From a search in this journal for "Chapter 11" —
Inner structure —
The above three images share the same
vector-space structure —
| 0 | c | d | c + d |
| a | a + c | a + d | a + c + d |
| b | b + c | b + d | b + c + d |
|
a + b |
a + b + c |
a + b + d |
a + b + |
(This vector-space a b c d diagram is from
Chapter 11 of Sphere Packings, Lattices
and Groups , by John Horton Conway and
N. J. A. Sloane, first published by Springer
in 1988.)
Note the three quadruplets of parallel edges in the 1984 figure above.
The above Gates article appeared earlier, in the June 2010 issue of
Physics World , with bigger illustrations. For instance —
Exercise: Describe, without seeing the rest of the article,
the rule used for connecting the balls above.
Wikipedia offers a much clearer picture of a (non-adinkra) tesseract —
And then, more simply, there is the Galois tesseract —
For parts of my own world in June 2010, see this journal for that month.
The above Galois tesseract appears there as follows:
See also the Klein correspondence in a paper from 1968
in yesterday's 2:54 PM ET post.
The Greek capital letter Omega, Ω, is customarily
used to denote a set that is acted upon by a group.
If the group is the affine group of 322,560
transformations of the four-dimensional
affine space over the two-element Galois field,
the appropriate Ω is the 4×4 grid above.
See the Cullinane diamond theorem .
If the group is the large Mathieu group of
244,823,040 permutations of 24 things,
the appropriate Ω is the 4×6 grid below.
See the Miracle Octad Generator of R. T. Curtis.
Or: Plato's Cave.
See also this journal on November 9, 2003 …
A post on Wittgenstein's "counting pattern" —
The walkerart.org passage above is from Feb. 17, 2011.
See also this journal on Feb. 17, 2011 —
"… Only by the form, the pattern,
Can words or music reach
The stillness…."
— T. S. Eliot,
Four Quartets
For further details, see Time Fold.
Some context for what Heidegger called
das Spiegel-Spiel des Gevierts
From Helen Lane's translation of El Mono Gramático ,
a book by Nobel winner Octavio Paz first published
in Barcelona by Seix Barral in 1974 —
|
Simultaneous perspective does not look upon language as a path because it is not the search for meaning that orients it. Poetry does not attempt to discover what there is at the end of the road; it conceives of the text as a series of transparent strata within which the various parts—the different verbal and semantic currents—produce momentary configurations as they intertwine or break apart, as they reflect each other or efface each other. Poetry contemplates itself, fuses with itself, and obliterates itself in the crystallizations of language. Apparitions, metamorphoses, volatilizations, precipitations of presences. These configurations are crystallized time: although they are perpetually in motion, they always point to the same hour—the hour of change. Each one of them contains all the others, each one is inside the others: change is only the oft-repeated and ever-different metaphor of identity.
— Paz, Octavio. The Monkey Grammarian |
A related 1960 meditation from Claude Lévi-Strauss taken from a
Log24 post of St. Andrew's Day 2017, "The Matrix for Quantum Mystics":
"In Vol. I of Structural Anthropology , p. 209, I have shown that
this analysis alone can account for the double aspect of time
representation in all mythical systems: the narrative is both
'in time' (it consists of a succession of events) and 'beyond'
(its value is permanent)." — Claude Lévi-Strauss
I prefer the earlier, better-known, remarks on time by T. S. Eliot
in Four Quartets , and the following four quartets
(from The Matrix Meets the Grid) —
.
Scholia on the title — See Quantum + Mystic in this journal.
"In Vol. I of Structural Anthropology , p. 209, I have shown that
this analysis alone can account for the double aspect of time
representation in all mythical systems: the narrative is both
'in time' (it consists of a succession of events) and 'beyond'
(its value is permanent)." — Claude Lévi-Strauss, 1976
I prefer the earlier, better-known, remarks on time by T. S. Eliot
in Four Quartets , and the following four quartets (from
The Matrix Meets the Grid) —
.
From a Log24 post of June 26-27, 2017:
A work of Eddington cited in 1974 by von Franz —
See also Dirac and Geometry and Kummer in this journal.
Ron Shaw on Eddington's triads "associated in conjugate pairs" —
For more about hyperbolic and isotropic lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.
For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.
The Matrix —
The Grid —
Picturing the Witt Construction —
"Read something that means something." — New Yorker ad
According to art historian Rosalind Krauss in 1979,
the grid's earliest employers
"can be seen to be participating in a drama
that extended well beyond the domain of art.
That drama, which took many forms, was staged
in many places. One of them was a courtroom,
where early in this century, science did battle with God,
and, reversing all earlier precedents, won."
The previous post discussed the 3×3 grid in the context of
Krauss's drama. In memory of T. S. Eliot, who died on this date
in 1965, an image of the next-largest square grid, the 4×4 array:
See instances of the above image.
See that phrase in this journal.
See also last night's post.
The Greek letter Ω is customarily used to
denote a set that is acted upon by a group.
If the group is the affine group of 322,560
transformations of the four-dimensional
affine space over the two-element Galois
field, the appropriate Ω is the 4×4 grid above.
Enotes.com on Herman Wouk's 1985 novel Inside, Outside :
"The 'outside' of the title is the goyish world
into which David’s profession has drawn him;
the 'inside' is the warm life of his Russian-
Jewish family on which he, as narrator, reflects
in the course of the novel."
For a different sort of 'inside' life, see this morning's post
Gesamtkunstwerk , and Nathan Shields's Feb. 8, 2011,
tribute to a serial composer "In Memoriam, Milton Babbitt."
Some other context for Shields's musical remarks —
Doctor Faustus and Dürer Square.
For a more interesting contrast of inside with outside
that has nothing to do with ethnicity, see the Feb. 10,
2014, post Mystery Box III: Inside, Outside, about
the following box:
.
This post is continued from a March 12, 2013, post titled
"Smoke and Mirrors" on art in Tromsø, Norway, and from
a June 22, 2014, post on the nineteenth-century
mathematicians Rosenhain and Göpel.
The latter day was the day of death for
mathematician Loren D. Olson, Harvard '64.
For some background on that June 22 post, see the tag
Rosenhain and Göpel in this journal.
Some background on Olson, who taught at the
University of Tromsø, from the American Mathematical
Society yesterday:
Olson died not long after attending the 50th reunion of the
Harvard Class of 1964.
For another connection between that class (also my own)
and Tromsø, see posts tagged "Elegantly Packaged."
This phrase was taken from today's (print)
New York Times review of a new play titled "Smoke."
The phrase refers here to the following "package" for
some mathematical objects that were named after
Rosenhain and Göpel — a 4×4 array —
For the way these objects were packaged within the array
in 1905 by British mathematician R. W. H. T. Hudson, see
a page at finitegometry.org/sc. For the connection to the art
in Tromsø mentioned above, see the diamond theorem.
"There is such a thing as a tesseract." — Madeleine L'Engle
An approach via the Omega Matrix:
See, too, Rosenhain and Göpel as The Shadow Guests .
(Continued from Mystery Box, Feb. 4, and Mystery Box II, Feb. 5.)
The Box
Inside the Box
Outside the Box
For the connection of the inside notation to the outside geometry,
see Desargues via Galois.
(For a related connection to curves and surfaces in the outside
geometry, see Hudson's classic Kummer's Quartic Surface and
Rosenhain and Göpel Tetrads in PG(3,2).)
Today’s 11 AM (ET) post was suggested by a New York Times
article, online yesterday, about art gallery owner Lisa Cooley.
A check of Cooley’s website yields the image below,
related to Beckett’s Molloy .
For the relevant passage from Molloy , click the following:
“I took advantage of being at the seaside
to lay in a store of sucking-stones.“
For posts on Molloy in this journal, click Beckett + Molloy .
Cynthia Daignault, 2011:
The one I shall now describe, if I can…
Oil on linen, in 2 parts: 40 x 30 inches, 96 x 75 inches
Related art theory —
The Whitney Museum of American Art has stated
that artist Frank Stella in 1959
"wanted to create work that was methodical,
intellectual and passionless."
Source: Whitney Museum, transcript of audio guide.
Related material:
A figure from this journal on July 13, 2003…
… and some properties of that figure.
The 16-point affine Galois space:
Further properties of this space:
In Configurations and Squares, see the
discusssion of the Kummer 166 configuration.
Some closely related material:
For the first two pages, click here.
Rock guitarist Alvin Lee, a founder of
the band Ten Years After , died
on March 6, 2013 (Michelangelo's
birthday). In his memory, a figure
from a post Ten Years Before —
Plato's reported motto for his Academy:
"Let no one ignorant of geometry enter."
For visual commentary by an artist ignorant
of geometry, see a work by Sol LeWitt.
For verbal commentary by an art critic ignorant
of geometry, see a review of LeWitt by
Robert Hughes—
"A Beauty Really Bare" (TIME, Feb. 6, 2001).
See also Ten Years Group and Four Gods.
"… Only by the form, the pattern,
Can words or music reach
The stillness…."
— T. S. Eliot,
Four Quartets
For further details, see Time Fold.
In the above view, four of the tesseract's 16
vertices are overlaid by other vertices.
For views that are more complete and
moveable, see Smith's tesseract page.
Four-Part Tesseract Divisions—
The above figure shows how four-part partitions
of the 16 vertices of a tesseract in an infinite
Euclidean space are related to four-part partitions
of the 16 points in a finite Galois space
|
Euclidean spaces versus Galois spaces in a larger context—
Infinite versus Finite 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." |
Another picture related to philosophy and religion—
Jung's Four-Diamond Figure from Aion—
This figure was devised by Jung
to represent the Self. Compare the
remarks of Paul Valéry on the Self—
|
Flight from Eden: The Origins of Modern Literary Criticism and Theory, by Steven Cassedy, U. of California Press, 1990, pages 156-157—
Valéry saw the mind as essentially a relational system whose operation he attempted to describe in the language of group mathematics. "Every act of understanding is based on a group," he says (C, 1:331). "My specialty— reducing everything to the study of a system closed on itself and finite" (C, 19: 645). The transformation model came into play, too. At each moment of mental life the mind is like a group, or relational system, but since mental life is continuous over time, one "group" undergoes a "transformation" and becomes a different group in the next moment. If the mind is constantly being transformed, how do we account for the continuity of the self? Simple; by invoking the notion of the invariant. And so we find passages like this one: "The S[elf] is invariant, origin, locus or field, it's a functional property of consciousness" (C, 15:170 [2:315]). Just as in transformational geometry, something remains fixed in all the projective transformations of the mind's momentary systems, and that something is the Self (le Moi, or just M, as Valéry notates it so that it will look like an algebraic variable). Transformation theory is all over the place. "Mathematical science… reduced to algebra, that is, to the analysis of the transformations of a purely differential being made up of homogeneous elements, is the most faithful document of the properties of grouping, disjunction, and variation in the mind" (O, 1:36). "Psychology is a theory of transformations, we just need to isolate the invariants and the groups" (C, 1:915). "Man is a system that transforms itself" (C, 2:896). O Paul Valéry, Oeuvres (Paris: Pléiade, 1957-60) C Valéry, Cahiers, 29 vols. (Paris: Centre National de le Recherche Scientifique, 1957-61) |
Note also the remarks of George David Birkhoff at Rice University
in 1940 (pdf) on Galois's theory of groups and the related
"theory of ambiguity" in Galois's testamentary letter—
|
… metaphysical reasoning always relies on the Principle of Sufficient Reason, and… the true meaning of this Principle is to be found in the “Theory of Ambiguity” and in the associated mathematical “Theory of Groups.” If I were a Leibnizian mystic, believing in his “preestablished harmony,” and the “best possible world” so satirized by Voltaire in “Candide,” I would say that the metaphysical importance of the Principle of Sufficient Reason and the cognate Theory of Groups arises from the fact that God thinks multi-dimensionally* whereas men can only think in linear syllogistic series, and the Theory of Groups is the appropriate instrument of thought to remedy our deficiency in this respect. * That is, uses multi-dimensional symbols beyond our grasp. |
Related material:
A medal designed by Leibniz to show how
binary arithmetic mirrors the creation by God
of something (1) from nothing (0).
Another array of 16 strings of 0's and 1's, this time
regarded as coordinates rather than binary numbers—
Some context by a British mathematician —
|
Imago by Wallace Stevens Who can pick up the weight of Britain, Who can move the German load Or say to the French here is France again? Imago. Imago. Imago. It is nothing, no great thing, nor man Of ten brilliancies of battered gold And fortunate stone. It moves its parade Of motions in the mind and heart, A gorgeous fortitude. Medium man In February hears the imagination's hymns And sees its images, its motions And multitudes of motions And feels the imagination's mercies, In a season more than sun and south wind, Something returning from a deeper quarter, A glacier running through delirium, Making this heavy rock a place, Which is not of our lives composed . . . Lightly and lightly, O my land, Move lightly through the air again. |
For a girl I saw once in USA Today
in 1995:
Top of the online front page, NY Times this morning–
Adapted song lyrics from “Colorado Trail“–
Kendra was a pretty girl God Almighty knows
Very like another girl 30 years earlier…
Text
Online Etymology Dictionary
|
“Discuss the geometry
underlying the above picture.”
— Log24, June 11, 2009
Geometry for Jews
(continued from Michelangelo’s birthday, 2003)
“Discuss the geometry underlying the above picture.”
Abstraction and the Holocaust (Mark Godfrey, Yale University Press, 2007) describes one approach to such a discussion: Bochner “took a photograph of a new arrangement of blocks, cut it up, reprinted it as a negative, and arranged the four corners in every possible configuration using the serial principles of rotation and reversal to make Sixteen Isomorphs (Negative) of 1967, which he later illustrated alongside works by Donald Judd, Sol LeWitt and Eva Hesse in his Artforum article ‘The Serial Attitude.’ [December 1967, pp. 28-33]” Bochner’s picture of “every possible configuration”–
Compare with the 24 figures in Frame Tales
(Log24, Nov. 10, 2008) and in Theme and Variations.
"My work is motivated by a hope that there may be a way to recapture the ancient and medieval vision of both Beauty and purpose in a way which is relevant to our own century. I even dare to hope that the two ideas may be related, that Beauty is actually part of the meaning and purpose of life."
"The Reverend T. P. Kirkman knew in 1862 that there exists a group of degree 16 and order 322560 with a normal, elementary abelian, subgroup of order 16 [1, p. 108]. Frobenius identified this group in 1904 as a subgroup of the Mathieu group M24 [4, p. 570]…."
1. Biggs N.L., "T. P. Kirkman, Mathematician," Bulletin of the London Mathematical Society 13, 97–120 (1981).
4. Frobenius G., "Über die Charaktere der mehrfach transitiven Gruppen," Sitzungsber. Königl. Preuss. Akad. Wiss. zu Berlin, 558–571 (1904). Reprinted in Frobenius, Gesammelte Abhandlungen III (J.-P. Serre, editor), pp. 335–348. Springer, Berlin (1968).
Olli Pottonen, "Classification of Steiner Quadruple Systems" (Master's thesis, Helsinki, 2005)–
"The concept of group actions is very useful in the study of isomorphisms of combinatorial structures."
"Simplify, simplify."
— Thoreau
"Beauty is bound up
with symmetry."
— Weyl
Pottonen's thesis is
dated Nov. 16, 2005.
For some remarks on
images and theology,
see Log24 on that date.
Click on the above image
for some further details.
Bertram Kostant, Professor Emeritus of Mathematics at MIT, on an object discussed in this week's New Yorker:
Hermann Weyl on the hard core of objectivity:
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"–
This puzzle shows
that the 4×4 array can
also be viewed in
thousands of ways.
"You can make 322,560
pairs of patterns. Each
pair pictures a different
symmetry of the underlying
16-point space."
— Steven H. Cullinane,
July 17, 2008
For other parts of the tale,
see Ashay Dharwadker,
the Four-Color Theorem,
and Usenet Postings.
Eight is a Gate (continued)
|
Tom Stoppard, Jumpers:
"Heaven, how can I believe in Heaven?" she sings at the finale. "Just a lying rhyme for seven!"
"To begin at the beginning: Is God?…" [very long pause]
|
|
From "Space," by Salomon Bochner
Makom. Our term “space” derives from the Latin, and is thus relatively late. The nearest to it among earlier terms in the West are the Hebrew makom and the Greek topos (τόπος). The literal meaning of these two terms is the same, namely “place,” and even the scope of connotations is virtually the same (Theol. Wörterbuch…, 1966). Either term denotes: area, region, province; the room occupied by a person or an object, or by a community of persons or arrangements of objects. But by first occurrences in extant sources, makom seems to be the earlier term and concept. Apparently, topos is attested for the first time in the early fifth century B.C., in plays of Aeschylus and fragments of Parmenides, and its meaning there is a rather literal one, even in Parmenides. Now, the Hebrew book Job is more or less contemporary with these Greek sources, but in chapter 16:18 occurs in a rather figurative sense:
Late antiquity was already debating whether this makom is meant to be a “hiding place” or a “resting place” (Dhorme, p. 217), and there have even been suggestions that it might have the logical meaning of “occasion,” “opportunity.” Long before it appears in Job, makom occurs in the very first chapter of Genesis, in:
This biblical account is more or less contemporary with Hesiod's Theogony, but the makom of the biblical account has a cosmological nuance as no corresponding term in Hesiod. Elsewhere in Genesis (for instance, 22:3; 28:11; 28:19), makom usually refers to a place of cultic significance, where God might be worshipped, eventually if not immediately. Similarly, in the Arabic language, which however has been a written one only since the seventh century A.D., the term makām designates the place of a saint or of a holy tomb (Jammer, p. 27). In post-biblical Hebrew and Aramaic, in the first centuries A.D., makom became a theological synonym for God, as expressed in the Talmudic sayings: “He is the place of His world,” and “His world is His place” (Jammer, p. 26). Pagan Hellenism of the same era did not identify God with place, not noticeably so; except that the One (τὸ ἕν) of Plotinus (third century A.D.) was conceived as something very comprehensive (see for instance J. M. Rist, pp. 21-27) and thus may have been intended to subsume God and place, among other concepts. In the much older One of Parmenides (early fifth century B.C.), from which the Plotinian One ultimately descended, the theological aspect was only faintly discernible. But the spatial aspect was clearly visible, even emphasized (Diels, frag. 8, lines 42-49). BIBLIOGRAPHY Paul Dhorme, Le livre de Job (Paris, 1926). H. Diels and W. Kranz, Die Fragmente der Vorsokratiker, 6th ed. (Berlin, 1938). Max Jammer, Concepts of Space… (Cambridge, Mass., 1954). J. M. Rist, Plotinus: The Road to Reality (Cambridge, 1967). Theologisches Wörterbuch zum Neuen Testament (1966), 8, 187-208, esp. 199ff.
|
Related material: In the previous entry — "Father Clark seizes at one place (page eight)
upon the fact that…."
Father Clark's reviewer (previous entry) called a remark by Father Clark "far fetched."
This use of "place" by the reviewer is, one might say, "near fetched."
Something Anonymous
— Simone Weil, Gravity and Grace
Nineteenth-century quilt design:
Related material:
Minimalism
"It's become our form of modern classicism."
— Nancy Spector in
the New York Times of April 23, 2004
Part I: Aesthetics
In honor of the current Guggenheim exhibition, "Singular Forms" — A quotation from the Guggenheim's own website:
"Minimalism refers to painting or sculpture
Discuss these seven points
in relation to the following:
Form,
by S. H. Cullinane
Mark Rothko's reference
to geometry as a "swamp"
and his talk of "the idea" in art
Michael Kimmelman's
remarks on ideas in art
ART WARS:
Judgment Day
(2003, 10/07)
Part II: Theology
Today's previous entry, "Skylark," concluded with an invocation of the Lord. Of course, the Lord one expects may not be the Lord that appears.
"… the idea that, in art at least, less is more.
It is an idea surely as old, as enduringly attractive and as ubiquitous as its opposite. In the beginning was the Word: only later came the Bible, not to mention the three-decker Victorian novel. The oracle at Delphi did not say, 'Exhaustive analysis and comprehension of one's own psyche may be prerequisite to an understanding of one's behavior and of the world at large'; it said, 'Know thyself.' Such inherently minimalist genres as oracles (from the Delphic shrine of Apollo to the modern fortune cookie), proverbs, maxims, aphorisms, epigrams, pensees, mottoes, slogans and quips are popular in every human century and culture–especially in oral cultures and subcultures, where mnemonic staying power has high priority–and many specimens of them are self-reflexive or self-demonstrative: minimalism about minimalism. 'Brevity is the soul of wit.' "
Another form of the oracle at Delphi, in minimalist prose that might make Hemingway proud:
"He would think about Bert. Bert was an interesting man. Bert had said something about the way a gambler wants to lose. That did not make sense. Anyway, he did not want to think about it. It was dark now, but the air was still hot. He realized that he was sweating, forced himself to slow down the walking. Some children were playing a game with a ball, in the street, hitting it against the side of a building. He wanted to see Sarah.
When he came in, she was reading a book, a tumbler of dark whiskey beside her on the end table. She did not seem to see him and he sat down before he spoke, looking at her and, at first, hardly seeing her. The room was hot; she had opened the windows, but the air was still. The street noises from outside seemed almost to be in the room with them, as if the shifting of gears were being done in the closet, the children playing in the bathroom. The only light in the room was from the lamp over the couch where she was reading.
He looked at her face. She was very drunk. Her eyes were swollen, pink at the corners. 'What's the book,' he said, trying to make his voice conversational. But it sounded loud in the room, and hard.
She blinked up at him, smiled sleepily, and said nothing.
'What's the book?' His voice had an edge now.
'Oh,' she said. 'It's Kierkegaard. Soren Kierkegaard.' She pushed her legs out straight on the couch, stretching her feet. Her skirt fell back a few inches from her knees. He looked away.
'What's that?' he said.
'Well, I don't exactly know, myself." Her voice was soft and thick.
He turned his face away from her again, not knowing what he was angry with. 'What does that mean, you don't know, yourself?'
She blinked at him. 'It means, Eddie, that I don't exactly know what the book is about. Somebody told me to read it once, and that's what I'm doing. Reading it.'
He looked at her, tried to grin at her — the old, meaningless, automatic grin, the grin that made everbody like him — but he could not. 'That's great,' he said, and it came out with more irritation than he had intended.
She closed the book, tucked it beside her on the couch. She folded her arms around her, hugging herself, smiling at him. 'I guess this isn't your night, Eddie. Why don't we have a drink?'
'No.' He did not like that, did not want her being nice to him, forgiving. Nor did he want a drink.
Her smile, her drunk, amused smile, did not change. 'Then let's talk about something else,' she said. 'What about that case you have? What's in it?' Her voice was not prying, only friendly, 'Pencils?'
'That's it,' he said. 'Pencils.'
She raised her eyebrows slightly. Her voice seemed thick. 'What's in it, Eddie?'
'Figure it out yourself.' He tossed the case on the couch."
— Walter Tevis, The Hustler, 1959,
Chapter 11
See, too, the invocation of Apollo in
A Mass for Lucero, as well as
GENERAL AUDIENCE OF JOHN PAUL II
Wednesday 15 January 2003:
"The invocation of the Lord is relentless…."
and
JOURNAL ENTRY OF S. H. CULLINANE
Wednesday 15 January 2003:
Karl Cullinane —
"I will fear no evil, for I am the
meanest son of a bitch in the valley."
11:11
“Why do we remember the past
but not the future?”
— Stephen Hawking,
A Brief History of Time,
Ch. 9, “The Arrow of Time”
For another look at
the arrow of time, see
“Imaginary time is a relatively simple concept that is rather difficult to visualize or conceptualize. In essence, it is another direction of time moving at right angles to ordinary time. In the image at right, the light gray lines represent ordinary time flowing from left to right – past to future. The dark gray lines depict imaginary time, moving at right angles to ordinary time.”
Is Time Quantized?
Yes.
Let us suppose, for the sake of argument, that time is in fact quantized and two-dimensional. Then the following picture,
from Time Fold, of “four quartets” time, of use in the study of poetry and myth, might, in fact, be of use also in theoretical physics.
In this event, last Sunday’s entry, on the symmetry group of a generic 4×4 array, might also have some physical significance.
At any rate, the Hawking quotation above suggests the following remarks from T. S. Eliot’s own brief history of time, Four Quartets:
“It seems, as one becomes older,
That the past has another pattern,
and ceases to be a mere sequence….
I sometimes wonder if that is
what Krishna meant—
Among other things—or one way
of putting the same thing:
That the future is a faded song,
a Royal Rose or a lavender spray
Of wistful regret for those who are
not yet here to regret,
Pressed between yellow leaves
of a book that has never been opened.
And the way up is the way down,
the way forward is the way back.”
Related reading:
For Hermann Weyl's Birthday:
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
Exercise: Apply Weyl's lesson to the following "structure-endowed entity."
What is the order of the resulting group of automorphisms? (The answer will, of course, depend on which aspects of the array's structure you choose to examine. It could be in the hundreds, or in the hundreds of thousands.)
The Form, the Pattern
"…the sort of organization that Eliot later called musical, in his lecture 'The Music of Poetry', delivered in 1942, just as he was completing Four Quartets: 'The use of recurrent themes is as natural to poetry as to music,' Eliot says:
There are possibilities for verse which bear some analogy to the development of a theme by different groups of instruments [‘different voices’, we might say]; there are possibilities of transitions in a poem comparable to the different movements of a symphony or a quartet; there are possibilities of contrapuntal arrangement of subject-matter."
— Louis L. Martz, from
"Origins of Form in Four Quartets,"
in Words in Time: New Essays on Eliot’s Four Quartets, ed. Edward Lobb, University of Michigan Press, 1993
"… Only by the form, the pattern,
Can words or music reach
The stillness…."
— T. S. Eliot,
Four Quartets
Four Quartets
For a discussion of the above
form, or pattern, click here.
ART WARS, 5:09
The Word in the Desert
For Harrison Ford in the desert.
(See previous entry.)
Words strain,
Crack and sometimes break,
under the burden,
Under the tension, slip, slide, perish,
Will not stay still. Shrieking voices
Scolding, mocking, or merely chattering,
Always assail them.
The Word in the desert
Is most attacked by voices of temptation,
The crying shadow in the funeral dance,
The loud lament of
the disconsolate chimera.— T. S. Eliot, Four Quartets
The link to the word "devilish" in the last entry leads to one of my previous journal entries, "A Mass for Lucero," that deals with the devilishness of postmodern philosophy. To hammer this point home, here is an attack on college English departments that begins as follows:
"William Faulkner's Snopes trilogy, which recounts the generation-long rise of the drily loathsome Flem Snopes from clerk in a country store to bank president in Jefferson, Mississippi, teems with analogies to what has happened to English departments over the past thirty years."
For more, see
The Word in the Desert,
by Glenn C. Arbery.
See also the link on the word "contemptible," applied to Jacques Derrida, in my Logos and Logic page.
This leads to an National Review essay on Derrida,
The Philosopher as King,
by Mark Goldblatt.
A reader's comment on my previous entry suggests the film "Scotland, PA" as viewing related to the Derrida/Macbeth link there.
I prefer the following notice of a 7-11 death, that of a powerful art museum curator who would have been well cast as Lady Macbeth:
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Die Fahne Hoch, |
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From the Whitney Museum site:
"Max Anderson: When artist Frank Stella first showed this painting at The Museum of Modern Art in 1959, people were baffled by its austerity. Stella responded, 'What you see is what you see. Painting to me is a brush in a bucket and you put it on a surface. There is no other reality for me than that.' He wanted to create work that was methodical, intellectual, and passionless. To some, it seemed to be nothing more than a repudiation of everything that had come before—a rational system devoid of pleasure and personality. But other viewers saw that the black paintings generated an aura of mystery and solemnity.
The title of this work, Die Fahne Hoch, literally means 'The banner raised.' It comes from the marching anthem of the Nazi youth organization. Stella pointed out that the proportions of this canvas are much the same as the large flags displayed by the Nazis.
But the content of the work makes no reference to anything outside of the painting itself. The pattern was deduced from the shape of the canvas—the width of the black bands is determined by the width of the stretcher bars. The white lines that separate the broad bands of black are created by the narrow areas of unpainted canvas. Stella's black paintings greatly influenced the development of Minimalism in the 1960s."
From Play It As It Lays:
She took his hand and held it. "Why are you here."
"Because you and I, we know something. Because we've been out there where nothing is. Because I wanted—you know why."
"Lie down here," she said after a while. "Just go to sleep."
When he lay down beside her the Seconal capsules rolled on the sheet. In the bar across the road somebody punched King of the Road on the jukebox again, and there was an argument outside, and the sound of a bottle breaking. Maria held onto BZ's hand.
"Listen to that," he said. "Try to think about having enough left to break a bottle over it."
"It would be very pretty," Maria said. "Go to sleep."
I smoke old stogies I have found…
Cigar Aficionado on artist Frank Stella:
" 'Frank actually makes the moment. He captures it and helps to define it.'
This was certainly true of Stella's 1958 New York debut. Fresh out of Princeton, he came to New York and rented a former jeweler's shop on Eldridge Street on the Lower East Side. He began using ordinary house paint to paint symmetrical black stripes on canvas. Called the Black Paintings, they are credited with paving the way for the minimal art movement of the 1960s. By the fall of 1959, Dorothy Miller of The Museum of Modern Art had chosen four of the austere pictures for inclusion in a show called Sixteen Americans."
For an even more austere picture, see
For more on art, Derrida, and devilishness, see Deborah Solomon's essay in the New York Times Magazine of Sunday, June 27, 1999:
"Blame Derrida and
his fellow French theorists…."
See, too, my site
Art Wars: Geometry as Conceptual Art.
For those who prefer a more traditional meditation, I recommend
("Behold the Wood of the Cross")
For more on the word "road" in the desert, see my "Dead Poet" entry of Epiphany 2003 (Tao means road) as well as the following scholarly bibliography of road-related cultural artifacts (a surprising number of which involve Harrison Ford):
ART WARS:
Geometry for Jews
Today is Michelangelo's birthday.
Those who prefer the Sistine Chapel to the Rothko Chapel may invite their Jewish friends to answer the following essay question:
Discuss the geometry underlying the above picture. How is this geometry related to the work of Jewish artist Sol LeWitt? How is it related to the work of Aryan artist Ernst Witt? How is it related to the Griess "Monster" sporadic simple group whose elements number
808 017 424 794 512 875 886 459 904 961 710 757 005 754 368 000 000 000?
Some background:
From Nobel Prize Women in Science, by Sharon Bertsch McGrayne, Second Edition (2001), Joseph Henry Press:
"Storm trooper Ernst Witt, resplendent in the Brownshirt uniform of Hitler's paramilitary, knocked on a Jew's apartment door in 1934. A short, rotund woman opened the door. Emmy Noether smiled, welcomed the young Nazi into her home, and started her underground math class. The Brownshirt was one of her favorite pupils."
On this date in 1962, Frank Sinatra recorded "I Gotta Right To Sing The Blues" for Capitol Records. This was his last recording for Capitol. He had already started recording for Reprise Records.
Related reading:
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