For art more closely related to the title "Alpha and Omega,"
see a different view of the above Hoyersten exhibition.
For art more closely related to the title "Alpha and Omega,"
see a different view of the above Hoyersten exhibition.
See Osterman and Brosterman.
Logline for Osterman Meets Brosterman! — See Super-8.
From "The Osterman Weekend" (1983) —
Counting symmetries of the R. T. Curtis Omega:
An Illustration from Shakespeare's birthday —
The reference to Vallega-Neu in posts that last night were tagged
The Ereignis Sanction leads to . . .
Heidegger’s ‘Contributions to Philosophy.’ An Introduction .
(Indiana University Press, 2003).
That book is about . . .
Martin Heidegger, Contributions to Philosophy (From Enowning) ,
trans. Parvis Emad and Kenneth Maly (Bloomington:
Indiana University Press, 1999). German edition:
Beiträge zur Philosophie (vom Ereignis) ,
ed. F.-W. von Herrmann, Gesamtausgabe, vol. 65
(Frankfurt a. M.: Klostermann, 1989).
* See today's news and a Log24 search for "Philippine."
… is the birth date of storytellers C.S. Lewis and Madeleine L'Engle.
Another perspective on this date —
In the context of mathematics, I prefer to think of it as Brosterman Day.
See, from last year on this date, Osterman Meets Brosterman . . .
and, more generally, Brosterman.
But seriously . . . LAST THOUGHTS ON DEVIL'S NIGHT :
"Omega is as real as we need it to be." — The Osterman Weekend
See also related material in The New Yorker and the National Review .
A nostalgia pill for Watchmen fans.
For Harvard Watchmen fans, a link to 2346:
From a news article featured on the American Mathematical Society
home page today —
A joint Vietnam-USA mathematical meeting in Vietnam on
June 10-13, 2019:
This journal on June 12, 2019:
Wednesday, June 12, 2019
|
See also the Twentieth of May, 2008 —
“… the utterly real thing in writing is the only thing that counts…."
— Maxwell Perkins to Ernest Hemingway, Aug. 30, 1935
"Omega is as real as we need it to be."
— Burt Lancaster in "The Osterman Weekend"
(Continued from yesterday's Sunday School Lesson Plan for Peculiar Children)
Novelist George Eliot and programming pioneer Ada Lovelace —
For an image that suggests a resurrected multifaceted
(specifically, 759-faceted) Osterman Omega (as in Sunday's afternoon
Log24 post), behold a photo from today's NY Times philosophy
column "The Stone" that was reproduced here in today's previous post —
For a New York Times view of George Eliot data, see a Log24 post
of September 20, 2016, on the diamond theorem as the Middlemarch
"key to all mythologies."
“Am I still on?” — Ending line of The Osterman Weekend (1983)
From a recent Gitterkrieg post:
"The motive for metaphor, shrinking from
The weight of primary noon,
The A B C of being…." — Wallace Stevens
See also the cover of the February 2015
Notices of the American Mathematical Society .
"Omega is as real as we need it to be."
— Burt Lancaster in The Osterman Weekend
Wallace Stevens in "An Ordinary Evening in New Haven"
(1950) on "The Ruler of Reality" —
"Again, 'He has thought it out, he thinks it out,
As he has been and is and, with the Queen
Of Fact, lies at his ease beside the sea.'"
One such scene, from 1953 —
Another perspective, from "The Osterman Weekend" (1983) —
In memory of radio personality Steve Post,
a link to some remarks on the date of his death.
“This is a divorce case that was before us on an earlier occasion.”
Wild:
From the director of The Wild Bunch —
Brady:
From The New York Times —
Dialogue from “The Osterman Weekend”—
01:57:22 “Why did he make us try to believe Omega existed?”
01:57:25 ….
01:57:26 “The existence of Omega has not been disproved.
01:57:28 Don’t you understand that?
01:57:31 Omega is as real as we need it to be.”
See also Omega elsewhere in this journal.
Update of 9:15 PM ET —
Continued from August 20, 2013
In honor of Sam Peckinpah, the closing shot of his last film:
“Am I still on?” — Ending line of The Osterman Weekend (1983)
The Philosopher's Gaze , by David Michael Levin, The post-metaphysical question—question for a post-metaphysical phenomenology—is therefore: Can the perceptual field, the ground of perception, be released from our historical compulsion to represent it in a way that accommodates our will to power and its need to totalize and reify the presencing of being? In other words: Can the ground be experienced as ground? Can its hermeneutical way of presencing, i.e., as a dynamic interplay of concealment and unconcealment, be given appropriate respect in the receptivity of a perception that lets itself be appropriated by the ground and accordingly lets the phenomenon of the ground be what and how it is? Can the coming-to-pass of the ontological difference that is constitutive of all the local figure-ground differences taking place in our perceptual field be made visible hermeneutically, and thus without violence to its withdrawal into concealment? But the question concerning the constellation of figure and ground cannot be separated from the question concerning the structure of subject and object. Hence the possibility of a movement beyond metaphysics must also think the historical possibility of breaking out of this structure into the spacing of the ontological difference: différance , the primordial, sensuous, ekstatic écart . As Heidegger states it in his Parmenides lectures, it is a question of "the way historical man belongs within the bestowal of being (Zufügung des Seins ), i.e., the way this order entitles him to acknowledge being and to be the only being among all beings to see the open" (PE* 150, PG** 223. Italics added). We might also say that it is a question of our response-ability, our capacity as beings gifted with vision, to measure up to the responsibility for perceptual responsiveness laid down for us in the "primordial de-cision" (Entscheid ) of the ontological difference (ibid.). To recognize the operation of the ontological difference taking place in the figure-ground difference of the perceptual Gestalt is to recognize the ontological difference as the primordial Riß , the primordial Ur-teil underlying all our perceptual syntheses and judgments—and recognize, moreover, that this rift, this division, decision, and scission, an ekstatic écart underlying and gathering all our so-called acts of perception, is also the only "norm" (ἀρχή ) by which our condition, our essential deciding and becoming as the ones who are gifted with sight, can ultimately be judged. * PE: Parmenides of Heidegger in English— Bloomington: Indiana University Press, 1992 ** PG: Parmenides of Heidegger in German— Gesamtausgabe , vol. 54— Frankfurt am Main: Vittorio Klostermann, 1992 |
Examples of "the primordial Riß " as ἀρχή —
For an explanation in terms of mathematics rather than philosophy,
see the diamond theorem. For more on the Riß as ἀρχή , see
Function Decomposition Over a Finite Field.
The December 2012 Notices of the American
Mathematical Society has an ad on page 1564
(in a review of two books on vulgarized mathematics)
for three workshops next year on “Low-dimensional
Topology, Geometry, and Dynamics”—
(Only the top part of the ad is shown; for further details
see an ICERM page.)
(ICERM stands for Institute for Computational
and Experimental Research in Mathematics.)
The ICERM logo displays seven subcubes of
a 2x2x2 eight-cube array with one cube missing—
The logo, apparently a stylized image of the architecture
of the Providence building housing ICERM, is not unlike
a picture of Froebel’s Third Gift—
© 2005 The Institute for Figuring
Photo by Norman Brosterman from the Inventing Kindergarten
exhibit at The Institute for Figuring (co-founded by Margaret Wertheim)
The eighth cube, missing in the ICERM logo and detached in the
Froebel Cubes photo, may be regarded as representing the origin
(0,0,0) in a coordinatized version of the 2x2x2 array—
in other words the cube invariant under linear , as opposed to
more general affine , permutations of the cubes in the array.
These cubes are not without relevance to the workshops’ topics—
low-dimensional exotic geometric structures, group theory, and dynamics.
See The Eightfold Cube, A Simple Reflection Group of Order 168, and
The Quaternion Group Acting on an Eightfold Cube.
Those who insist on vulgarizing their mathematics may regard linear
and affine group actions on the eight cubes as the dance of
Snow White (representing (0,0,0)) and the Seven Dwarfs—
.
The following picture provides a new visual approach to
the order-8 quaternion group's automorphisms.
Click the above image for some context.
Here the cube is called "eightfold" because the eight vertices,
like the eight subcubes of a 2×2×2 cube,* are thought of as
independently movable. See The Eightfold Cube.
See also…
Related material: Robin Chapman and Karen E. Smith
on the quaternion group's automorphisms.
* See Margaret Wertheim's Christmas Eve remarks on mathematics
and the following eightfold cube from an institute she co-founded—
Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)
Non-Euclidean
Blocks
Passages from a classic story:
… he took from his pocket a gadget he had found in the box, and began to unfold it. The result resembled a tesseract, strung with beads…. Tesseract
"Your mind has been conditioned to Euclid," Holloway said. "So this– thing– bores us, and seems pointless. But a child knows nothing of Euclid. A different sort of geometry from ours wouldn't impress him as being illogical. He believes what he sees."
"Are you trying to tell me that this gadget's got a fourth dimensional extension?" Paradine demanded. "Hardening of the thought-arteries," Jane interjected. Paradine was not convinced. "Then a baby could work calculus better than Einstein? No, I don't mean that. I can see your point, more or less clearly. Only–" "Well, look. Let's suppose there are two kinds of geometry– we'll limit it, for the sake of the example. Our kind, Euclidean, and another, which we'll call x. X hasn't much relationship to Euclid. It's based on different theorems. Two and two needn't equal four in it; they could equal y, or they might not even equal. A baby's mind is not yet conditioned, except by certain questionable factors of heredity and environment. Start the infant on Euclid–" "Poor kid," Jane said. Holloway shot her a quick glance. "The basis of Euclid. Alphabet blocks. Math, geometry, algebra– they come much later. We're familiar with that development. On the other hand, start the baby with the basic principles of our x logic–" "Blocks? What kind?" Holloway looked at the abacus. "It wouldn't make much sense to us. But we've been conditioned to Euclid." — "Mimsy Were the Borogoves," Lewis Padgett, 1943 |
For the intuitive basis of one type of non-Euclidean* geometry– finite geometry over the two-element Galois field– see the work of…
Friedrich Froebel
(1782-1852), who
invented kindergarten.
His "third gift" —
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:
“through a glass”—
[di’ esoptrou].
By means of
a mirror [esoptron].
Childish things:
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 — 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 — 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: “We have now reached “Space: what you — James Joyce, Ulysses |
Wikipedia on Rubik's 2×2×2 "Pocket Cube"–
"Any permutation of the 8 corner cubies is possible (8! positions)."
Some pages related to this claim–
Analyzing Rubik's Cube with GAP
Online JavaScript Pocket Cube.
The claim is of course trivially true for the unconnected subcubes of Froebel's Third Gift:
See also:
and
"Ad Reinhardt and Tony Smith:
A Dialogue,"
an exhibition opening today
at Pace Wildenstein.
For a different sort
of dialogue, click on the
artists' names above.
For a different
approach to S8,
see Symmetries.
"With humor, my dear Zilkov.
Always with a little humor."
-- The Manchurian Candidate
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.
This photo may serve as an
introduction to a different
sort of space.
See The Eightfold Cube.
For the religious meaning
of this small space, see
For a related larger space,
see the entry and links of
St. Augustine’s Day, 2006.
TIME magazine, issue dated June 12, 2006, item posted Sunday, June 4, 2006:
IF AT FIRST YOU DON'T SUCCEED …
By JULIE RAWE
"Nervous kids and obscure words are not the stuff of big-time TV, but this year's Scripps National Spelling Bee was an improbable nail-biter. One of the 13 finalists got reinstated after judges made a spelling error, a Canadian came in second–who knew foreign kids could compete?–and KATHARINE CLOSE, 13, prevailed in her fifth year. The eighth-grader from Spring Lake, N.J., won with ursprache. It means protolanguage. Now try to use it in conversation."
— Heidegger, Erlauterungen
zu Holderlins Dichtung.
Frankfurt am Main:
Klostermann, 1971: 41.
(Skewed Mirrors,
Sept. 14, 2003)
"Evil did not have
the last word."
— Richard John Neuhaus,
April 4, 2005
"This is the exact opposite
of what echthroi do in
their X-ing or un-naming."
— Wikipedia on
A Wind in the Door
"Lps. The keys to. Given! A way a lone a last a loved a long the PARIS, 1922-1939" — James Joyce, Finnegans Wake |
"There is never any ending
to Paris."
— Ernest Hemingway
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