzzle is found in a dialogue
between Socrates and Meno written by the Greek philosopher,
Plato, where a square is drawn inside a square such that
the blue square is twice the area of the yellow square.
Colouring the triangles produces a starting pattern
which is a one-diamond figure made up of four tiles
and there are 24 different possible arrangements."
The King and the Corpse —
"The king asked, in compensation for his toils during this strangest
of all the nights he had ever known, that the twenty-four riddle tales
told him by the specter, together with the story of the night itself,
should be made known over the whole earth
and remain eternally famous among men."
Frame Tale Three:
Finnegans Wake —
"The quad gospellers may own the targum
but any of the Zingari shoolerim may pick a peck
of kindlings yet from the sack of auld hensyne."
Comments Off on Monday November 10, 2008
Saturday, November 8, 2008
From a
Cartoon Graveyard
“That corpse you planted
last year in your garden,
Has it begun to sprout?
Will it bloom this year?
Or has the sudden frost
disturbed its bed?”
— T. S. Eliot, “The Waste Land“
Wikipedia:
“In the Roman Catholic tradition, the term ‘Body of Christ’ refers not only to the body of Christ in the spiritual realm, but also to two distinct though related things: the Church and the reality of the transubstantiated bread of the Eucharist….
According to the Catechism of the Catholic Church, ‘the comparison of the Church with the body casts light on the intimate bond between Christ and his Church. Not only is she gathered around him; she is united in him, in his body….’
….To distinguish the Body of Christ in this sense from his physical body, the term ‘Mystical Body of Christ’ is often used. This term was used as the first words, and so as the title, of the encyclical Mystici Corporis Christi of Pope Pius XII.”
Pope Pius XII:
“83. The Sacrament of the Eucharist is itself a striking and wonderful figure of the unity of the Church, if we consider how in the bread to be consecrated many grains go to form one whole, and that in it the very Author of supernatural grace is given to us, so that through Him we may receive the spirit of charity in which we are bidden to live now no longer our own life but the life of Christ, and to love the Redeemer Himself in all the members of His social Body.”
Comments Off on Saturday November 8, 2008
Friday, November 7, 2008
The Sincerest Form
of Flattery
At a British puzzle website today I found this, titled “Tiles Puzzle by Steven H. Cullinane”–
The version there states that
“there are 322,560 patterns made by swapping rows, swapping columns and swapping the four 2×2 quadrants!”
Comments Off on Friday November 7, 2008
Thursday, November 6, 2008
Death of a Classmate
Michael Crichton,
Harvard College, 1964
Authors Michael Crichton and
David Foster Wallace in today’s
New York Times obituaries
The Times’s remarks above
on the prose styles of
Crichton and Wallace–
“compelling formula” vs.
“intricate complexity”–
suggest the following works
of visual art in memory
of Crichton.
“Crystal”—
“Dragon”
(from Crichton’s
Jurassic Park)–
For the mathematics
(dyadic harmonic analysis)
relating these two figures,
see
Crystal and Dragon.
Some philosophical
remarks related to
the Harvard background
that Crichton and I share–
Hitler’s Still Point
and
The Crimson Passion.
Comments Off on Thursday November 6, 2008
Thursday, October 30, 2008
Readings for
Devil’s Night
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Friday, October 24, 2008
“The Cube Space” is a name given to the eightfold cube in a vulgarized mathematics text, Discrete Mathematics: Elementary and Beyond, by Laszlo Lovasz et al., published by Springer in 2003. The identification in a natural way of the eight points of the linear 3-space over the 2-element field GF(2) with the eight vertices of a cube is an elementary and rather obvious construction, doubtless found in a number of discussions of discrete mathematics. But the less-obvious generation of the affine group AGL(3,2) of order 1344 by permutations of parallel edges in such a cube may (or may not) have originated with me. For descriptions of this process I wrote in 1984, see Diamonds and Whirls and Binary Coordinate Systems. For a vulgarized description of this process by Lovasz, without any acknowledgement of his sources, see an excerpt from his book.
Comments Off on Friday October 24, 2008
Wednesday, October 22, 2008
Euclid vs. Galois
On May 4, 2005, I wrote a note about how to visualize the 7-point Fano plane within a cube.
Last month, John Baez showed slides that touched on the same topic. This note is to clear up possible confusion between our two approaches.
From Baez’s Rankin Lectures at the University of Glasgow:
Note that Baez’s statement (
pdf) “Lines in the Fano plane correspond to planes through the origin [the vertex labeled ‘1’] in this cube” is, if taken (wrongly) as a statement about a cube in Euclidean 3-space, false.
The statement is, however, true of the eightfold cube, whose eight subcubes correspond to points of the linear 3-space over the two-element field, if “planes through the origin” is interpreted as planes within that linear 3-space, as in Galois geometry, rather than within the Euclidean cube that Baez’s slides seem to picture.
This Galois-geometry interpretation is, as an article of his from 2001 shows, actually what Baez was driving at. His remarks, however, both in 2001 and 2008, on the plane-cube relationship are both somewhat trivial– since “planes through the origin” is a standard definition of lines in projective geometry– and also unrelated– apart from the possibility of confusion– to my own efforts in this area. For further details, see The Eightfold Cube.
Comments Off on Wednesday October 22, 2008
Wednesday, October 8, 2008
Serious Numbers
A Yom Kippur
Meditation
"When times are mysterious
Serious numbers
Will always be heard."
— Paul Simon,
"When Numbers Get Serious"
"There is a pleasantly discursive treatment of Pontius Pilate's unanswered question 'What is truth?'"
— H. S. M. Coxeter, introduction to Richard J. Trudeau's remarks on the "story theory" of truth as opposed to the "diamond theory" of truth in The Non-Euclidean Revolution
Trudeau's 1987 book uses the phrase "diamond theory" to denote the philosophical theory, common since Plato and Euclid, that there exist truths (which Trudeau calls "diamonds") that are certain and eternal– for instance, the truth in Euclidean geometry that the sum of a triangle's angles is 180 degrees. As the excerpt below shows, Trudeau prefers what he calls the "story theory" of truth–
"There are no diamonds. People make up stories about what they experience. Stories that catch on are called 'true.'"
(By the way, the phrase "diamond theory" was used earlier, in 1976, as the title of a monograph on geometry of which Coxeter was aware.)
Excerpt from
The Non-Euclidean RevolutionWhat does this have to do with numbers?
Pilate's skeptical tone suggests he may have shared a certain confusion about geometric truth with thinkers like Trudeau and the slave boy in Plato's Meno. Truth in a different part of mathematics– elementary arithmetic– is perhaps more easily understood, although even there, the existence of what might be called "non-Euclidean number theory"– i.e., arithmetic over finite fields, in which 1+1 can equal zero– might prove baffling to thinkers like Trudeau.
Trudeau's book exhibits, though it does not discuss, a less confusing use of numbers– to mark the location of pages. For some philosophical background on this version of numerical truth that may be of interest to devotees of the Semitic religions on this evening's High Holiday, see Zen and Language Games.
For uses of numbers that are more confusing, see– for instance– the new website The Daily Beast and the old website Story Theory and the Number of the Beast.
Comments Off on Wednesday October 8, 2008
Friday, September 26, 2008
Christmas Knotfor T.S. Eliot’s birthday
(Continued from Sept. 22–
“A Rose for Ecclesiastes.”)
From Kibler’s
“Variations on a Theme of
Heisenberg, Pauli, and Weyl,”
July 17, 2008:
“It is to be emphasized
that the 15 operators…
are underlaid by the geometry
of the generalized quadrangle
of order 2…. In this geometry,
the five sets… correspond to
a spread of this quadrangle,
i.e., to a set of 5 pairwise
skew lines….”
— Maurice R. Kibler,
July 17, 2008
For ways to visualize
this quadrangle,
see Inscapes.
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Related material
A remark of Heisenberg
quoted here on Christmas 2005:
“… die Schönheit… [ist] die
richtige Übereinstimmung
der Teile miteinander
und mit dem Ganzen.”
“Beauty is the proper conformity
of the parts to one another
and to the whole.”
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Thursday, August 28, 2008
Associations
for the writer
known as UD
"Have liberty not as
the air within a grave
Or down a well. Breathe freedom,
oh, my native,
In the space of horizons
that neither love nor hate."
— Wallace Stevens,
"Things of August"
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A related visual
association of ideas —
("The association is the idea"
— Ian Lee, The Third Word War)
From UD Jewelry:
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by John Braheny
"Hook" is the term you'll hear most often in the business and craft of commercial songwriting. (Well, maybe not as much as "Sorry, we can't use your song," but it's possible that the more you hear about hooks now, the less you'll hear "we can't use it" later.)
The hook has been described as "the part(s) you remember after the song is over," "the part that reaches out and grabs you," "the part you can't stop singing (even when you hate it)" and "the catchy repeated chorus…."
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See also UD's recent
A Must-Read and In My Day*
as well as the five
Log24 entries ending
Sept. 20, 2002.
More seriously:
Stevens's "space of horizons" may, if one likes, be interpreted as a reference to
projective geometry. Despite the bleak physicist's view of mathematics quoted above, this discipline is– thanks to
Blaise Pascal— not totally lacking in literary and spiritual
associations.
* Hey Hey
Comments Off on Thursday August 28, 2008
Friday, August 22, 2008
Tentative movie title:
Blockheads
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.
Other related material:
1. Wechsler Cubes (intelligence testing cubes derived from the Kohs cubes shown above). See…
Harvard psychiatry and…
The Montessori Method;
The Crimson Passion;
The Lottery Covenant.
2. Wechsler Cubes of a different sort (Log24, May 25, 2008)
3. Manohla Dargis in today's New York Times:
"… 'Momma’s Man' is a touchingly true film, part weepie, part comedy, about the agonies of navigating that slippery slope called adulthood. It was written and directed by Azazel Jacobs, a native New Yorker who has set his modestly scaled movie with a heart the size of the Ritz in the same downtown warren where he was raised. Being a child of the avant-garde as well as an A student, he cast his parents, the filmmaker Ken Jacobs and the artist Flo Jacobs, as the puzzled progenitors of his centerpiece, a wayward son of bohemia….
In American movies, growing up tends to be a job for either Aryan Youth types or the oddballs and outsiders…."
4. The bohemian who named his son
Azazel:
"… I think that the deeper opportunity, the greater opportunity film can offer us is as an exercise of the mind. But an exercise, I hate to use the word, I won't say 'soul,' I won't say 'soul' and I won't say 'spirit,' but that it can really put our deepest psychological existence through stuff. It can be a powerful exercise. It can make us think, but I don't mean think about this and think about that. The very, very process of powerful thinking, in a way that it can afford, is I think very, very valuable. I basically think that the mind is not complete yet, that we are working on creating the mind. Okay. And that the higher function of art for me is its contribution to the making of mind."
— Interview with Ken Jacobs, UC Berkeley, October 1999
5. For Dargis's "Aryan Youth types"–
Comments Off on Friday August 22, 2008
Tuesday, August 19, 2008
Three Times
"Credences of Summer," VII,
by Wallace Stevens, from
Transport to Summer (1947)
"Three times the concentred
self takes hold, three times
The thrice concentred self,
having possessed
The object, grips it
in savage scrutiny,
Once to make captive,
once to subjugate
Or yield to subjugation,
once to proclaim
The meaning of the capture,
this hard prize,
Fully made, fully apparent,
fully found." |
Stevens does not say what object he is discussing.
One possibility —
Bertram Kostant, Professor Emeritus of Mathematics at MIT, on an object discussed in a recent New Yorker:
"A word about E(8). In my opinion, and shared by others, E(8) is the most magnificent 'object' in all of mathematics. It is like a diamond with thousands of facets. Each facet offering a different view of its unbelievable intricate internal structure."
Another possibility —
A more modest object —
the 4×4 square.
Update of Aug. 20-21 —
Symmetries and Facets
Kostant's poetic comparison might be applied also to this object.
The natural rearrangements (symmetries) of the 4×4 array might also be described poetically as "thousands of facets, each facet offering a different view of… internal structure."
More precisely, there are 322,560 natural rearrangements– which a poet might call facets*— of the array, each offering a different view of the array's internal structure– encoded as a unique ordered pair of symmetric graphic designs. The symmetry of the array's internal structure is reflected in the symmetry of the graphic designs. For examples, see the Diamond 16 Puzzle.
For an instance of Stevens's "three times" process, see the three parts of the 2004 web page Ideas and Art.
* For the metaphor of rearrangements as facets, note that each symmetry (rearrangement) of a Platonic solid corresponds to a rotated facet: the number of symmetries equals the number of facets times the number of rotations (edges) of each facet–
The metaphor of rearrangements as facets breaks down, however, when we try to use it to compute, as above with the Platonic solids, the
number of natural rearrangements, or symmetries, of the 4×4 array. Actually, the true analogy is between the 16 unit squares of the 4×4 array, regarded as the 16 points of a finite 4-space (which has finitely many symmetries), and the infinitely many points of Euclidean 4-space (which has infinitely many symmetries).
If Greek geometers had started with a finite space (as in The Eightfold Cube), the history of mathematics might have dramatically illustrated Halmos's saying (Aug. 16) that
"The problem is– the genius is– given an infinite question, to think of the right finite question to ask. Once you thought of the finite answer, then you would know the right answer to the infinite question."
The Greeks, of course, answered the infinite questions first– at least for Euclidean space. Halmos was concerned with more general modern infinite spaces (such as Hilbert space) where the intuition to be gained from finite questions is still of value.
Comments Off on Tuesday August 19, 2008
Saturday, August 16, 2008
Seeing the Finite Structure
The following supplies some context for remarks of Halmos on combinatorics.
From Paul Halmos: Celebrating 50 years of Mathematics, by John H. Ewing, Paul Richard Halmos, Frederick W. Gehring, published by Springer, 1991–
Interviews with Halmos, “Paul Halmos by Parts,” by Donald J. Albers–
“Part II: In Touch with God*“– on pp. 27-28:
The Root of All Deep Mathematics
“Albers. In the conclusion of ‘Fifty Years of Linear Algebra,’ you wrote: ‘I am inclined to believe that at the root of all deep mathematics there is a combinatorial insight… I think that in this subject (in every subject?) the really original, really deep insights are always combinatorial, and I think for the new discoveries that we need– the pendulum needs– to swing back, and will swing back in the combinatorial direction.’ I always thought of you as an analyst.
Halmos: People call me an analyst, but I think I’m a born algebraist, and I mean the same thing, analytic versus combinatorial-algebraic. I think the finite case illustrates and guides and simplifies the infinite.
Some people called me full of baloney when I asserted that the deep problems of operator theory could all be solved if we knew the answer to every finite dimensional matrix question. I still have this religion that if you knew the answer to every matrix question, somehow you could answer every operator question. But the ‘somehow’ would require genius. The problem is not, given an operator question, to ask the same question in finite dimensions– that’s silly. The problem is– the genius is– given an infinite question, to think of the right finite question to ask. Once you thought of the finite answer, then you would know the right answer to the infinite question.
Combinatorics, the finite case, is where the genuine, deep insight is. Generalizing, making it infinite, is sometimes intricate and sometimes difficult, and I might even be willing to say that it’s sometimes deep, but it is nowhere near as fundamental as seeing the finite structure.”
Finite Structure
on a Book Cover:
Walsh Series: An Introduction
to Dyadic Harmonic Analysis,
by F. Schipp
et al.,
Taylor & Francis, 1990
Halmos’s above remarks on combinatorics as a source of “deep mathematics” were in the context of operator theory. For connections between operator theory and harmonic analysis, see (for instance) H.S. Shapiro, “Operator Theory and Harmonic Analysis,” pp. 31-56 in
Twentieth Century Harmonic Analysis– A Celebration, ed. by J.S. Byrnes, published by Springer, 2001.
Walsh Series states that Walsh functions provide “the simplest non-trivial model for harmonic analysis.”
The patterns on the faces of the cube on the cover of
Walsh Series above illustrate both the
Walsh functions of order 3 and the same structure in a different guise,
subspaces of the affine 3-space over the binary field. For a note on the relationship of Walsh functions to finite geometry, see
Symmetry of Walsh Functions.
Whether the above sketch of the passage from operator theory to harmonic analysis to Walsh functions to finite geometry can ever help find “the right finite question to ask,” I do not know. It at least suggests that finite geometry (and my own work on models in finite geometry) may not be completely irrelevant to mathematics generally regarded as more deep.
* See the Log24 entries following Halmos’s death.
Comments Off on Saturday August 16, 2008
Thursday, August 14, 2008
Magister Ludi(
The Glass Bead Game)
is now available for
download in pdf or
text format at
Scribd.
“How far back the historian wishes to place the origins and antecedents of the Glass Bead Game is, ultimately, a matter of his personal choice. For like every great idea it has no real beginning; rather, it has always been, at least the idea of it. We find it foreshadowed, as a dim anticipation and hope, in a good many earlier ages. There are hints of it in Pythagoras, for example, and then among Hellenistic Gnostic circles in the late period of classical civilization. We find it equally among the ancient Chinese, then again at the several pinnacles of Arabic-Moorish culture; and the path of its prehistory leads on through Scholasticism and Humanism to the academies of mathematicians of the seventeenth and eighteenth centuries and on to the Romantic philosophies and the runes of Novalis’s hallucinatory visions. This same eternal idea, which for us has been embodied in the Glass Bead Game, has underlain every movement of Mind toward the ideal goal of a
universitas litterarum, every Platonic academy, every league of an intellectual elite, every
rapprochement between the exact and the more liberal disciplines, every effort toward reconciliation between science and art or science and religion. Men like Abelard, Leibniz, and Hegel unquestionably were familiar with the dream of capturing the universe of the intellect in concentric systems, and pairing the living beauty of thought and art with the magical expressiveness of the exact sciences. In that age in which music and mathematics almost simultaneously attained classical heights, approaches and cross-fertilizations between the two disciplines occurred frequently.”
— Hermann Hesse
Author’s dedication:
to the Journeyers
to the East
Related material:
Ring Theory
Comments Off on Thursday August 14, 2008
Monday, August 11, 2008
Comments Off on Monday August 11, 2008
Sunday, August 10, 2008
“Duration is… not a state of rest, for mere standstill is regression. Duration is rather the self-contained and therefore self-renewing movement of an organized, firmly integrated whole [click on link for an example], taking place in accordance with immutable laws and beginning anew at every ending.”
Globe at the
Beijing 2008 Olympics
Opening Ceremony
The eight trigrams
were perhaps implied in
the opening’s date,
8/8/8.
Comments Off on Sunday August 10, 2008
Friday, August 8, 2008
Click on image for details.
Comments Off on Friday August 8, 2008
Sunday, August 3, 2008
Kindergarten
Geometry
Preview of a Tom Stoppard play presented at Town Hall in Manhattan on March 14, 2008 (Pi Day and Einstein’s birthday):
The play’s title, “Every Good Boy Deserves Favour,” is a mnemonic for the notes of the treble clef EGBDF.
The place, Town Hall, West 43rd Street. The time, 8 p.m., Friday, March 14. One single performance only, to the tinkle– or the clang?– of a triangle. Echoing perhaps the clang-clack of Warsaw Pact tanks muscling into Prague in August 1968.
The “u” in favour is the British way, the Stoppard way, “EGBDF” being “a Play for Actors and Orchestra” by Tom Stoppard (words) and André Previn (music).
And what a play!– as luminescent as always where Stoppard is concerned. The music component of the one-nighter at Town Hall– a showcase for the Boston University College of Fine Arts– is by a 47-piece live orchestra, the significant instrument being, well, a triangle.
When, in 1974, André Previn, then principal conductor of the London Symphony, invited Stoppard “to write something which had the need of a live full-time orchestra onstage,” the 36-year-old playwright jumped at the chance.
One hitch: Stoppard at the time knew “very little about ‘serious’ music… My qualifications for writing about an orchestra,” he says in his introduction to the 1978 Grove Press edition of “EGBDF,” “amounted to a spell as a triangle player in a kindergarten percussion band.”
— Jerry Tallmer in The Villager, March 12-18, 2008
Review of the same play as presented at Chautauqua Institution on July 24, 2008:
“Stoppard’s modus operandi– to teasingly introduce numerous clever tidbits designed to challenge the audience.”
— Jane Vranish, Pittsburgh Post-Gazette, Saturday, August 2, 2008
“The leader of the band is tired
And his eyes are growing old
But his blood runs through
My instrument
And his song is in my soul.”
— Dan Fogelberg
“He’s watching us all the time.”
— Lucia Joyce
Finnegans Wake,
Book II, Episode 2, pp. 296-297:
I’ll make you to see figuratleavely the whome of your eternal geomater. And if you flung her headdress on her from under her highlows you’d wheeze whyse Salmonson set his seel on a hexengown.1 Hissss!, Arrah, go on! Fin for fun!
1 The chape of Doña Speranza of the Nacion. |
| ReciprocityFrom my entry of Sept. 1, 2003:
“…the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity….
… E. M. Forster famously advised his readers, ‘Only connect.’ ‘Reciprocity’ would be Michael Kruger’s succinct philosophy, with all that the word implies.”
— William Boyd, review of Himmelfarb, a novel by Michael Kruger, in The New York Times Book Review, October 30, 1994
Last year’s entry on this date:
The picture above is of the complete graph K6 … Six points with an edge connecting every pair of points… Fifteen edges in all.
Diamond theory describes how the 15 two-element subsets of a six-element set (represented by edges in the picture above) may be arranged as 15 of the 16 parts of a 4×4 array, and how such an array relates to group-theoretic concepts, including Sylvester’s synthematic totals as they relate to constructions of the Mathieu group M24.
If diamond theory illustrates any general philosophical principle, it is probably the interplay of opposites…. “Reciprocity” in the sense of Lao Tzu. See
Reciprocity and Reversal in Lao Tzu.
For a sense of “reciprocity” more closely related to Michael Kruger’s alleged philosophy, see the Confucian concept of Shu (Analects 15:23 or 24) described in
Shu: Reciprocity.
Kruger’s novel is in part about a Jew: the quintessential Jewish symbol, the star of David, embedded in the K6 graph above, expresses the reciprocity of male and female, as my May 2003 archives illustrate. The star of David also appears as part of a graphic design for cubes that illustrate the concepts of diamond theory:
Click on the design for details.
Those who prefer a Jewish approach to physics can find the star of David, in the form of K6, applied to the sixteen 4×4 Dirac matrices, in
A Graphical Representation
of the Dirac Algebra.
The star of David also appears, if only as a heuristic arrangement, in a note that shows generating partitions of the affine group on 64 points arranged in two opposing triplets.
Having thus, as the New York Times advises, paid tribute to a Jewish symbol, we may note, in closing, a much more sophisticated and subtle concept of reciprocity due to Euler, Legendre, and Gauss. See
The Jewel of Arithmetic and
|
FinnegansWiki:
Salmonson set his seel:
“Finn MacCool ate the Salmon of Knowledge.”
Wikipedia:
“George Salmon spent his boyhood in Cork City, Ireland. His father was a linen merchant. He graduated from Trinity College Dublin at the age of 19 with exceptionally high honours in mathematics. In 1841 at age 21 he was appointed to a position in the mathematics department at Trinity College Dublin. In 1845 he was appointed concurrently to a position in the theology department at Trinity College Dublin, having been confirmed in that year as an Anglican priest.”
Comments Off on Sunday August 3, 2008
Saturday, August 2, 2008
Comments Off on Saturday August 2, 2008
Thursday, July 31, 2008
Symmetry in Review
“Put bluntly, who is kidding whom?”
— Anthony Judge, draft of
“Potential Psychosocial Significance
of Monstrous Moonshine:
An Exceptional Form of Symmetry
as a Rosetta Stone for
Cognitive Frameworks,”
dated September 6, 2007.
Good question.
Also from
September 6, 2007 —
the date of
Madeleine L’Engle‘s death —
Related material:
1. The performance of a work by
Richard Strauss,
“Death and Transfiguration,”
(Tod und Verklärung, Opus 24)
by the Chautauqua Symphony
at Chautauqua Institution on
July 24, 2008
2. Headline of a music review
in today’s New York Times:
Welcoming a Fresh Season of
Transformation and Death
3. The picture of the R. T. Curtis
Miracle Octad Generator
on the cover of the book
Twelve Sporadic Groups:
4. Freeman Dyson’s hope, quoted by
Gorenstein in 1986, Ronan in 2006,
and Judge in 2007, that the Monster
group is “built in some way into
the structure of the universe.”
5. Symmetry from Plato to
the Four-Color Conjecture
6. Geometry of the 4×4 Square
7. Yesterday’s entry,
“Theories of Everything“
Coda:
“There is such a thing
as a tesseract.“
— Madeleine L’Engle
For a profile of
L’Engle, click on
the Easter eggs.
Comments Off on Thursday July 31, 2008
Wednesday, July 30, 2008
Like
Garrett Lisi’s, it is based on an unusual and highly symmetric mathematical structure. Lisi’s approach is related to the exceptional simple Lie group E
8.* Dharwadker uses
a structure long associated with the sporadic simple Mathieu group M
24.
| GRAND UNIFICATION
OF THE STANDARD MODEL WITH QUANTUM GRAVITY
by Ashay Dharwadker
Abstract
“We show that the mathematical proof of the four colour theorem [1] directly implies the existence of the standard model, together with quantum gravity, in its physical interpretation. Conversely, the experimentally observable standard model and quantum gravity show that nature applies the mathematical proof of the four colour theorem, at the most fundamental level. We preserve all the established working theories of physics: Quantum Mechanics, Special and General Relativity, Quantum Electrodynamics (QED), the Electroweak model and Quantum Chromodynamics (QCD). We build upon these theories, unifying all of them with Einstein’s law of gravity. Quantum gravity is a direct and unavoidable consequence of the theory. The main construction of the Steiner system in the proof of the four colour theorem already defines the gravitational fields of all the particles of the standard model. Our first goal is to construct all the particles constituting the classic standard model, in exact agreement with t’Hooft’s table [8]. We are able to predict the exact mass of the Higgs particle and the CP violation and mixing angle of weak interactions. Our second goal is to construct the gauge groups and explicitly calculate the gauge coupling constants of the force fields. We show how the gauge groups are embedded in a sequence along the cosmological timeline in the grand unification. Finally, we calculate the mass ratios of the particles of the standard model. Thus, the mathematical proof of the four colour theorem shows that the grand unification of the standard model with quantum gravity is complete, and rules out the possibility of finding any other kinds of particles.” |
* See, for instance, “The Scientific Promise of Perfect Symmetry” in The New York Times of March 20, 2007.
Comments Off on Wednesday July 30, 2008
Friday, July 25, 2008
56 Triangles
"This wonderful picture was drawn by Greg Egan with the help of ideas from Mike Stay and Gerard Westendorp. It's probably the best way for a nonmathematician to appreciate the symmetry of Klein's quartic. It's a 3-holed torus, but drawn in a way that emphasizes the tetrahedral symmetry lurking in this surface! You can see there are 56 triangles: 2 for each of the tetrahedron's 4 corners, and 8 for each of its 6 edges."
Exercise:
Click on image for further details.
Note that if eight points are arranged
in a cube (like the centers of the
eight subcubes in the figure above),
there are 56 triangles formed by
the 8 points taken 3 at a time.
Baez's discussion says that the Klein quartic's 56 triangles can be partitioned into 7 eight-triangle Egan "cubes" that correspond to the 7 points of the Fano plane in such a way that automorphisms of the Klein quartic correspond to automorphisms of the Fano plane. Show that the 56 triangles within the eightfold cube can also be partitioned into 7 eight-triangle sets that correspond to the 7 points of the Fano plane in such a way that (affine) transformations of the eightfold cube induce (projective) automorphisms of the Fano plane.
Comments Off on Friday July 25, 2008
Monday, July 21, 2008
Knight Moves:
The Relativity Theory
of Kindergarten Blocks
(Continued from
January 16, 2008)
Something:
From Friedrich Froebel,
who invented kindergarten:
Click on image for details.
An Unusually
Complicated Theory:
From Christmas 2005:
Click on image for details.
For the eightfold cube
as it relates to Klein's
simple group, see
"A Reflection Group
of Order 168."
For an even more
complicated theory of
Klein's simple group, see
Click on image for details.
Comments Off on Monday July 21, 2008
Saturday, July 19, 2008
Bertram Kostant, Professor Emeritus of Mathematics at MIT, on an object discussed in this week's New Yorker:
"
A word about E(8). In my opinion, and shared by others, E(8) is the most magnificent 'object' in all of mathematics. It is like a diamond with thousands of facets. Each facet offering a different view of its unbelievable intricate internal structure."
Hermann Weyl on the hard core of objectivity:
"Perhaps the philosophically most relevant feature of modern science is the emergence of abstract symbolic structures as the hard core of objectivity behind– as Eddington puts it– the colorful tale of the subjective storyteller mind." (Philosophy of Mathematics and Natural Science, Princeton, 1949, p. 237)
Steven H. Cullinane on the symmetries of a 4×4 array of points:
|
A Structure-Endowed Entity
"A guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity S, try to determine its group of automorphisms, the group of those element-wise transformations which leave all structural relations undisturbed. You can expect to gain a deep insight into the constitution of S in this way."
— Hermann Weyl in Symmetry
Let us apply Weyl's lesson to the following "structure-endowed entity."
What is the order of the resulting group of automorphisms?
|
The above group of
automorphisms plays
a role in what Weyl,
following Eddington,
called a "colorful tale"–
Comments Off on Saturday July 19, 2008
Thursday, July 17, 2008
CHANGE
FEW CAN BELIEVE IN
|
Continued from June 18.
Jungian Symbols
of the Self —
Compare and contrast:
Jung's four-diamond figure from
Aion — a symbol of the self —
Jung's Map of the Soul,
by Murray Stein:
"… Jung thinks of the self as undergoing continual transformation during the course of a lifetime…. At the end of his late work
Aion, Jung presents a diagram to illustrate the dynamic movements of the self…."
Comments Off on Thursday July 17, 2008
Wednesday, July 9, 2008
God, Time, Epiphany
8:28:32 AM
Anthony Hopkins, from
All Hallows' Eve
last year:
"For me time is God,
God is time. It's an equation,
like an Einstein equation."
James Joyce, from
June 26 (the day after
AntiChristmas) this year:
"… he glanced up at the clock
of the Ballast Office and smiled:
— It has not epiphanised yet,
he said."
Ezra Pound (from a page
linked to yesterday morning):
"It seems quite natural to me
that an artist should have
just as much pleasure in an
arrangement of planes
or in a pattern of figures,
as in painting portraits…."
From Epiphany 2008:
An arrangement of planes:
From May 10, 2008:
A pattern of figures:
See also Richard Wilhelm on
Hexagram 32 of the I Ching:
"Duration is a state whose movement is not worn down by hindrances. It is not a state of rest, for mere standstill is regression. Duration is rather the self-contained and therefore self-renewing movement of an organized, firmly integrated whole, taking place in accordance with immutable laws and beginning anew at every ending. The end is reached by an inward movement, by inhalation, systole, contraction, and this movement turns into a new beginning, in which the movement is directed outward, in exhalation, diastole, expansion."
— The Middle-English
Harrowing of Hell…
by Hulme, 1907, page 64
Comments Off on Wednesday July 9, 2008
Wednesday, June 25, 2008
“The Renaissance thinkers liked to
organize the four elements using
a chain of analogies running
from light to heavy:
fire : air :: air : water :: water : earth
They also organized them
in a diamond, like this:”
This figure of Baez
is related to a saying
attributed to Heraclitus:
For related thoughts by Jung,
see Aion, which contains the
following diagram:
“The formula reproduces exactly the essential features of the symbolic process of transformation. It shows the rotation of the mandala, the antithetical play of complementary (or compensatory) processes, then the apocatastasis, i.e., the restoration of an original state of wholeness, which the alchemists expressed through the symbol of the uroboros, and finally the formula repeats the ancient alchemical tetrameria, which is implicit in the fourfold structure of unity.”
— Carl Gustav Jung
That the words Maximus of Tyre (second century A.D.) attributed to Heraclitus imply a cycle of the elements (analogous to the rotation in Jung’s diagram) is not a new concept. For further details, see “The Rotation of the Elements,” a 1995 webpage by one “John Opsopaus.”
Related material:
Log24 entries of June 9, 2008, and
“Quintessence: A Glass Bead Game,”
by Charles Cameron.
Comments Off on Wednesday June 25, 2008
Tuesday, June 24, 2008
Plato’s Cave, continued:
… we know that we use
Only the eye as faculty, that the mind
Is the eye, and that this landscape of the mind
Is a landscape only of the eye; and that
We are ignorant men incapable
Of the least, minor, vital metaphor….
— Wallace Stevens, “Crude Foyer”
… So, so,
O son of man, the ignorant night, the travail
Of early morning, the mystery of the beginning
Again and again,
while history is unforgiven.
— Delmore Schwartz,
“In the Naked Bed, in Plato’s Cave“
Comments Off on Tuesday June 24, 2008
Thursday, June 19, 2008
Soul Theorem
“The soul of the commonest object,
the structure of which is so adjusted,
seems to us radiant. The object
achieves its epiphany.”
— James Joyce, Stephen Hero
Above: Screenshot of today’s
New York Times obituary for
mathematician Detlef Gromoll,
known for the “soul theorem.”
Gromoll died on May 31
according to his son
Hans Christian.
From his obituary:
“Detlef Gromoll was born in Berlin
in 1938, and his childhood
was disrupted by the falling
bombs of World War II.”
Related material:
The discussion here
on June 1 of a lottery number
from the date of Gromoll’s death,
childhood, mathematics,
and prewar Berlin.
Comments Off on Thursday June 19, 2008
Wednesday, June 18, 2008
CHANGE
FEW CAN BELIEVE IN |
What I Loved, a novel by Siri Hustvedt (New York, Macmillan, 2003), contains a paragraph on the marriage of a fictional artist named Wechsler–
Page 67 —
“… Bill and Violet were married. The wedding was held in the Bowery loft on June 16th, the same day Joyce’s Jewish Ulysses had wandered around Dublin. A few minutes before the exchange of vows, I noted that Violet’s last name, Blom, was only an o away from Bloom, and that meaningless link led me to reflect on Bill’s name, Wechsler, which carries the German root for change, changing, and making change. Blooming and changing, I thought.”
For Hustvedt’s discussion of Wechsler’s art– sculptured cubes, which she calls “tightly orchestrated semantic bombs” (p. 169)– see Log24, May 25, 2008.
Related material:
Wechsler cubes
(after David Wechsler,
1896-1981, chief
psychologist at Bellevue)
These cubes are used to
make 3×3 patterns for
psychological testing.
Related 3×3 patterns appear
in “nine-patch” quilt blocks
and in the following–
Don Park at docuverse.com, Jan. 19, 2007:
“How to draw an Identicon
A 9-block is a small quilt using only 3 types of patches, out of 16 available, in 9 positions. Using the identicon code, 3 patches are selected: one for center position, one for 4 sides, and one for 4 corners.
Positions and Rotations
For center position, only a symmetric patch is selected (patch 1, 5, 9, and 16). For corner and side positions, patch is rotated by 90 degree moving clock-wise starting from top-left position and top position respectively.” |
| Jared Tarbell at levitated.net, May 15, 2002:
“The nine block is a common design pattern among quilters. Its construction methods and primitive building shapes are simple, yet produce millions of interesting variations.
Figure A. Four 9 block patterns, arbitrarily assembled, show the
grid composition of the block.
Each block is composed of 9 squares, arranged in a 3 x 3 grid. Each square is composed of one of 16 primitive shapes. Shapes are arranged such that the block is radially symmetric. Color is modified and assigned arbitrarily to each new block.
The basic building blocks of the nine block are limited to 16 unique geometric shapes. Each shape is allowed to rotate in 90 degree increments. Only 4 shapes are allowed in the center position to maintain radial symmetry.
Figure B. The 16 possible shapes allowed
for each grid space. The 4 shapes allowed
in the center have bold numbers.”
|
Such designs become of mathematical interest when their size is increased slightly, from square arrays of nine blocks to square arrays of sixteen. See Block Designs in Art and Mathematics.
(This entry was suggested by examples of 4×4 Identicons in use at Secret Blogging Seminar.)
Comments Off on Wednesday June 18, 2008
Monday, June 16, 2008
Bloomsday for Nash:
The Revelation Game
(American Mathematical Society Feb. 2008
review of Steven Brams’s Superior Beings:
If They Exist, How Would We Know?)
(pdf, 15 megabytes)
"Brams does not attempt to prove or disprove God. He uses elementary ideas from game theory to create situations between a Person (P) and God (Supreme Being, SB) and discusses how each reacts to the other in these model scenarios….
In the 'Revelation Game,' for example, the Person (P) has two options:
1) P can believe in SB's existence
2) P can not believe in SB's existence
The Supreme Being also has two options:
1) SB can reveal Himself
2) SB can not reveal Himself
Each player also has a primary and secondary goal. For the Person, the primary goal is to have his belief (or non-belief) confirmed by evidence (or lack thereof). The secondary goal is to 'prefer to believe in SB’s existence.' For the Supreme Being, the primary goal is to have P believe in His existence, while the secondary goal is to not reveal Himself. These goals allow us to rank all the outcomes for each player from best (4) to worst (1). We end up with a matrix as follows (the first number in the parentheses represents the SB's ranking for that box; the second number represents P's ranking):
The question we must answer is: what is the Nash equilibrium in this case?"
Analogously:
Lotteries on
Bloomsday,
June 16,
2008
|
Pennsylvania
(No revelation)
|
New York
(Revelation)
|
Mid-day
(No belief)
|
418
No belief,
no revelation
|
064
Revelation
without belief
|
Evening
(Belief)
|
709
Belief without
revelation
|
198
(A Cheap
Epiphany)
Belief and
revelation
|
The holy image
denoting belief and revelation
may be interpreted as
a black hole or as a
symbol by James Joyce:
|
When?
Going to dark bed there was a square round Sinbad the Sailor roc's auk's egg in the night of the bed of all the auks of the rocs of Darkinbad the Brightdayler.
Where?
— Ulysses, conclusion of Chapter 17
|
Comments Off on Monday June 16, 2008
Saturday, June 14, 2008
Cross and Wheel
An online tribute to Tim Russert
this morning had a song by a
Russert favorite, Bruce Springsteen:
"Wearin' the cross
of my calling,
on wheels of fire
I come rollin' down here."
— "The Rising"
Related material:
Hard Lessons
and the
five Log24 entries
ending on July 20, 2006,
which contain the following
example of what might be
caled "sacred order"
(see yesterday's entries)–
See also "Grave Matters" here
on November 8, 2006, and
the same date four years earlier,
as well as
"O Grave, Where Is Thy Victory?"
(pdf), a lecture by Jack Miles
at Clark Art Institute
(see Oct. 7-9, 2002)
on November 9, 2002.
The Miles lecture may be of
more comfort to Russert's
mourners than the
cross/wheel symbolism,
which has its dark side.
The cross, the wheel,
the Catholic faith, and
Russert's field of expertise,
politics, are of course
notably combined in the
crux gammata, discussed
here in a 2002 entry on
the Triumph of the Cross
and the Death of Grace
(Princess of Monaco).
Comments Off on Saturday June 14, 2008
Tuesday, June 10, 2008
Return to Paradise
Edward Rothstein's review in yesterday's New York Times–
Museum’s Vision:
West Coast Paradise —
seems to me more a description of Hell.
My own concept of paradise is closer to the Gary Cooper film "Return to Paradise," which impressed me greatly when I saw it on TV when I was in 10th grade.
A related vision: two frames from the Jodie Foster film "Contact"–
Comments Off on Tuesday June 10, 2008
Monday, June 9, 2008
Lying Rhymes
Readers of the previous entry
who wish to practice their pardes
may contemplate the following:
The evening 563 may, as in other recent entries, be interpreted as a page number in
Gravity’s Rainbow (Penguin Classics, 1995). From
that page:
“He brings out the mandala he found.
‘What’s it mean?’
[….]
Slothrop gives him the mandala. He hopes it will work like the mantra that Enzian told him once, mba-kayere (I am passed over), mba-kayere… a spell […]. A mezuzah. Safe passage through a bad night….”
Christ and the Four Elements
This 1495 image is found in
The Janus Faces of Genius:
The Role of Alchemy
in Newton’s Thought,
by B. J. T. Dobbs,
Cambridge University Press,
2002, p. 85
Related mandalas:
and
For further details,
click on any of the
three mandalas above.
“For every kind of vampire,
there is a kind of cross.”
— Thomas Pynchon, quoted
here on 9/13, 2007
(As for today’s New York Lottery midday number 007, see (for instance) Edward Rothstein in today’s New York Times on paradise, and also Tom Stoppard on heaven as “just a lying rhyme” for seven.)
Time of entry: 10:20:55 PM
Comments Off on Monday June 9, 2008
Interpret This
"With respect, you only interpret."
"Countries have gone to war
after misinterpreting one another."
— The Interpreter
"Once upon a time (
say, for Dante),
it must have been a revolutionary
and creative move to design works
of art so that they might be
experienced on several levels."
— Susan Sontag,
"Against Interpretation"
Edward Rothstein in today's
New York Times review of San Francisco's new Contemporary Jewish Museum:
"An introductory wall panel tells us that in the Jewish mystical tradition the four letters [in Hebrew] of pardes each stand for a level of biblical interpretation: very roughly, the literal, the allusive, the allegorical and the hidden. Pardes, we are told, became the museum’s symbol because it reflected the museum’s intention to cultivate different levels of interpretation: 'to create an environment for exploring multiple perspectives, encouraging open-mindedness' and 'acknowledging diverse backgrounds.' Pardes is treated as a form of mystical multiculturalism.
But even the most elaborate interpretations of a text or tradition require more rigor and must begin with the literal. What is being said? What does it mean? Where does it come from and where else is it used? Yet those are the types of questions– fundamental ones– that are not being asked or examined […].
How can multiple perspectives and open-mindedness and diverse backgrounds be celebrated without a grounding in knowledge, without history, detail, object and belief?"
"It's the system that matters.
How the data arrange
themselves inside it."
— Gravity's Rainbow
"Examples are the stained-
glass windows of knowledge."
— Vladimir Nabokov
Click on image to enlarge.
Comments Off on Monday June 9, 2008
Sunday, June 1, 2008
The conclusion of yesterday’s commentary on the May 30-31 Pennsylvania Lottery numbers:
Thomas Pynchon, Gravity’s Rainbow:
“The fear balloons again inside his brain. It will not be kept down with a simple Fuck You…. A smell, a forbidden room, at the bottom edge of his memory. He can’t see it, can’t make it out. Doesn’t want to. It is allied with the Worst Thing.
He knows what the smell has to be: though according to these papers it would have been too early for it, though he has never come across any of the stuff among the daytime coordinates of his life, still, down here, back here in the warm dark, among early shapes where the clocks and calendars don’t mean too much, he knows that’s what haunting him now will prove to be the smell of Imipolex G.
Then there’s this recent dream he is afraid of having again. He was in his old room, back home. A summer afternoon of lilacs and bees and
286”
What are we to make of this enigmatic 286? (No fair peeking at page 287.)
One possible meaning, given The Archivist‘s claim that “existence is infinitely cross-referenced”–
Page 286 of Ernest G. Schachtel, Metamorphosis: On the Conflict of Human Development and the Psychology of Creativity (first published in 1959), Hillsdale NJ and London, The Analytic Press, 2001 (chapter– “On Memory and Childhood Amnesia”):
“Both Freud and Proust speak of the autobiographical [my italics] memory, and it is only with regard to this memory that the striking phenomenon of childhood amnesia and the less obvious difficulty of recovering any past experience may be observed.”
The concluding “summer afternoon of lilacs and bees” suggests that 286 may also be a chance allusion to the golden afternoon of Disney’s Alice in Wonderland. (Cf. St. Sarah’s Day, 2008)
Some may find the Disney afternoon charming; others may see it as yet another of Paul Simon’s dreaded cartoon graveyards.
More tastefully, there is poem 286 in the 1919 Oxford Book of English Verse– “Love.”
For a midrash on this poem, see Simone Weil, who became acquainted with the poem by chance:
“I always prefer saying chance rather than Providence.”
— Simone Weil, letter of about May 15, 1942
Weil’s brother André might prefer Providence (source of the Bulletin of the American Mathematical Society.)
Related material:
Log24, December 20, 2003–
White, Geometric, and Eternal—
A description in
Gravity’s Rainbow of prewar Berlin as “white and geometric” suggested, in combination with a reference elsewhere to “the eternal,” a citation of the following
illustration of the concept “white, geometric, and eternal”–
For more on the mathematical significance of this figure, see (for instance) Happy Birthday, Hassler Whitney, and Combinatorics of Coxeter Groups, by Anders Björner and Francesco Brenti, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005.
This book is reviewed in the current issue (July 2008) of the above-mentioned Providence Bulletin.
The review in the Bulletin discusses reflection groups in continuous spaces.
Comments Off on Sunday June 1, 2008
Monday, May 26, 2008
Comments Off on Monday May 26, 2008
Sunday, May 25, 2008
Hall of Mirrors
Epigraph to
“Deploying the Glass Bead Game, Part II,”
by Robert de Marrais:
“For a complete logical argument,”
Arthur began
with admirable solemnity,
“we need two prim Misses –”
“Of course!” she interrupted.
“I remember that word now.
And they produce — ?”
“A Delusion,” said Arthur.
— Lewis Carroll,
Sylvie and Bruno
Prim Miss 1:
Erin O’Connor’s weblog “Critical Mass” on May 24:
Roger Rosenblatt’s Beet [Ecco hardcover, Jan. 29, 2008] is the latest addition to the noble sub-genre of campus fiction….
Curricular questions and the behavior of committees are at once dry as dust subjects and areas ripe for sarcastic send-up– not least because, as dull as they are, they are really both quite vital to the credibility and viability of higher education.
Here’s an excerpt from the first meeting, in which committee members propose their personal plans for a new, improved curriculum:
“… Once the students really got into playing with toy soldiers, they would understand history with hands-on excitement.”
To demonstrate his idea, he’d brought along a shoe box full of toy doughboys and grenadiers, and was about to reenact the Battle of Verdun on the committee table when Heilbrun stayed his hand. “We get it,” he said.
“That’s quite interesting, Molton,” said Booth [a chemist]. “But is it rigorous enough?”
At the mention of the word, everyone, save Peace, sat up straight.
“Rigor is so important,” said Kettlegorf.
“We must have rigor,” said Booth.
“You may be sure,” said the offended Kramer. “I never would propose anything lacking rigor.”
Smythe inhaled and looked at the ceiling. “I think I may have something of interest,” he said, as if he were at a poker game and was about to disclose a royal flush. “My proposal is called ‘Icons of Taste.’ It would consist of a galaxy of courses affixed to several departments consisting of lectures on examples of music, art, architecture, literature, and other cultural areas a student needed to indicate that he or she was sophisticated.”
“Why would a student want to do that?” asked Booth.
“Perhaps sophistication is not a problem for chemists,” said Smythe. Lipman tittered.
“What’s the subject matter?” asked Heilbrun. “Would it have rigor?”
“Of course it would have rigor. Yet it would also attract those additional students Bollovate is talking about.” Smythe inhaled again. “The material would be carefully selected,” he said. “One would need to pick out cultural icons the students were likely to bring up in conversation for the rest of their lives, so that when they spoke, others would recognize their taste as being exquisite yet eclectic and unpredictable.”
“You mean Rembrandt?” said Kramer.
Smythe smiled with weary contempt. “No, I do not mean Rembrandt. I don’t mean Beethoven or Shakespeare, either, unless something iconic has emerged about them to justify their more general appeal.”
“You mean, if they appeared on posters,” said Lipman.
“That’s it, precisely.”
Lipman blushed with pride.
“The subject matter would be fairly easy to amass,” Smythe said. “We could all make up a list off the top of our heads. Einstein–who does have a poster.” He nodded to the ecstatic Lipman. “Auden, for the same reason. Students would need to be able to quote ‘September 1939[ or at least the last lines. And it would be good to teach ‘Musee des Beaux Arts’ as well, which is off the beaten path, but not garishly. Mahler certainly. But Cole Porter too. And Sondheim, I think. Goya. Warhol, it goes without saying, Stephen Hawking, Kurosawa, Bergman, Bette Davis. They’d have to come up with some lines from Dark Victory, or better still, Jezebel. La Dolce Vita. Casablanca. King of Hearts. And Orson, naturally. Citizen Kane, I suppose, though personally I prefer F for Fake.”
“Judy!” cried Heilbrun.
“Yes, Judy too. But not ‘Over the Rainbow.’ It would be more impressive for them to do ‘The Trolley Song,’ don’t you think?” Kettlegorf hummed the intro.
“Guernica,” said Kramer. “Robert Capa.” 
“Edward R. Murrow,” said Lipman.
“No! Don’t be ridiculous!” said Smythe, ending Lipman’s brief foray into the world of respectable thought.
“Marilyn Monroe!” said Kettlegorf.
“Absolutely!” said Smythe, clapping to indicate his approval.
“And the Brooklyn Bridge,” said Booth, catching on. “And the Chrysler Building.”
“Maybe,” said Smythe. “But I wonder if the Chrysler Building isn’t becoming something of a cliche.”
Peace had had enough. “And you want students to nail this stuff so they’ll do well at cocktail parties?”
Smythe sniffed criticism, always a tetchy moment for him. “You make it sound so superficial,” he said.
Prim Miss 2:
Siri Hustvedt speaks at Adelaide Writers’ Week– a story dated March 24, 2008 †—
“I have come to think of my books as echo chambers or halls of mirrors in which themes, ideas, associations continually reflect and reverberate inside a text. There is always point and counterpoint, to use a musical illustration. There is always repetition with difference.”
A Delusion:
Exercise — Identify in the following article the sentence that one might (by unfairly taking it out of context) argue is a delusion.
(Hint: See Reflection Groups in Finite Geometry.)
Why Borovik’s Figure 4
is included above:
Comments Off on Sunday May 25, 2008
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) –
Part IV:
A Magic Gallery
|
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
|
Comments Off on Sunday May 25, 2008
Thursday, May 22, 2008
The Undertaking:
An Exercise in
Conceptual Art
Hexagram 54:
THE JUDGMENT
Undertakings bring misfortune.
Nothing that would further.
“Brian O’Doherty, an Irish-born artist,
before the [Tuesday, May 20] wake
of his alter ego* ‘Patrick Ireland’
on the grounds of the
Irish Museum of Modern Art.”
— New York Times, May 22, 2008
THE IMAGE
Thus the superior man
understands the transitory
in the light of
the eternity of the end.
Another version of
the image:
See 2/22/08
and 4/19/08.
Related material:
Michael Kimmelman in today’s New York Times—
“An essay from the ’70s by Mr. O’Doherty, ‘Inside the White Cube,’ became famous in art circles for describing how modern art interacted with the gallery spaces in which it was shown.”
Brian O’Doherty, “Inside the White Cube,” 1976 Artforum essays on the gallery space and 20th-century art:
“The history of modernism is intimately framed by that space. Or rather the history of modern art can be correlated with changes in that space and in the way we see it. We have now reached a point where we see not the art but the space first…. An image comes to mind of a white, ideal space that, more than any single picture, may be the archetypal image of 20th-century art.”
An archetypal image
THE SPACE:
A non-archetypal image
THE ART:
Natasha Wescoat, 2004
See also
Epiphany 2008:
“Nothing that would further.”
— Hexagram 54
|
Lear’s fool:
…. Now thou art an 0
without a figure. I am better
than thou art, now. I am a fool;
thou art nothing….
|
“…. in the last mystery of all the single figure of what is called the World goes joyously dancing in a state beyond moon and sun, and the number of the Trumps is done. Save only for that which has no number and is called the Fool, because mankind finds it folly till it is known. It is sovereign or it is nothing, and if it is nothing then man was born dead.”
— The Greater Trumps,
by Charles Williams, Ch. 14
Comments Off on Thursday May 22, 2008
Sunday, May 18, 2008
From the Grave
DENNIS OVERBYE
in yesterday's New York Times:
"From the grave, Albert Einstein
poured gasoline on the culture wars
between science and religion this week…."
An announcement of a
colloquium at Princeton:
Above: a cartoon,
"Coxeter exhuming Geometry,"
with the latter's tombstone inscribed
"GEOMETRY
600 B.C. —
1900 A.D.
R.I.P."
The above is from
The Paradise of Childhood,
a work first published in 1869.
"I need a photo-opportunity,
I want a shot at redemption.
Don't want to end up a cartoon
In a cartoon graveyard."
— Paul Simon
Albert Einstein,
1879-1955:
"It is quite clear to me that the religious paradise of youth, which was thus lost, was a first attempt to free myself from the chains of the 'merely-personal,' from an existence which is dominated by wishes, hopes and primitive feelings. Out yonder there was this huge world, which exists independently of us human beings and which stands before us like a great, eternal riddle, at least partially accessible to our inspection and thinking. The contemplation of this world beckoned like a liberation…."
— Autobiographical Notes, 1949
Related material:
A commentary on Tom Wolfe's
"Sorry, but Your Soul Just Died"–
"The Neural Buddhists," by David Brooks,
in the May 13 New York Times:
"The mind seems to have
the ability to transcend itself
and merge with a larger
presence that feels more real."
A New Yorker commentary on
a new translation of the Psalms:
"Suddenly, in a world without
Heaven, Hell, the soul, and
eternal salvation or redemption,
the theological stakes seem
more local and temporal:
'So teach us to number our days.'"
and a May 13 Log24 commentary
on Thomas Wolfe's
"Only the Dead Know Brooklyn"–
"… all good things — trout as well as
eternal salvation — come by grace
and grace comes by art
and art does not come easy."
— A River Runs Through It
"Art isn't easy."
— Stephen Sondheim,
quoted in
Solomon's Cube.
For further religious remarks,
consult Indiana Jones and the
Kingdom of the Crystal Skull
and The Librarian:
Return to King Solomon's Mines.
Comments Off on Sunday May 18, 2008
Friday, May 16, 2008
“From the grave, Albert Einstein poured gasoline on the culture wars between science and religion this week.
A letter the physicist wrote in 1954 to the philosopher Eric Gutkind, in which he described the Bible as ‘pretty childish’ and scoffed at the notion that the Jews could be a ‘chosen people,’ sold for $404,000 at an auction in London. That was 25 times the presale estimate.”
A less controversial Einstein-related remark:
“The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time.”
— Hermann Weyl, “Relativity Theory as a Stimulus in Mathematical Research,” Proceedings of the American Philosophical Society, Vol. 93, No. 7, Theory of Relativity in Contemporary Science: Papers Read at the Celebration of the Seventieth Birthday of Professor Albert Einstein in Princeton, March 19, 1949 (Dec. 30, 1949), pp. 535-541
Comments Off on Friday May 16, 2008
Saturday, May 10, 2008
MoMA Goes to
Kindergarten
"… the startling thesis of Mr. Brosterman's new book, 'Inventing Kindergarten' (Harry N. Abrams, $39.95): that everything the giants of modern art and architecture knew about abstraction they learned in kindergarten, thanks to building blocks and other educational toys designed by Friedrich Froebel, a German educator, who coined the term 'kindergarten' in the 1830's."
— "Was Modernism Born
in Toddler Toolboxes?"
by Trip Gabriel, New York Times,
April 10, 1997
RELATED MATERIAL
Figure 1 —
Concept from 1819:
(Footnotes 1 and 2)
Figure 2 —
The Third Gift, 1837:
Froebel's Third Gift
Froebel, the inventor of
kindergarten, worked as
an assistant to the
crystallographer Weiss
mentioned in Fig. 1.
(Footnote 3)
Figure 3 —
The Third Gift, 1906:
Figure 4 —
Solomon's Cube,
1981 and 1983:
Figure 5 —
Design Cube, 2006:
For some mathematical background, see
Footnotes:
Comments Off on Saturday May 10, 2008
Friday, May 9, 2008
Comments Off on Friday May 9, 2008
Wednesday, May 7, 2008
Forms of the Rock
“point A / In a perspective
that begins again / At B”
— Wallace Stevens,
The Rock
See also
August 2, 2002
January 20, 2003
April 8, 2003
December 5, 2004
December 10, 2004
January 11, 2006
April 30, 2006
August 25, 2006
August 26, 2006
February 6, 2007
July 23, 2007
July 24, 2007
September 30, 2007
April 14, 2008
Christmas Eve, 1981
Comments Off on Wednesday May 7, 2008
Friday, May 2, 2008
A Balliol Star
In memory of
mathematician
Graham Higman of
Balliol College and
Magdalen College,
Oxford,
Jan. 19, 1917 –
April 8, 2008
From a biography of an earlier Balliol student,
Gerard Manley Hopkins (1844-1889):
"In 1867 he won First-Class degrees in Classics
and 'Greats' (a rare 'double-first') and was
considered by Jowett to be the star of Balliol."
Hopkins, a poet who coined the term "inscape," was a member of the Society of Jesus.
According to a biography, Higman was the founder of Oxford's Invariant Society.
From a publication of that society, The Invariant, Issue 15– undated but (according to Issue 16, of 2005) from 1996 (pdf):
Taking the square root
of a function
by Ian Collier
"David Singmaster once gave a talk at the Invariants and afterwards asked this question:
What is the square root of the exponential function? In other words, can you define a function f such that for all x, f 2(x) — that is, f (f (x)) — is equal to e x ? He did not give the answer straight away so I attempted it…." |
Another approach to the expression f(f(x)), by myself in 1982:
For further details,
see Inscapes.
For more about Higman, see an interview in the September 2001 newsletter of the European Mathematical Society (pdf).
"Philosophers ponder the idea
of identity: what it is to give
something a name
on Monday
and have it respond to
that name on Friday…."
— Bernard Holland
Comments Off on Friday May 2, 2008
Monday, April 28, 2008
Religious Art
The black monolith of
Kubrick's 2001 is, in
its way, an example
of religious art.
One artistic shortcoming
(or strength– it is, after
all, monolithic) of
that artifact is its
resistance to being
analyzed as a whole
consisting of parts, as
in a Joycean epiphany.
The following
figure does
allow such
an epiphany.
One approach to
the epiphany:
"Transformations play
a major role in
modern mathematics."
– A biography of
Felix Christian Klein
The above 2×4 array
(2 columns, 4 rows)
furnishes an example of
a transformation acting
on the parts of
an organized whole:
For other transformations
acting on the eight parts,
hence on the 35 partitions, see
"Geometry of the 4×4 Square,"
as well as Peter J. Cameron's
"The Klein Quadric
and Triality" (pdf),
and (for added context)
"The Klein Correspondence,
Penrose Space-Time, and
a Finite Model."
For a related structure–
not rectangle but cube–
see Epiphany 2008.
Comments Off on Monday April 28, 2008
Sunday, April 27, 2008
Happy Birthday
to the late
Gian-Carlo Rota,
mathematician and
scholar of philosophy
Rota* on his favorite philosopher:
“I believe Husserl to be the greatest philosopher of all times….
Intellectual honesty is the striking quality of Husserl’s writings. He wrote what he honestly believed to be true, neither more nor less. However, honesty is not clarity; as a matter of fact, honesty and clarity are at opposite ends. Husserl proudly refused to stoop to the demands of showmanship that are indispensable in effective communication.”
Related material:
The Diamond Theorem
* Gian-Carlo Rota, “Ten Remarks on Husserl and Phenomenology,” in O.K. Wiegand et al. (eds.), Phenomenology on Kant, German Idealism, Hermeneutics and Logic, pp. 89-97, Kluwer Academic Publishers, 2000
Comments Off on Sunday April 27, 2008
Saturday, April 26, 2008
Mere Philosophy
In Memory of
Edmund Husserl
“Mereology (from the Greek μερος, ‘part’)
is the theory of parthood relations:
of the relations of part to whole and the
relations of part to part within a whole.
Its roots can be traced back to
the early days of philosophy….”
— Stanford Encyclopedia of Philosophy
“Beauty is the proper conformity
of the parts to one another
and to the whole.”
— Classic definition quoted
by Werner Heisenberg
(Log24, May 18-20, 2005)
“It seems, as one becomes older,
That the past has another pattern,
and ceases to be a mere sequence….”
— T. S. Eliot, Four Quartets
See also Time Fold
and Theme and Variations.
Comments Off on Saturday April 26, 2008
Saturday, April 19, 2008
A Midrash for Benedict
On April 16, the Pope’s birthday, the evening lottery number in Pennsylvania was 441. The Log24 entries of April 17 and April 18 supplied commentaries based on 441’s incarnation as a page number in an edition of Heidegger’s writings. Here is a related commentary on a different incarnation of 441. (For a context that includes both today’s commentary and those of April 17 and 18, see Gian-Carlo Rota– a Heidegger scholar as well as a mathematician– on mathematical Lichtung.)
From R. D. Carmichael, Introduction to the Theory of Groups of Finite Order (Boston, Ginn and Co., 1937)– an exercise from the final page, 441, of the final chapter, “Tactical Configurations”–
“23. Let G be a multiply transitive group of degree n whose degree of transitivity is k; and let G have the property that a set S of m elements exists in G such that when k of the elements S are changed by a permutation of G into k of these elements, then all these m elements are permuted among themselves; moreover, let G have the property P, namely, that the identity is the only element in G which leaves fixed the n – m elements not in S. Then show that G permutes the m elements S into
n(
n -1) … (
n –
k + 1)
____________________
m(m – 1) … (m – k + 1)
sets of
m elements each, these sets forming a configuration having the property that any (whatever) set of
k elements appears in one and just one of these sets of
m elements each. Discuss necessary conditions on
m,
n,
k in order that the foregoing conditions may be realized. Exhibit groups illustrating the theorem.”
This exercise concerns an important mathematical structure said to have been discovered independently by the American Carmichael and by the German Ernst Witt.
For some perhaps more comprehensible material from the preceding page in Carmichael– 440– see Diamond Theory in 1937.
Comments Off on Saturday April 19, 2008
Monday, April 14, 2008
Classical Quantum
From this morning's
New York Times:
"John A. Wheeler, a visionary physicist… died Sunday morning [April 13, 2008]….
… Dr. Wheeler set the agenda for generations of theoretical physicists, using metaphor as effectively as calculus to capture the imaginations of his students and colleagues and to pose questions that would send them, minds blazing, to the barricades to confront nature….
'He rejuvenated general relativity; he made it an experimental subject and took it away from the mathematicians,' said Freeman Dyson, a theorist at the Institute for Advanced Study….
… he [Wheeler] sailed to Copenhagen to work with Bohr, the godfather of the quantum revolution, which had shaken modern science with paradoxical statements about the nature of reality.
'You can talk about people like Buddha, Jesus, Moses, Confucius, but the thing that convinced me that such people existed were the conversations with Bohr,' Dr. Wheeler said….
… Dr. Wheeler was swept up in the Manhattan Project to build an atomic bomb. To his lasting regret, the bomb was not ready in time to change the course of the war in Europe….
Dr. Wheeler continued to do government work after the war, interrupting his research to help develop the hydrogen bomb, promote the building of fallout shelters and support the Vietnam War….
… Dr. Wheeler wondered if this quantum uncertainty somehow applied to the universe and its whole history, whether it was the key to understanding why anything exists at all.
'We are no longer satisfied with insights only into particles, or fields of force, or geometry, or even space and time,' Dr. Wheeler wrote in 1981. 'Today we demand of physics some understanding of existence itself.'
At a 90th birthday celebration in 2003, Dr. Dyson said that Dr. Wheeler was part prosaic calculator, a 'master craftsman,' who decoded nuclear fission, and part poet. 'The poetic Wheeler is a prophet,' he said, 'standing like Moses on the top of Mount Pisgah, looking out over the promised land that his people will one day inherit.'"
— Dennis Overbye, The New York Times,
Monday, April 14, 2008
As prophets go, I prefer
the poet Wallace Stevens:
"point A / In a perspective
that begins again / At B"
— Wallace Stevens,
"The Rock"
Comments Off on Monday April 14, 2008
Sunday, April 13, 2008
The Echo
in Plato’s Cave
“It is said that the students of medieval Paris came to blows in the streets over the question of universals. The stakes are high, for at issue is our whole conception of our ability to describe the world truly or falsely, and the objectivity of any opinions we frame to ourselves. It is arguable that this is always the deepest, most profound problem of philosophy.”
— Simon Blackburn, Think (Oxford, 1999)
Michael Harris, mathematician at the University of Paris:
“… three ‘parts’ of tragedy identified by Aristotle that transpose to fiction of all types– plot (mythos), character (ethos), and ‘thought’ (dianoia)….”
— paper (pdf) to appear in Mathematics and Narrative, A. Doxiadis and B. Mazur, eds.
Mythos —
A visitor from France this morning viewed the entry of Jan. 23, 2006: “In Defense of Hilbert (On His Birthday).” That entry concerns a remark of Michael Harris.
A check of Harris’s website reveals a new article:
“Do Androids Prove Theorems in Their Sleep?” (slighly longer version of article to appear in Mathematics and Narrative, A. Doxiadis and B. Mazur, eds.) (pdf).
From that article:
“The word ‘key’ functions here to structure the reading of the article, to draw the reader’s attention initially to the element of the proof the author considers most important. Compare E.M. Forster in Aspects of the Novel:
[plot is] something which is measured not be minutes or hours, but by intensity, so that when we look at our past it does not stretch back evenly but piles up into a few notable pinnacles.”
Ethos —
“Forster took pains to widen and deepen the enigmatic character of his novel, to make it a puzzle insoluble within its own terms, or without. Early drafts of A Passage to India reveal a number of false starts. Forster repeatedly revised drafts of chapters thirteen through sixteen, which comprise the crux of the novel, the visit to the Marabar Caves. When he began writing the novel, his intention was to make the cave scene central and significant, but he did not yet know how:
When I began a A Passage to India, I knew something important happened in the Malabar (sic) Caves, and that it would have a central place in the novel– but I didn’t know what it would be… The Malabar Caves represented an area in which concentration can take place. They were to engender an event like an egg.”
— E. M. Forster: A Passage to India, by Betty Jay
Dianoia —
Flagrant Triviality
or Resplendent Trinity?
“Despite the flagrant triviality of the proof… this result is the key point in the paper.”
— Michael Harris, op. cit., quoting a mathematical paper
Online Etymology Dictionary:
flagrant c.1500, “resplendent,” from L. flagrantem (nom. flagrans) “burning,” prp. of flagrare “to burn,” from L. root *flag-, corresponding to PIE *bhleg– (cf. Gk. phlegein “to burn, scorch,” O.E. blæc “black”). Sense of “glaringly offensive” first recorded 1706, probably from common legalese phrase in flagrante delicto “red-handed,” lit. “with the crime still blazing.”
A related use of “resplendent”– applied to a Trinity, not a triviality– appears in the Liturgy of Malabar:
— The Liturgies of SS. Mark, James, Clement, Chrysostom, and Basil, and the Church of Malabar, by the Rev. J.M. Neale and the Rev. R.F. Littledale, reprinted by Gorgias Press, 2002
On Universals and
A Passage to India:
“”The universe, then, is less intimation than cipher: a mask rather than a revelation in the romantic sense. Does love meet with love? Do we receive but what we give? The answer is surely a paradox, the paradox that there are Platonic universals beyond, but that the glass is too dark to see them. Is there a light beyond the glass, or is it a mirror only to the self? The Platonic cave is even darker than Plato made it, for it introduces the echo, and so leaves us back in the world of men, which does not carry total meaning, is just a story of events.”
Comments Off on Sunday April 13, 2008
Tuesday, April 8, 2008
Comments Off on Tuesday April 8, 2008
Monday, April 7, 2008
A year ago…
(Holy Saturday, 2007) —
From Friedrich Froebel,
who invented kindergarten:
For further details, see
Gift of the Third Kind
and
Kindergarten Relativity.
Related material:
“… There was a problem laid out on the board, a six-mover. I couldn’t solve it, like a lot of my problems. I reached down and moved a knight…. I looked down at the chessboard. The move with the knight was wrong. I put it back where I had moved it from. Knights had no meaning in this game. It wasn’t a game for knights.”
— Raymond Chandler, The Big Sleep
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Saturday, April 5, 2008
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Monday, March 31, 2008
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Thursday, March 27, 2008
Back to the Garden
Film star Richard Widmark
died on Monday, March 24.
From Log24 on that date:
"Hanging from the highest limb
of the apple tree are
the three God's Eyes…"
— Ken Kesey
Related material:
The Beauty Test, 5/23/07–
H.S.M. Coxeter's classic
Introduction to Geometry (2nd ed.):
Note the resemblance of
the central part to
a magical counterpart–
the Ojo de Dios
of Mexico's Sierra Madre.
From a Richard Widmark film festival:
GARDEN OF EVIL
Henry Hathaway, 1954
"A severely underrated Scope western, shot in breathtaking mountain locations near Cuernavaca. Widmark, Gary Cooper and Cameron Mitchell are a trio of fortune hunters stranded in Mexico, when they are approached by Susan Hayward to rescue her husband (Hugh Marlowe) from a caved-in gold mine in Indian country. When they arrive at the 'Garden of Evil,' they must first battle with one another before they have to stave off their bloodthirsty Indian attackers. Widmark gives a tough, moving performance as Fiske, the one who sacrifices himself to save his friends. 'Every day it goes, and somebody goes with it,' he says as he watches the setting sun. 'Today it's me.' This was one of the best of Hollywood veteran Henry Hathaway's later films. With a brilliant score by Bernard Herrmann."
See also
the apple-tree
entries from Monday
(the date of Widmark's death)
and Tuesday, as well as
today's previous entry and
previous Log24
entries on Cuernavaca.
Comments Off on Thursday March 27, 2008
Thursday, March 6, 2008
This note is prompted by the March 4 death of Richard D. Anderson, writer on geometry, President (1981-82) of the Mathematical Association of America (MAA), and member of the MAA's Icosahedron Society.
"The historical road
from the Platonic solids
to the finite simple groups
is well known."
— Steven H. Cullinane,
November 2000,
Symmetry from Plato to
the Four-Color Conjecture
Euclid is said to have remarked that "there is no royal road to geometry." The road to the end of the four-color conjecture may, however, be viewed as a royal road
from geometry to the wasteland of mathematical recreations.* (See, for instance, Ch. VIII, "Map-Colouring Problems," in
Mathematical Recreations and Essays, by
W. W. Rouse Ball and
H. S. M. Coxeter.) That road
ended in 1976 at the AMS-MAA summer meeting in Toronto– home of
H. S. M. Coxeter, a.k.a. "the king of geometry."
See also Log24, May 21, 2007.
A different road– from Plato to the finite simple groups– is, as I noted in November 2000, well known. But new roadside attractions continue to appear. One such attraction is the role played by a Platonic solid– the icosahedron– in design theory, coding theory, and the construction of the sporadic simple group M
24.
"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."
— "Block Designs," by Andries E. Brouwer (Ch. 14 (pp. 693-746) of
Handbook of Combinatorics, Vol. I, MIT Press, 1995, edited by Ronald L. Graham, Martin Grötschel, and László Lovász, Section 16 (p. 716))
This Steiner system is closely connected to M24 and to the extended binary Golay code. Brouwer gives an elegant construction of that code (and therefore of M24):
"Let N be the adjacency matrix of the icosahedron (points: 12 vertices, adjacent: joined by an edge). Then the rows of the 12×24 matrix (I J-N) generate the extended binary Golay code." [Here I is the identity matrix and J is the matrix of all 1's.]
— Op. cit., p. 719
Related material:
Finite Geometry of
the Square and Cube
and
Jewel in the Crown
"There is a pleasantly discursive
treatment of Pontius Pilate's
unanswered question
'What is truth?'"
— H. S. M. Coxeter, 1987,
introduction to Trudeau's
"story theory" of truth
Those who prefer stories to truth
may consult the Log24 entries
of March 1, 2, 3, 4, and 5.
They may also consult
the poet Rubén Darío:
… Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte.
Comments Off on Thursday March 6, 2008
Wednesday, March 5, 2008
For CENTRAL
Central Intelligence:
"God does not play dice."
— Paraphrase of a remark
by Albert Einstein
Another Nobel Prize winner,
Isaac Bashevis Singer—
"a God who speaks in deeds,
not in words, and whose
vocabulary is the Cosmos"
From "The Escapist:
The Reality of Fantasy Games"–
Dungeons & Dragons Dice
From today's New York Times:
A Kaddish for Gygax:
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"I was reading Durant's section on Plato, struggling to understand his theory of the ideal Forms that lay in inviolable perfection out beyond the phantasmagoria. (That was the first, and I think the last, time that I encountered that word.)"
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Related material:
For more on the word
"phantasmagoria," see
Log24 on Dec. 12, 2004
and on Sept. 23, 2006.
For phantasmagoria in action,
see Dungeons & Dragons
and Singer's (and others')
Jewish fiction.
For non-phantasmagoria,
see (for instance) the Elements
of Euclid, which culminates
in the construction of the
Platonic solids illustrated above.
See also Geometry for Jews.
Comments Off on Wednesday March 5, 2008
Tuesday, March 4, 2008
… And for a
Swiftly Tilting
Shadowed Planet …
John Trever, Albuquerque Journal, 2/29/08
The pen's point:
Log24, Dec. 11, 2006
SINGER, ISAAC:
"Are Children the
Ultimate Literary Critics?"
— Top of the News 29
(Nov. 1972): 32-36.
"Sets forth his own aims in writing for children and laments 'slice of life' and chaos in children's literature. Maintains that children like good plots, logic, and clarity, and that they have a concern for 'so-called eternal questions.'"
— An Annotated Listing
of Criticism
by Linnea Hendrickson
"She returned the smile, then looked across the room to her youngest brother, Charles Wallace, and to their father, who were deep in concentration, bent over the model they were building of a tesseract: the square squared, and squared again: a construction of the dimension of time."
— A Swiftly Tilting Planet,
by Madeleine L'Engle
A Swiftly Tilting Planet is a fantasy for children set partly in Vespugia, a fictional country bordered by Chile and Argentina.
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Comments Off on Tuesday March 4, 2008
Thursday, February 28, 2008
Popularity of MUB’s
From an entry today at the weblog of Lieven Le Bruyn (U. of Antwerp):
“MUBs (for Mutually Unbiased Bases) are quite popular at the moment. Kea is running a mini-series Mutual Unbias….”
The link to Kea (Marni Dee Sheppeard (pdf) of New Zealand) and a link in her Mutual Unbias III (Feb. 13) lead to the following illustration, from a talk, “Discrete phase space based on finite fields,” by William Wootters at the Perimeter Institute in 2005:
This illustration makes clear the
close relationship of MUB’s to the
finite geometry of the 4×4 square.
The Wootters talk was on
July 20, 2005. For related material from that July which some will find more entertaining, see “Steven Cullinane is a Crank,” conveniently reproduced as
a five-page thread in the Mathematics Forum at groupsrv.com.
Comments Off on Thursday February 28, 2008
Tuesday, February 26, 2008
The Just Word
The title of the previous entry, "Where Entertainment is God," comes (via Log24, Nov. 26, 2004) from Frank Rich.
The previous entry dealt, in part, with a dead Jesuit whose obituary appears in today's Los Angeles Times. The online obituaries page places the Jesuit, without a photo, beneath a picture of a dead sitcom writer and to the left of a picture of a dead guru.
"Walter John Burghardt was born July 10, 1914, in New York, the son of immigrants from what is now Poland. He entered a Jesuit seminary in Poughkeepsie, N.Y., at 16, and in 1937 received a master's degree from Woodstock College in Maryland. He was ordained in 1941." He died, by the way, on Saturday, Feb. 16, 2008.
The reference to Woodstock College brings to mind a fellow Jesuit, Joseph T. Clark, who wrote a book on logic published by that college.
From a review of the book:
"In order to show that Aristotelian logicians were at least vaguely aware of a kind of analogy or possible isomorphism between logical relations and mathematical relations, Father Clark seizes at one place (p. 8) upon the fact that Aristotle uses the word, 'figure' (schema), in describing the syllogism and concludes from this that 'it is obvious that the schema of the syllogism is to serve the logician precisely as the figure serves the geometer.' On the face of it, this strikes one as a bit far fetched…."
— Henry Veatch in Speculum, Vol. 29, No. 2, Part 1 (Apr., 1954), pp. 266-268 (review of Conventional Logic and Modern Logic: A Prelude to Transition (1952), by Joseph T. Clark, Society of Jesus)
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Monday, February 25, 2008
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Saturday, February 16, 2008
Bridges
Between Two Worlds
From the world of mathematics…
“… my advisor once told me, ‘If you ever find yourself drawing one of those meaningless diagrams with arrows connecting different areas of mathematics, it’s a good sign that you’re going senile.'”
— Scott Carnahan at Secret Blogging Seminar, December 14, 2007
Carnahan’s remark in context:
“About five years ago, Cheewhye Chin gave a great year-long seminar on Langlands correspondence for GLr over function fields…. In the beginning, he drew a diagram….
If we remove all of the explanatory text, the diagram looks like this:
I was a bit hesitant to draw this, because my advisor once told me, ‘If you ever find yourself drawing one of those meaningless diagrams with arrows connecting different areas of mathematics, it’s a good sign that you’re going senile.’ Anyway, I’ll explain roughly how it works.
Langlands correspondence is a ‘bridge between two worlds,’ or more specifically, an assertion of a bijection….”
Compare and contrast the above…
… to the world of Rudolf Kaehr:
The above reference to “diamond theory” is from Rudolf Kaehr‘s paper titled Double Cross Playing Diamonds.
Another bridge…
Carnahan’s advisor, referring to “meaningless diagrams with arrows connecting different areas of mathematics,” probably did not have in mind diagrams like the two above, but rather diagrams like the two below–
From the world of mathematics…
“A rough sketch of
how diamond theory is
related to some other
fields of mathematics”
— Steven H. Cullinane
Related material:
For further details on
the “diamond theory” of
Cullinane, see
Finite Geometry of the
Square and Cube.
For further details on
the “diamond theory” of
Kaehr, see
Rudy’s Diamond Strategies.
Those who prefer entertainment
may enjoy an excerpt
from Log24, October 2007:
Comments Off on Saturday February 16, 2008
Friday, February 1, 2008
Kindergarten Theology
On the late
James Edwin Loder,
a Presbyterian minister and
a professor of Christian education
at Princeton Theological Seminary,
co-author of
The Knight’s Move (1992):
“At his memorial service his daughter Tami told the story of ‘little Jimmy,’ whose kindergarten teacher recognized a special quality of mind that set him apart. ‘Every day we read a story, and after the story is over, Jimmy gets up and wants to tell us what the story means.'” —
Dana R. WrightFor a related story about
knight moves and kindergarten,
see Knight Moves: The Relativity
Theory of Kindergarten Blocks,
and Log24, Jan. 16, 17, and 18.
See also Loder’s book
(poorly written, but of some
interest in light of the above):
Opening of
The Knight’s Move —
“In a game of chess, the knight’s move is unique because it alone goes around corners. In this way, it combines the continuity of a set sequence with the discontinuity of an unpredictable turn in the middle. This meaningful combination of continuity and discontinuity in an otherwise linear set of possibilities has led some to refer to the creative act of discovery in any field of research as a ‘knight’s move’ in intelligence.
The significance of the title of this volume might stop there but for Kierkegaard’s use of the ‘knight’ image. The force of Kierkegaards’s usage might be described in relation to the chess metaphor by saying that not merely does Kierkegaard’s ‘knight of faith’ undertake a unique move within the rules of the human game, but faith transposes the whole idea of a ‘knight’s move’ into the mind of the Chess Master Himself. That is to say, chess is a game of multiple possibilities and interlocking strategies, so a chess master must combine the continuity represented by the whole complex of the game with the unpredictable decision he must make every time it is his turn. A master chess player, then, does not merely follow the rules; in him the game becomes a construct of consciousness. The better the player the more fully the game comes into its own as a creation of human intelligence. Similarly, for Kierkegaard, the knight of faith is a unique figure in human experience. The knight shows how, by existing in faith as a creative act of Christ’s Spirit, human existence comes into its own as an expression of the mind of Christ. Thus, the ultimate form of a ‘knight’s move’ is a creative act raised to the nth power by
Spiritus Creator, but it still partakes fully in the concrete pieces and patterns that comprise the nature of the human game and the game of nature.”
— James E. Loder and W. Jim Neidhardt (Helmers & Howard Publishing, 1992)
For a discussion, see Triplett’s
“Thinking Critically as a Christian.”
Many would deny that such
a thing is possible; let them
read the works of T. S. Eliot.
Related material:
The Knight’s Move
discusses (badly) Hofstadter’s
“strange loop” concept; see
Not Mathematics but Theology
(Log24, July 12, 2007).
Comments Off on Friday February 1, 2008
Wednesday, January 16, 2008
Knight Moves:
Geometry of the
Eightfold Cube
Click on the image for a larger version
and an expansion of some remarks
quoted here on Christmas 2005.
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Sunday, January 6, 2008
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Thursday, October 25, 2007
Something Anonymous
Yesterday:
Nineteenth-century quilt design:
Related material:
Battlefield Geometry
Comments Off on Thursday October 25, 2007
Wednesday, October 24, 2007
Descartes’s Twelfth Step
An earlier entry today (“Hollywood Midrash continued“) on a father and son suggests we might look for an appropriate holy ghost. In that context…
Descartes
A search for further background on Emmanuel Levinas, a favorite philosopher of the late R. B. Kitaj (previous two entries), led (somewhat indirectly) to the following figures of Descartes:
This trinity of figures is taken from Descartes’
Rule Twelve in
Rules for the Direction of the Mind. It seems to be meant to suggest an analogy between superposition of colors and superposition of shapes.Note that the first figure is made up of vertical lines, the second of vertical and horizontal lines, and the third of vertical, horizontal, and diagonal lines.
Leon R. Kass recently suggested that the Descartes figures might be replaced by a more modern concept– colors as wavelengths. (
Commentary, April 2007). This in turn suggests an analogy to Fourier series decomposition of a waveform in harmonic analysis. See the Kass essay for
a discussion of the Descartes figures in the context of (pdf)
Science, Religion, and the Human Future (not to be confused with
Life, the Universe, and Everything).
Compare and contrast:
The harmonic-analysis analogy suggests a review of an earlier entry’s
link today to 4/30– Structure and Logic— as well as
re-examination of Symmetry and a Trinity
(Dec. 4, 2002).
See also —
A Four-Color Theorem,
The Diamond Theorem, and
The Most Violent Poem,
from Mike Nichols’s birthday, 2003.
Comments Off on Wednesday October 24, 2007
Wednesday, October 3, 2007
Janitor Monitor
Will Hunting may be
interested in the following
vacant editorships at
The Open Directory:
Graph Theory
and
Combinatorics.
Related material:
The Long Hello and
On the Holy Trinity —
"Hey, Carrie-Anne, what's
your game now….?"
Picture sources:
azstarnet.com,
vibrationdata.com.
Personally, I prefer
Carol Ann:
From Criticism, Fall, 2001,
by Carol Ann Johnston—
"Drawing upon Platonic thought, Augustine argues that ideas are actually God's objective pattern and as such exist in God's mind. These ideas appear in the mirror of the soul. (35)."
(35.) In Augustine, De Trinitate, trans., Stephen McKenna (Washington, D.C.: Catholic University Press, 1970). See A. B. Acton, "Idealism," in The Encyclopedia of Philosophy, ed., Paul Edwards. Vol. 4 (New York: Macmillan, 1967): 110-118; Robert McRae, "`Idea' as a Philosophical Term in the Seventeenth Century," JHI 26 (1965): 175-190, and Erwin Panofsky, Idea: A Concept in Art History, trans., Joseph J. S. Peake (Columbia, S.C.: University of South Carolina Press, 1968) for explications of this term.
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Comments Off on Wednesday October 3, 2007
Monday, October 1, 2007
Bright as Magnesium
"Definitive"
— The New York Times,
Sept. 30, 2007, on
Blade Runner:
The Final Cut
Institute for Advanced Study, Princeton, N.J.—
"The art historian Kirk Varnedoe died on August 14, 2003, after a long and valiant battle with cancer. He was 57. He was a faculty member in the Institute for Advanced Study’s School of Historical Studies, where he was the fourth art historian to hold this prestigious position, first held by the German Renaissance scholar Erwin Panofsky in the 1930s."
Hal Crowther—
"His final lecture was an eloquent, prophetic flight of free association….
Varnedoe chose to introduce his final lecture with the less-quoted last words of the android Roy Batty (Rutger Hauer) in Ridley Scott's film Blade Runner: 'I've seen things you people wouldn't believe– attack ships on fire off the shoulder of Orion, bright as magnesium; I rode on the back decks of a blinker and watched C-beams glitter in the dark near the Tannhauser Gate. All those moments will be lost in time, like tears in the rain. Time to die.'"
Related material:
tears in the rain–
Game Over
(Nov. 5, 2003):
Comments Off on Monday October 1, 2007
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