The location in the title is the opening scene of the new version
of "All Creatures Great and Small." The year in the title suggests
a look at (for instance) The Pentagram Papers.
A "support provided by" credit suggests some related images —
A Scotland-related post: The Blue Path and the Red Path
(Log24, Aug. 24, 2014).
Wendelboe's games included chess sets.
See as well a related purchase from further north on the same block.
|
Ratan Naval Tata was born on Dec. 28, 1937, in Bombay, now Mumbai, during the British Raj. His family belonged to the Parsi religion, a small Zoroastrian community that originated in Persia, fled persecution by the Muslim majority there centuries ago and found refuge in India. Mr. Tata became a leader of that community. — New York Times obituary on 9 October 2024 |
See also theta functions in this journal.
For those who prefer narratives to mathematics . . .
Tiger at the Fire Temple
The conclusion of a Hungarian political figure's obituary in
tonight's online New York Times, written by Clay Risen —
"A quietly religious man, he spent his last years translating
works dealing with Roman Catholic canon law."
This journal on the Hungarian's date of death, October 8,
a Sunday, dealt in part with the submission to Wikipedia of
the following brief article . . . and its prompt rejection.
|
The Cullinane diamond theorem is a theorem
The theorem also explains symmetry properties of the Reference
1. Cullinane diamond theorem at |
Some quotations I prefer to Catholic canon law —
|
Ludwig Wittgenstein,
97. Thought is surrounded by a halo. * See the post Wittgenstein's Diamond. Related language in Łukasiewicz (1937)—
|
See as well Diamond Theory in 1937.
Click the above to search this journal for "Dead End."
In memory of a political figure who reportedly died on Sunday —
Note the approximate target of the holy nib.
Personally, I prefer To Catch a Thief —
“Breast or Thigh”?
" In 2017, Milioti appeared in the fourth season of
the sci-fi anthology show Black Mirror in the episode
"USS Callister" as Nanette Cole, a newly employed
game developer whose digital recreation becomes
trapped in a virtual simulator game.[16] "
16. Strause, Jackie (December 30, 2017).
"'Black Mirror': Cristin Milioti on Battling a "Misogynistic Bully"
in Empowering Space Epic". The Hollywood Reporter.
Retrieved June 22, 2021.
"If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
See as well this journal on the above Sundance photo date —
“We keep coming back and coming back
To the real: to the hotel instead of the hymns . . . .”
— Wallace Stevens, quoted in posts tagged Portal1937
Update of 12:35 PM ET the same day —
The above author reportedly died on August 22, 2009.
A check of that date in this journal yields . . .
A passage from the “Assumption” link target above, with a title
suggested by the preceding “Making Mathematics” piece —
Making Money: The Communist Connection
“He and his fellow monks were cut off from the rest of the world
during the second Sino-Japanese War, which lasted from 1937 to
1945. When hostilities ended, the monastery relocated to the
provincial capital of Chengdu, where the monks opened the
Institute of Chinese and Western Cultural Studies with a library
that eventually had 10,000 volumes.
Chengdu, the last stronghold of the Nationalist Party, was overrun
by Mao’s army on Christmas Day 1949. The Communists closed the
institute, confiscated the books and forced Winance and the other
monks to attend indoctrination sessions on Marxism.”
Another article by Wieschenberg might, to some, actually be worth $49 —
In memory of Ralph Winter and Barbara Windsor.
See also . . .
Personally, I prefer To Catch a Thief —
“Breast or Thigh”?
“If you have built castles in the air, your work need not be lost;
that is where they should be. Now put the foundations under them.”
— Henry David Thoreau
Between, or: Interality Illustrated (according to Sherald)
“We live in between the lines;
in between the cracks . . . .
That’s where the bonds are formed. ”
"We learned so much about singing from each other because you get to sort of be them for a second when you're shadowing them in harmony. It's like getting on an eagle and getting to see the world through that eagle's experience." Read more: https://www.springfieldspringfield.co.uk/ movie_script.php?movie=linda-ronstadt-the-sound-of-my-voice
See also Aguila de Oro.
Poet Wallace Stevens was born 140 years ago today.
For another 140, see Diamond Theory in 1937.
For some notes related to a Stevens poem from 1937,
see "arrowy, still strings" in this journal.
"The area is home to many artists and people who work in
the media, including many journalists, writers and professionals
working in film and television." — Wikipedia
Tusen takk to My Square Lady —
The title abbreviates* that of a collection of Wittgenstein's remarks:
|
Ludwig Wittgenstein — Culture and Value Showing 20 results for spirit — page 18, rubble & finally a heap of ashes; but spirits will hover over the ashes. MS 107 229: page 18, Page 5 Only something supernatural can expre page 20, contemplating it from above in its†c flight.† page 21, spirit in which it is written.†f This spirit is, I believe, different from that of t page 21, and American civilization. The spirit of this civilization the expression of page 21, day†h fascism & socialism, is a spirit that is alien & uncongenial†i to the au page 21, he Page Break 9 can work in the spirit of the whole, and his strength can with page 21, straight for what is concrete. Which is chara page 22, danger in a long foreword is that the spirit of a book has to be evident in the book page 22, It is all one to me whether the typical weste page 23, a great temptation to want to make the spirit explicit. MS 109 204: 6-7.11.1930 Page page 23, readers that will be clear just from the fact page 28, Foggy day. Grey autumn haunts us. Laughter se page 42, If one wanted to characterize the essence of page 51, attention from what matters.) The Spirit puts what is essential, essential for y page 51, how far all this is exactly in the spirit of Kierkegaard.) MS 119 151: 22.10.1937 page 51, something feminine about this outlook?) MS 11 page 100, comfortable, clearer expression, but cannot b page 106, act otherwise."–Perhaps, though, one might s page 210, Page 7 †b function Page 7 †c from its Page **************************************************************** |
The above "spirit guide" was suggested by yesterday's post
on Knuth as Yoda and by the paper in today's previous post,
"Shadowhunter Tales."
This post's title, "CV," is from . . .
From a Log24 post of March 4, 2008 —
|
SINGER, ISAAC:
"Sets forth his own aims in writing for children and laments
— An Annotated Listing of Criticism
"She returned the smile, then looked across the room to
— A Swiftly Tilting Planet,
For "the dimension of time," see A Fold in Time, Time Fold,
A Swiftly Tilting Planet is a fantasy for children |
Ibid. —
The pen's point:
John Trever, Albuquerque Journal, 2/29/08
Note the figure on the cover of National Review above —
A related figure from Pentagram Design —
See, more generally, Isaac Singer in this journal.
From the Web this morning —
A different 35-year wait:
A monograph of August 1976 —
Thirty-five years later, in a post of August 2011, "Coordinated Steps" —
"SEE HEAR READ" — Walt Disney Productions
Some other diamond-mine productions —
"It's not a lie if you believe it."
— Poster for "Operation Avalanche"
“We keep coming back and coming back
To the real: to the hotel instead of the hymns . . . .”
— Wallace Stevens, quoted in posts tagged Portal1937
The title is that of a presentation by Arnold Emch
at the 1928 International Congress of Mathematicians:
See also yesterday's "Emch as a Forerunner of S(5, 8, 24)."
Related material: Diamond Theory in 1937.
Further remarks: Christmas 2013 and the fact that
759 × 322,560 = the order of the large Mathieu group M24 .
The author of the review in the previous post, Dara Horn, supplies
below a midrash on "desmic," a term derived from the Greek desmé
( δέσμη: bundle, sheaf , or, in the mathematical sense, pencil —
French faisceau ), which is related to the term desmos , bond …
(The term "desmic," as noted earlier, is relevant to the structure of
Heidegger's Sternwürfel .)
The Horn midrash —
(The "medieval philosopher" here is not the remembered pre-Christian
Ben Sirah (Ecclesiasticus ) but the philosopher being read — Maimonides:
Guide for the Perplexed , 3:51.)
Here of course "that bond" may be interpreted as corresponding to the
Greek desmos above, thus also to the desmic structure of the
stellated octahedron, a sort of three-dimensional Star of David.
See "desmic" in this journal.
Today in History —
"On December 21, 1937, 'Snow White and the Seven Dwarfs'
premiered to a record-breaking audience at the Carthay Circle
Theatre in Los Angeles."
Related material: Today's previous post and the Red Book.
Under the Volcano:
A Bottle, a Door, a Box
See also Glory Season (Nov. 12, 2005) and Unique Figure (April 12, 2011).
Update of 11:22 AM —
Today in History —
"On December 21, 1937, 'Snow White and the Seven Dwarfs'
premiered to a record-breaking audience at the Carthay Circle
Theatre in Los Angeles."
Related material: The Red Book.
(A Prequel to Dirac and Geometry)
"So Einstein went back to the blackboard.
And on Nov. 25, 1915, he set down
the equation that rules the universe.
As compact and mysterious as a Viking rune,
it describes space-time as a kind of sagging mattress…."
— Dennis Overbye in The New York Times online,
November 24, 2015
Some pure mathematics I prefer to the sagging Viking mattress —
Readings closely related to the above passage —
Thomas Hawkins, "From General Relativity to Group Representations:
the Background to Weyl's Papers of 1925-26," in Matériaux pour
l'histoire des mathématiques au XXe siècle: Actes du colloque
à la mémoire de Jean Dieudonné, Nice, 1996 (Soc. Math.
de France, Paris, 1998), pp. 69-100.
The 19th-century algebraic theory of invariants is discussed
as what Weitzenböck called a guide "through the thicket
of formulas of general relativity."
Wallace Givens, "Tensor Coordinates of Linear Spaces," in
Annals of Mathematics Second Series, Vol. 38, No. 2, April 1937,
pp. 355-385.
Tensors (also used by Einstein in 1915) are related to
the theory of line complexes in three-dimensional
projective space and to the matrices used by Dirac
in his 1928 work on quantum mechanics.
For those who prefer metaphors to mathematics —
Rota fails to cite the source of his metaphor.
|
Part I: Magic Moonlight
Part II: To Walk the Night
Cover from a 1944 edition of
the 1937 novel by William Sloane —
Part III: Sept. 18, 2015, review by Stephen King
of the works of William Sloane
Nan Tucker McEvoy, last of founding family
to run Chronicle, dies
By Sam Whiting at SFGate.com, Friday, March 27, 2015
From the story —
"After graduating from Dominican Convent Upper School
in San Rafael in 1937, she was discouraged from attending college
by family members who wanted her to be a socialite."
Related material —
"There is such a thing as a tesseract." — Madeleine L'Engle
An approach via the Omega Matrix:
See, too, Rosenhain and Göpel as The Shadow Guests .
"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
|
"The theory of poetry, that is to say, the total of the theories of poetry, often seems to become in time a mystical theology or, more simply, a mystique. The reason for this must by now be clear. The reason is the same reason why the pictures in a museum of modern art often seem to become in time a mystical aesthetic, a prodigious search of appearance, as if to find a way of saying and of establishing that all things, whether below or above appearance, are one and that it is only through reality, in which they are reflected or, it may be, joined together, that we can reach them. Under such stress, reality changes from substance to subtlety, a subtlety in which it was natural for Cézanne to say: 'I see planes bestriding each other and sometimes straight lines seem to me to fall' or 'Planes in color…. The colored area where shimmer the souls of the planes, in the blaze of the kindled prism, the meeting of planes in the sunlight.' The conversion of our Lumpenwelt went far beyond this. It was from the point of view of another subtlety that Klee could write: 'But he is one chosen that today comes near to the secret places where original law fosters all evolution. And what artist would not establish himself there where the organic center of all movement in time and space– which he calls the mind or heart of creation– determines every function.' Conceding that this sounds a bit like sacerdotal jargon, that is not too much to allow to those that have helped to create a new reality, a modern reality, since what has been created is nothing less.
— Wallace Stevens, Harvard College Class of 1901, "The Relations between Poetry and Painting" in The Necessary Angel (Knopf, 1951) |
For background on the planes illustrated above,
see Diamond theory in 1937.
The summoning of the spirit of Bertrand Russell
yesterday by Peter J. Cameron at his weblog
suggests a review of this weblog’s posts of
Christmas Eve, December 24-25, 2013.
(Recall that Robert D. Carmichael, who, in a book
linked to at midnight last Christmas Eve discusses
some “magic” mathematical structures,
reportedly was trained as a Presbyterian minister.
See also The Presbyterian Exorcist.)
“… her mind rotated the facts….”
Related material— hypercube rotation,* in the context
of rotational symmetries of the Platonic solids:
“I’ve heard of affairs that are strictly Platonic”
* Footnote added on Dec. 26, 2013 —
See Arnold Emch, “Triple and Multiple Systems, Their Geometric
Configurations and Groups,” Trans. Amer. Math. Soc. 31 (1929),
No. 1, 25–42.
On page 42, Emch describes the above method of rotating a
hypercube’s 8 facets (i.e., three-dimensional cubes) to count
rotational symmetries —
See also Diamond Theory in 1937.
Also on p. 42, Emch mentions work of Carmichael on a
Steiner system with the Mathieu group M11 as automorphism
group, and poses the problem of finding such systems and
groups that are larger. This may have inspired the 1931
discovery by Carmichael of the Steiner system S(5, 8, 24),
which has as automorphisms the Mathieu group M24 .
Mathematics:
A review of posts from earlier this month —
Wednesday, September 4, 2013
|
Narrative:
Aooo.
Happy birthday to Stephen King.
"How do you get young people excited
about space? How do you get them interested
not just in watching movies about space,
or in playing video games set in space …
but in space itself?"
— Megan Garber in The Atlantic , Aug. 16, 2012
One approach:
"There is such a thing as a tesseract" and
Diamond Theory in 1937.
See, too, Baez in this journal.
Harvard's President Faust:
Last evening's post Moondance was suggested by a check
in this journal of the date October 10, 2012. That date was
in turn suggested by the date of the above remarks.
Wer immer strebend sich bemüht,
Den können wir erlösen.
Who always striving efforts makes,
For him there is salvation.
— Faust Part 2, Act V, Scene 7: Mountain Gorges.
The finite (i.e., Galois) field GF(16),
according to J. J. Seidel in 1974—
The same field according to Steven H. Cullinane in 1986,
in its guise as the affine 4-space over GF(2)—
The same field, again disguised as an affine 4-space,
according to John H. Conway and N.J.A. Sloane in
Sphere Packings, Lattices, and Groups , first published in 1988—
The above figure by Conway and Sloane summarizes, using
a 4×4 array, the additive vector-space structure of the finite
field GF(16).
This structure embodies what in Euclidean space is called
the parallelogram rule for vector addition—
(Thanks to June Lester for the 3D (uvw) part of the above figure.)
For the transition from this colored Euclidean hypercube
(used above to illustrate the parallelogram rule) to the
4×4 Galois space (illustrated by Cullinane in 1979 and
Conway and Sloane in 1988— or later… I do not have
their book’s first edition), see Diamond Theory in 1937,
Vertex Adjacency in a Tesseract and in a 4×4 Array,
Spaces as Hypercubes, and The Galois Tesseract.
For some related narrative, see tesseract in this journal.
(This post has been added to finitegeometry.org.)
Update of August 9, 2013—
Coordinates for hypercube vertices derived from the
parallelogram rule in four dimensions were better
illustrated by Jürgen Köller in a web page archived in 2002.
Update of August 13, 2013—
The four basis vectors in the 2002 Köller hypercube figure
are also visible at the bottom of the hypercube figure on
page 7 of “Diamond Theory,” excerpts from a 1976 preprint
in Computer Graphics and Art , Vol. 2, No. 1, February 1977.
A predecessor: Coxeter’s 1950 hypercube figure from
“Self-Dual Configurations and Regular Graphs.”
(An episode of Mathematics and Narrative )
A report on the August 9th opening of Sondheim's Into the Woods—
Amy Adams… explained why she decided to take on the role of the Baker’s Wife.
“It’s the ‘Be careful what you wish’ part,” she said. “Since having a child, I’m really aware that we’re all under a social responsibility to understand the consequences of our actions.” —Amanda Gordon at businessweek.com
Related material—
Amy Adams in Sunshine Cleaning "quickly learns the rules and ropes of her unlikely new market. (For instance, there are products out there specially formulated for cleaning up a 'decomp.')" —David Savage at Cinema Retro
Compare and contrast…
1. The following item from Walpurgisnacht 2012—
2. The six partitions of a tesseract's 16 vertices
into four parallel faces in Diamond Theory in 1937—
(Click logo for details.)
NeXT in action:
This morning's post Opening Act suggests the following scholium—
To Purgatory fire you'll come at last;
And Christ receive your soul.
If ever you gave meat or drink,
Every night and all,
The fire will never make you shrink;
And Christ receive your soul.
See also The Wall Street Journal 's Ice Water in Hell story.
Followup scholium — "Vague but exciting …" —
“Design is how it works.” — Steven Jobs (See Symmetry and Design.)
“By far the most important structure in design theory is the Steiner system S(5, 8, 24).”
— “Block Designs,” by Andries E. Brouwer
The name Carmichael is not to be found in Booher’s thesis. A book he does cite for the history of S(5,8,24) gives the date of Carmichael’s construction of this design as 1937. It should be dated 1931, as the following quotation shows—
From Log24 on Feb. 20, 2010—
“The linear fractional group modulo 23 of order 24•23•11 is often represented as a doubly transitive group of degree 24 on the symbols ∞, 0, 1, 2,…, 22. This transitive group contains a subgroup of order 8 each element of which transforms into itself the set ∞, 0, 1, 3, 12, 15, 21, 22 of eight elements, while the whole group transforms this set into 3•23•11 sets of eight each. This configuration of octuples has the remarkable property that any given set of five of the 24 symbols occurs in one and just one of these octuples. The largest permutation group Γ on the 24 symbols, each element of which leaves this configuration invariant, is a five-fold transitive group of degree 24 and order 24•23•22•21•20•48. This is the Mathieu group of degree 24.”
– R. D. Carmichael, “Tactical Configurations of Rank Two,” in American Journal of Mathematics, Vol. 53, No. 1 (Jan., 1931), pp. 217-240
Epigraph from Ch. 4 of Design Theory , Vol. I:
“Es is eine alte Geschichte,
doch bleibt sie immer neu ”
—Heine (Lyrisches Intermezzo XXXIX)
See also “Do you like apples?“
Today's online New York Times on the conclusion of the Roman Catholic Church's "World Youth" week—
"At the end of Sunday’s Mass, the pope announced that the next such event would be in Rio de Janeiro in 2013. Until then, he told those at the service, in Portuguese, that they 'will be swimming against the tide in a society with a relativistic culture, which wishes neither to seek nor hold on to the truth.'*"
* Fact check— This agrees with the Vatican Radio version.
Related material: Relativity Blues and Portal to 1937…
The "Portal" link above is in honor of the May 2 dies natalis of Salomon Bochner (pdf).
For some background, see yesterday's Castles in the Air and Bochner in this journal.
"… the Jews have discovered a way to access a fourth spatial dimension."
— Clifford Pickover, description of his novel Jews in Hyperspace
"If you have built castles in the air, your work need not be lost;
that is where they should be. Now put the foundations under them.”
— Henry David Thoreau
"King Solomon's Mines," 1937—
The image above is an illustration from "Romancing the Hyperspace," May 4, 2010.
Happy birthday to the late Salomon Bochner.
Philosophical Investigations (1953)—
97. Thought is surrounded by a halo.
—Its essence, logic, presents an order,
in fact the a priori order of the world:
that is, the order of possibilities * ,
which must be common to both world and thought.
But this order, it seems, must be
utterly simple . It is prior to all experience,
must run through all experience;
no empirical cloudiness or uncertainty can be allowed to affect it
——It must rather be of the purest crystal.
But this crystal does not appear as an abstraction;
but as something concrete, indeed, as the most concrete,
as it were the hardest thing there is
(Tractatus Logico-Philosophicus No. 5.5563).
— Translation by G.E.M. Anscombe
All propositions of our colloquial language
are actually, just as they are, logically completely in order.
That simple thing which we ought to give here is not
a model of the truth but the complete truth itself.
(Our problems are not abstract but perhaps
the most concrete that there are.)
97. Das Denken ist mit einem Nimbus umgeben.
—Sein Wesen, die Logik, stellt eine Ordnung dar,
und zwar die Ordnung a priori der Welt,
d.i. die Ordnung der Möglichkeiten ,
die Welt und Denken gemeinsam sein muß.
Diese Ordnung aber, scheint es, muß
höchst einfach sein. Sie ist vor aller Erfahrung;
muß sich durch die ganze Erfahrung hindurchziehen;
ihr selbst darf keine erfahrungsmäßige Trübe oder Unsicherheit anhaften.
——Sie muß vielmehr vom reinsten Kristall sein.
Dieser Kristall aber erscheint nicht als eine Abstraktion;
sondern als etwas Konkretes, ja als das Konkreteste,
gleichsam Härteste . (Log. Phil. Abh. No. 5.5563.)
Related language in Łukasiewicz (1937)—
* Updates of 9:29 PM ET July 10, 2011—
A mnemonic from a course titled “Galois Connections and Modal Logics“—
“Traditionally, there are two modalities, namely, possibility and necessity.
The basic modal operators are usually written 
(square) for necessarily
and 
(diamond) for possibly. Then, for example, P can be read as
‘it is possibly the case that P .'”
| See also Intensional Semantics , lecture notes by Kai von Fintel and Irene Heim, MIT, Spring 2007 edition—
“The diamond ⋄ symbol for possibility is due to C.I. Lewis, first introduced in Lewis & Langford (1932), but he made no use of a symbol for the dual combination ¬⋄¬. The dual symbol □ was later devised by F.B. Fitch and first appeared in print in 1946 in a paper by his doctoral student Barcan (1946). See footnote 425 of Hughes & Cresswell (1968). Another notation one finds is L for necessity and M for possibility, the latter from the German möglich ‘possible.’” Barcan, Ruth C.: 1946. “A Functional Calculus of First Order Based on Strict Implication.” Journal of Symbolic Logic, 11(1): 1–16. URL http://www.jstor.org/pss/2269159. Hughes, G.E. & Cresswell, M.J.: 1968. An Introduction to Modal Logic. London: Methuen. Lewis, Clarence Irving & Langford, Cooper Harold: 1932. Symbolic Logic. New York: Century. |
Literary remarks for Maundy Thursday—
— C. P. Snow, foreword to G. H. Hardy's A Mathematician's Apology
Related material—
Emory University press release of January 20th, 2011:
"In 1937, Hans Rademacher found an exact formula for calculating partition values. While the method was a big improvement over Euler's exact formula, it required adding together infinitely many numbers that have infinitely many decimal places. 'These numbers are gruesome,' Ono says….
… The final eureka moment occurred near another Georgia landmark: Spaghetti Junction. Ono and Jan Bruinier were stuck in traffic near the notorious Atlanta interchange. While chatting in the car, they hit upon a way to overcome the infinite complexity of Rademacher's method. They went on to prove a formula that requires only finitely many simple numbers.
'We found a function, that we call P, that is like a magical oracle,' Ono says. 'I can take any number, plug it into P, and instantly calculate the partitions of that number….'"
See also this journal on April 15 and a Google Groups [sage-devel] thread, Ono-Bruinier partition formula. That thread started on April 15 and was last updated this morning.
The On-Line Encyclopedia of Integer Sequences has an article titled "Number of combinatorial configurations of type (n_3)," by N.J.A. Sloane and D. Glynn.
From that article:
The following corrects the word "unique" in the example.
* This post corrects an earlier post, also numbered 14660 and dated 7 PM March 18, 2011, that was in error.
The correction was made at about 11:50 AM on March 20, 2011.
_____________________________________________________________
Update of March 21
The problem here is of course with the definition. Sloane and Glynn failed to include in their definition a condition that is common in other definitions of configurations, even abstract or purely "combinatorial" configurations. See, for instance, Configurations of Points and Lines , by Branko Grunbaum (American Mathematical Society, 2009), p. 17—
In the most general sense we shall consider combinatorial (or abstract) configurations; we shall use the term set-configurations as well. In this setting "points" are interpreted as any symbols (usually letters or integers), and "lines" are families of such symbols; "incidence" means that a "point" is an element of a "line". It follows that combinatorial configurations are special kinds of general incidence structures. Occasionally, in order to simplify and clarify the language, for "points" we shall use the term marks, and for "lines" we shall use blocks. The main property of geometric configurations that is preserved in the generalization to set-configurations (and that characterizes such configurations) is that two marks are incident with at most one block, and two blocks with at most one mark.
Whether or not omitting this "at most one" condition from the definition is aesthetically the best choice, it dramatically changes the number of configurations in the resulting theory, as the above (8_3) examples show.
Update of March 22 (itself updated on March 25)
For further background on configurations, see Dolgachev—
Note that the two examples Dolgachev mentions here, with 16 points and 9 points, are not unrelated to the geometry of 4×4 and 3×3 square arrays. For the Kummer and related 16-point configurations, see section 10.3, "The Three Biplanes of Order 4," in Burkard Polster's A Geometrical Picture Book (Springer, 1998). See also the 4×4 array described by Gordon Royle in an undated web page and in 1980 by Assmus and Sardi. For the Hesse configuration, see (for instance) the passage from Coxeter quoted in Quaternions in an Affine Galois Plane.
Update of March 27
See the above link to the (16,6) 4×4 array and the (16,6) exercises using this array in R.D. Carmichael's classic Introduction to the Theory of Groups of Finite Order (1937), pp. 42-43. For a connection of this sort of 4×4 geometry to the geometry of the diamond theorem, read "The 2-subsets of a 6-set are the points of a PG(3,2)" (a note from 1986) in light of R.W.H.T. Hudson's 1905 classic Kummer's Quartic Surface , pages 8-9, 16-17, 44-45, 76-77, 78-79, and 80.
Part 3 of 5 (See also Part 1 and Part 2) begins as follows…
"Incommensurable. It is a strange word. I wondered, why did Kuhn choose it? What was the attraction?
Here’s one clue. At the very end of 'The Road Since Structure,' a compendium of essays on Kuhn’s work, there is an interview with three Greek philosophers of science, Aristides Baltas, Kostas Gavroglu and Vassiliki Kindi. Kuhn provides a brief account of the historical origins of his idea. Here is the relevant segment of the interview.
T. KUHN: Look, 'incommensurability' is easy.
V. KINDI: You mean in mathematics?
T. KUHN: …When I was a bright high school mathematician and beginning to learn Calculus, somebody gave me—or maybe I asked for it because I’d heard about it—there was sort of a big two-volume Calculus book by, I can’t remember whom. And then I never really read it. I read the early parts of it. And early on it gives the proof of the irrationality of the square root of 2. And I thought it was beautiful. That was terribly exciting, and I learned what incommensurability was then and there. So, it was all ready for me, I mean, it was a metaphor but it got at nicely what I was after. So, that’s where I got it.
'It was all ready for me.' I thought, 'Wow.' The language was suggestive. I imagined √2 provocatively dressed, its lips rouged. But there was an unexpected surprise. The idea didn’t come from the physical sciences or philosophy or linguistics, but from mathematics ."
A footnote from Morris (no. 29)—
"Those who are familiar with the proof [of irrationality] certainly don’t want me to explain it here; likewise, those who are unfamiliar with it don’t want me to explain it here, either. There are many simple proofs in many histories of mathematics — E.T. Bell, Sir Thomas Heath, Morris Kline, etc., etc. Barry Mazur offers a proof in his book, 'Imagining Numbers (particularly the square root of minus fifteen),' New York, NY: Farrar, Straus and Giroux. 2003, 26ff. And there are two proofs in his essay, 'How Did Theaetetus Prove His Theorem?', available on Mazur’s Harvard Web site."
There may, actually, be a few who do want the proof. They may consult the sources Morris gives, or the excellent description by G.H. Hardy in A Mathematician's Apology , or, perhaps best of all for present purposes, the proof as described in a "sort of a big two-volume Calculus book" (perhaps the one Kuhn mentioned)… See page 6 and page 7 of Volume One of Richard Courant's classic Differential and Integral Calculus (second edition, 1937, reprinted many times through 1970, and again in a Wiley Classics Library Edition in 1988).
From May 11, 2010, an image—
See also the same date in 2005 in light of
The late mathematician V.I. Arnold was born on this date in 1937.
"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
Light
Choosing light rather than darkness, we observe Arnold's birthday with a quotation from his 1997 Paris talk 'On Teaching Mathematics.'
"The Jacobi identity (which forces the heights of a triangle to cross at one point) is an experimental fact…."
The "experimental fact" part, perhaps offered with tongue in cheek, is of less interest than the assertion that the Jacobi identity forces the altitude-intersection theorem.
Albert Einstein on that theorem in the "holy geometry book" he read at the age of 12—
"Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which– though by no means evident– could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty made an indescribable impression upon me.”
Arnold's much less evident assertion about altitudes and the Jacobi identity is discussed in "Arnol'd, Jacobi identity, and orthocenters" (pdf) by Nikolai V. Ivanov.
Ivanov says, without giving a source, that the altitudes theorem "was known to Euclid." Alexander Bogomolny, on the other hand, says it is "a matter of real wonderment that the fact of the concurrency of altitudes is not mentioned in either Euclid's Elements or subsequent writings of the Greek scholars. The timing of the first proof is still an open question."
For other remarks on geometry, search this journal for the year of Arnold's birth.
William Lubtchansky, a cinematographer, was born on October 26, 1937, and died on May 4, 2010.
Yesterday's post included an illustration from this journal on the date of his death.
Here is a Log24 entry from last year on the date of his birth—
| Monday, October 26, 2009 The Keys Enigma
Related material: |
Clicking on the Shift Lock key leads to the following page—
—The Philosopher's Gaze,
by David Michael Levin,
University of California Press, 1999
Related images—
Detail from May 4 image:
Holocaust Museum, Washington, DC:
(http://www.scrapbookpages.com/USHMM/Exterior.html)
See also Lubtchansky's Duelle and
Art Wars for Trotsky's Birthday, 2003.
Paul Robeson in
"King Solomon's Mines," 1937—
The image above is an illustration from
"Romancing the Hyperspace," May 4, 2010.
This illustration, along with Georgia Brown's
song from "Cabin in the Sky"—
"There's honey in the honeycomb"—
suggests the following picture.
"What might have been and what has been
Point to one end, which is always present."
— Four Quartets
Romancing the
Non-Euclidean Hyperspace
Backstory —
Mere Geometry, Types of Ambiguity,
Dream Time, and Diamond Theory, 1937
For the 1937 grid, see Diamond Theory, 1937.
The grid is, as Mere Geometry points out, a non-Euclidean hyperspace.
For the diamonds of 2010, see Galois Geometry and Solomon’s Cube.
“…geometrically organized, with the parts labeled”
— Ursula K. Le Guin on what she calls “the Euclidean utopia”
“There is such a thing as a tesseract.”
Related material– Diamond Theory, 1937
This journal’s Christmas Day entry, Brightness at Noon, was in response to the Orwellian headline “Arthur Koestler, Man of Darkness,” at the top of the online New York Times front page on Christmas morning.
The entry offered, as an example of brightness, some thoughts of Leibniz on his discovery of binary arithmetic.
Related material:
“To make all things from nothing, unity suffices.” So it is written on a medal entitled Imago Creationis and designed by Leibniz to “exhibit to posterity in silver” his discovery of the binary system. Baron Gottfried Wilhelm von Leibniz (also Leibnitz) 1646-1716. Philosopher and mathematician. Invented calculus independently of Newton. Proposed the metaphysical theory that we live in “the best of all possible worlds.” He also discovered binary number system and believed in its profound metaphysical significance. He noticed similarity with the ancient Chinese divination system “I Ching.” We chose him for our patron, for Krawtchuk polynomials can be understood as a sophistication of the simple counting of 0 and 1… — Philip Feinsilver and Jerzy Kocik, 17 July 2001 |
From Mikhail Krawtchouk: Short Biography—
Anyone knowing even a little Soviet history of the thirties can conclude that Krawtchouk could not avoid the Great Terror. During the Orwellian “hours of hatred” in 1937 he was denounced as a “Polish spy,” “bourgeois nationalist,” etc. In 1938, he was arrested and sentenced to 20 years of confinement and 5 years of exile.
Academician Krawtchouk, the author of results which became part of the world’s mathematical knowledge, outstanding lecturer, member of the French, German, and other mathematical societies, died on March 9, 1942, in Kolyma branch of the GULAG (North-Eastern Siberia) more than 6 months short of his 50th birthday.
Incidentally, happy birthday
to John von Neumann.
From the September 1953 Bulletin of the American Mathematical Society—
Emil Artin, in a review of Éléments de mathématique, by N. Bourbaki, Book II, Algebra, Chaps. I-VII–
"We all believe that mathematics is an art. The author of a book, the lecturer in a classroom tries to convey the structural beauty of mathematics to his readers, to his listeners. In this attempt he must always fail. Mathematics is logical to be sure; each conclusion is drawn from previously derived statements. Yet the whole of it, the real piece of art, is not linear; worse than that its perception should be instantaneous. We all have experienced on some rare occasions the feeling of elation in realizing that we have enabled our listeners to see at a moment's glance the whole architecture and all its ramifications. How can this be achieved? Clinging stubbornly to the logical sequence inhibits the visualization of the whole, and yet this logical structure must predominate or chaos would result."
Art Versus Chaos
From an exhibit,
"Reimagining Space"
The above tesseract (4-D hypercube)
sculpted in 1967 by Peter Forakis
provides an example of what Artin
called "the visualization of the whole."
For related mathematical details see
Diamond Theory in 1937.
"'The test?' I faltered, staring at the thing.
'Yes, to determine whether you can live
in the fourth dimension or only die in it.'"
— Fritz Leiber, 1959
See also the Log24 entry for
Nov. 26, 2009, the date that
Forakis died.
"There is such a thing
as a tesseract."
— Madeleine L'Engle, 1962
The Shining
of Dec. 13
continued from
Dec. 13, 2003
“There is a place for a hint
somewhere of a big agent
to complete the picture.”
— Notes for an unfinished novel,
The Last Tycoon,
by F. Scott Fitzgerald
The Good Earth (1937)
casting: Chinese extras
(uncredited)
(From the book Tangram)
See also
yesterday’s entries
as well as…
Serpent’s Eyes Shine,
Alice’s Tea Party,
Janet’s Tea Party,
Hollywood Memory,
and
Hope of Heaven.
“… it’s going to be
accomplished in steps,
this establishment of
the Talented
in the scheme of things.”
From Braque's birthday, 2006:
"The senses deform, the mind forms. Work to perfect the mind. There is no certitude but in what the mind conceives."
— Georges Braque,
Reflections on Painting, 1917
Those who wish to follow Braque's advice may try the following exercise from a book first published in 1937:
For a different view
of the square and cube
see yesterday's entry
Abstraction and Faith.
On April 16, the Pope’s birthday, the evening lottery number in Pennsylvania was 441. The Log24 entries of April 17 and April 18 supplied commentaries based on 441’s incarnation as a page number in an edition of Heidegger’s writings. Here is a related commentary on a different incarnation of 441. (For a context that includes both today’s commentary and those of April 17 and 18, see Gian-Carlo Rota– a Heidegger scholar as well as a mathematician– on mathematical Lichtung.)
From R. D. Carmichael, Introduction to the Theory of Groups of Finite Order (Boston, Ginn and Co., 1937)– an exercise from the final page, 441, of the final chapter, “Tactical Configurations”–
“23. Let G be a multiply transitive group of degree n whose degree of transitivity is k; and let G have the property that a set S of m elements exists in G such that when k of the elements S are changed by a permutation of G into k of these elements, then all these m elements are permuted among themselves; moreover, let G have the property P, namely, that the identity is the only element in G which leaves fixed the n – m elements not in S. Then show that G permutes the m elements S into
m(m – 1) … (m – k + 1)
This exercise concerns an important mathematical structure said to have been discovered independently by the American Carmichael and by the German Ernst Witt.
For some perhaps more comprehensible material from the preceding page in Carmichael– 440– see Diamond Theory in 1937.
John Trever, Albuquerque Journal, 2/29/08
The pen's point:
|
Log24, Dec. 11, 2006
SINGER, ISAAC:
"Sets forth his own aims in writing for children and laments 'slice of life' and chaos in children's literature. Maintains that children like good plots, logic, and clarity, and that they have a concern for 'so-called eternal questions.'"
— An Annotated Listing
"She returned the smile, then looked across the room to her youngest brother, Charles Wallace, and to their father, who were deep in concentration, bent over the model they were building of a tesseract: the square squared, and squared again: a construction of the dimension of time."
— A Swiftly Tilting Planet,
A Swiftly Tilting Planet is a fantasy for children set partly in Vespugia, a fictional country bordered by Chile and Argentina.
|
The title of the previous entry, "Where Entertainment is God," comes (via Log24, Nov. 26, 2004) from Frank Rich.
The previous entry dealt, in part, with a dead Jesuit whose obituary appears in today's Los Angeles Times. The online obituaries page places the Jesuit, without a photo, beneath a picture of a dead sitcom writer and to the left of a picture of a dead guru.
"Walter John Burghardt was born July 10, 1914, in New York, the son of immigrants from what is now Poland. He entered a Jesuit seminary in Poughkeepsie, N.Y., at 16, and in 1937 received a master's degree from Woodstock College in Maryland. He was ordained in 1941." He died, by the way, on Saturday, Feb. 16, 2008.
The reference to Woodstock College brings to mind a fellow Jesuit, Joseph T. Clark, who wrote a book on logic published by that college.
From a review of the book:
"In order to show that Aristotelian logicians were at least vaguely aware of a kind of analogy or possible isomorphism between logical relations and mathematical relations, Father Clark seizes at one place (p. 8) upon the fact that Aristotle uses the word, 'figure' (schema), in describing the syllogism and concludes from this that 'it is obvious that the schema of the syllogism is to serve the logician precisely as the figure serves the geometer.' On the face of it, this strikes one as a bit far fetched…."
— Henry Veatch in Speculum, Vol. 29, No. 2, Part 1 (Apr., 1954), pp. 266-268 (review of Conventional Logic and Modern Logic: A Prelude to Transition (1952), by Joseph T. Clark, Society of Jesus)
The New York Times
November 6, 2007
More on the Career of
the Genius Who Boldly
Compared Himself to God
“Picasso… once said…
‘… No wonder his [Picasso’s] style is so ambiguous. It’s like God’s. God is really only another artist. He invented the giraffe, the elephant and the cat. He has no real style. He just keeps on trying other things. The same with this sculptor….’
The comparison to God, like the use of the third person, was deliberate, of course.”
Of Modern Poetry
The poem of the mind
in the act of finding
What will suffice ….
… It has
To construct a new stage.
It has to be on that stage,
And, like an insatiable actor,
slowly and
With meditation, speak words
that in the ear,
In the delicatest ear
of the mind, repeat,
Exactly, that which it
wants to hear, at the sound
Of which, an invisible
audience listens,
Not to the play, but to
itself, expressed
In an emotion as of
two people, as of two
Emotions becoming one.
The actor is
A metaphysician in the dark….
— Wallace Stevens in
Parts of a World, 1942
Of Modern Metaphysics
“For every work [or act] of creation is threefold, an earthly trinity to match the heavenly.
First, [not in time, but merely in order of enumeration] there is the Creative Idea, passionless, timeless, beholding the whole work complete at once, the end in the beginning: and this is the image of the Father.
Second, there is the Creative Energy [or Activity] begotten of that idea, working in time from the beginning to the end, with sweat and passion, being incarnate in the bonds of matter: and this is the image of the Word.
Third, there is the Creative Power, the meaning of the work and its response in the lively soul: and this is the image of the indwelling Spirit.
And these three are one, each equally in itself the whole work, whereof none can exist without other: and this is the image of the Trinity.”
— Concluding speech of St. Michael the Archangel in a 1937 play, “The Zeal of Thy House,” by Dorothy Sayers, as quoted in her 1941 book The Mind of the Maker. That entire book was, she wrote, an expansion of St. Michael’s speech.
Related material:
|
57:
“The author takes the place of the omniscient narrator. He heightens the tension by using striking dialogue. To decrease the tension he uses some light forms of comedy, like the commands for the Dobermans of the little boy: ‘Ketchup’ for retreating, ‘Pickles’ for attacking, and ‘Mustard’ for killing.” |
|
225:
 |
J. G. Ballard on “the architecture of death“:
“… a huge system of German fortifications that included the Siegfried line, submarine pens and huge flak towers that threatened the surrounding land like lines of Teutonic knights. Almost all had survived the war and seemed to be waiting for the next one, left behind by a race of warrior scientists obsessed with geometry and death.”
— The Guardian, March 20, 2006
“For him, writing is a struggle both with geometry and death.”
— “The Duende,” American Poetry Review, July/August 1999
— Harper’s Magazine review
quoted on back cover of
Cubism and Twentieth-Century Art,
by Robert Rosenblum
(Abrams paperback, 2001)
— An Annotated Listing of Criticism
by Linnea Hendrickson
“She returned the smile, then looked
across the room to her youngest brother,
Charles Wallace, and to their father,
who were deep in concentration, bent
over the model they were building
of a tesseract: the square squared,
and squared again: a construction
of the dimension of time.”
— A Swiftly Tilting Planet,
by Madeleine L’Engle
For “the dimension of time,”
see A Fold in Time,
Time Fold, and
Diamond Theory in 1937.
For a more adult audience —
In memory of General Augusto Pinochet, who died yesterday in Santiago, Chile, a quotation from Federico Garcia Lorca‘s lecture on “the Duende” (Buenos Aires, Argentina, 1933):
Casino Royale
and
Time in the Rock
In today’s cognitive blend,
the role of Casino Royale
is played by the
Pennsylvania Lottery,
which points to 7/26,
Venus at St. Anne’s
(title of the closing chapter
of That Hideous Strength).
The role of
Time in the Rock
is played by a
Log24 entry of 3/29,
Diamond Theory in 1937.
“There is such a thing
as a tesseract.“
For clues to interpreting
today’s Keystone State
mid-day lottery number,
023, see
The Prime Cut Gospel.
For clues to interpreting
today’s Keystone State
evening lottery number,
666, see
the “Apocalypse Now”
quotations on
All Saints’ Day, 2006.
From 7/07, an art review from The New York Times:
Endgame Art?
It's Borrow, Sample and Multiply
in an Exhibition at Bard College
"The show has an endgame, end-time mood….
I would call all these strategies fear of form…. the dismissal of originality is perhaps the oldest ploy in the postmodern playbook. To call yourself an artist at all is by definition to announce a faith, however unacknowledged, in some form of originality, first for yourself, second, perhaps, for the rest of us.
Fear of form above all means fear of compression– of an artistic focus that condenses experiences, ideas and feelings into something whole, committed and visually comprehensible."
— Roberta Smith
It nevertheless does
"announce a faith."
"First for yourself"
Today's mid-day
Pennsylvania number:
707
See Log24 on 7/07
and the above review.
"Second, perhaps,
for the rest of us"
Today's evening
Pennsylvania number:
384
This number is an
example of what the
reviewer calls "compression"–
"an artistic focus that condenses
experiences, ideas and feelings
into something
whole, committed
and visually comprehensible."
"Experiences"
See (for instance)
Joan Didion's writings
(1160 pages, 2.35 pounds)
on "the shifting phantasmagoria
which is our actual experience."
"Ideas"
"Feelings"
See A Wrinkle in Time.
"Whole"
The automorphisms
of the tesseract
form a group
of order 384.
"Committed"
See the discussions of
groups of degree 16 in
R. D. Carmichael's classic
Introduction to the Theory
of Groups of Finite Order.
"Visually comprehensible"
See "Diamond Theory in 1937,"
an excerpt from which
is shown below.
The "faith" announced by
the above lottery numbers
on All Hallows' Eve is
perhaps that of the artist
Madeleine L'Engle:
ART WARS continued…
|
|
 |
“The senses deform, the mind forms. Work to perfect the mind. There is no certitude but in what the mind conceives.”
— Georges Braque, Reflections on Painting, 1917
Those who wish to follow Braque’s advice may try the following exercise from a book first published in 1937:
Hint: See the above picture of
Braque and the construction of
a tesseract.
— Cynthia Zarin on Madeleine L’Engle,
“The Storyteller,” in The New Yorker,
issue dated April 12, 2004
"Does the word 'tesseract'
mean anything to you?"
— Robert A. Heinlein in
The Number of the Beast
(1980)
My reply–
Part I:
A Wrinkle in Time, by
Madeleine L'Engle
(first published in 1962)
Part II:
Diamond Theory in 1937
and
Geometry of the 4×4 Square
Part III:
"Wells and trees were dedicated to saints. But the offerings at many wells and trees were to something other than the saint; had it not been so they would not have been, as we find they often were, forbidden. Within this double and intertwined life existed those other capacities, of which we know more now, but of which we still know little– clairvoyance, clairaudience, foresight, telepathy."
— Charles Williams, Witchcraft, Faber and Faber, London, 1941
A New Yorker profile of Madeleine L'Engle from April 2004, which I found tonight online for the first time. For a related reflection on truth, stories, and values, see Saint's Day. For a wider context, see the Log24 entries of February 1-15, 2003 and February 1-15, 2006.
Holy Geometry
What was “the holy geometry book” (“das heilige Geometrie-Büchlein,” p. 10 in the Schilpp book below) that so impressed the young Albert Einstein?
“At the age of 12 I experienced a second wonder of a totally different nature: in a little book dealing with Euclidian plane geometry, which came into my hands at the beginning of a schoolyear. Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which– though by no means evident– could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty made an indescribable impression upon me.”
(“Im Alter von 12 Jahren erlebte ich ein zweites Wunder ganz verschiedener Art: An einem Büchlein über Euklidische Geometrie der Ebene, das ich am Anfang eines Schuljahres in die Hand bekam. Da waren Aussagen wie z.B. das Sich-Schneiden der drei Höhen eines Dreieckes in einem Punkt, die– obwohl an sich keineswegs evident– doch mit solcher Sicherheit bewiesen werden konnten, dass ein Zweifel ausgeschlossen zu sein schien. Diese Klarheit und Sicherheit machte einen unbeschreiblichen Eindruck auf mich.”)
— Albert Einstein, Autobiographical Notes, pages 8 and 9 in Albert Einstein: Philosopher-Scientist, ed. by Paul A. Schilpp
From a website by Hans-Josef Küpper:
“Today it cannot be said with certainty which book is Einstein’s ‘holy geometry book.’ There are three different titles that come into question:
Theodor Spieker, 1890
Lehrbuch der ebenen Geometrie. Mit Übungsaufgaben für höhere Lehranstalten.
Heinrich Borchert Lübsen, 1870
Ausführliches Lehrbuch der ebenen und sphärischen Trigonometrie. Zum Selbstunterricht. Mit Rücksicht auf die Zwecke des praktischen Lebens.
Adolf Sickenberger, 1888
Leitfaden der elementaren Mathematik.
Young Albert Einstein owned all of these three books. The book by T. Spieker was given to him by Max Talmud (later: Talmey), a Jewish medic. The book by H. B. Lübsen was from the library of his uncle Jakob Einstein and the one of A. Sickenberger was from his parents.”
Küpper does not state clearly his source for the geometry-book information.
According to Banesh Hoffman and Helen Dukas in Albert Einstein, Creator and Rebel, the holy geometry book was Lehrbuch der Geometrie zum Gebrauch an höheren Lehranstalten, by Eduard Heis (Catholic astronomer and textbook writer) and Thomas Joseph Eschweiler.
An argument for Sickenberger from The Young Einstein: The Advent of Relativity (pdf), by Lewis Pyenson, published by Adam Hilger Ltd., 1985:
“Throughout Einstein’s five and a half years at the Luitpold Gymnasium, he was taught mathematics from one or another edition of the separately published parts of Sickenberger’s Textbook of Elementary Mathematics. When it first appeared in 1888 the book constituted a major contribution to reform pedagogy. Sickenberger based his book on twenty years of experience that in his view necessarily took precedence over ‘theoretical doubts and systematic scruples.’ At the same time Sickenberger made much use of the recent pedagogical literature, especially that published in the pages of Immanuel Carl Volkmar Hoffmann’s Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht, the leading pedagogical mathematics journal of the day. Following in the tradition of the reform movement, he sought to present everything in the simplest, most intuitive way possible. He opposed introducing scientific rigour and higher approaches in an elementary text. He emphasised that he would follow neither the synthesis of Euclidean geometry nor the so-called analytical-genetic approach. He opted for a great deal of freedom in the form of presentation because he believed that a textbook was no more than a crutch for oral instruction. The spoken word, in Sickenberger’s view, could infuse life into the dead forms of the printed text. Too often, he insisted in the preface to his text, mathematics was seen and valued ‘as the pure science of reason.’ In reality, he continued, mathematics was also ‘an essential tool for daily work.’ In view of the practical dimension of mathematics Sickenberger sought most of all to present basic propositions clearly rather than to arrive at formal conciseness. Numerous examples took the place of long, complicated, and boring generalities. In addition to the usual rules of arithmetic Sickenberger introduced diophantine equations. To solve three linear, homogeneous, first-order equations with three unknowns he specified determinants and determinant algebra. Then he went on to quadratic equations and logarithms. In the second part of his book, Sickenberger treated plane geometry.
According to a biography of Einstein written by his step-son-in-law, Rudolf Kayser– one that the theoretical physicist described as ‘duly accurate’– when he was twelve years old Einstein fell into possession of the ‘small geometry book’ used in the Luitpold Gymnasium before this subject was formally presented to him. Einstein corroborated Kayser’s passage in autobiographical notes of 1949, when he described how at the age of twelve ‘a little book dealing with Euclidean plane geometry’ came into his hands ‘at the beginning of a school year.’ The ‘lucidity and certainty’ of plane geometry according to this ‘holy geometry booklet’ made, Einstein wrote, ‘an indescribable impression on me.’ Einstein saw here what he found in other texts that he enjoyed: it was ‘not too particular’ in logical rigour but ‘made up for this by permitting the main thoughts to stand out clearly and synoptically.’ Upon working his way through this text, Einstein was then presented with one of the many editions of Theodor Spieker’s geometry by Max Talmey, a medical student at the University of Munich who dined with the Einsteins and who was young Einstein’s friend when Einstein was between the ages of ten and fifteen. We can only infer from Einstein’s retrospective judgment that the first geometry book exerted an impact greater than that produced by Spieker’s treatment, by the popular science expositions of Aaron Bernstein and Ludwig Büchner also given to him by Talmey, or by the texts of Heinrich Borchert Lübsen from which Einstein had by the age of fourteen taught himself differential and integral calculus.
Which text constituted the ‘holy geometry booklet’? In his will Einstein gave ‘all his books’ to his long-time secretary Helen Dukas. Present in this collection are three bearing the signature ‘J Einstein’: a logarithmic and trigonometric handbook, a textbook on analysis, and an introduction to infinitesimal calculus. The signature is that of Einstein’s father’s brother Jakob, a business partner and member of Einstein’s household in Ulm and Munich. He presented the books to his nephew Albert. A fourth book in Miss Dukas’s collection, which does not bear Jakob Einstein’s name, is the second part of a textbook on geometry, a work of astronomer Eduard Heis’s which was rewritten after his death by the Cologne schoolteacher Thomas Joseph Eschweiler. Without offering reasons for his choice Banesh Hoffmann has recently identified Heis and Eschweiler’s text as the geometry book that made such an impression on Einstein. Yet, assuming that Kayser’s unambiguous reporting is correct, it is far more likely that the geometrical part of Sickenberger’s text was what Einstein referred to in his autobiographical notes. Sickenberger’s exposition was published seven years after that of Heis and Eschweiler, and unlike the latter it appeared with a Munich press. Because it was used in the Luitpold Gymnasium, copies would have been readily available to Uncle Jakob or to whoever first acquainted Einstein with Euclidean geometry.”
What might be the modern version of a “holy geometry book”?
Click on picture for details.
“Who’s got the last laugh now?”
— “They All Laughed,”
words by Ira Gershwin,
music by George Gershwin,
from the 1937 film Shall We Dance
(sung by Ginger Rogers,
then danced by
Fred Astaire and Ginger Rogers)
See also
the entry of June 4 last year,
The Four Last Things.
Clarity and Certainty
“At the age of 12 I experienced a second wonder of a totally different nature: in a little book* dealing with Euclidean plane geometry, which came into my hands at the beginning of a schoolyear. Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which — though by no means evident — could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty [Klarheit und Sicherheit] made an indescribable impression upon me…. For example I remember that an uncle told me the Pythagorean theorem before the holy geometry booklet* had come into my hands. After much effort I succeeded in ‘proving’ this theorem on the basis of the similarity of triangles … for anyone who experiences [these feelings] for the first time, it is marvellous enough that man is capable at all to reach such a degree of certainty and purity [Sicherheit und Reinheit] in pure thinking as the Greeks showed us for the first time to be possible in geometry.”
— from “Autobiographical Notes” in Albert Einstein: Philosopher-Scientist, edited by Paul Arthur Schilpp
“Although our intellect always longs for clarity and certainty, our nature often finds uncertainty fascinating.”
— Carl von Clausewitz at Quotes by Clausewitz
For clarity and certainty, consult All About Altitudes (and be sure to click the “pop it up” button).
For murkiness and uncertainty, consult The Fog of War.
Happy birthday, Albert.
* Einstein’s “holy geometry booklet” was, according to Banesh Hoffman, Lehrbuch der Geometrie zum Gebrauch an höheren Lehranstalten, by Eduard Heis (Catholic astronomer and textbook writer) and Thomas Joseph Eschweiler.
— ART WARS —
Mental Health Month, Day 25:
Matrix of the Death God
Having dealt yesterday with the Death Goddess Sarah, we turn today to the Death God Abraham. (See Jacques Derrida, The Gift of Death, University of Chicago Press, 1996.) For a lengthy list of pictures of this damned homicidal lunatic about to murder his son, see The Text This Week.
See, too, The Matrix of Abraham, illustrated below. This is taken from a book by R. M. Abraham, Diversions and Pastimes, published by Constable and Company, London, in 1933.
The Matrix of Abraham
A summary of the religious import of the above from Princeton University Press:
“Moslems of the Middle Ages were fascinated by pandiagonal squares with 1 in the center…. The Moslems thought of the central 1 as being symbolic of the unity of Allah. Indeed, they were so awed by that symbol that they often left blank the central cell on which the 1 should be positioned.”
— Clifford A. Pickover, The Zen of Magic Squares, Circles, and Stars, Princeton U. Press, 2002, pp. 71-72
Other appearances of this religious icon on the Web:
A less religious approach to the icon may be found on page 393 of R. D. Carmichael’s Introduction to the Theory of Groups of Finite Order (Ginn, Boston, 1937, reprinted by Dover, 1956).
This matrix did not originate with Abraham but, unlike Neo, I have not yet found its Architect.
Mental Health Month, Day 23:
The Prime Cut Gospel
On Christmas Day, 1949,
Mary Elizabeth Spacek was born in Texas.
Lee Marvin, Sissy Spacek in “Prime Cut”
Exercises for Mental Health Month:
Read this discussion of the phrase, suggested by Spacek’s date of birth, “God’s gift to men.”
Read this discussion of the phrase “the same yesterday, today, and forever,” suggested by the previous reading.
Read the more interesting of these discussions of the phrase “the eternal in the temporal.”
Read this discussion of eternal, or “necessary,” truths versus other sorts of alleged “truths.”
Read this discussion of unimportant mathematical properties of the prime number 23.
Read these discussions of important properties of 23:
Introduction to the Theory of Groups of Finite Order, Ginn, Boston, 1937 (reprinted by Dover in 1956), final chapter, “Tactical Configurations,” and
J. H. Conway, “Three Lectures on Exceptional Groups,” pp. 215-247 in Finite Simple Groups (Oxford, 1969), edited by M. B. Powell and G. Higman, Academic Press, London, 1971….. Reprinted as Ch. 10 in Sphere Packings, Lattices, and Groups
Read this discussion of what might be called “contingent,” or “literary,” properties of the number 23.
Read also the more interesting of these discussions of the phrase “the 23 enigma.”
Having thus acquired some familiarity with both contingent and necessary properties of 23…
Read this discussion of Aquinas’s third proof of the existence of God.
Note that the classic Spacek film “Prime Cut” was released in 1972, the year that Spacek turned 23:
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1949
+ 23 1972 |
2:23 PM
Sequel
to the previous two entries
"This world is not conclusion;
A sequel stands beyond…."
— Emily Dickinson
Today's birthday: dancer/actress Ann Miller.
"In 1937, she was discovered by Lucille Ball…."
Lucille Ball, Desi Arnaz,
and Ann Miller, cast photo
from Too Many Girls (1940)
"Just goes to show star quality shines through…."
— Website on Too Many Girls"It'll shine when it shines."
— Folk saying, epigraph to The Shining"Shine on, you crazy diamond."
— Pink Floyd"Well we all shine on…"
— John Lennon, "Instant Karma"
On This Date
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In 1937, composer Our site music for today For “Bolero” purposes, some may prefer Kylie Minogue’s rendition of “Locomotion.” |
Zen meditation: “Kylie Eleison!”
(For evidence that this is a valid Japanese religious exclamation, click here.)
Nightmare Alley
Tonight’s site music in the garden of good and evil is “Hooray for Hollywood,” with lyrics by Johnny Mercer:
Hooray for Hollywood.
You may be homely in your neighborhood,
But if you think you can be an actor,
see Mr. Factor,
he’d make a monkey look good.
Within a half an hour,
you look like Tyrone Power!
Hooray for Hollywood!
From Pif magazine:
Nightmare Alley (1947)
Directed by Edmund Goulding
Reviewed by Nick Burton
“Edmund Goulding’s film of William Lindsay Gresham’s 1946 novel Nightmare Alley may just be the great forgotten American film; it is certainly the darkest film that came from the Hollywood studio system in the ’40s….
A never better Tyrone Power stars as Stan Carlisle, a small-time carny shill…. Stan shills for mind reader Zeena…. The… pretty ‘electric girl’… tells Stan that Zeena… had a ‘code’ for the mind-reading act… Stan… decides to seduce… Zeena in hopes of luring the code from her.”
The rest of this review is well worth reading, though less relevant to my present theme — that of my
which points out that the article on “nothing” is on page 265 of The Oxford Dictionary of Philosophy. (This is also the theme of yesterday’s journal entry “Last-Minute Shopping.”) Here is another work that prominently features “nothing” on page 265… As it happens, this is a web page describing a mind-reading act, titled simply
“Imagine this: A spectator is invited to take a readable and 100% examinable, 400 page, 160,000 word novel, open it to any page and think of any word on that page. Without touching the book or approaching the spectator, you reveal the word in the simplest, most startlingly direct manner ever! It truly must be seen to be believed.
The ultimate any-word-on-any-page method that makes all other book tests obsolete….
All pages are different.
Nothing is written down.
There are no stooges of any kind. Everything may be examined….
‘Throw away your Key. This is direct mindreading at its best.'”
|
From Finnegans Wake, page 265: “…the winnerful wonnerful wanders off, with hedges of ivy and |
Hooray.
Mercer’s lyrics are from the 1937 film
A Logocentric Archetype
Today we examine the relativist, nominalist, leftist, nihilist, despairing, depressing, absurd, and abominable work of Samuel Beckett, darling of the postmodernists.
One lens through which to view Beckett is an essay by Jennifer Martin, "Beckettian Drama as Protest: A Postmodern Examination of the 'Delogocentering' of Language." Martin begins her essay with two quotations: one from the contemptible French twerp Jacques Derrida, and one from Beckett's masterpiece of stupidity, Molloy. For a logocentric deconstruction of Derrida, see my note, "The Shining of May 29," which demonstrates how Derrida attempts to convert a rather important mathematical result to his brand of nauseating and pretentious nonsense, and of course gets it wrong. For a logocentric deconstruction of Molloy, consider the following passage:
"I took advantage of being at the seaside to lay in a store of sucking-stones. They were pebbles but I call them stones…. I distributed them equally among my four pockets, and sucked them turn and turn about. This raised a problem which I first solved in the following way. I had say sixteen stones, four in each of my four pockets these being the two pockets of my trousers and the two pockets of my greatcoat. Taking a stone from the right pocket of my greatcoat, and putting it in my mouth, I replaced it in the right pocket of my greatcoat by a stone from the right pocket of my trousers, which I replaced by a stone from the left pocket of my trousers, which I replaced by a stone from the left pocket of my greatcoat, which I replaced by the stone which was in my mouth, as soon as I had finished sucking it. Thus there were still four stones in each of my four pockets, but not quite the same stones….But this solution did not satisfy me fully. For it did not escape me that, by an extraordinary hazard, the four stones circulating thus might always be the same four."
Beckett is describing, in great detail, how a damned moron might approach the extraordinarily beautiful mathematical discipline known as group theory, founded by the French anticleric and leftist Evariste Galois. Disciples of Derrida may play at mimicking the politics of Galois, but will never come close to imitating his genius. For a worthwhile discussion of permutation groups acting on a set of 16 elements, see R. D. Carmichael's masterly work, Introduction to the Theory of Groups of Finite Order, Ginn, Boston, 1937, reprinted by Dover, New York, 1956.
There are at least two ways of approaching permutations on 16 elements in what Pascal calls "l'esprit géométrique." My website Diamond Theory discusses the action of the affine group in a four-dimensional finite geometry of 16 points. For a four-dimensional euclidean hypercube, or tesseract, with 16 vertices, see the highly logocentric movable illustration by Harry J. Smith. The concept of a tesseract was made famous, though seen through a glass darkly, by the Christian writer Madeleine L'Engle in her novel for children and young adults, A Wrinkle in Tme.
This tesseract may serve as an archetype for what Pascal, Simone Weil (see my earlier notes), Harry J. Smith, and Madeleine L'Engle might, borrowing their enemies' language, call their "logocentric" philosophy.
For a more literary antidote to postmodernist nihilism, see Archetypal Theory and Criticism, by Glen R. Gill.
For a discussion of the full range of meaning of the word "logos," which has rational as well as religious connotations, click here.
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Initial Xanga entry. Updated Nov. 18, 2006.
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
3115052 2:16 PM
Sunday, July 28, 2002
3114730 1:56 PM
Friday, July 26, 2002
3077091 1:59 PM
Thursday, July 25, 2002
3061170 9:18 PM
Saturday, July 20, 2002
2947581 10:13 PM
Saturday, August 31, 2002
3995905 3:36 AM
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3966763 2:30 AM
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Tuesday, August 27, 2002
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Tuesday, August 06, 2002
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Monday, June 23, 2003
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Sunday, June 22, 2003
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Friday, June 20, 2003
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Wednesday, June 18, 2003
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Tuesday, June 17, 2003
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Monday, June 16, 2003
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Sunday, June 15, 2003
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Saturday, June 14, 2003
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Friday, June 13, 2003
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Wednesday, June 11, 2003
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Tuesday, June 10, 2003
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Sunday, June 08, 2003
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Saturday, June 07, 2003
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Friday, June 06, 2003
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Thursday, June 05, 2003
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Wednesday, June 04, 2003
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Tuesday, June 03, 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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Thursday, July 31, 2003
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Wednesday, July 30, 2003
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Tuesday, July 29, 2003
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Monday, July 28, 2003
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Monday, July 28, 2003
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Sunday, July 27, 2003
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Saturday, July 26, 2003
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Friday, July 25, 2003
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Thursday, July 24, 2003
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Wednesday, July 23, 2003
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Tuesday, July 22, 2003
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Monday, July 21, 2003
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Friday, July 18, 2003
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Wednesday, July 16, 2003
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Sunday, July 13, 2003
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Sunday, July 13, 2003
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Saturday, July 12, 2003
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Friday, July 11, 2003
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Friday, July 11, 2003
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Thursday, July 10, 2003
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Wednesday, July 09, 2003
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Monday, July 07, 2003
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Monday, July 07, 2003
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Sunday, July 06, 2003
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Saturday, July 05, 2003
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Friday, July 04, 2003
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Thursday, July 03, 2003
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Wednesday, July 02, 2003
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Tuesday, July 01, 2003
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Thursday, August 28, 2003
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Wednesday, August 27, 2003
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Monday, August 25, 2003
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Sunday, August 24, 2003
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Saturday, August 23, 2003
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Friday, August 22, 2003
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Tuesday, August 19, 2003
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Monday, August 18, 2003
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Sunday, August 17, 2003
Saturday, August 16, 2003
Tuesday, August 12, 2003
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Saturday, August 09, 2003
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Friday, August 08, 2003
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Thursday, August 07, 2003
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Wednesday, August 06, 2003
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Tuesday, August 05, 2003
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Monday, August 04, 2003
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Sunday, August 03, 2003
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Friday, August 01, 2003
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Friday, August 01, 2003
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Tuesday, September 30, 2003
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Monday, September 29, 2003
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Sunday, September 28, 2003
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Friday, September 26, 2003
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Thursday, September 25, 2003
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Wednesday, September 24, 2003
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Tuesday, September 23, 2003
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Monday, September 22, 2003
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Thursday, September 18, 2003
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Wednesday, September 17, 2003
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Monday, September 15, 2003
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Sunday, September 14, 2003
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Saturday, September 13, 2003
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Friday, September 12, 2003
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Thursday, September 11, 2003
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Wednesday, September 10, 2003
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Tuesday, September 09, 2003
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Tuesday, September 09, 2003
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Monday, September 08, 2003
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Monday, September 08, 2003
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Sunday, September 07, 2003
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Friday, September 05, 2003
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Friday, September 05, 2003
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Thursday, September 04, 2003
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Wednesday, September 03, 2003
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Tuesday, September 02, 2003
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Friday, October 31, 2003
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Wednesday, October 29, 2003
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Tuesday, October 28, 2003
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Monday, October 27, 2003
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Sunday, October 26, 2003
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Saturday, October 25, 2003
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Saturday, October 25, 2003
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Thursday, October 23, 2003
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Tuesday, October 21, 2003
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Saturday, October 18, 2003
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Friday, October 17, 2003
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Thursday, October 16, 2003
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Wednesday, October 15, 2003
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Tuesday, October 14, 2003
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Sunday, October 12, 2003
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Saturday, October 11, 2003
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Friday, October 10, 2003
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Friday, October 10, 2003
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Wednesday, October 08, 2003
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Tuesday, October 07, 2003
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Sunday, October 05, 2003
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Saturday, October 04, 2003
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Friday, October 03, 2003
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Thursday, October 02, 2003
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Sunday, November 30, 2003
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Saturday, November 29, 2003
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Friday, November 28, 2003
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Tuesday, November 25, 2003
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Tuesday, November 25, 2003
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Monday, November 24, 2003
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Sunday, November 23, 2003
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Saturday, November 22, 2003
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Friday, November 21, 2003
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Thursday, November 20, 2003
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Wednesday, November 19, 2003
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Monday, November 17, 2003
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Sunday, November 16, 2003
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Saturday, November 15, 2003
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Friday, November 14, 2003
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Thursday, November 13, 2003
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Thursday, November 13, 2003
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Wednesday, November 12, 2003
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Tuesday, November 11, 2003
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Sunday, November 09, 2003
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Friday, November 07, 2003
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Thursday, November 06, 2003
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Wednesday, November 05, 2003
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Wednesday, November 05, 2003
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Tuesday, November 04, 2003
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Monday, November 03, 2003
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Wednesday, December 31, 2003
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Saturday, December 27, 2003
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Friday, December 26, 2003
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Monday, December 22, 2003
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Monday, December 22, 2003
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Sunday, December 21, 2003
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Saturday, December 20, 2003
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Friday, December 19, 2003
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Wednesday, December 17, 2003
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Tuesday, December 16, 2003
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Sunday, December 14, 2003
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Saturday, December 13, 2003
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Thursday, December 11, 2003
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Wednesday, December 10, 2003
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Tuesday, December 09, 2003
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Sunday, December 07, 2003
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Sunday, December 07, 2003
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Friday, December 05, 2003
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Saturday, January 31, 2004
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Thursday, January 29, 2004
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Wednesday, January 28, 2004
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Monday, January 26, 2004
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Monday, January 26, 2004
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Saturday, January 24, 2004
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Thursday, January 22, 2004
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Tuesday, January 20, 2004
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Sunday, January 18, 2004
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Saturday, January 17, 2004
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Friday, January 16, 2004
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Wednesday, January 14, 2004
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Wednesday, January 14, 2004
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Tuesday, January 13, 2004
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Sunday, January 11, 2004
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Sunday, January 11, 2004
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Friday, January 09, 2004
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Thursday, January 08, 2004
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Monday, January 05, 2004
Sunday, January 04, 2004
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Sunday, January 04, 2004
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Friday, January 02, 2004
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Thursday, February 26, 2004
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Wednesday, February 25, 2004
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Tuesday, February 24, 2004
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Thursday, February 12, 2004
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Wednesday, February 11, 2004
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Tuesday, February 10, 2004
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Friday, February 06, 2004
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Thursday, February 05, 2004
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Monday, February 02, 2004
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Wednesday, March 31, 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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Monday, March 22, 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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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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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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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
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Saturday, May 08, 2004
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Wednesday, May 05, 2004
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Monday, May 03, 2004
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Saturday, May 01, 2004
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Saturday, May 01, 2004
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Tuesday, June 29, 2004
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Sunday, June 27, 2004
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Friday, June 25, 2004
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Tuesday, June 22, 2004
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Thursday, June 17, 2004
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Wednesday, June 16, 2004
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Tuesday, June 15, 2004
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Friday, June 11, 2004
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Tuesday, June 08, 2004
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Sunday, June 06, 2004
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Sunday, June 06, 2004
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Tuesday, June 01, 2004
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Saturday, July 31, 2004
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Thursday, July 29, 2004
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Wednesday, July 28, 2004
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Tuesday, July 27, 2004
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Monday, July 26, 2004
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Sunday, July 25, 2004
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Saturday, July 24, 2004
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Friday, July 23, 2004
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Sunday, July 18, 2004
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Saturday, July 17, 2004
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Thursday, July 15, 2004
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Monday, July 12, 2004
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Sunday, July 11, 2004
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Friday, July 09, 2004
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Thursday, July 08, 2004
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Thursday, July 08, 2004
Wednesday, July 07, 2004
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Tuesday, August 31, 2004
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Monday, August 30, 2004
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Sunday, August 29, 2004
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Sunday, August 22, 2004
Thursday, August 19, 2004
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Wednesday, August 18, 2004
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Tuesday, August 17, 2004
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Monday, August 16, 2004
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Thursday, August 12, 2004
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Wednesday, August 11, 2004
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Monday, August 09, 2004
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Monday, August 09, 2004
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Friday, August 06, 2004
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Tuesday, August 03, 2004
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Monday, August 02, 2004
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Tuesday, September 14, 2004
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Monday, September 13, 2004
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Friday, September 10, 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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Sunday, October 24, 2004
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Friday, October 01, 2004
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Friday, November 26, 2004
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Wednesday, November 24, 2004
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Sunday, November 21, 2004
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Friday, December 03, 2004
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Wednesday, January 26, 2005
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Tuesday, January 18, 2005
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Wednesday, May 18, 2005
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Monday, July 18, 2005
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Wednesday, July 06, 2005
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Thursday, September 22, 2005
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Wednesday, September 21, 2005
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Monday, September 19, 2005
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Sunday, September 18, 2005
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Saturday, September 17, 2005
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Thursday, September 15, 2005
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Wednesday, September 14, 2005
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Wednesday, September 14, 2005
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Tuesday, September 13, 2005
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Monday, September 12, 2005
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Saturday, September 10, 2005
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Saturday, September 10, 2005
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Sunday, September 04, 2005
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Friday, September 02, 2005
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Monday, October 31, 2005
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Sunday, October 30, 2005
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Saturday, October 29, 2005
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Saturday, October 29, 2005
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Friday, October 28, 2005
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Thursday, October 27, 2005
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Wednesday, October 26, 2005
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Tuesday, October 25, 2005
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Saturday, October 22, 2005
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Thursday, October 20, 2005
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Wednesday, October 19, 2005
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Tuesday, October 18, 2005
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Monday, October 17, 2005
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Sunday, October 16, 2005
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Saturday, October 15, 2005
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Friday, October 14, 2005
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Thursday, October 13, 2005
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Wednesday, October 12, 2005
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Tuesday, October 11, 2005
Tuesday, October 11, 2005
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Monday, October 10, 2005
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Sunday, October 09, 2005
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Saturday, October 08, 2005
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Saturday, October 08, 2005
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Friday, October 07, 2005
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Thursday, October 06, 2005
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Wednesday, October 05, 2005
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Monday, October 03, 2005
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Sunday, October 02, 2005
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Wednesday, November 30, 2005
397658830 8:20 PM
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Tuesday, November 29, 2005
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Monday, November 28, 2005
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Friday, November 25, 2005
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Thursday, November 24, 2005
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Thursday, November 24, 2005
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Wednesday, November 23, 2005
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Tuesday, November 22, 2005
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Monday, November 21, 2005
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Sunday, November 20, 2005
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Friday, November 18, 2005
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Thursday, November 17, 2005
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Thursday, November 17, 2005
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Wednesday, November 16, 2005
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Tuesday, November 15, 2005
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Monday, November 14, 2005
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Sunday, November 13, 2005
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Saturday, November 12, 2005
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<a
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March 22, 2006</a><br>
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Tuesday, November 28, 2006
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<span>
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