See also Page 168.
Saturday, October 5, 2024
Friday, August 31, 2018
Perception of Number
Review of yesterday's post Perception of Space —
From Harry Potter and the Philosopher's Stone (1997),
republished as "… and the Sorcerer's Stone ," Kindle edition:
In a print edition from Bloomsbury (2004), and perhaps in the
earliest editions, the above word "movements" is the first word
on page 168:
Click the above ellipse for some Log24 posts on the eightfold cube,
the source of the 168 automorphisms ("movements") of the Fano plane.
"Refined interpretation requires that you know that
someone once said the offspring of reality and illusion
is only a staggering confusion."
— Poem, "The Game of Roles," by Mary Jo Bang
Related material on reality and illusion —
an ad on the back cover of the current New Yorker —
"Hey, the stars might lie, but the numbers never do." — Song lyric
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Tuesday, July 10, 2018
A Dark and Stormy May 29th
He had come a long way to this blue lawn,
and his dream must have seemed so close
that he could hardly fail to grasp it.
He did not know that it was already behind him,
somewhere back in that vast obscurity
beyond the city, where the dark fields of the republic
rolled on under the night.
F. Scott Fitzgerald, The Great Gatsby
— Epigraph to Limitless: A Novel , by Alan Glynn
Glynn's novel was originally published in 2002 under the title
The Dark Fields .
Compare and contrast —
Stephen King's IT was first published by Viking in 1986.
See as well the May 29th date mentioned by King.
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Sunday, April 29, 2018
Amusement
From the online New York Times this afternoon:
Disney now holds nine of the top 10
domestic openings of all time —
six of which are part of the Marvel
Cinematic Universe. “The result is
a reflection of 10 years of work:
of developing this universe, creating
stakes as big as they were, characters
that matter and stories and worlds that
people have come to love,” Dave Hollis,
Disney’s president of distribution, said
in a phone interview.
From this journal this morning:
"But she felt there must be more to this
than just the sensation of folding space
over on itself. Surely the Centaurs hadn't
spent ten years telling humanity how to
make a fancy amusement-park ride.
There had to be more—"
— Factoring Humanity , by Robert J. Sawyer,
Tom Doherty Associates, 2004 Orb edition,
page 168
"The sensation of folding space . . . ."
Or unfolding:
Click the above unfolded space for some background.
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Monday, May 4, 2015
Light to Light
From yesterday —
Another remark on "still light" —
" . . . After the kingfisher's wing
Has answered light to light, and is silent, the light is still
At the still point of the turning world." — Four Quartets
Note the page number, 168, in the above quote from Capobianco.
From another page 168,* a reproduction of a title page —
"In quella parte del libro…."
* In Jewel Spears Brooker's book
T.S. Eliot: The Contemporary Reviews ,
Cambridge University Press, 2004
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Monday, February 7, 2011
Reappearing All Over Again
A sequel to "A Reappearing Number," "Reappearing Continued," and "Classical Requiem"—
A connection to the numerology of today's date, The Seventh— "Frame Tale." (Click, then scroll down.)
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Sunday, May 25, 2008
Sunday May 25, 2008
Wechsler Cubes
"Confusion is nothing new."
— Song lyric, Cyndi Lauper
— Song lyric, Cyndi Lauper
Part I:
Magister Ludi
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
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) –
(named not for
Bill Wechsler, the
fictional artist above,
but for the non-fictional
David Wechsler) –
|
From 2002:
Above: Dr. Harrison Pope, Harvard professor of psychiatry, demonstrates the use of the Wechsler Adult Intelligence Scale "block design" subtest. |
Part IV:
A Magic Gallery
A Magic Gallery
|
ZZ
Figures from the
Poem by Eugen Jost:
Mit Zeichen und Zahlen
Numbers and Names,
With numbers and names English translation A related poem:
Alphabets
From time to time
But if a savage
— Hermann Hesse (1943), |
Comments Off on Sunday May 25, 2008
Tuesday, December 19, 2023
Copilot Report
|
From Microsoft Copilot (with "Precise" setting)
on Dec. 18, 2023 —
How is the 1910 work of G. M. Conwell on PG(3,2) related to the Miracle Octad Generator of R. T. Curtis?
Searching for:
Searching for: Generating answers for you… The work of George M. Conwell on the finite projective 3-space PG(3,2) and the Miracle Octad Generator (MOG) of R. T. Curtis are both significant contributions to the field of combinatorial mathematics12. Conwell’s work on PG(3,2) provided a detailed study of the structure of this finite projective space1. On the other hand, the MOG is a tool for exploring the binary Golay code and its automorphism group M242. The connection between these two works lies in the underlying combinatorial structures they explore. The Rosenhain and Göpel tetrads, which are key structures in the finite geometry PG(3,2), play a significant role in the Curtis Miracle Octad Generator3. These structures provide a link between the finite geometric structures studied by Conwell and the combinatorial structures generated by the MOG3. For a more detailed understanding of this connection, I recommend checking out the resources mentioned in the references123. If you have any specific questions about the concepts in these works, feel free to ask! I’ll do my best to help. Learn more
5
_________________________________________________________
__________________________________________________________
Here is a summary of the document:
Reference:
|
Correction:
"Subsets of four points in a projective 3-space" above is a Copilot error, and not
from the document being summarized. It should be "subsets of four points in an
affine 4-space."
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Monday, October 17, 2022
Wednesday, March 9, 2022
Supercube Space
The new URL supercube.space forwards to http://box759.wordpress.com/.
The term supercube is from a 1982 article by Solomon W. Golomb.
The related new URL supercube.group forwards to a page that
describes how the 2x2x2 (or eightfold, or "super") cube's natural
underlying automorphism group is Klein's simple group of order 168.
For further context, see the new URL supercube.art.
For some background, see the phrase Cube Space in this journal.
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Monday, September 13, 2021
Cube Space Revisited
The above Quanta article mentions
"Maryna Viazovska’s 2016 discovery of the most efficient
ways of packing spheres in dimensions eight and 24."
From a course to be taught by Viazovska next spring:
The Lovasz reference suggests a review of my own webpage
Cube Space, 1984-2003.
See as well a review of Log24 posts on Packing.
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Wednesday, November 7, 2018
Defense Against the Dark Arts
"What is the Necropastoral?" was published on April 29th, 2014.
Also on April 29th, 2014 —
See as well Christianity and Culture by T. S. Eliot.
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Sunday, August 19, 2018
Possible Permutations
John Calder, an independent British publisher who built a prestigious list
of authors like Samuel Beckett and Heinrich Böll and spiritedly defended
writers like Henry Miller against censorship, died on Aug. 13 in Edinburgh.
He was 91.
— Richard Sandomir in the online New York Times this evening
On Beckett —
Also on August 13th —
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Sunday, July 29, 2018
The Materialization
McCarthy's "materialization of plot and character" does not,
for me, constitute a proof that "there is being, after all,
beyond the arbitrary flux of existence."
Neither does the above materialization of 281 as the page
number of her philosophical remark.
See also the materialization of 281 as a page number in
the book Witchcraft by Charles Williams —
The materialization of 168 as a page number in a
Stephen King novel is somewhat more convincing,
but less convincing than the materialization of Klein's
simple group of of 168 elements in the eightfold cube.
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Wednesday, July 11, 2018
Clarity and Precision
"The whole meaning of the word is
looking into something with clarity and precision,
seeing each component as distinct,
and piercing all the way through
so as to perceive the most fundamental reality
of that thing."
For the word itself, try a Web search on
noteworthy phrases above.
“. . . the utterly real thing in writing is
the only thing that counts . . . ."
— Maxwell Perkins to Ernest Hemingway, Aug. 30, 1935
"168"
— Page number in a 2016 Scribner edition
of Stephen King's IT
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Monday, June 4, 2018
The Trinity Stone Defined
“Unsheathe your dagger definitions.” — James Joyce, Ulysses
The “triple cross” link in the previous post referenced the eightfold cube
as a structure that might be called the trinity stone .
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Tuesday, March 27, 2018
Compare and Contrast
Related material on automorphism groups —
The "Eightfold Cube" structure shown above with Weyl
competes rather directly with the "Eightfold Way" sculpture
shown above with Bryant. The structure and the sculpture
each illustrate Klein's order-168 simple group.
Perhaps in part because of this competition, fans of the Mathematical
Sciences Research Institute (MSRI, pronounced "Misery') are less likely
to enjoy, and discuss, the eight-cube mathematical structure above
than they are an eight-cube mechanical puzzle like the one below.
Note also the earlier (2006) "Design Cube 2x2x2" webpage
illustrating graphic designs on the eightfold cube. This is visually,
if not mathematically, related to the (2010) "Expert's Cube."
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Monday, October 19, 2015
Borromean Generators
From slides dated June 28, 2008 —
Compare to my own later note, from March 4, 2010 —
It seems that Guitart discovered these "A, B, C" generators first,
though he did not display them in their natural setting,
the eightfold cube.
Some context: The epigraph to my webpage
"A Simple Reflection Group of Order 168" —
"Let G be a finite, primitive subgroup of GL(V) = GL(n,D) ,
where V is an n-dimensional vector space over the
division ring D . Assume that G is generated by 'nice'
transformations. The problem is then to try to determine
(up to GL(V) -conjugacy) all possibilities for G . Of course,
this problem is very vague. But it is a classical one,
going back 150 years, and yet very much alive today."
— William M. Kantor, "Generation of Linear Groups,"
pp. 497-509 in The Geometric Vein: The Coxeter Festschrift ,
published by Springer, 1981
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Tuesday, January 20, 2015
State of the NOW:
The Malfunctioning TARDIS continues…
The New York Times this evening claims that
this is a photo from the Year of Our Lord 1970:
It clearly is not. [See correction below.]
Related material:
See the reference to 1970 in a post from last Saturday night
and an image from the 2002 film Minority Report :
Update to Log24 at 6:52 PM ET Jan. 21
copied from an earlier correction at the Times :
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Friday, October 4, 2013
Roll Credits
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Tuesday, September 10, 2013
Moss on the Wall
The following page from the Sept. 16, 2013, issue of The New Yorker
deals with current trends in paranoid schizophrenia. It may interest
fans of Philip K. Dick.
(Click for a larger, clearer image.)
As for the poem by Stanley Moss on the above page (35, by the way),
a quote from Wallace Stevens seems appropriate —
"It was a language he spoke, because he must, yet did not know.
It was a page he had found in the handbook of heartbreak."
For the "wall" theme of Moss, see (for instance), this journal
on June 3, 2013 — New Yorker Art.
"All in all…." — Pink Floyd
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Saturday, March 30, 2013
Bright Star
An earlier (noon) version of this post showed a photo of a
heartbreakingly beautiful young woman I encountered on
the Web today by chance. Perhaps it is best if she remains
anonymous here.
A detail taken today from her Facebook page:
Location of a nightclub in Bucharest
Related material: Shining Forth by myself and Malcolm Lowry,
and Lucifer by Mihai Eminescu.
See also this journal "about 3 weeks ago."
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Sunday, March 10, 2013
Sunday School
The previous post described briefly a 2002
book on magic and religion, Golems of Gotham .
For a Sunday New York Times review, click here.
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Tuesday, November 27, 2012
Counterexample
The non-Coxeter simple reflection group of order 168
is a counterexample to the statement that
"Every finite reflection group is a Coxeter group."
The counterexample is based on a definition of "reflection group"
that includes reflections defined over finite fields.
Today I came across a 1911 paper that discusses the counterexample.
Of course, Coxeter groups were undefined in 1911, but the paper, by
Howard H. Mitchell, discusses the simple order-168 group as a reflection group .
(Naturally, Mitchell's definition of "reflection" and his statement that
"The discussion of the binary groups
applies also to the case p = 2."
should be approached with care.)
A review of this topic might be appropriate for Jessica Fintzen's 2012 fall tutorial at Harvard
on reflection groups and Coxeter groups. The syllabus for the tutorial states that
"finite Coxeter groups correspond precisely to finite reflection groups." This statement
is based on Fintzen's definition of "reflection group"—
"Reflection groups are— as their name indicates—
groups generated by reflections across
hyperplanes of Rn which contain the origin."
For some background, see William Kantor's 1981 paper "Generation of Linear Groups"
(quoted at the finitegeometry.org page on the simple order-168 counterexample).
Kantor discusses Mitchell's work in some detail, but does not mention the
simple order-168 group explicitly.
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Wednesday, November 14, 2012
Group Actions
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—
.
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Monday, June 18, 2012
Surface
"Poetry is an illumination of a surface…."
— Wallace Stevens
Some poetic remarks related to a different surface, Klein's Quartic—
This link between the Klein map κ and the Mathieu group M24
is a source of great delight to the author. Both objects were
found in the 1870s, but no connection between them was
known. Indeed, the class of maximal subgroups of M24
isomorphic to the simple group of order 168 (often known,
especially to geometers, as the Klein group; see Baker [8])
remained undiscovered until the 1960s. That generators for
the group can be read off so easily from the map is
immensely pleasing.
— R. T. Curtis, Symmetric Generation of Groups ,
Cambridge University Press, 2007, page 39
Other poetic remarks related to the simple group of order 168—
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Wednesday, April 18, 2012
Adam in Eden
…. and John Golding, an authority on Cubism who "courted abstraction"—
"Adam in Eden was the father of Descartes." — Wallace Stevens
Fictional symbologist Robert Langdon and a cube—
From a Log24 post, "Eightfold Cube Revisited,"
on the date of Golding's death—
A related quotation—
"… quaternions provide a useful paradigm
for studying the phenomenon of 'triality.'"
— David A. Richter's webpage Zometool Triality
See also quaternions in another Log24 post
from the date of Golding's death— Easter Act.
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Sunday, September 18, 2011
Anatomy of a Cube
R.D. Carmichael's seminal 1931 paper on tactical configurations suggests
a search for later material relating such configurations to block designs.
Such a search yields the following—
"… it seems that the relationship between
BIB [balanced incomplete block ] designs
and tactical configurations, and in particular,
the Steiner system, has been overlooked."
— D. A. Sprott, U. of Toronto, 1955
The figure by Cullinane included above shows a way to visualize Sprott's remarks.
For the group actions described by Cullinane, see "The Eightfold Cube" and
"A Simple Reflection Group of Order 168."
Update of 7:42 PM Sept. 18, 2011—
From a Summer 2011 course on discrete structures at a Berlin website—
A different illustration of the eightfold cube as the Steiner system S(3, 4, 8)—
Note that only the static structure is described by Felsner, not the
168 group actions discussed (as above) by Cullinane. For remarks on
such group actions in the literature, see "Cube Space, 1984-2003."
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Monday, September 5, 2011
Illuminata*
At Heaven’s Gate
Sunday, Sept. 4, 2011, RSS at 23:59 EDT:
Peter Woit's weblog Not Even Wrong—
"Lisa Randall’s new book is about to come out, it’s entitled
Knocking on Heaven’s Door: How Physics and Scientific Thinking
Illuminate the Universe and the Modern World."
Angels & Demons (the film)—
As she enters the lab she reacts in horror
as she sees an eyeball lying on the floor…
Click images for some backstory —
|
|
 |
"She has taken on the role of a public face of physics,
and has written a book which is in part a very general defense
of science and the materialist, rationalist world-view
that modern science is based on."
See also yesterday's "The Stone" column in The New York Times—
I prefer philosophy enacted by Reba.
* A reference to Dan Brown, not Marianne Williamson
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Monday, July 25, 2011
The Game
Virginia Heffernan in Sunday's online New York Times—
"… In the past, information on paper was something to read. Bricks and mortar were a place to be. But, since the first appearance of the Web in 1990, we have come to accept that information in pixels is something to read— and also a place to be . That familiar and yet still jaw-dropping metaphor takes energy to maintain. The odd shared sense that there’s three-dimensionality and immersion and real-world consequences on the Web as in no book or board game— that’s the Web’s sine qua non. Hence, cyberspace . And 'being on' the Internet….
… The dominant social networks are fantasy games built around rigged avatars, outright fictions and a silent— and often unconscious— agreement among players that the game and its somewhat creaky conceits influence the real world…."
— "The Confidence Game at Google+"
"It's just another manic Monday
I wish it was Sunday
'Cause that's my funday"
— The Bangles
"Accentuate the Positive"
— Clint Eastwood, soundtrack album
for "Midnight in the Garden of Good and Evil"
This journal on All Saints' Day, Sunday, November 1, 2009—
|
Suggested by the New York State lottery numbers on All Hallows’ Eve [2009]— 430 (mid-day) and 168 (evening)… From 430 as a date, 4/30— Beyond Grief and Nothing: A Reading of Don DeLillo , by Joseph Dewey, University of South Carolina Press, 2006, page 123: “It is as if DeLillo himself had moved to an endgame….” For such an endgame, see yesterday’s link to a Mira Sorvino drama. The number 168 suggested by the Halloween lottery deals with the properties of space itself and requires a more detailed exegesis… For the full picture, consider the Log24 entries of Feb. 16-28 this year, esp. the entries of Feb. 27 and the phrase they suggest— Flores, flores para los muertos. |
See also Pearly Gates of Cyberspace in this journal.
For flores para los muertos , see today's Times .
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Wednesday, July 6, 2011
Nordstrom-Robinson Automorphisms
A 2008 statement on the order of the automorphism group of the Nordstrom-Robinson code—
"The Nordstrom-Robinson code has an unusually large group of automorphisms (of order 8! = 40,320) and is optimal in many respects. It can be found inside the binary Golay code."
— Jürgen Bierbrauer and Jessica Fridrich, preprint of "Constructing Good Covering Codes for Applications in Steganography," Transactions on Data Hiding and Multimedia Security III, Springer Lecture Notes in Computer Science, 2008, Volume 4920/2008, 1-22
A statement by Bierbrauer from 2004 has an error that doubles the above figure—
The automorphism group of the binary Golay code G is the simple Mathieu group M24 of order
— Jürgen Bierbrauer, "Nordstrom-Robinson Code and A7-Geometry," preprint dated April 14, 2004, published in Finite Fields and Their Applications , Volume 13, Issue 1, January 2007, Pages 158-170
The error is corrected (though not detected) later in the same 2004 paper—
In fact the symmetry group of the octacode is a semidirect product of an elementary abelian group of order 16 and the simple group GL(3, 2) of order 168. This constitutes a large automorphism group (of order 2688), but the automorphism group of NR is larger yet as we saw earlier (order 40,320).
For some background, see a well-known construction of the code from the Miracle Octad Generator of R.T. Curtis—
For some context, see the group of order 322,560 in Geometry of the 4×4 Square.
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Thursday, June 23, 2011
Script Magic…
In a Jewish Cathedral
From The New York Times Magazine of Sunday, April 6, 1986—
"David Rayfiel's Script Magic" by Alex Ward
WHEN THE CALL came last year to revise ''The Morning After,'' Rayfiel was working on a screenplay about the 1911 Triangle Shirtwaist Company fire for Barbra Streisand and Jane Fonda. He has now resumed work— as the principal writer, not the reviser— on that script. But chances are good that he will have further interruptions. Pollack will probably call and say, as he usually does, ''David, I need access to your brain.'' And Rayfiel will probably say, as he usually does, ''That's O.K., I'm not using it.'' He will revise another script, and be reluctant about taking credit for it.
''I guess it's like the medieval stonecutters who worked on the cathedrals,'' he says. ''There's all that elaborate work. The saints were carved by one guy, the cherubs by someone else. They didn't care about getting credit, they knew what they'd done. I'm like that. I'm the guy who does the cherubs.''
Related material:
Proginoskes in this journal and Abracadabra from the midnight of June 18-19.
See also Rayfiel's obituary in today"s Times .
For some quite different work, also from April 1986, see—
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Tuesday, June 21, 2011
The Longest Day Continues
The AND Publishing weblog page referred to in
a Sunday post has been changed to reflect the
source— my finite-geometry website— of pages
copied and altered by London artist Steve Richards
that are a large part of his contribution to the
AND Publishing Piracy Project.
The new version is as follows—
Note, however, that the cover page is a figure titled
by Richards "metalibrarianship" that has nothing
whatever to do with the concepts in the pages he copied
from my site, finitegeometry.org/sc.
Other pages within Richards's contribution to the
Piracy Project are similarly completely unrelated to
the content of my own site, which deals with geometry.
The image on the cover page also appears, it turns out,
on a website called intertwining.org.
At that site, it occurs in the following resume item:
The links in the resume item do not work,
but some background is available at a page titled
"Circularity, Practicality and Philosophy of Librarianship, or
The Making of 'The Nitecki Trilogy'" by Joanne Twining.
Other images in Richards's contribution to the Piracy Project also occur
in Twining's webpage "Dimensional Advances for Information Architecture."
I never heard of Twining or Nitecki before I encountered Richards's
Piracy Project contribution, and I do not wish to be associated
again in any way with Twining, with Nitecki, or with Richards.
Comments Off on The Longest Day Continues
Sunday, June 19, 2011
Abracadabra (continued)
Yesterday's post Ad Meld featured Harry Potter (succeeding in business),
a 4×6 array from a video of the song "Abracadabra," and a link to a post
with some background on the 4×6 Miracle Octad Generator of R.T. Curtis.
A search tonight for related material on the Web yielded…
Weblog post by Steve Richards titled "The Search for Invariants:
The Diamond Theory of Truth, the Miracle Octad Generator
and Metalibrarianship." The artwork is by Steven H. Cullinane.
Richards has omitted Cullinane's name and retitled the artwork.
The author of the post is an artist who seems to be interested in the occult.
His post continues with photos of pages, some from my own work (as above), some not.
My own work does not deal with the occult, but some enthusiasts of "sacred geometry" may imagine otherwise.
The artist's post concludes with the following (note also the beginning of the preceding post)—
"The Struggle of the Magicians" is a 1914 ballet by Gurdjieff. Perhaps it would interest Harry.
Comments Off on Abracadabra (continued)
Saturday, June 18, 2011
Midnight in the Garden (continued)–
Tracking Shot
Related material—
See also this journal's September 2009 posts.
This post was suggested by today's previous post and by today's NY Lottery.
For some background to the ioncinema.com post numbered 4210 above,
see, in conjunction with the page headed "Azazel" linked to here earlier today,
the ioncinema.com post numbered 5601.
|
“Stranger, dreams verily are baffling and unclear of meaning, For two are the gates of shadowy dreams, But in my case it was not from thence, methinks, Translation by A.T. Murray, in two volumes. Quoted in a press release for the film "Two Gates of Sleep." |
From the post numbered 460 in this journal—
At the still point… from the film "Absolute Power" :
Photo credit – Graham Kuhn
I’ve heard of affairs that are strictly plutonic,
But diamonds are a girl’s best friend!
Comments Off on Midnight in the Garden (continued)–
Friday, June 17, 2011
Bloomsday Lottery
This morning's exercise in lottery hermeneutics is unusually difficult.
Yesterday was Bloomsday (the date described in
James Joyce's Ulysses ) and the New York Lottery numbers were…
Midday numbers: 3-digit 181, 4-digit 9219.
Evening numbers: 3-digit 478, 4-digit 6449.
For 181 and 9219, see the following—
"With respect to every event, we must ask
which element has been subjected directly to change."
— Ferdinand de Saussure, Course in General Linguistics
(New York, The Philosophical Library, Inc., 1959), page 181
That Saussure page number was referenced in the following thesis
on James Joyce's other major novel, Finnegans Wake—
The thesis is from the University of Vienna (Universität Wien ).
The word Wien , in the derived form denoting an inhabitant of that city,
figured prominently in yesterday's news.
As for the evening numbers—
478 perhaps signifies the year 478 BC,
cited in Lawrence Durrell's Sicilian Carousel as the year
the ruler Gelon died.
For the evening 6449, note that the poem by Wallace Stevens quoted
here on June 15 in A for Anastasios deals with "the river of rivers"…
perhaps signifying time.
Interpreting 6449 chronologically yields 6/4/49.
The film artist John Huston, discussed in an essay from that date,
might appreciate the representation of the ancient Sicilian
river god Gelas as a man-headed bull on a coin from
around the year 478 BC.
For some perceptive remarks about Durrell, see the
article by Nigel Dennis in LIFE magazine's Nov. 21, 1960
issue (with cover noting Kennedy's victory in that year's
presidential election).
All of the above may be viewed as an approach to the aesthetic
problem posed by Dennis in yesterday's Bloomsday post—
"The problem that arises with this sort of writing is
one of form, i.e. , how to make one strong parcel
out of so many differently shaped commodities,
how to impose method on what would otherwise
be madness."
"The world has gone mad today…." — Cole Porter
For some related remarks, see page 161 of
Joyce's Catholic Comedy of Language *
by Beryl Schlossman (U. of Wisconsin Press, 1985)
and James Joyce in the final pages of The Left Hand of God
by Adolf Holl.
* Update of July 6, 2011—
This title is a correction from the previous title
given here, Moral Language by Mary Gore Forrester.
Google Books had Schlossman's content previewed
under Forrester's title.
Comments Off on Bloomsday Lottery
Monday, May 30, 2011
Darkness at Noon
A Meditation on the NY Lottery of May 29
Yesterday's NY Lottery— Midday 981, Evening 275.
As noted in yesterday morning's linked-to post,
The Shining of May 29…
"By groping toward the light we are made to realize
how deep the darkness is around us."
— Arthur Koestler, The Call Girls: A Tragi-Comedy ,
Random House, 1973, page 118
One interpretation of the mystic numbers revealed by the Lottery yesterday—
981 as the final page* of David Foster Wallace's famed novel Infinite Jest …
275 as a page in Wallace's non-fiction book about infinity Everything and More …
Gregory Chaitin points out that this is nonsense …
As noted elsewhere in this journal, I have a different concept of "math's absolute
Prince of Darkness"— and, indeed, of a "quest for Omega." (See posts of May 2010.)
Yesterday's numbers indicate a different struggle between darkness and light—
Light —
Darkness —
* From infinitesummer.org/archives/168 — "A note about editions:
As it turns out, all (physical) editions of Infinite Jest have 981 pages:
the one from 1996, the one from 2004, the paperback, the hardcover, etc.
A big thank you to the men and women in the publishing industry who
were kind and/or lazy enough to keep things consistent."
Comments Off on Darkness at Noon
Wednesday, April 20, 2011
Romancing the Cube
It was a dark and stormy night…
— Page 180, Logicomix
“… the class of reflections is larger in some sense over an arbitrary field than over a characteristic zero field.”
– Julia Hartmann and Anne V. Shepler, “Jacobians of Reflection Groups”
For some context, see the small cube in “A Simple Reflection Group of Order 168.”
See also the larger cube in “Many Dimensions” + Whitehead in this journal (scroll down to get past the current post).
That search refers to a work by Whitehead published in 1906, the year at the top of the Logicomix page above—
A related remark on axiomatics that has metaphysical overtones suitable for a dark and stormy night—
“An adequate understanding of mathematical identity requires a missing theory that will account for the relationships between formal systems that describe the same items. At present, such relationships can at best be heuristically described in terms that invoke some notion of an ‘intelligent user standing outside the system.'”
— Gian-Carlo Rota, “Syntax, Semantics, and…” in Indiscrete Thoughts . See also the original 1988 article.
Comments Off on Romancing the Cube
Tuesday, November 23, 2010
Art Object
There is more than one way
to look at a cube.
From Cambridge U. Press on Feb. 20, 2006 —
and from this journal on June 30, 2010 —
In memory of Wu Guanzhong, Chinese artist
who died in Beijing on June 25, 2010 —
See also this journal on Feb. 20, 2006
(the day The Cube was published).
Comments Off on Art Object
Sunday, May 30, 2010
A Post for Galois
Evariste Galois, 1811-1832 (Vita Mathematica, V. 11)
- Paperback: 168 pages
- Publisher: Birkhäuser Basel; 1 edition (December 6, 1996)
- Language: English
- ISBN-10: 3764354100
- ISBN-13: 978-3764354107
- Product Dimensions: 9.1 x 6 x 0.4 inches
- Shipping Weight: 9.1 ounces
- Average Customer Review: 5.0 out of 5 stars (1 customer review)
- Amazon Bestsellers Rank: #933,939 in Books
Awarded 5 stars by Christopher G. Robinson (Cambridge, MA USA).
See also other reviews by Robinson.
Galois was shot in a duel on today's date, May 30, in 1832. Related material for those who prefer entertainment to scholarship—
"It is a melancholy pleasure that what may be [Martin] Gardner’s last published piece, a review of Amir Alexander’s Duel at Dawn: Heroes, Martyrs & the Rise of Modern Mathematics, will appear next week in our June issue." —Roger Kimball of The New Criterion, May 23, 2010.
Today is, incidentally, the feast day of St. Joan of Arc, Die Jungfrau von Orleans. (See "against stupidity" in this journal.)
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Thursday, February 18, 2010
Theories: An Outline
Truth, Geometry, Algebra
The following notes are related to A Simple Reflection Group of Order 168.
1. According to H.S.M. Coxeter and Richard J. Trudeau
“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?’.”
— Coxeter, 1987, introduction to Trudeau’s The Non-Euclidean Revolution
1.1 Trudeau’s Diamond Theory of Truth
1.2 Trudeau’s Story Theory of Truth
2. According to Alexandre Borovik and Steven H. Cullinane
2.1 Coxeter Theory according to Borovik
2.1.1 The Geometry–
Mirror Systems in Coxeter Theory
2.1.2 The Algebra–
Coxeter Languages in Coxeter Theory
2.2 Diamond Theory according to Cullinane
2.2.1 The Geometry–
Examples: Eightfold Cube and Solomon’s Cube
2.2.2 The Algebra–
Examples: Cullinane and (rather indirectly related) Gerhard Grams
Summary of the story thus far:
Diamond theory and Coxeter theory are to some extent analogous– both deal with reflection groups and both have a visual (i.e., geometric) side and a verbal (i.e., algebraic) side. Coxeter theory is of course highly developed on both sides. Diamond theory is, on the geometric side, currently restricted to examples in at most three Euclidean (and six binary) dimensions. On the algebraic side, it is woefully underdeveloped. For material related to the algebraic side, search the Web for generators+relations+”characteristic two” (or “2“) and for generators+relations+”GF(2)”. (This last search is the source of the Grams reference in 2.2.2 above.)
Comments Off on Theories: An Outline
Sunday, February 14, 2010
Sunday School
"Simplify, simplify." — Henry David Thoreau
"Because of their truly fundamental role in mathematics, even the simplest diagrams concerning finite reflection groups (or finite mirror systems, or root systems– the languages are equivalent) have interpretations of cosmological proportions."
— Alexandre Borovik, 2010 (See previous entry.)
Exercise: Discuss Borovik's remark
that "the languages are equivalent"
in light of the web page
A Simple Reflection Group
of Order 168.
Background:
Theorems 15.1 and 15.2 of Borovik's book (1st ed. Nov. 10, 2009)
Mirrors and Reflections: The Geometry of Finite Reflection Groups—
15.1 (p. 114): Every finite reflection group is a Coxeter group.
15.2 (p. 114): Every finite Coxeter group is isomorphic to a finite reflection group.
Consider in this context the above simple reflection group of order 168.
(Recall that "…there is only one simple Coxeter group (up to isomorphism); it has order 2…" —A.M. Cohen.)
Comments Off on Sunday School
Wednesday, December 16, 2009
The Passion of the Beast
A thought from yesterday by David Brooks:
“Life is a struggle to push back against the evils of the world without succumbing to the passions of the beast lurking inside.”
Associated Press Thought for Today:
“Any sufficiently advanced technology is indistinguishable from magic.”
— British science-fiction author Arthur C. Clarke (born this date in 1917, died in 2008).
Photo by Amy Marash
Arthur C. Clarke at his home office
in Sri Lanka, 28 March 2005
A search for that date
in this journal yields…
Bright Star and Happy Six.
Comments Off on The Passion of the Beast
Sunday, November 1, 2009
October Endgame
Suggested by the New York State lottery numbers on All Hallows' Eve–
430 (mid-day) and 168 (evening)…
From 430 as a date, 4/30— Beyond Grief and Nothing: A Reading of Don DeLillo, by Joseph Dewey, University of South Carolina Press, 2006, page 123:
"It is as if DeLillo himself had moved to an endgame…."
For such an endgame, see yesterday's link to a Mira Sorvino drama. The number 168 suggested by the Halloween lottery deals with the properties of space itself and requires a more detailed exegesis… For the full picture, consider the Log24 entries of Feb. 16-28 this year, esp. the entries of Feb. 27 and the phrase they suggest–
Flores, Flores para los muertos.
Consider also Xinhua today, with its discussion of rocket science and seal-cutting:
Click image for context.
For space technology, see the above link to Feb. 16-28 this year as well as the following (click on image for details)–
As for seal-cutting, see the following seal from a Korean Christian site:
See Mizian Translation Service for some background on the seal's designer.
Comments Off on October Endgame
Tuesday, July 21, 2009
Tuesday July 21, 2009
Today's Readings:
-
The White Itself
Plato and the "concrete universal"-- Log24 on Thursday, July 16, 2009
-
Edged with Brown
Context for a Log24 entry of July 16: "So we moved, and they, in a formal pattern, Along the empty alley, into the box circle, To look down into the drained pool. Dry the pool, dry concrete, brown edged, And the pool was filled with water out of sunlight, And the lotos rose, quietly, quietly, The surface glittered out of heart of light...."
-
Signifying Nothing
An essay on Harvard professor Henry Louis Gates, Jr.
-
Professor Gates Arrested
A racial incident in Cambridge on Thursday, July 16, 2009
-
New England White
Race relations in Academia
-
Notes on Mathematics
and Narrative"... the glue that binds the brotherhood is ultimately made not of love and interracial harmony, but of something stronger and more enduring: shame, fear, and greed." -- Review of New England White
-
Don't Forget Hate
Cf. Eugene Burdick and The Word, 1966. More recently, Tom Wolfe and The Word and Pig and Rat Get Lost.
Comments Off on Tuesday July 21, 2009
Friday, February 27, 2009
Friday February 27, 2009
Lasting Significance
Wittgenstein’s Lasting Significance, edited by Max Kölbel and Bernhard Weiss, published by Routledge, 2004–
Page 168:
“Wittgenstein told Norman Malcolm that ‘a serious and good philosophical work could be written that would consist entirely of jokes (without being facetious)’ (Malcolm 1999: 64).”
Malcolm, N. (1999) “Wittgenstein: A Memoir,” in F.A. Flowers (ed.) Portraits of Wittgenstein, vol. 3, Bristol: Thoemmes Press, pp. 60-112
The lasting significance here is perhaps in the page numbers.
Comments Off on Friday February 27, 2009
Tuesday, October 7, 2008
Tuesday October 7, 2008
The Color Grey
The previous two entries mention,
and illustrate, the color grey.
Another illustration, on the cover
of one of my favorite books:
"A colour is eternal.
It haunts time like a spirit."
— Alfred North Whitehead
From John Lahr's
winter 2002 review
of "Our Town"–
"We all know that something is eternal," the Stage Manager says. "And it ain't houses and it ain't names, and it ain't earth, and it ain't even stars– everybody knows in their bones that something is eternal, and that something has to do with human beings."
The Stage Manager was played by Paul Newman. The review was subtitled "Getting the Spirit Onstage."
Comments Off on Tuesday October 7, 2008
Saturday, February 9, 2008
Saturday February 9, 2008
The timestamp of this entry, 4:23 AM, may be regarded as a memorial to Fra’ Andrew Bertie (see Andrew Cusack’s journal). It was at about this time that I heard of Fra’ Andrew’s death. The timestamp is a reference to Shakespeare’s birthday and to the following thought:
Page 162 of Orthodoxy, by G. K. Chesterton (1908), reprinted in 1995 by Ignatius Press, San Francisco–
The man who lives in contact with what he believes to be a living Church is a man always expecting to meet Plato and Shakespeare tomorrow at breakfast. He is always expecting to see some truth that he has never seen before.
The entry itself was written later… on the morning of Monday, Feb. 11, 2008. For a similar reference of sorts, to Plato, see “Epiphany for Roy, Part II” (timestamped 7:59 AM Sunday, Feb. 10, 2008).
Comments Off on Saturday February 9, 2008
Saturday, August 11, 2007
Saturday August 11, 2007
Four Colours
The previous entry dealt with Plato’s myth of the ring of Gyges that conferred invisibility. Another legendary ring, from Hermann Hesse, with some background from Carl Jung:
From C. G. Jung, Collected Works (Princeton U. Press), Volume 12– Psychology and Alchemy (1944)– Part II– “Individual Dream Symbolism in Relation to Alchemy”– Chapter 3, “The Symbolism of the Mandala”– as quoted in Jung, Dreams, published by Routledge, 2001– Page 265–
“… the dreamer is wandering about in a dark cave, where a battle is going on between good and evil. But there is also a prince who knows everything. He gives the dreamer a ring set with a diamond….
Visual impression (waking dream):
The dreamer is falling into the abyss. At the bottom there is a bear whose eyes gleam alternately in four colours: red, yellow, green, and blue. Actually it has four eyes that change into four lights. The bear disappears and the dreamer goes through a long dark tunnel. Light is shimmering at the far end. A treasure is there, and on top of it the ring with the diamond. It is said that this ring will lead him on a long journey to the east.”
Hermann Hesse, The Journey to the East (1932):
“‘… Despair is the result of each earnest attempt to go through life with virtue, justice, and understanding and to fulfil their requirements. Children live on one side of despair, the awakened on the other side. Defendant H. is no longer a child and is not yet fully awakened. He is still in the midst of despair. He will overcome it and thereby go through his second novitiate. We welcome him anew into the League, the meaning of which he no longer claims to understand. We give back to him his lost ring, which the servant Leo has kept for him.’
The Speaker then brought the ring, kissed me on the cheek and placed the ring on my finger. Hardly had I looked at the ring, hardly had I felt its metallic coolness on my fingers, when a thousand things occurred to me, a thousand inconceivable acts of neglect. Above all, it occurred to me that the ring had four stones at equal distances apart, and that it was a rule of the League and part of the vow to turn the ring slowly on the finger at least once a day, and at each of the four stones to bring to mind one of the four basic precepts of the vow. I had not only lost the ring and had not once missed it, but during all those dreadful years I had also no longer repeated the four basic precepts or thought of them. Immediately, I tried to say them again inwardly. I had an idea what they were, they were still within me, they belonged to me as does a name which one will remember in a moment but at that particular moment cannot be recalled. No, it remained silent within me, I could not repeat the rules, I had forgotten the wording. I had forgotten the rules; for many years I had not repeated them, for many years I had not observed them and held them sacred– and yet I had considered myself a loyal League brother.
The Speaker patted my arm kindly when he observed my dismay and deep shame.”
Comments Off on Saturday August 11, 2007
Thursday, March 30, 2006
Thursday March 30, 2006
| Galatians 4:4
But when the fulness |
Luke 2:13
And suddenly |
|
Inscape: The Christology and
Poetry of Gerard Manley Hopkins, by James Finn Cotter, University of Pittsburgh Press, 1972. See esp. the references to pleroma 40-48, 51, 65, 70, 81, 85, 92, 93, |
Comments Off on Thursday March 30, 2006
Wednesday, December 7, 2005
Wednesday December 7, 2005
Magical Thinking
(continued)
(continued)
1:00:19 EST
The Lion, the Witch
and the Wardrobe
premieres tonight at
the Royal Albert Hall.
Hexagram 19 in the
Cullinane series:
— Katherine Neville, The Eight
|
“What does this have to do with why we’re here?”
“I saw it in a chess book Mordecai showed me. The most ancient chess service ever discovered was found at the palace of King Minos on Crete– the place where the famous Labyrinth was built, named after this sacred axe. The chess service dates to 2000 B.C. It was made of gold and silver and jewels…. And in the center was carved a labrys.” … “But I thought chess wasn’t even invented until six or seven hundred A.D.,” I added. “They always say it came from Persia or India. How could this Minoan chess service be so old?” “Mordecai’s written a lot himself on the history of chess,” said Lily…. “He thinks that chess set in Crete was designed by the same guy who built the Labyrinth– the sculptor Daedalus….” Now things were beginning to click into place…. “Why was this axe carved on the chessboard?” I asked Lily, knowing the answer in my heart before she spoke. “What did Mordecai say was the connection?”…. “That’s what it’s all about,” she said quietly. “To kill the King.” The sacred axe was used to kill the King. The ritual had been the same since the beginning of time. The game of chess was merely a reenactment. Why hadn’t I recognized it before? |
|
“But what does it all mean?” asked Susan when they were somewhat calmer. “It means,” said Aslan, “that though the Witch knew the Deep Magic, there is a magic deeper still which she did not know. Her knowledge goes back only to the dawn of Time. But if she could have looked a little further back, into the stillness and the darkness before Time dawned, she would have read there a different incantation. She would have known that when a willing victim who had committed no treachery was killed in a traitor’s stead, the Table would crack and Death itself would start working backward.” |
Comments Off on Wednesday December 7, 2005
Saturday, October 8, 2005
Wednesday, July 13, 2005
Wednesday July 13, 2005
Today's birthday: Harrison Ford
Location, Location, Location
Wikipedia on Temple of Doom:
Recent Messages:
"Most of the filming was done
on location in Sri Lanka."
| Math Forum | |
|---|---|
| UserID: | 226278 |
| Name: | |
| Email: | |
| Registered: | 7/5/05 |
| Occupation: | Teacher |
| Location: | Srilanka |
| Homepage: | http://www.math16.com/ |
| Biography: | Mathematics teacher in Srilanka. |
| Total Posts: | 10 |
Recent Messages:
1. Re: Steven Cullinane is a Crank
Jul 12, 2005
2. Re: Steven Cullinane is a Crank
Jul 12, 2005
3. Re: Steven Cullinane is a Crank
Jul 12, 2005
4. Re: Steven Cullinane's "Diamond Theory"
Jul 8, 2005
5. Steven Cullinane's "Diamond Theory"
Jul 7, 2005
6. Re: Steven Cullinane is a Crank
Jul 7, 2005
7. Re: Steven Cullinane is a Crank
Jul 5, 2005
8. Re: Steven Cullinane is a Crank
Jul 5, 2005
9. Steven Cullinane is a Crank
Jul 5, 2005
10. Steven Cullinane is a Crank
Jul 5, 2005
Jul 12, 2005
2. Re: Steven Cullinane is a Crank
Jul 12, 2005
3. Re: Steven Cullinane is a Crank
Jul 12, 2005
4. Re: Steven Cullinane's "Diamond Theory"
Jul 8, 2005
5. Steven Cullinane's "Diamond Theory"
Jul 7, 2005
6. Re: Steven Cullinane is a Crank
Jul 7, 2005
7. Re: Steven Cullinane is a Crank
Jul 5, 2005
8. Re: Steven Cullinane is a Crank
Jul 5, 2005
9. Steven Cullinane is a Crank
Jul 5, 2005
10. Steven Cullinane is a Crank
Jul 5, 2005
Google Groups view of
the main thread (at sci.math)
to which crankbuster has posted
Comments Off on Wednesday July 13, 2005
Wednesday, May 4, 2005
Wednesday May 4, 2005
The Fano Plane
Revisualized:
Revisualized:
or, The Eightfold Cube
Here is the usual model of the seven points and seven lines (including the circle) of the smallest finite projective plane (the Fano plane):
Every permutation of the plane's points that preserves collinearity is a symmetry of the plane. The group of symmetries of the Fano plane is of order 168 and is isomorphic to the group PSL(2,7) = PSL(3,2) = GL(3,2). (See Cameron on linear groups (pdf).)
The above model indicates with great clarity six symmetries of the plane– those it shares with the equilateral triangle. It does not, however, indicate where the other 162 symmetries come from.
Shown below is a new model of this same projective plane, using partitions of cubes to represent points:
The cubes' partitioning planes are added in binary (1+1=0) fashion. Three partitioned cubes are collinear if and only if their partitioning planes' binary sum equals zero.
The second model is useful because it lets us generate naturally all 168 symmetries of the Fano plane by splitting a cube into a set of four parallel 1x1x2 slices in the three ways possible, then arbitrarily permuting the slices in each of the three sets of four. See examples below.
For a proof that such permutations generate the 168 symmetries, see Binary Coordinate Systems.
(Note that this procedure, if regarded as acting on the set of eight individual subcubes of each cube in the diagram, actually generates a group of 168*8 = 1,344 permutations. But the group's action on the diagram's seven partitions of the subcubes yields only 168 distinct results. This illustrates the difference between affine and projective spaces over the binary field GF(2). In a related 2x2x2 cubic model of the affine 3-space over GF(2) whose "points" are individual subcubes, the group of eight translations is generated by interchanges of parallel 2x2x1 cube-slices. This is clearly a subgroup of the group generated by permuting 1x1x2 cube-slices. Such translations in the affine 3-space have no effect on the projective plane, since they leave each of the plane model's seven partitions– the "points" of the plane– invariant.)
To view the cubes model in a wider context, see Galois Geometry, Block Designs, and Finite-Geometry Models.
For another application of the points-as-partitions technique, see Latin-Square Geometry: Orthogonal Latin Squares as Skew Lines.
For more on the plane's symmetry group in another guise, see John Baez on Klein's Quartic Curve and the online book The Eightfold Way. For more on the mathematics of cubic models, see Solomon's Cube.
For a large downloadable folder with many other related web pages, see Notes on Finite Geometry.
Comments Off on Wednesday May 4, 2005
Thursday, March 3, 2005
Thursday March 3, 2005
Necessity, Possibility, Symmetry
Matrix group actions,
March 26, 1985
"We symbolize logical necessity
with the box 
)
and logical possibility
with the diamond 
).
— Keith Allen Korcz,
(Log24.net, 1/25/05)
And what do we
symbolize by 
?
"The possibilia that exist,
and out of which
the Universe arose,
are located in
a necessary being…."
— Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)
Tuesday, March 1, 2005
Tuesday March 1, 2005
3/16 Continued
The New Yorker, issue dated March 7, 2005, on Hunter S. Thompson:
“… his true model and hero was F. Scott Fitzgerald. He used to type out pages from ‘The Great Gatsby,’ just to get the feeling, he said, of what it was like to write that way, and Fitzgerald’s novel was continually on his mind while he was working on ‘Fear and Loathing in Las Vegas,’ which was published, after a prolonged and agonizing compositional nightmare, in 1972. That book was supposed to be called ‘The Death of the American Dream,’ a portentous age-of-Aquarius cliché that won Thompson a nice advance but that he naturally came to consider, as he sat wretchedly before his typewriter night after night, a millstone around his neck.”
for St. Patrick’s Eve
by Steven H. Cullinane
on March 16, 2001
“I hope she’ll be a fool — that’s the best thing a girl can be in this world, a beautiful little fool.”
— Daisy Buchanan in Chapter I of The Great Gatsby
“Thanks for the tip, American Dream.”
— Spider-Girl, in Vol. 1, No. 30, March 2001
Comments Off on Tuesday March 1, 2005
Sunday, February 20, 2005
Sunday February 20, 2005
Relativity Blues
Today, February 20, is the 19th anniversary of my note The Relativity Problem in Finite Geometry. Here is some related material.
In 1931, the Christian writer Charles Williams grappled with the theology of time, space, free will, and the many-worlds interpretation of quantum mechanics (anticipating by many years the discussion of this topic by physicists beginning in the 1950's).
(Some pure mathematics — untainted by physics or theology — that is nevertheless related, if only by poetic analogy, to Williams's 1931 novel, Many Dimensions, is discussed in the above-mentioned note and in a generalization, Solomon's Cube.)
On the back cover of Williams's 1931 novel, the current publisher, William B. Eerdmans Publishing Company of Grand Rapids, Michigan, makes the following statement:
"Replete with rich religious imagery, Many Dimensions explores the relation between predestination and free will as it depicts different human responses to redemptive transcendence."
One possible response to such statements was recently provided in some detail by a Princeton philosophy professor. See On Bullshit, by Harry G. Frankfurt, Princeton University Press, 2005.
A more thoughtful response would take into account the following:
1. The arguments presented in favor of philosopher John Calvin, who discussed predestination, in The Death of Adam: Essays on Modern Thought, by Marilynne Robinson
2. The physics underlying Einstein's remarks on free will, God, and dice
3. The physics underlying Rebecca Goldstein's novel Properties of Light and Paul Preuss's novels Secret Passages and Broken Symmetries
4. The physics underlying the recent so-called "free will theorem" of John Conway and Simon Kochen of Princeton University
5. The recent novel Gilead, by Marilynne Robinson, which deals not with philosophy, but with lives influenced by philosophy — indirectly, by the philosophy of the aforementioned John Calvin.
From a review of Gilead by Jane Vandenburgh:
"In The Death of Adam, Robinson shows Jean Cauvin to be the foremost prophet of humanism whose Protestant teachings against the hierarchies of the Roman church set in motion the intellectual movements that promoted widespread literacy among the middle and lower classes, led to both the American and French revolutions, and not only freed African slaves in the United States but brought about suffrage for women. It's odd then that through our culture's reverse historicism, the term 'Calvinism' has come to mean 'moralistic repression.'"
For more on what the Calvinist publishing firm Eerdmans calls "redemptive transcendence," see various July 2003 Log24.net entries. If these entries include a fair amount of what Princeton philosophers call bullshit, let the Princeton philosophers meditate on the summary of Harvard philosophy quoted here on November 5 of last year, as well as the remarks of November 5, 2003, and those of November 5, 2002.
From Many Dimensions (Eerdmans paperback, 1963, page 53):
"Lord Arglay had a suspicion that the Stone would be purely logical. Yes, he thought, but what, in that sense, were the rules of its pure logic?"
A recent answer:
Modal Theology
"We symbolize logical necessity
with the box )
and logical possibility
with the diamond ).
— Keith Allen Korcz,
(Log24.net, 1/25/05)
And what do we
symbolize by ?
"The possibilia that exist,
and out of which
the Universe arose,
are located in
a necessary being…."
— Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)
Comments Off on Sunday February 20, 2005
Friday, December 10, 2004
Friday December 10, 2004
Gray Particular
in Hartford
From Wallace Stevens,
"The Rock, Part III:
Forms of the Rock in a Night-Hymn" —
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…
From this morning's
New York Times obituaries—
leve Gray, a painter admired for his large-scale, vividly colorful and lyrically gestural abstract compositions, died on Wednesday in Hartford. He was 86.
The cause was a massive subdural hematoma suffered after he fell on ice and hit his head on Tuesday outside his home in Warren, Conn., said his wife, the writer Francine du Plessix Gray.
*******************************
Jackson Mac Low, a poet, composer and performance artist whose work reveled in what happens when the process of composition is left to carefully calibrated chance, died on Wednesday….
… in 1999 [he] received the Wallace Stevens Award, which carries a $100,000 prize, from the Academy of American Poets.
A Wallace Stevens Award,
in Seven Parts:
I. From a page linked to in
Tuesday's entry White Christmas:
"A bemused Plato reasoned that nonbeing must in some sense be, otherwise what is it that there is not? In our own day Martin Heidegger ventured that das Nichts nichtet — 'the nothing nothings' — evidently still sensing a problem."
— W. V. Quine in Quiddities
II. "As if nothingness
contained a métier…"
— Wallace Stevens, "The Rock"
III. "Massive subdural hematoma"
— Three-word poem
performed on Tuesday
in Connecticut
IV. mé·tier n.
- An occupation, a trade, or a profession.
- Work or activity for which a person is particularly suited; one's specialty.
[French, from Old French mestier, from Vulgar Latin misterium, from Latin ministerium. See ministry.]
Source: The American Heritage® Dictionary of the English Language, Fourth Edition
V. "ho"
— Wallace Stevens, "The Rock"
VI. Francine du Plessix Gray…
From the
Archives of the
New York Review of Books:
July 16, 1992: Splendor and Miseries, review of
Women for Hire: Prostitution and Sexuality in France after 1850 by Alain Corbin, translated by Alan Sheridan
La Vie quotidienne dans les maisons closes, 1830–1930 by Laure Adler
Figures of Ill Repute: Representing Prostitution in Nineteenth-Century France by Charles Bernheimer
Painted Love: Prostitution in French Art of the Impressionist Era by Hollis Clayson
VII. From an entry of April 29, 2004:
"… a 'dead shepherd who brought
tremendous chords from hell
And bade the sheep carouse' "
— Wallace Stevens
as quoted by Michael Bryson
as quoted by Michael Bryson
(p. 227, The Palm
at the End of the Mind:
Selected Poems and a Play.
Ed. Holly Stevens.
New York: Vintage Books, 1990)
Comments Off on Friday December 10, 2004
Tuesday, June 29, 2004
Tuesday June 29, 2004
|
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 Ch. 17 |
Part II: Badcoc
A Visual Meditation for
the Feast of St. Peter
For further details on this structure, see
Magic Squares, Finite Planes,
and Points of Inflection
on Elliptic Curves,
by Ezra Brown, and
Visualizing GL(2, p)
by Steven H. Cullinane.
For a more literary approach
to this structure, see
Balanchine's Birthday (Jan. 9, 2003),
Art Theory for Yom Kippur (Oct. 5, 2003),
A Form (May 22, 2004),
Ineluctable (May 27, 2004),
A Form, continued (June 5, 2004),
Parallelisms (June 6, 2004),
Deep Game (June 26, 2004), and
Gameplayers of Zen (June 27, 2004).
To appreciate fully this last entry
on Gameplayers,
one must understand
the concept of "suicide"
in the game of Go
and be reminded
by the fatuous phrase of the
Institute of Contemporary Art
quoted in Gameplayers —
"encompassed by 'nothing' " —
of John 1:5.
Comments Off on Tuesday June 29, 2004
Thursday, January 8, 2004
Thursday January 8, 2004
Natasha’s Dance
| “… at the still point, there the dance is….”
“… to apprehend — T. S. Eliot, Four Quartets |
It seems, according to Eliot’s criterion, that the late author John Gregory Dunne may be a saint.
Pursuing further information on the modular group, a topic on which I did a web page Dec. 30, 2003, the date of Dunne’s death, I came across a review of Apostol’s work on that subject (i.e., the modular group, not Dunne’s death, although there is a connection). The review:
“A clean, elegant,
absolutely lovely text…”
Searching further at Amazon for a newer edition of the Apostol text, I entered the search phrase “Apostol modular functions” and got a list that included the following as number four:
Natasha’s Dance:
A Cultural History of Russia,
which, by coincidence, includes all three words of the search.
For a connection — purely subjective and coincidental, of course — with Dunne’s death, see The Dark Lady (Jan. 1, 2004), which concerns another Natasha… the actress Natalie Wood, the subject of an essay (“Star!“) by Dunne in the current issue of the New York Review of Books.
The Review’s archives offer another essay, on science and religion, that includes the following relevant questions:
“Have the gates of death
been opened unto thee?
Or hast thou seen the doors
of the shadow of death?”
From my December 31 entry:
In memory of
John Gregory Dunne,
who died on
Dec. 30, 2003:
For further details, click
on the black monolith.
Monday, March 24, 2003
Monday March 24, 2003
Orwell’s question, according to
an admirer of leftist Noam Chomsky:
“When so much of the BS is right out in the open,
why is it that we know so little about it?
Why don’t we see what’s right in front of our eyes?”
 Oscar |
Lying, Truth-Telling, and the Social Order |
 Michael Moore |
“First of all, I’d like to thank the Academy….”
— Quotation attributed to Plato
The New Yorker of March 31, 2003, discusses leftist academic Noam Chomsky. The online edition provides a web page listing pro-Chomsky links.
Chomsky’s influence is based in part on the popularity of his half-baked theories on linguistics, starting in the 1950’s with “deep structure” and “transformational,” or “generative,” grammar.
Chomsky has abandoned many of his previous ideas and currently touts what he calls The Minimalist Program.
For some background on Chomsky’s recent linguistic notions, see the expository essay “Syntactic Theory,” by Elly van Gelderen of the Arizona State University English Department. Van Gelderen lists her leftist political agenda on her “Other Interests” page. Her department may serve as an example of how leftists have converted many English departments in American universities to propaganda factories.
Some attacks on Chomsky’s scholarship:
Forty-four Reasons Why the Chomskians Are Mistaken
Chomsky’s (Mis)Understanding of Human Thinking
Anatomy of a Revolution… Chomsky in 1962
…Linguistic Theory: The Rationality of Noam Chomsky
Some attacks on Chomsky’s propaganda:
Destructive Generation excerpt
Partners in Hate: Noam Chomsky and the Holocaust Deniers
Chomsky and Plato’s Diamond
Like another purveyor of leftist nonsense, Jacques Derrida, Chomsky is fond of citing Plato as a precedent. In particular, what Chomsky calls “Plato’s problem” is discussed in Plato’s Meno. For a look at the diamond figure that plays a central role in that dialogue, see Diamond Theory. For an excellent overview of related material in Plato, see Theory of Forms.
Wednesday, February 26, 2003
Wednesday February 26, 2003
The Eight Revisited
“…search for thirty-three and three…”
— The Black Queen in The Eight, by Katherine Neville, Ballantine Books, January 1989, page 140
Samuel Beckett on Dante and Joyce:
“Another point of comparison is the preoccupation with the significance of numbers…. Thus the poem is divided into three Cantiche, each composed of 33 Canti….”
— “Dante… Bruno. Vico.. Joyce,” in James Joyce/Finnegans Wake: A Symposium (1929), New Directions paperback, 1972
“– Nel mezzo del bloody cammin di nostra vita mi ritrovai in…”
— Under the Volcano, by Malcolm Lowry, 1947, beginning of Chapter VI
“‘The Divine Comedy’ celebrates Dante’s journey of knowledge to God through life: hell, purgatory and paradise. Dante Alighieri Academy continues Dante’s Christian philosophy of education….”
Chorus of the Damned:
I don’t know where it is we’re goin’
and God knows if I ever will,
but what a way this is to get there.
I got those archetypal, rubber-room,
astral-plane Moebius strip blues.
I got those in-and-out, round-about,
which way’s out Moebius strip blues.
© 1997 by C.K. Latham
Added March 3, 2003, 6:00 AM:
For a less confused song, click this Glasgow site.
Comments Off on Wednesday February 26, 2003
Wednesday, January 8, 2003
Wednesday January 8, 2003
In the Labyrinth of Memory
Taking a cue from Danny in the labyrinth of Kubrick's film "The Shining," today I retraced my steps.
My Jan. 6 entry, "Dead Poet in the City of Angels," links to a set of five December 21, 2002, entries. In the last of these, "Irish Lament," is a link to a site appropriate for Maud Gonne's birthday — a discussion of Yeats's "Among School Children."
Those who recall a young woman named Patricia Collinge (Radcliffe '64) might agree that her image is aptly described by Yeats:
Hollow of cheek as though it drank the wind
And took a mess of shadows for its meat
This meditation leads in turn to a Sept. 20, 2002, entry, "Music for Patricias," and a tune familiar to James Joyce, "Finnegan's Wake," the lyrics of which lead back to images in my entries of Dec. 20, 2002, "Last-Minute Shopping," and of Dec. 28, 2002, "Solace from Hell's Kitchen." The latter entry is in memory of George Roy Hill, director of "The Sting," who died Dec. 27, 2002.
The Dec. 28 image from "The Sting" leads us back to more recent events — in particular, to the death of a cinematographer who won an Oscar for picturing Newman and Redford in another film — Conrad L. Hall, who died Saturday, Jan. 4, 2003.
For a 3-minute documentary on Hall's career, click here.
Hall won Oscars for "Butch Cassidy and the Sundance Kid" and "American Beauty," and may win a posthumous Oscar for "Road to Perdition," last year's Irish-American mob saga:
"Tom Hanks plays Angel of Death Michael Sullivan. An orphan 'adopted' by crime boss John Rooney (Paul Newman), Sullivan worships Rooney above his own family. Rooney gave Sullivan a home when he had none. Rooney is the father Sullivan never knew. Too bad Rooney is the
Rock Island
branch of Capone's mob."
In keeping with this Irish connection, here is a set of images.
|
American Beauty |
|
|
A Game of Chess |
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 |
"Like a chess player, he knows that to win a tournament, it is sometimes wise to offer a draw in a game even when you think you can win it."
— Roger Ebert on Robert Duvall's character in "A Civil Action"
Director Steven Zaillian will take part in a tribute to Conrad L. Hall at the Palm Springs International Film Festival awards ceremony on Jan 11. Hall was the cinematographer for Zaillian's films "A Civil Action" and "Searching for Bobby Fischer."
"A Civil Action" was cast by the Boston firm Collinge/Pickman Casting, named in part for that same Patricia Collinge ("hollow of cheek") mentioned above.
See also "Conrad Hall looks back and forward to a Work in Progress." ("Work in Progress" was for a time the title of Joyce's Finnegans Wake.)
What is the moral of all this remembrance?
An 8-page (paper) journal note I compiled on November 14, 1995 (feast day of St. Lawrence O'Toole, patron saint of Dublin, allegedly born in 1132) supplies an answer in the Catholic tradition that might have satisfied Joyce (to whom 1132 was a rather significant number):
How can you tell there's an Irishman present
at a cockfight?
He enters a duck.
How can you tell a Pole is present?
He bets on the duck.
How can you tell an Italian is present?
The duck wins.
|
Every picture tells a story. |
|
Comments Off on Wednesday January 8, 2003
Friday, July 19, 2002
Old Xanga post numbers
This WordPress page from 9:22 PM ET on Aug. 17, 2016,
gives id numbers of old Xanga posts for Log24 and user m759.
It is backdated to July 19, 2002, the day before the first post in
this WordPress weblog, so it will not appear before other posts
in searches of the weblog.
Wednesday, July 31, 2002
3201621 11:29 PM
Tuesday, July 30, 2002
3152201 12:12 AM
Monday, July 29, 2002
3146028 8:34 PM
Sunday, July 28, 2002
3115928 3:07 PM
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Sunday, July 28, 2002
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Friday, July 26, 2002
3077091 1:59 PM
Thursday, July 25, 2002
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Monday, March 10, 2003
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Sunday, March 09, 2003
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Thursday, March 06, 2003
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Tuesday, March 04, 2003
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Sunday, March 02, 2003
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Wednesday, April 30, 2003
Tuesday, April 29, 2003
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Friday, April 25, 2003
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Thursday, April 24, 2003
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Wednesday, April 23, 2003
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Tuesday, April 22, 2003
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Monday, April 21, 2003
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Sunday, April 20, 2003
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Saturday, April 19, 2003
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Friday, April 18, 2003
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Wednesday, April 16, 2003
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Tuesday, April 15, 2003
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Sunday, April 13, 2003
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Saturday, April 12, 2003
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Friday, April 11, 2003
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Wednesday, April 09, 2003
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Tuesday, April 08, 2003
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Monday, April 07, 2003
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Saturday, April 05, 2003
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Thursday, April 03, 2003
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Wednesday, April 02, 2003
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Wednesday, May 28, 2003
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Sunday, May 25, 2003
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Saturday, May 24, 2003
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Wednesday, May 21, 2003
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Monday, May 19, 2003
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Friday, May 02, 2003
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Thursday, May 01, 2003
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Thursday, June 26, 2003
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Wednesday, June 25, 2003
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Friday, June 20, 2003
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Wednesday, June 18, 2003
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Monday, June 02, 2003
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Sunday, June 01, 2003
<strong><font size="3">Thursday, May 29,
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Tuesday, July 29, 2003
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Friday, July 25, 2003
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Thursday, July 24, 2003
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Saturday, October 11, 2003
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Friday, October 10, 2003
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Saturday, October 04, 2003
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Tuesday, November 25, 2003
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Saturday, November 22, 2003
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Thursday, November 20, 2003
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Sunday, November 16, 2003
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Friday, November 14, 2003
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Wednesday, December 31, 2003
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Sunday, December 21, 2003
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Friday, December 05, 2003
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Thursday, February 05, 2004
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Sunday, March 28, 2004
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Saturday, March 27, 2004
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Tuesday, March 23, 2004
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Friday, March 19, 2004
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Thursday, March 18, 2004
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Wednesday, March 17, 2004
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Tuesday, March 16, 2004
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Monday, March 15, 2004
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Monday, March 15, 2004
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Sunday, March 14, 2004
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Saturday, March 13, 2004
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Friday, March 12, 2004
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Thursday, March 11, 2004
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Wednesday, March 10, 2004
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Sunday, March 07, 2004
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Saturday, March 06, 2004
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Thursday, March 04, 2004
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Friday, April 30, 2004
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Thursday, April 29, 2004
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Tuesday, April 27, 2004
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Monday, April 26, 2004
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Sunday, April 25, 2004
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Friday, April 23, 2004
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Thursday, April 22, 2004
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Wednesday, April 21, 2004
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Tuesday, April 20, 2004
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Monday, April 19, 2004
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Sunday, April 18, 2004
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Friday, April 16, 2004
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Thursday, April 15, 2004
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Wednesday, April 14, 2004
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Tuesday, April 13, 2004
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Sunday, April 11, 2004
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Saturday, April 10, 2004
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Friday, April 09, 2004
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Friday, April 09, 2004
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Thursday, April 08, 2004
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Wednesday, April 07, 2004
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Tuesday, April 06, 2004
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Monday, April 05, 2004
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Sunday, April 04, 2004
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Friday, April 02, 2004
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Thursday, April 01, 2004
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Monday, May 31, 2004
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Thursday, May 27, 2004
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Saturday, May 22, 2004
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Friday, May 21, 2004
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Thursday, May 20, 2004
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Wednesday, May 19, 2004
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Saturday, May 15, 2004
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Friday, May 14, 2004
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Thursday, May 13, 2004
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Wednesday, May 12, 2004
Tuesday, May 11, 2004
Monday, May 10, 2004
Sunday, May 09, 2004
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Saturday, May 08, 2004
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Saturday, May 08, 2004
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Friday, May 07, 2004
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Wednesday, May 05, 2004
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Tuesday, May 04, 2004
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Monday, May 03, 2004
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Sunday, May 02, 2004
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Saturday, May 01, 2004
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<a name="1" target="_new"></a><big><b><font
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<a name="3" target="_new"></a><big><b><font
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<a
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532660984 8:48 AM
Monday, September 25, 2006
532380406 10:00 AM
Saturday, September 23, 2006
531815254 12:24 PM
531727140 9:00 AM
Friday, September 22, 2006
531430219 7:00 AM
Thursday, September 21, 2006
531150877 8:00 AM
Wednesday, September 20, 2006
530853075 8:00 AM
Tuesday, September 19, 2006
530561687 10:00 AM
Monday, September 18, 2006
530343679 1:00 PM
530234670 9:14 AM
Sunday, September 17, 2006
529840151 12:00 AM
Saturday, September 16, 2006
529765780 4:00 PM
529674926 11:07 AM
Friday, September 15, 2006
529309099 7:11 AM
Thursday, September 14, 2006
529191445 7:11 PM
Wednesday, September 13, 2006
528913004 9:28 PM
528661802 2:56 AM
Monday, September 11, 2006
528311247 11:00 PM
528053481 6:00 AM
Sunday, September 10, 2006
527898122 2:56 PM
Sunday, September 10, 2006
527733946 4:00 AM
Saturday, September 09, 2006
527607606 4:07 PM
Friday, September 08, 2006
527153018 7:20 AM
Thursday, September 07, 2006
526980592 2:56 PM
Thursday, September 07, 2006
526818663 4:04 AM
Wednesday, September 06, 2006
526698920 5:26 PM
526505874 4:00 AM
Tuesday, September 05, 2006
526151900 1:00 AM
Monday, September 04, 2006
526081092 7:20 PM
Monday, September 04, 2006
526005744 1:00 PM
Sunday, September 03, 2006
525632213 1:00 PM
Saturday, September 02, 2006
525274301 7:31 AM
Friday, September 01, 2006
524956461 7:59 AM
Tuesday, October 31, 2006
543220072 11:00 PM
543036518 9:00 AM
Monday, October 30, 2006
542741247 11:30 AM
Sunday, October 29, 2006
542226821 1:00 AM
Saturday, October 28, 2006
542077417 12:00 PM
Saturday, October 28, 2006
541993048 8:00 AM
Friday, October 27, 2006
541788632 12:31 PM
Thursday, October 26, 2006
541473071 12:00 PM
Wednesday, October 25, 2006
541099967 9:00 AM
Tuesday, October 24, 2006
540945257 9:00 PM
Tuesday, October 24, 2006
540889575 2:56 PM
540711551 2:02 AM
Monday, October 23, 2006
540399609 12:00 AM
Sunday, October 22, 2006
540398938 11:59 PM
540352447 7:45 PM
Sunday, October 22, 2006
540329304 5:01 PM
540206189 9:00 AM
540137184 2:45 AM
Saturday, October 21, 2006
539915249 8:23 AM
Friday, October 20, 2006
539714144 12:00 PM
Friday, October 20, 2006
539580608 2:00 AM
Thursday, October 19, 2006
539547041 10:31 PM
539520075 7:59 PM
539352522 7:59 AM
Wednesday, October 18, 2006
539208436 5:11 PM
Tuesday, October 17, 2006
538885252 2:00 PM
538696036 12:00 AM
Monday, October 16, 2006
538536197 11:00 AM
Sunday, October 15, 2006
538292977 2:00 PM
Saturday, October 14, 2006
538028186 7:00 PM
Saturday, October 14, 2006
537923433 10:31 AM
Friday, October 13, 2006
537539006 12:00 AM
Thursday, October 12, 2006
537374965 10:31 AM
Wednesday, October 11, 2006
537098380 11:07 AM
Tuesday, October 10, 2006
536907708 8:00 PM
Tuesday, October 10, 2006
536821801 12:00 PM
Monday, October 09, 2006
536449718 9:00 AM
Sunday, October 08, 2006
536056986 12:00 AM
Saturday, October 07, 2006
535902024 10:31 AM
Friday, October 06, 2006
535718439 5:00 PM
Friday, October 06, 2006
535656412 12:00 PM
Thursday, October 05, 2006
535321554 9:11 AM
Wednesday, October 04, 2006
534985203 6:15 AM
Tuesday, October 03, 2006
534793806 12:00 PM
534740724 9:26 AM
Monday, October 02, 2006
534424864 8:00 AM
Sunday, October 01, 2006
534264268 6:00 PM
534112245 8:00 AM
Tuesday, November 28, 2006
551316090 9:00 AM
Monday, November 27, 2006
550936988 2:45 AM
Sunday, November 26, 2006
550708497 7:26 AM
550629210 12:25 AM
Saturday, November 25, 2006
550530933 1:25 PM
Friday, November 24, 2006
550261390 1:06 PM
Wednesday, November 22, 2006
549773712 9:00 PM
549690171 12:00 PM
Tuesday, November 21, 2006
549314127 7:07 AM
Monday, November 20, 2006
549185488 7:20 PM
Sunday, November 19, 2006
548850626 2:02 PM
Saturday, November 18, 2006
548430419 4:09 AM
Thursday, November 16, 2006
547986596 12:00 PM
Thursday, November 16, 2006
547846160 2:00 AM
Wednesday, November 15, 2006
547568133 3:10 AM
Monday, November 13, 2006
547212241 8:23 PM
547118652 11:07 AM
Sunday, November 12, 2006
546911992 8:00 PM
Sunday, November 12, 2006
546871765 3:10 PM
546860204 2:00 PM
546670969 12:25 AM
Friday, November 10, 2006
546379803 11:20 PM
546337563 7:00 PM
Friday, November 10, 2006
546310170 3:31 PM
Thursday, November 09, 2006
546056034 7:20 PM
545846641 3:00 AM
Wednesday, November 08, 2006
545731395 8:00 PM
Tuesday, November 07, 2006
545493984 9:00 PM
Tuesday, November 07, 2006
545237466 12:00 AM
Monday, November 06, 2006
545087018 11:00 AM
Sunday, November 05, 2006
544822953 8:00 PM
544747439 12:00 PM
Saturday, November 04, 2006
544435730 2:15 PM
Friday, November 03, 2006
544014222 9:00 AM
Thursday, November 02, 2006
543837012 4:28 PM
Wednesday, November 01, 2006
543566440 9:48 PM
543364389 8:24 AM
Sunday, December 31, 2006
559841270 8:00 PM
559837227 7:15 PM
559837123 7:14 PM
559836583 7:13 PM
559723926 9:00 AM
Friday, December 29, 2006
559345518 11:01 AM
Wednesday, December 27, 2006
559022945 7:14 AM
Tuesday, December 26, 2006
558838974 8:00 AM
558838771 7:59 AM
Monday, December 25, 2006
558574412 8:00 AM
Monday, December 25, 2006
Sunday, December 24, 2006
558307364 7:15 AM
558305180 7:00 AM
Saturday, December 23, 2006
558075208 9:00 AM
Thursday, December 21, 2006
557568187 10:02 AM
Wednesday, December 20, 2006
557296561 9:26 AM
Tuesday, December 19, 2006
557027497 9:00 AM
557009854 8:00 AM
Monday, December 18, 2006
556733648 7:20 AM
Saturday, December 16, 2006
556278563 10:31 AM
Friday, December 15, 2006
555928858 2:02 AM
Thursday, December 14, 2006
555701770 6:06 AM
Wednesday, December 13, 2006
555488381 9:29 AM
Tuesday, December 12, 2006
555245307 11:22 AM
Monday, December 11, 2006
554871237 7:20 AM
Sunday, December 10, 2006
554758961 9:00 PM
554678660 12:00 PM
554620343 9:00 AM
Sunday, December 10, 2006
554579130 6:00 AM
Saturday, December 09, 2006
554293023 4:00 AM
Friday, December 08, 2006
554080818 9:00 AM
Thursday, December 07, 2006
553734328 3:30 AM
Wednesday, December 06, 2006
553457007 3:15 AM
Tuesday, December 05, 2006
553189510 5:01 AM
Monday, December 04, 2006
553009937 11:01 AM
552891193 2:45 AM
Sunday, December 03, 2006
552618347 2:22 AM
Saturday, December 02, 2006
552479112 12:00 PM
Saturday, December 02, 2006
552346255 1:29 AM
Friday, December 01, 2006
552089558 4:07 AM
Wednesday, January 31, 2007
567016831 3:09 PM
Monday, January 29, 2007
566464410 9:00 AM
Saturday, January 27, 2007
565989258 9:00 AM
Friday, January 26, 2007
565850351 2:48 PM
Wednesday, January 24, 2007
565235023 1:00 AM
Tuesday, January 23, 2007
565129903 12:00 PM
Monday, January 22, 2007
564874899 11:11 AM
Sunday, January 21, 2007
564631387 11:22 AM
Saturday, January 20, 2007
564436026 2:14 PM
564422848 1:00 PM
Friday, January 19, 2007
564270554 11:07 PM
564235389 7:00 PM
564163988 11:11 AM
Tuesday, January 09, 2007
561821658 12:00 PM
Sunday, January 07, 2007
561364149 12:00 PM
561351192 11:00 AM
561338932 10:00 AM
Saturday, January 06, 2007
561221807 9:00 PM
561135694 10:31 AM
Friday, January 05, 2007
561006783 7:59 PM
560949022 12:00 PM
560908207 9:26 AM
Thursday, January 04, 2007
560724702 12:00 PM
Wednesday, January 03, 2007
560503430 11:32 AM
Tuesday, January 02, 2007
560213061 7:00 AM
Monday, January 01, 2007
560063883 3:00 PM
Wednesday, February 28, 2007
573564673 7:59 AM
Tuesday, February 27, 2007
573457454 9:25 PM
573325144 7:59 AM
Monday, February 26, 2007
573114879 9:29 AM
Sunday, February 25, 2007
572892280 10:31 AM
Sunday, February 25, 2007
572684236 10:30 AM
Tuesday, February 20, 2007
571734254 10:15 AM
571699878 7:09 AM
Sunday, February 18, 2007
571295304 10:30 AM
571215316 2:00 AM
Saturday, February 17, 2007
571049774 9:00 AM
Friday, February 16, 2007
570797968 6:16 AM
Thursday, February 15, 2007
570546492 6:25 AM
Tuesday, February 13, 2007
569967901 5:24 AM
Monday, February 12, 2007
569725030 5:24 AM
Sunday, February 11, 2007
569566943 11:00 AM
569472193 2:56 AM
Saturday, February 10, 2007
569311138 10:00 AM
569234801 2:00 AM
Friday, February 09, 2007
569157082 3:24 PM
Wednesday, February 07, 2007
568528949 3:00 AM
Tuesday, February 06, 2007
568339753 8:00 AM
Sunday, February 04, 2007
567865161 9:00 AM
Saturday, February 03, 2007
567767470 11:07 PM
567696177 1:00 PM
Saturday, February 03, 2007
Friday, February 02, 2007
567381550 7:11 AM
567368370 6:00 AM
567359201 5:00 AM
Thursday, February 01, 2007
567158486 7:59 AM
567102919 2:00 AM
Saturday, March 31, 2007
580786073 3:00 PM
Friday, March 30, 2007
580598485 9:48 PM
580387534 12:07 AM
Wednesday, March 28, 2007
580143213 10:10 PM
580082245 1:00 PM
Saturday, March 24, 2007
579168922 3:09 PM
579023567 2:12 AM
579023248 2:09 AM
Thursday, March 22, 2007
578685858 12:00 PM
Wednesday, March 21, 2007
578517739 7:29 PM
Wednesday, March 21, 2007
578493506 3:18 PM
Monday, March 19, 2007
577965290 10:31 AM
Sunday, March 18, 2007
577790905 2:20 PM
Saturday, March 17, 2007
577503480 10:48 AM
Friday, March 16, 2007
577282355 10:48 AM
Thursday, March 15, 2007
577086728 12:07 PM
Wednesday, March 14, 2007
576781050 8:00 AM
Saturday, March 10, 2007
575879752 9:00 AM
575820044 2:00 AM
Friday, March 09, 2007
575657188 9:00 AM
Thursday, March 08, 2007
575537141 7:13 PM
575501585 1:00 PM
575439331 9:00 AM
Wednesday, March 07, 2007
575310629 7:00 PM
575205306 8:35 AM
575202353 8:24 AM
575201901 8:23 AM
Sunday, March 04, 2007
574538429 11:00 AM
Saturday, March 03, 2007
574217419 3:00 AM
Friday, March 02, 2007
574146124 6:00 PM
574017308 7:00 AM
Thursday, March 01, 2007
573816566 9:00 AM
573779445 6:29 AM
Monday, April 30, 2007
587459040 6:24 PM
Sunday, April 29, 2007
587227693 7:00 AM
Saturday, April 28, 2007
587160680 11:07 PM
Friday, April 27, 2007
586932514 9:48 PM
586874639 1:00 PM
Wednesday, April 25, 2007
586483728 10:30 PM
Tuesday, April 24, 2007
586118063 9:00 AM
Monday, April 23, 2007
586015501 10:15 PM
Monday, April 23, 2007
585815078 2:45 AM
Sunday, April 22, 2007
585765994 8:31 PM
585672266 11:09 AM
Saturday, April 21, 2007
585540638 8:23 PM
Friday, April 20, 2007
585340404 10:31 PM
Friday, April 20, 2007
585245439 11:07 AM
Thursday, April 19, 2007
584912891 12:06 AM
Wednesday, April 18, 2007
584845189 2:00 PM
Monday, April 16, 2007
584392474 4:01 PM
Sunday, April 15, 2007
584086087 10:18 AM
Sunday, April 15, 2007
583999845 1:23 AM
Saturday, April 14, 2007
583975355 10:31 PM
583964622 8:30 PM
583929813 2:12 PM
583813104 4:30 AM
Saturday, April 14, 2007
583810285 4:06 AM
Friday, April 13, 2007
583740412 6:23 PM
Thursday, April 12, 2007
583346347 1:09 AM
Wednesday, April 11, 2007
583234663 12:00 PM
Monday, April 09, 2007
582841593 9:00 PM
Monday, April 09, 2007
582830764 7:20 PM
582781043 1:00 PM
Sunday, April 08, 2007
582621813 11:09 PM
582511597 11:00 AM
582410289 12:00 AM
Saturday, April 07, 2007
582326723 12:25 PM
Friday, April 06, 2007
582153702 2:56 PM
Thursday, April 05, 2007
581929567 2:02 PM
Wednesday, April 04, 2007
581713085 2:02 PM
Tuesday, April 03, 2007
581535713 10:10 PM
581328446 1:00 AM
Monday, April 02, 2007
581103868 4:00 AM
Sunday, April 01, 2007
580974600 12:00 PM
580839905 12:00 AM
Thursday, May 31, 2007
594667854 8:06 PM
Wednesday, May 30, 2007
594446265 10:00 PM
Monday, May 28, 2007
593909298 5:00 PM
Sunday, May 27, 2007
Saturday, May 26, 2007
593447364 1:09 PM
593411247 9:13 AM
Friday, May 25, 2007
593103496 7:11 AM
Thursday, May 24, 2007
592916009 9:29 AM
592863568 4:00 AM
Wednesday, May 23, 2007
592653180 7:00 AM
592635600 5:15 AM
592631459 4:29 AM
Tuesday, May 22, 2007
592428946 7:11 AM
Monday, May 21, 2007
592323806 4:00 PM
Monday, May 21, 2007
592174031 4:48 AM
Sunday, May 20, 2007
592108855 7:00 PM
591989432 8:00 AM
Saturday, May 19, 2007
591784476 9:29 AM
Friday, May 18, 2007
591663855 3:00 PM
591559533 8:00 AM
591540971 6:29 AM
591528334 5:18 AM
Thursday, May 17, 2007
591329564 7:31 AM
Wednesday, May 16, 2007
591219429 3:37 PM
591165206 11:22 AM
591070738 4:01 AM
Tuesday, May 15, 2007
590980779 2:11 PM
590855957 5:55 AM
Monday, May 14, 2007
590713401 11:30 AM
590605707 3:09 AM
590601109 2:22 AM
Sunday, May 13, 2007
590501771 12:31 PM
590499319 12:23 PM
Sunday, May 13, 2007
590391246 5:01 AM
590360827 1:00 AM
Saturday, May 12, 2007
590237112 11:07 AM
Friday, May 11, 2007
590129837 11:30 PM
590094253 5:11 PM
Friday, May 11, 2007
589958582 6:29 AM
Thursday, May 10, 2007
589898333 10:00 PM
589795765 10:31 AM
589708467 2:45 AM
Wednesday, May 09, 2007
589484681 3:09 AM
Tuesday, May 08, 2007
589422941 6:29 PM
589404948 2:56 PM
Monday, May 07, 2007
589038535 5:55 AM
Sunday, May 06, 2007
588945045 4:04 PM
588939049 3:09 PM
Sunday, May 06, 2007
588804755 4:02 AM
Friday, May 04, 2007
588513891 5:01 PM
Thursday, May 03, 2007
588267343 3:00 PM
Tuesday, May 01, 2007
587848094 10:20 PM
Saturday, June 30, 2007
601100365 10:04 PM
601043057 1:00 PM
Thursday, June 28, 2007
600688241 9:00 PM
600625531 12:06 PM
Wednesday, June 27, 2007
600439774 3:33 PM
Monday, June 25, 2007
599992402 3:00 PM
Sunday, June 24, 2007
599728145 12:00 PM
599608180 12:07 AM
Saturday, June 23, 2007
599580117 6:00 PM
599478844 9:00 AM
Friday, June 22, 2007
599358531 2:22 PM
Thursday, June 21, 2007
599162672 9:57 PM
599135588 4:30 PM
599084420 12:07 PM
Wednesday, June 20, 2007
598852470 1:06 AM
Tuesday, June 19, 2007
598791123 3:17 PM
598768155 1:00 PM
598749235 12:00 PM
598748596 11:49 AM
Monday, June 18, 2007
598545405 1:00 PM
Monday, June 18, 2007
598414235 2:00 AM
Sunday, June 17, 2007
598362900 7:00 PM
598343169 3:00 PM
598334186 2:02 PM
Saturday, June 16, 2007
598096835 12:00 PM
Saturday, June 16, 2007
597981712 12:00 AM
Friday, June 15, 2007
597968099 10:31 PM
597909781 1:00 PM
Thursday, June 14, 2007
597737455 9:00 PM
597716324 4:00 PM
Thursday, June 14, 2007
597648395 10:35 AM
597562687 2:00 AM
597553082 1:06 AM
Wednesday, June 13, 2007
597510431 6:29 PM
Wednesday, June 13, 2007
597430891 12:00 PM
Tuesday, June 12, 2007
597318382 11:07 PM
597252546 2:00 PM
Monday, June 11, 2007
596883736 12:00 AM
Sunday, June 10, 2007
596798177 12:00 PM
Sunday, June 10, 2007
596694631 2:00 AM
Saturday, June 09, 2007
596544097 9:00 AM
Friday, June 08, 2007
596433149 4:00 PM
596416184 2:00 PM
Thursday, June 07, 2007
596210744 4:15 PM
Thursday, June 07, 2007
596157264 11:30 AM
Wednesday, June 06, 2007
595925649 11:07 AM
Tuesday, June 05, 2007
595770434 6:01 PM
595746674 2:00 PM
595701814 11:08 AM
Sunday, June 03, 2007
595329455 9:29 PM
595234110 10:31 AM
Saturday, June 02, 2007
594983767 8:00 AM
Tuesday, July 31, 2007
607246376 7:11 AM
607238023 6:00 AM
Monday, July 30, 2007
607164258 7:00 PM
607059660 9:00 AM
Monday, July 30, 2007
607050168 8:00 AM
607049312 7:59 AM
607042018 7:00 AM
Sunday, July 29, 2007
606867417 9:00 AM
606858987 8:00 AM
Sunday, July 29, 2007
606858846 7:59 AM
Saturday, July 28, 2007
606672190 6:15 AM
Thursday, July 26, 2007
606271827 4:00 AM
Wednesday, July 25, 2007
606111937 9:00 AM
Tuesday, July 24, 2007
605893580 7:11 AM
Monday, July 23, 2007
605698535 8:00 AM
605698073 7:59 AM
Saturday, July 21, 2007
605340686 9:45 AM
Thursday, July 19, 2007
604965427 10:31 AM
Thursday, July 19, 2007
604894106 2:00 AM
Wednesday, July 18, 2007
604847943 6:28 PM
604724768 7:03 AM
Tuesday, July 17, 2007
604515850 7:00 AM
Monday, July 16, 2007
604316589 8:06 AM
Sunday, July 15, 2007
604100066 6:20 AM
Saturday, July 14, 2007
603896105 4:07 AM
Friday, July 13, 2007
603712325 7:00 AM
Thursday, July 12, 2007
603629730 7:00 PM
603588597 1:00 PM
Wednesday, July 11, 2007
603247518 2:45 AM
Tuesday, July 10, 2007
603081833 9:00 AM
603070402 8:00 AM
603069912 7:59 AM
603003210 12:01 AM
Monday, July 09, 2007
603003081 11:59 PM
602729011 2:35 AM
Saturday, July 07, 2007
602529877 2:22 PM
602398088 1:48 AM
602383121 12:00 AM
Friday, July 06, 2007
602375889 11:07 PM
602359542 8:00 PM
602357485 7:47 PM
602175079 12:26 AM
602173734 12:18 AM
Friday, July 06, 2007
602172268 12:06 AM
Thursday, July 05, 2007
602143009 7:11 PM
602097762 12:48 PM
Tuesday, July 03, 2007
601723043 9:29 PM
Monday, July 02, 2007
601494979 8:28 PM
Monday, July 02, 2007
601432920 12:03 PM
Sunday, July 01, 2007
601293940 10:31 PM
601243649 1:27 PM
601130231 2:06 AM
<span>
<div class="blogheader">Friday, August 31, 2007
613358969 10:10 PM
Wednesday, August 22, 2007
611673223 11:00 PM
611572141 10:31 AM
Tuesday, August 21, 2007
611446544 3:29 PM
611350451 8:14 AM
611344571 8:00 AM
Monday, August 20, 2007
611155585 8:01 AM
Sunday, August 19, 2007
610972092 8:19 AM
Saturday, August 18, 2007
610897230 7:20 PM
Tuesday, August 14, 2007
610054662 12:00 PM
Monday, August 13, 2007
609832401 11:07 AM
Sunday, August 12, 2007
609619788 9:00 AM
Saturday, August 11, 2007
609532971 10:00 PM
Friday, August 10, 2007
609282854 10:31 AM
Thursday, August 09, 2007
609085797 12:00 PM
Tuesday, August 07, 2007
608725736 6:25 PM
Tuesday, August 07, 2007
608610911 8:00 AM
Monday, August 06, 2007
608410039 8:00 AM
Sunday, August 05, 2007
608321060 7:00 PM
608227620 9:00 AM
Saturday, August 04, 2007
608037417 8:00 AM
Friday, August 03, 2007
607979954 10:09 PM
607941337 2:02 PM
607861289 8:09 AM
Wednesday, August 01, 2007
607579945 9:29 PM
607568556 6:19 PM
Wednesday, August 01, 2007
607455961 8:00 AM
Sunday, September 30, 2007
618944496 2:00 PM
618870750 8:28 AM
618834293 3:14 AM
Saturday, September 29, 2007
618659874 3:09 AM
Friday, September 28, 2007
618490463 6:25 AM
Thursday, September 27, 2007
618423240 4:00 PM
618292623 6:29 AM
Tuesday, September 25, 2007
617947916 8:00 AM
Monday, September 24, 2007
617704736 1:06 AM
Sunday, September 23, 2007
617692818 11:07 PM
Sunday, September 23, 2007
617555085 5:51 AM
617548823 5:01 AM
Saturday, September 22, 2007
617377554 6:25 AM
617377300 6:23 AM
617377167 6:22 AM
Friday, September 21, 2007
617206110 8:28 AM
Wednesday, September 19, 2007
616810021 5:00 AM
Saturday, September 15, 2007
616192373 8:00 PM
Thursday, September 13, 2007
615683763 3:57 AM
Thursday, September 13, 2007
615675861 2:02 AM
Wednesday, September 12, 2007
615630908 5:01 PM
Tuesday, September 11, 2007
615287767 12:07 AM
Monday, September 10, 2007
615195800 11:07 AM
Saturday, September 08, 2007
614881336 7:11 PM
Saturday, September 08, 2007
614857110 2:02 PM
Friday, September 07, 2007
614712034 9:29 PM
614680955 2:02 PM
Thursday, September 06, 2007
614448043 11:00 AM
Sunday, September 02, 2007
613711378 5:11 PM
Wednesday, October 31, 2007
624616664 8:28 PM
Tuesday, October 30, 2007
Monday, October 29, 2007
624137721 7:20 AM
Sunday, October 28, 2007
623967292 7:59 AM
Thursday, October 25, 2007
623439304 9:19 AM
Wednesday, October 24, 2007
623365906 11:11 PM
623285567 11:01 AM
623258985 9:26 AM
Sunday, October 21, 2007
622727625 10:31 AM
Thursday, October 18, 2007
622251512 3:14 PM
622198809 11:07 AM
Wednesday, October 17, 2007
622007807 10:00 AM
Tuesday, October 16, 2007
621824383 10:00 AM
Monday, October 15, 2007
Sunday, October 14, 2007
621464914 11:00 AM
Saturday, October 13, 2007
621253189 9:22 AM
Friday, October 12, 2007
621072729 9:10 AM
621011874 12:00 AM
Thursday, October 11, 2007
620997954 9:26 PM
620981910 5:01 PM
620943547 12:00 PM
Tuesday, October 09, 2007
620482237 4:09 AM
Monday, October 08, 2007
Sunday, October 07, 2007
620194402 12:07 PM
Friday, October 05, 2007
619784656 9:00 AM
Thursday, October 04, 2007
Wednesday, October 03, 2007
619494781 3:09 PM
Monday, October 01, 2007
619055443 7:20 AM
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