"I need a photo opportunity…" —Paul Simon
Tuesday, February 21, 2012
Photo Opportunity
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Monday, February 20, 2012
Coxeter and the Relativity Problem
In the Beginning…
"As is well known, the Aleph is the first letter of the Hebrew alphabet."
– Borges, "The Aleph" (1945)
From some 1949 remarks of Weyl—
"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."
— Hermann Weyl, "Relativity Theory as a Stimulus in Mathematical Research," Proceedings of the American Philosophical Society , Vol. 93, No. 7, Theory of Relativity in Contemporary Science: Papers Read at the Celebration of the Seventieth Birthday of Professor Albert Einstein in Princeton, March 19, 1949 (Dec. 30, 1949), pp. 535-541
Weyl in 1946—:
"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."
— Hermann Weyl, The Classical Groups , Princeton University Press, 1946, p. 16
Coxeter in 1950 described the elements of the Galois field GF(9) as powers of a primitive root and as ordered pairs of the field of residue-classes modulo 3—
"… the successive powers of the primitive root λ or 10 are
λ = 10, λ2 = 21, λ3 = 22, λ4 = 02,
λ5 = 20, λ6 = 12, λ7 = 11, λ8 = 01.
These are the proper coordinate symbols….
(See Fig. 10, where the points are represented in the Euclidean plane as if the coordinate residue 2 were the ordinary number -1. This representation naturally obscures the collinearity of such points as λ4, λ5, λ7.)"
Coxeter's Figure 10 yields...
The Aleph
The details:
Coxeter's phrase "in the Euclidean plane" obscures the noncontinuous nature of the transformations that are automorphisms of the above linear 2-space over GF(3).
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Saturday, February 18, 2012
Symmetry
From the current Wikipedia article "Symmetry (physics)"—
"In physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are 'unchanged', according to a particular observation. A symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is 'preserved' under some change.
A family of particular transformations may be continuous (such as rotation of a circle) or discrete (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group)."….
"A discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance."
Note the confusion here between continuous (or discontinuous) transformations and "continuous" (or "discontinuous," i.e. "discrete") groups .
This confusion may impede efforts to think clearly about some pure mathematics related to current physics— in particular, about the geometry of spaces made up of individual units ("points") that are not joined together in a continuous manifold.
For an attempt to forestall such confusion, see Noncontinuous Groups.
For related material, see Erlanger and Galois as well as the opening paragraphs of Diamond Theory—
Symmetry is often described as invariance under a group of transformations. An unspoken assumption about symmetry in Euclidean 3-space is that the transformations involved are continuous.
Diamond theory rejects this assumption, and in so doing reveals that Euclidean symmetry may itself be invariant under rather interesting groups of non-continuous (and a-symmetric) transformations. (These might be called noncontinuous groups, as opposed to so-called discontinuous (or discrete ) symmetry groups. See Weyl's Symmetry .)
For example, the affine group A on the 4-space over the 2-element field has a natural noncontinuous and asymmetric but symmetry-preserving action on the elements of a 4×4 array. (Details)
(Version first archived on March 27, 2002)
Update of Sunday, February 19, 2012—
The abuse of language by the anonymous authors
of the above Wikipedia article occurs also in more
reputable sources. For instance—
Some transformations referred to by Brading and Castellani
and their editees as "discrete symmetries" are, in fact, as
linear transformations of continuous spaces, themselves
continuous transformations.
This unfortunate abuse of language is at least made explicit
in a 2003 text, Mathematical Perspectives on Theoretical
Physics (Nirmala Prakash, Imperial College Press)—
"… associated[*] with any given symmetry there always exists
a continuous or a discrete group of transformations….
A symmetry whose associated group is continuous (discrete)
is called a continuous (discrete ) symmetry ." — Pp. 235, 236
[* Associated how?]
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Logo
Pentagram design agency on the new Windows 8 logo—
"… the logo re-imagines the familiar four-color symbol
as a modern geometric shape"—
Sam Moreau, Principal Director of User Experience for Windows,
yesterday—
On Redesigning the Windows Logo—
"To see what is in front of one's nose
needs a constant struggle." —George Orwell
That is the feeling we had when Paula Scher
(from the renowned Pentagram design agency)
showed us her sketches for the new Windows logo.
Related material:
- A Four-Color Theorem,
- The Galois Window, and
-
An illustration from
Finite Geometry of the Square and Cube—
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Friday, February 17, 2012
Pregeometry and Finite Geometry
Today's previous post, on the Feb. 2012 Scientific American
article "Is Space Digital?", suggested a review of a notion
that the theoretical physicist John Archibald Wheeler called
pregeometry .
From a paper on that topic—
"… the idea that geometry should constitute
'the magic building material of the universe'
had to collapse on behalf of what Wheeler
has called pregeometry (see Misner et al. 1973,
pp. 1203-1212; Wheeler 1980), a somewhat
indefinite term which expresses “a combination
of hope and need, of philosophy and physics
and mathematics and logic” (Misner et al. 1973,
p. 1203)."
— Jacques Demaret, Michael Heller, and
Dominique Lambert, "Local and Global Properties
of the World," preprint of paper published in
Foundations of Science 2 (1): 137-176
Misner, C. W., Thorne, K. S. and Wheeler, J. A.
1973, Gravitation , W.H. Freeman and Company:
San Francisco.
Wheeler, J.A. 1980, "Pregeometry: Motivations
and Prospects," in: Quantum Theory and Gravitation ,
ed. A.R. Marlow, Academic Press: New York, pp. 1-11.
Some related material from pure mathematics—
Click image for further details.
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Physics vs. Geometry
Physics
The February 2012 issue of Scientific American
has a cover article titled "Is Space Digital?".
The article discusses whether physical space
"is made of chunks. Blocks. Bits."
Maybe it is, maybe it isn't.
Geometry
The word "space" in pure mathematics
(as opposed to physics) applies to
a great variety of structures.
Some are continuous, some are not.
For some purely mathematical structures
that are not continuous, (i.e., are made of
"chunks, blocks, bits") see finitegeometry.org/sc —
in particular, the pages on Finite Geometry and Physical Space
and on Noncontinuous Groups.
The geometry of these structures may or may not eventually
be relevant to the "21st-century physics" discussed
in the February Scientific American.
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Thursday, February 16, 2012
Truth and Fiction
Readings—
- Facts Are Stupid (Dan Kois at Slate.com yesterday)
- Tim O'Brien Quotes (cached from Goodreads.com)
- The Story Theory of Truth
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Midnight in the Garden
(Continued from February 10.)
A passage suggested by the T.S. Eliot epigraphs in
Parallelisms of Compete Designs , by a weblog post
of Peter J. Cameron yesterday, and by this journal's
"Within You Without You" posts—
— Joseph Campbell, The Inner Reaches of Outer Space:
Metaphor as Myth and as Religion , New World Library,
Second Edition, St. Bridget's Day 2002, page 106
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Wednesday, February 15, 2012
A Job for St. Valentine
Maureen Dowd's NY Times column today is on exorcism.
Related material— This morning's update at the end of
yesterday morning's Valentine's Day post Notable Transitions.
See also another post for St. Valentine — The Ninth Configuration.
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Tuesday, February 14, 2012
The Ninth Configuration
The showmanship of Nicki Minaj at Sunday's
Grammy Awards suggested the above title,
that of a novel by the author of The Exorcist .
The Ninth Configuration —
The ninth* in a list of configurations—
"There is a (2d-1)d configuration
known as the Cox configuration."
— MathWorld article on "Configuration"
For further details on the Cox 326 configuration's Levi graph,
a model of the 64 vertices of the six-dimensional hypercube γ6 ,
see Coxeter, "Self-Dual Configurations and Regular Graphs,"
Bull. Amer. Math. Soc. Vol. 56, pages 413-455, 1950.
This contains a discussion of Kummer's 166 as it
relates to γ6 , another form of the 4×4×4 Galois cube.
See also Solomon's Cube.
* Or tenth, if the fleeting reference to 113 configurations is counted as the seventh—
and then the ninth would be a 153 and some related material would be Inscapes.
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Notable Transitions
This morning's New York Times gives a folklorist's
view of The Great Gatsby—
"Daisy Buchanan, he argued in a 1960 article,
is a Jazz Age incarnation of the beautiful,
seductive Fairy Queen of Celtic lore."
— Margalit Fox, obituary of Tristram P. Coffin,
who died at 89 on January 31st, 2012
See also…
Two screenshots in memory of fashion and fine-art photographer
Lillian Bassman, who died yesterday at 94—
Update of 10:10 AM EST Wed., Feb. 15, 2012…
In memory of Dory Previn, a song for "Hanna" and "Lord of the Rings" star Cate Blanchett.
Previn died yesterday, on Valentine's Day. Perhaps an inspiration for a lyric by Leonard Cohen?
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Monday, February 13, 2012
Why Now?
From the 2011 film "Hanna"—
Marissa: Why now, Erik?
Erik: Kids grow up.
From the 50th Anniversary Edition
(January 31, 2012) of A Wrinkle in Time—
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Ms. Lincoln
Suggested by a USA Today story on last night's Grammy ratings,
by the showmanship of Nicki Minaj, and by…
"Lincoln was one of many singers influenced by Billie Holiday." —Wikipedia
Related material—
- St. Bridget's Day, 2012 (a Roman Catholic holiday)
- Egress (in memory of Etta James)
- Feast of the Assumption, 2010 (another Roman Catholic holiday)
"Apart from that, Ms. Lincoln…"
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Sunday, February 12, 2012
Rhetorical Answer
NOW ENJOY BRILLIANT COLLEGE COURSES
IN YOUR HOME OR CAR!
The sun was burning down….
There was a trembling in the air as the unnamed colors
and landforms took on definition, a clarity of outline and extent….
This is where we sat through his hushed hour, a torchlit sky,
the closeness of hills barely visible at high white noon.
— DeLillo, Don, Point Omega
Midi là-haut, Midi sans mouvement
En soi se pense et convient à soi-même…
Tête complète et parfait diadème,
Je suis en toi le secret changement.
— Valéry, Paul, "Le Cimetière Marin"
… Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte.
— Darío, Rubén, "Los Tres Reyes Magos"
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Saturday, February 11, 2012
Sports
"News and Traffic. Sports and Weather. These were his acid terms
for the life he’d left behind, more than two years of living with
the tight minds that made the war. It was all background noise,
he said, waving a hand. He liked to wave a hand in dismissal."
— DeLillo, Don (2010-02-02), Point Omega
Send in the Clowns. (Click to enlarge.)
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Grids
See Notes for a Haiku.
Related material—
A novel published on Groundhog Day, 2010—
— as well as Conceptual Art, Josefine Lyche's
"Grids, You Say?" and The Speed of Thought.
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Friday, February 10, 2012
24 Hour Psycho
From "Kill Bill: Vol. 1"—
The Bride: [Japanese] I need Japanese steel.
Related material —
Yodogawa, 1982 (see yesterday evening's Psycho) and…
See also this afternoon's 5:01 post.
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Pensée
Wednesday, February 1, 2012
Politics
|
Related material—
See also At the Still Point (a post in memory of film editor Sally Menke).
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Garden of Good and Evil
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Thursday, February 9, 2012
ART WARS continued
On the Complexity of Combat—
The above article (see original pdf), clearly of more
theoretical than practical interest, uses the concept
of "symmetropy" developed by some Japanese
researchers.
For some background from finite geometry, see
Symmetry of Walsh Functions. For related posts
in this journal, see Smallest Perfect Universe.
Update of 7:00 PM EST Feb. 9, 2012—
Background on Walsh-function symmetry in 1982—
(Click image to enlarge. See also original pdf.)
Note the somewhat confusing resemblance to
a four-color decomposition theorem
used in the proof of the diamond theorem.
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Wednesday, February 8, 2012
Beach Boy
(Continued from March 28, 2006, and February 6, 2012)
Paris—
Sylvia Beach and James Joyce at Shakespeare and Company
See also Walking into Eternity.
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Old Sport
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Lichtung!
(Continued from Dec. 5, 2002)
From Bad…
|
Braucht´s noch Text? |
To Verse—
|
manche meinen |
|
by Ernst Jandl |
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Tuesday, February 7, 2012
Zip Enter Get a Quote
![IMAGE- Top of NY Times obits page, morning of Feb. 7th, 2012, with ad saying ZIP [ENTER] GET A QUOTE](http://www.log24.com/log/pix12/120207-DarknessSearch.jpg)
Okay… http://m759.net/wordpress/?p=16365.
See also last night's 11:48 post and Erin Burnett in "Glad Rags."
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Random Walk
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Barcelona
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Monday, February 6, 2012
Straight Talk
Tonight's TV, in brief—
AUDITION:
At 10, Smash. At 11, Erin Burnett OutFront (repeat).
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Beach Boy
| Cached from artnet.com |
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In memory of a dealer in artists' ephemera,
Steven Leiber, who died on January 28, 2012—
a link to a post from the date of Leiber's death—
See also Me and My Shadow, a post from
the date the above photo was offered for sale.
Related ephemeral art— a post titled, with irony,
Introduction to Harmonic Analysis.
Related non -ephemeral art—
Mathematical Imagery.
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Savage Logic
|
Saturday, October 16, 2010 m759 @ 12:00 PM Brightness at Noon continued – Sir William Rowan Hamilton, See also this journal on 1/09, 2010. |
This post was suggested by the date
of a user comment in Wikipedia.
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Sunday, February 5, 2012
Sermon Highlight
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Sunday Morning
CBS Sunday Morning 's piece on
the number pi today suggested…
Hexagram 20, Contemplation/View,
from the website Rightreading.com
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Saturday, February 4, 2012
Ein Satz
|
4 × 5 = ̲2̲0̲ . Related material— The link "Ten is a Hen" "She's told a detective story without violence, Judy Davis, shown above in |
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Cock Tale (continued)
The late Ben Gazzara as Jackie Treehorn
in "The Big Lebowski"—
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Office Visit
From the screenplay of "The Number 23"—
INT. NATHANIEL'S INSTITUTE, STAIRWELL – NIGHT
Agatha climbs a dark staircase. Layers of dust
testify to years of neglect.
INT. 3RD FLOOR CORRIDOR – CONTINUOUS
Agatha finds ROOM 318. A rusting door plaque reads,
"DR. SIRIUS LEARY, M.D. PSYCHIATRY."
For related material, see "Leary + Cuernavaca" and "Prime Cut."
Happy belated 2/3 birthday to Walter Sparrow.
Related material— Two other occurrences of "318" in this journal—
in another horror story, "The Sweet Smell of Avon,"
and in a quote from the Feast of St. Nicholas, 2010—
"When Novelists Become Cubists," by Andre Furlani—
"A symbol comes into being when an artist sees that
it is the only way to get all the meaning in.
Genius always proceeds by faith" (312).
The unparaphrasable architectonic text
"differs from other narrative in that the meaning
shapes into a web, or globe, rather than along a line" (318).
[The references are to page numbers in
Guy Davenport's The Geography of the Imagination .]
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Friday, February 3, 2012
Souvenir
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Unusual Suspects
"… 'Mes chers frères, n'oubliez jamais,
quand vous entendrez vanter le progrès des lumières,
que la plus belle des ruses du diable
est de vous persuader qu'il n'existe pas!'
Le souvenir de ce célèbre orateur
nous conduisit naturellement vers le sujet des académies,
et mon étrange convive m'affirma qu'il ne dédaignait pas,
en beaucoup de cas, d'inspirer la plume, la parole et la conscience
des pédagogues, et qu'il assistait presque toujours en personne,
quoique invisible, à toutes les séances académiques."
— Baudelaire, "Le Joueur Généreux"
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Thursday, February 2, 2012
Rien de Rien*
In memory of the longtime co-editor of Series A of the
Journal of Combinatorial Theory , who died on January 12th,
2012, here is a link to a Log24 post of that date.
* For the title, see January 19th.
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Die Nichtung
"It seems that Hilbert had no taste for philosophers….
Hans Rademacher told this reviewer that, after Heidegger
once lectured in Göttingen, Hilbert gleefully repeated
to everyone the phrase "Das Nichts nichtet die Nichtung ."
— Gian-Carlo Rota, Discrete Thoughts , 2nd ed., p. 233
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Die Lichtung
See January 4th, 2012.
(This link resulted from an application of Heidegger's
philosophy of "the opening" and "the shining" (Die Lichtung ).)
See also The Shining of May 29.
Update of 12:19 AM Feb. 3, 2012—
The undated (but cached by Google on January 4th, 2012)
unsigned post from a deleted weblog linked to above as
"an application" is also available in a version that is signed
(but still undated).
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The Opening
From ShiftLock in this journal—
"Philosophy knows nothing of the opening."
— Heidegger
See also a post of September 25, 2009,
and a film whose opening was on that date.
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Wednesday, February 1, 2012
Politics
"Should we arbitrate life and death
at a round table or a square one?"
See also the two previous posts,
Disturbing Archimedes and Tesseract.
Update—
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Disturbing Archimedes
Princeton University Press on a book it will publish in March—
Circles Disturbed: The Interplay of
Mathematics and Narrative
"… brings together important thinkers in mathematics,
history, and philosophy to explore the relationship
between mathematics and narrative.
The book's title recalls the last words of the great
Greek mathematician Archimedes before he was
slain by a Roman soldier— 'Don't disturb my circles' …."
Related material—
Archimedes's Vicious Circle
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Tuesday, January 31, 2012
Tesseract
|
"… a finite set with n elements Tesseract formed from a 4-set—
The same 16 subsets or points can
"There is such a thing as a 4-set." |
Update of August 12, 2012:
Figures like the above, with adjacent vertices differing in only one coordinate,
appear in a 1950 paper of H. S. M. Coxeter—
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Monday, January 30, 2012
Enda’s Game*
The following passage by Tolkien was suggested by a copy of next Sunday's New York Times Book Review that arrived in the mail today. (See Orson Scott Card's remarks on page 26— "Uncle Orson"— and the Review 's concluding essay "Grand Allusion.")
"Lastly, tengwesta [system or code of signs] has also become an impediment. It is in Incarnates clearer and more precise than their direct reception of thought. By it also they can communicate easily with others, when no strength is added to their thought: as, for example, when strangers first meet. And, as we have seen, the use of 'language' soon becomes habitual, so that the practice of ósanwe (interchange of thought) is neglected and becomes more difficult. Thus we see that the Incarnate tend more and more to use or to endeavour to use ósanwe only in great need and urgency, and especially when lambe is unavailing. As when the voice cannot be heard, which comes most often because of distance. For distance in itself offers no impediment whatever to ósanwe . But those who by affinity might well use ósanwe will use lambe when in proximity, by habit or preference. Yet we may mark also how the 'affine' may more quickly understand the lambe that they use between them, and indeed all that they would say is not put into words. With fewer words they come swifter to a better understanding. There can be no doubt that here ósanwe is also often taking place; for the will to converse in lambe is a will to communicate thought, and lays the minds open. It may be, of course, that the two that converse know already part of the matter and the thought of the other upon it, so that only allusions dark to the stranger need be made; but this is not always so. The affine** will reach an understanding more swiftly than strangers upon matters that neither have before discussed, and they will more quickly perceive the import of words that, however numerous, well-chosen, and precise, must remain inadequate."
* "If a poem catches a student's interest at all, he or she should damned well be able to look up an unfamiliar word in the dictionary…."
— Elizabeth Bishop, quoted in the essay "Grand Allusion" mentioned above. For a brief dictionary of most of the unfamiliar words in this post's title and in the above passage, see Vinyar Tengwar 39 (July 1998). This is copyrighted but freely available on the Web.
** The word "affine" has connotations not intended by Tolkien. See that word in this journal. See also page 5 of next Sunday's Times Book Review , which contains a full-page ad for the 50th anniversary edition of A Wrinkle in Time . "There is such a thing as a tesseract."
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Sunday, January 29, 2012
Declarations
Weblog posts of two prominent mathematicians today discussed
what appears to be a revolution inspired by the business practices
of some commercial publishers of mathematics.
My own concern is more with the so-called "Non-Euclidean Revolution"
described by Richard Trudeau in a book of that title (Birkhäuser, 1987).
A 1976 document relevant to the concerns in the Trudeau book—
Though not as well known as another document discussing
"self-evident" truths, Cameron's remarks are also of some
philosophical interest.
They apply to finite geometry, a topic unknown to Euclid,
but nevertheless of considerable significance for the foundations
of mathematics.
"The hand of the creative artist, laid upon the major premise,
rocks the foundations of the world." — Dorothy Sayers
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Sunday School
Beach with Palms
The following images were suggested by the
"other things" link in yesterday's "287501346" post.
From June 10, 2008—
Two frames from the Jodie Foster film "Contact"—
 |
 |
See Storyline and Time Fold.
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Saturday, January 28, 2012
The Sweet Smell of Avon
The twin topics of autism and of narrowing definitions
suggested the following remarks.
The mystical number "318" in the pilot episode
of Kiefer Sutherland's new series about autism, "Touch,"
is so small that it can easily apply (as the pilot
illustrated) to many different things: a date, a
time, a bus number, an address, etc.
The last 3/18 Log24 post— Defining Configurations—
led, after a false start and some further research,
to the writing of the webpage Configurations and Squares.
An image from that page—
Interpreting this, in an autistic manner, as the number
287501346 lets us search for more specific items
than those labeled simply 318.
The search yields, among other things, an offer of
Night Magic Cologne (unsold)—
For further mystery and magic, see, from the date
the Night Magic offer closed— May 8, 2010— "A Better Story."
See also the next day's followup, "The Ninth Gate."
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Friday, January 27, 2012
Narrative
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Mathematics and Narrative (continued)
Princeton University Press on a book it will publish in March—
Circles Disturbed: The Interplay of Mathematics and Narrative
"Circles Disturbed brings together important thinkers in mathematics, history, and philosophy to explore the relationship between mathematics and narrative. The book's title recalls the last words of the great Greek mathematician Archimedes before he was slain by a Roman soldier— 'Don't disturb my circles'— words that seem to refer to two radically different concerns: that of the practical person living in the concrete world of reality, and that of the theoretician lost in a world of abstraction. Stories and theorems are, in a sense, the natural languages of these two worlds–stories representing the way we act and interact, and theorems giving us pure thought, distilled from the hustle and bustle of reality. Yet, though the voices of stories and theorems seem totally different, they share profound connections and similarities."
Timeline of the Marvel Cinematic Universe — Norway, March 1942—
"The Red Skull finds the Tesseract, a cube of strange power,
said to be the jewel of Odin’s treasure room, in Tonsberg Norway.
(Captain America: The First Avenger)"
Tesseracts Disturbed — (Click to enlarge)
Detail of Tesseracts Disturbed —
Narrative of the detail—
See Tesseract in this journal and Norway, May 2010—
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Thursday, January 26, 2012
Change Illustrated
"Something is happening to our town."
— Mayor of Pleasantville
Related material— Alicia Keys's birthday,
the ending of Midnight in Paris , and Rocket Billie.
See also a Sinatra song uploaded on this date
(Paul Newman's birthday) last year.
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Found in Space
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Lamedvavnik
Dick Tufeld, Robot Voice in TV’s ‘Lost in Space,’ Dies at 85
Wed Jan 25, 2012 23:42 from NYT Obituaries By Bruce Weber
"Mr. Tufeld possessed one of Hollywood’s most often-heard
disembodied voices, especially from the 1950s through the 1970s."
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Wednesday, January 25, 2012
A Larger City
| BOOKS OF THE TIMES
HOW IT ALL BEGAN Review by Michiko Kakutani As a historian, Henry acknowledges that he has “a soft spot for what is known as the Cleopatra’s nose theory of history— the proposal that had the nose of Cleopatra been an inch longer, the fortunes of Rome would have been different.” It’s a bit of a reductio ad absurdum, he admits, but nonetheless “a reference to random causality that makes a lot of sense when we think about the erratic sequence of events that we call history.” What Ms. Lively has done in this captivating volume is to use all her copious storytelling gifts to show how a similar kind of random causality rules individual lives, how one unlucky event can set off unexpected chain reactions, how the so-called butterfly effect— whereby the flapping of a tiny butterfly’s wings can supposedly lead to a huge storm elsewhere in the world— ripples through the ebb and flow of daily life. |
Rhetorical question—
"Why walk when you can fly?"
— Mary Chapin Carpenter
Rhetorical answer—
Two excerpts from a webpage on random walks—
A drunk man will find his way home,
but a drunk bird may get lost forever.
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Tuesday, January 24, 2012
The Infinity Point
From Labyrinth of the Line (March 2, 2011)—
"… construct the Golay code by taking the 24 points
to be the points of the projective line F23 ∪ {∞}…."
— Robert A. Wilson
A simpler projective line— a Galois geometry
model of the line F2 ∪ {∞}—
Here we may consider ∞ to be modeled*
by the third square above— the Galois window .
* Update of about 1 AM Jan. 25, 2012—
This infinity-modeling is of course a poetic conceit,
not to be taken too seriously. For a serious
discussion of points at infinity and finite fields,
see (for instance) Daniel Bump's "The Group GL(2)."
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The Screwing
"Debates about canonicity have been raging in my field
(literary studies) for as long as the field has been
around. Who's in? Who's out? How do we decide?"
— Stephen Ramsay, "The Hermeneutics of Screwing Around"
An example of canonicity in geometry—
"There are eight heptads of 7 mutually azygetic screws, each consisting of the screws having a fixed subscript (from 0 to 7) in common. The transformations of LF(4,2) correspond in a one-to-one manner with the even permutations on these heptads, and this establishes the isomorphism of LF(4,2) and A8. The 35 lines in S3 correspond uniquely to the separations of the eight heptads into two complementary sets of 4…."
— J.S. Frame, 1955 review of a 1954 paper by W.L. Edge,
"The Geometry of the Linear Fractional Group LF(4,2)"
Thanks for the Ramsay link are due to Stanley Fish
(last evening's online New York Times ).
For further details, see The Galois Tesseract.
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Monday, January 23, 2012
How It Works
J. H. Conway in 1971 discussed the role of an elementary abelian group
of order 16 in the Mathieu group M24. His approach at that time was
purely algebraic, not geometric—
For earlier (and later) discussions of the geometry (not the algebra )
of that order-16 group (i.e., the group of translations of the affine space
of 4 dimensions over the 2-element field), see The Galois Tesseract.
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How Stuff Works
"Design is how it works." —Steve Jobs
Website logo—
Screenshot from How Stuff Works—
(Click image for details.)
From "A Device Worthy of a Gothic Novel,"
Chapter XVI of The Club Dumas,
by Arturo Perez-Reverte (1993),
Vintage International, April 1998….
the basis of the 1999 Roman Polanski film
The Ninth Gate —
Aren't you going to give me a document to sign?"
"A document?"
"Yes. It used to be called a pact. Now it would be a contract
with lots of small print, wouldn't it? 'In the event of litigation,
the parties are to submit to the jurisdiction of the courts of…'
That's a funny thing. I wonder which court covers this."
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Labyrinth
"Yo sé de un laberinto griego que es una línea única, recta."
—Borges, "La Muerte y la Brújula"
"I know of one Greek labyrinth which is a single straight line."
—Borges, "Death and the Compass"
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Sunday, January 22, 2012
Souvenir*
From life's box of chocolates…
Happy birthday to Piper Laurie.
* Those who prefer their
souvenirs without sentiment
may consult the quaternions.
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Year of the Dragon
In China, the Year of the Dragon
has now begun. See Holy Field
in this journal.
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Pound Sign
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Egress
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Card Trick
An image suggested by a book at
Princeton University Press—
Click image for details.
See also a somewhat deeper book from Princeton.
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The Presbyterian Exorcist
(Backstory— Presbyterian in this journal)
Princeton University Press on a book it will publish in March—
Circles Disturbed brings together important thinkers in mathematics, history, and philosophy to explore the relationship between mathematics and narrative. The book's title recalls the last words of the great Greek mathematician Archimedes before he was slain by a Roman soldier–"Don't disturb my circles"–words that seem to refer to two radically different concerns: that of the practical person living in the concrete world of reality, and that of the theoretician lost in a world of abstraction. Stories and theorems are, in a sense, the natural languages of these two worlds–stories representing the way we act and interact, and theorems giving us pure thought, distilled from the hustle and bustle of reality. Yet, though the voices of stories and theorems seem totally different, they share profound connections and similarities.
Exercise— Discuss the above paragraph's vulgarity.
Discuss also the more robust vulgarity of Marvel Entertainment…
Context— "Marvel" in this journal, and The Cosmic Cube.
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Saturday, January 21, 2012
The Oxford Murders
Blame It on Trajan
Wikipedia on the 2008 film The Oxford Murders—
Christmas Eve image search
suggested by Stevens's phrase
"diamond globe."
(Larger version: 2 MB)
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Friday, January 20, 2012
Cock Tale
(Continued from November 12, 2005 and June 7, 2011)
Related material— "Labyrinth," a fiction
by the late Roberto Bolaño
in the current New Yorker —
"There's no photo credit."
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The Nothing That Is
"The 'one' with whom the reader has identified himself
has now become 'the listener, who listens in the snow';
he has become the snow man, and he knows winter
with a mind of winter, knows it in its strictest reality,
stripped of all imagination and human feeling.
But at that point when he sees the winter scene
reduced to absolute fact, as the object not of the mind,
but of the perfect perceptual eye that sees
'nothing that is not there,' then the scene,
devoid of its imaginative correspondences,
has become 'the nothing that is.'"
—Robert Pack, Wallace Stevens:
An Approach to His Poetry and Thought.
New Brunswick: Rutgers UP, 1958.
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Chess
| Saturday, November 12, 2005
— m759 @ 8:00 PM (continued)
“… problems are the poetry of chess. |
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Soul Souvenirs
"If the fault is with the soul, the sovereigns
Of the soul must likewise be at fault, and first.
If the fault is with the souvenirs, yet these
Are the soul itself. And the whole of the soul, Swenson,
As every man in Sweden will concede,
Still hankers after lions, or, to shift,
Still hankers after sovereign images."
— Wallace Stevens
"Friday night and the lights are low"
— ABBA
See also American Music Award.
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Brightness at Noon
See "harmonic analysis" in Mathematical Imagery and elsewhere in this journal.
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Poetry and Thought*
* Title courtesy of George Steiner.
For the "thought" part, see Plato's diamond
in last night's Mathematical Imagery.
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Thursday, January 19, 2012
Mathematical Imagery
From the Crafoord Prize website—
Related meta -mathematical image from Diamond Theory—
Mathematical image related to combinatorics—
See also permutahedron in this journal.
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Hegel
Those impressed by George Steiner's remark on Hegel in the previous post may consult…
(The Christian Examiner. Volume LXXX. New Series, Volume I. January, March, May, 1866.
New York: James Miller, Publisher, 522, Broadway. Boston: Walker, Fuller, & Co.
No. CCLIV, Art. IV.– THE SECRET OF HEGEL.
By C. C. Everett, pp. 196-207.
A review of…
The Secret of Hegel, being the Hegelian System in Origin, Principle, Form, and Matter.
By James Hutchinson Sterling. In two volumes.
London: Longman, Green, Longman, Roberts, & Green. 1865. 8vo, 2 vols.)
On Hegel, from the review—
"He starts not from the beginning, but from the heart, of the world.
There never was a time when this pure Being— which, in its
undivided absoluteness, is indistinguishable from nothing;
as pure, unbroken light is indistinguishable from darkness—
was by itself alone; but this absolute Being is yet the foundation
and the groundwork of whatever is."
For more on Hegel's logic, see Marxists.org.
See also Steiner on chess and Lenin in The New Yorker
(September 7, 1968, page 133).
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’Ceptions
Perception vs. Inception
Question—
"Where philosophy and literature mesh, where they are litigious toward one another in form or matter, these echoes of origin can be heard. The poetic genius of abstract thought is lit, is made audible. Argument, even analytic, has its drumbeat. It is made ode. What voices the closing movements of Hegel’s Phenomenology better than Edith Piaf’s non de non , a twofold negation which Hegel would have prized?
This essay is an attempt to listen more closely."
— George Steiner, The Poetry of Thought
Answer—
Edith Piaf's rien de rien . See also Is Nothing Sacred?
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Square Triangles
MathWorld.Wolfram.com has an article titled "Square-Triangle Theorem."
An article of my own, whose HTML title was previously "Triangles are Square," has been retitled accordingly.
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What Rough Beast
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Wednesday, January 18, 2012
Tuesday, January 17, 2012
Khora as Synchronicity
A search for khora + tao yields a paper on Derrida—
A check of the above date— Nov. 18, 2010— yields…
| Thursday, November 18, 2010
m759 @ 8:02 AM
Peter Woit has a post on Scientific American 's new Garrett Lisi article, "A Geometric Theory of Everything." The Scientific American subtitle is "Deep down, the particles and forces of the universe are a manifestation of exquisite geometry." See also Rhetoric (Nov. 4, 2010) and Exquisite Geometries (May 19, 2009). |
Related material on the temptation of physics
for a pure mathematician—
This morning's post on khora and Cardinal Manning, and,
from Hawking's birthday this year, Big Apple.
Within this post, by leading us to the apple,
Derrida as usual plays the role of Serpent.
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Manning and Khora
A weblog post from Saturday, Jan. 14, 2012—
"Today is the 120th anniversary of Cardinal Henry Edward Manning's death."
— A Reluctant Sinner (Thanks to Andrew Cusack for the link.)
If Manning is a saint, then Saturday was his feast day.
Some background— Manning in this journal.
See also Saturday's Derrida at Villanova. The link there to
previous posts on that topic leads to a post on Derrida's promotion
of his neologism différance as a version of Plato's khôra.
I prefer Manning's discussion of a closely related concept,
the scholastic philosophers' materia prima .
See Hugh R. King's 1956 paper sneering at the scholastics'
concept, and Heisenberg's much better-informed remarks
on the related concept of potentia —
For a related fictional account of a religious quest for "possibilities"
and "excluded middles" between "zeroes and ones," see
Ingraffia on The Crying of Lot 49 .
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Monday, January 16, 2012
Mapping Problem
Thursday's post Triangles Are Square posed the problem of
finding "natural" maps from the 16 subsquares of a 4×4 square
to the 16 equilateral subtriangles of an edge-4 equilateral triangle.
Here is a trial solution of the inverse problem—
Exercise— Devise a test for "naturality" of
such mappings and apply it to the above.
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Sunday, January 15, 2012
Sermon
"Accentuate the Positive."
— Clint Eastwood, soundtrack album for
"Midnight in the Garden of Good and Evil."
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Requiem Maas
Headline from yesterday evening's New York Times obituaries—
Frederica Sagor Maas, Silent-Era Scriptwriter, Dies at 111.
For Maas… Past Tense (Jan. 7, 2012) and its link to Dogma.
Related material—
Last night's SNL, this morning's Entertainment Break,
and — in the context of Dogma — Catholics Believe.
From an LA Times story about Maas on January 7—
"Many of the screenplays she and her husband wrote between 1938 and 1950 were never produced. Hopeless, humiliated and having little money, the couple drove to a hilltop overlooking Hollywood with the intention of committing suicide in their Plymouth. Clutching each other, they started sobbing and realized that 'none of these things mattered. We had each other,' wrote Maas…."
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Entertainment Break
Related material– Saturday night's Derrida at Villanova and Villanueva.
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Saturday, January 14, 2012
Derrida at Villanova
"As Derrida said at Villanova,
"We wait for something we would not like to wait for.
That is another name for death."
— Brian D. Ingraffia, "Is the Postmodern Post-Secular?,"
p. 50 in Postmodern Philosophy and Christian Thought ,
ed. by Merold Westphal, Indiana University Press, 1999, pp. 44-68
See also Derrida at Villanova in this journal.
The link to Ingraffia's remarks was suggested by
this evening's New York Times obituaries—
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Defining Form (continued)
Detail of Sylvie Donmoyer picture discussed
here on January 10—
The "13" tile may refer to the 13 symmetry axes
in the 3x3x3 Galois cube, or the corresponding
13 planes through the center in that cube. (See
this morning's post and Cubist Geometries.)
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Damnation Morning*
The following is adapted from a 2011 post—
* The title, that of a Fritz Leiber story, is suggested by
the above picture of the symmetry axes of the square.
Click "Continued" above for further details. See also
last Wednesday's Cuber.
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Friday, January 13, 2012
Mysteries of Faith
… he was already recognised as the ablest of the group of young English mathematicians who have been inspired by the genius of G H Hardy and J E Littlewood. In a group notable for its brilliant technique, no one had developed this technique to a higher degree than Paley. Nevertheless he should not be though of primarily as a technician, for with this ability he combined creative power of the first order. As he himself was wont to say, technique without 'rugger tactics' will not get one far, and these rugger tactics he practised to a degree that was characteristic of his forthright and vigorous nature.
The Telegraph today on British mystery author Reginald Hill—
"After National Service between 1955 and 1957,
he went up on a scholarship to St Catherine’s College, Oxford,
where he played rugby…."
Further details—
"Unsheathe your dagger definitions." — James Joyce
Some context— St. Catherine in this journal and her feast day last year.
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Logos
Click logos for related persons.
Background from this journal—
Collegiality, That Hideous Strength , and The Oxford Murders .
See also…
"The heart of the book is the conveying of a meaningful understanding
of where mathematical results originated…."
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Thursday, January 12, 2012
Collegiality in Action
"The transcripts of the 2006 meetings, released after a standard five-year delay,
clearly show some of the nation’s pre-eminent economic minds did not fully understand
the basic mechanics of the economy that they were charged with supervising."
— Binyamin Appelbaum in the Jan. 12 online New York Times
Academics may recall other examples of comfortably ignorant collegiality.
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Triangles Are Square
Coming across John H. Conway's 1991*
pinwheel triangle decomposition this morning—
|
|
— suggested a review of a triangle decomposition result from 1984:
Figure A
(Click the below image to enlarge.)
The above 1985 note immediately suggests a problem—
What mappings of a square with c 2 congruent parts
to a triangle with c 2 congruent parts are "natural"?**
(In Figure A above, whether the 322,560 natural transformations
of the 16-part square map in any natural way to transformations
of the 16-part triangle is not immediately apparent.)
* Communicated to Charles Radin in January 1991. The Conway
decomposition may, of course, have been discovered much earlier.
** Update of Jan. 18, 2012— For a trial solution to the inverse
problem, see the "Triangles are Square" page at finitegeometry.org.
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Wednesday, January 11, 2012
Cuber
"Examples galore of this feeling must have arisen in the minds of the people who extended the Magic Cube concept to other polyhedra, other dimensions, other ways of slicing. And once you have made or acquired a new 'cube'… you will want to know how to export a known algorithm , broken up into its fundamental operators , from a familiar cube. What is the essence of each operator? One senses a deep invariant lying somehow 'down underneath' it all, something that one can’t quite verbalize but that one recognizes so clearly and unmistakably in each new example, even though that example might violate some feature one had thought necessary up to that very moment. In fact, sometimes that violation is what makes you sure you’re seeing the same thing , because it reveals slippabilities you hadn’t sensed up till that time….
… example: There is clearly only one sensible 4 × 4 × 4 Magic Cube. It is the answer; it simply has the right spirit ."
— Douglas R. Hofstadter, 1985, Metamagical Themas: Questing for the Essence of Mind and Pattern (Kindle edition, locations 11557-11572)
See also Many Dimensions in this journal and Solomon's Cube.
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Language Game
Tension in the Common Room—
In memory of population geneticist James F. Crow,
who died at 95 on January 4th.
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Tuesday, January 10, 2012
Frye’s
"'Interpenetration'" — Stanley Fish in yesterday evening's online New York Times
"You want Frye's with that?" — A recent humanities graduate
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Defining Form
(Continued from Epiphany and from yesterday.)
Detail from the current American Mathematical Society homepage—
Further detail, with a comparison to Dürer’s magic square—
 |
 |
The three interpenetrating planes in the foreground of Donmoyer‘s picture
provide a clue to the structure of the the magic square array behind them.
Group the 16 elements of Donmoyer’s array into four 4-sets corresponding to the
four rows of Dürer’s square, and apply the 4-color decomposition theorem.
Note the symmetry of the set of 3 line diagrams that result.
Now consider the 4-sets 1-4, 5-8, 9-12, and 13-16, and note that these
occupy the same positions in the Donmoyer square that 4-sets of
like elements occupy in the diamond-puzzle figure below—
Thus the Donmoyer array also enjoys the structural symmetry,
invariant under 322,560 transformations, of the diamond-puzzle figure.
Just as the decomposition theorem’s interpenetrating lines explain the structure
of a 4×4 square , the foreground’s interpenetrating planes explain the structure
of a 2x2x2 cube .
For an application to theology, recall that interpenetration is a technical term
in that field, and see the following post from last year—
| Saturday, June 25, 2011
— m759 @ 12:00 PM “… the formula ‘Three Hypostases in one Ousia ‘
Ousia
|
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Monday, January 9, 2012
M Theory
Yesterday's All About Eve post featured Pope John Paul II
with his close friend and confidant Jerzy Kluger.
Their counterparts Xavier and Magneto in the recent film
"X-Men: First Class," together with Catholic doctrine on telepathy,
suggest the following meditations.
Douglas Hofstadter on interpenetration—
— as well as Trinity in this journal.
First the punchline—
Then the joke.
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Sunday, January 8, 2012
Big Apple
“…the nonlinear characterization of Billy Pilgrim
emphasizes that he is not simply an established
identity who undergoes a series of changes but
all the different things he is at different times.”
This suggests that the above structure
be viewed as illustrating not eight parts
but rather 8! = 40,320 parts.
"The Cardinal seemed a little preoccupied today."
The New Yorker , May 13, 2002
See also a note of May 14 , 2002.
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All About Eve (continued)
Literary symbolism, offered without comment—
 |
 |
|
Log24 post of January 14, 2011 |
 |
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Saturday, January 7, 2012
Doodles
Today's Google Doodle for the 100th birthday of Charles Addams—
A doodle from this year's Feast of the Epiphany—
A doodle based on today's previous post and on a post for Twelfth Night, 2003—
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Past Tense
From a post that was written for Twelfth Night—
Bernhard Weiss on the philosophy of Michael Dummett—
" … debates about realism, that is, those debates that ask
whether or not one or another aspect of the world is independent
of the way we represent that aspect to ourselves. For example,
is there a realm of mathematical entities that exists fully formed
independently of our mathematical activity? Are there facts about
the past that our use of the past tense aims to capture?"
Yes and Yes.
-
In memory of Anne Tyng, who died last year on Dec. 27,
see Number and Form. -
In memory of Frederica Sagor Maas, who
died on Twelfth Night, 2012, see Dogma. -
In memory of time I spent on Wilcox Avenue
in the Hollywood of the early sixties, see
Hans Reichenbach on the use of the past tense
(section 51 of Elements of Symbolic Logic ,
New York: The Macmillan Company, 1947, 287–98.)
See also The Whirligig of Time in this journal.
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Fearful Cold Intelligence
"Dreams are sleep's watchful brother, of death's fraternity,
heralds, watchmen of that coming night, and our attitude
toward them may be modeled upon Hades, receiving, hospitable,
yet relentlessly deepening, attuned to the nocturne, dusky, and
with a fearful cold intelligence that gives permanent shelter
in his house to the incurable conditions of human being."
— James Hillman, conclusion of
The Dream and the Underworld (Harper & Row, 1979)
In memory of Raymond Edward Alan Christopher Paley—
Related material— Mathieu Symmetry.
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Friday, January 6, 2012
Form
An example for the Feast of the Epiphany*
For one approach to defining this form, see Diamond Star.
* And for Pomona College
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Defining Form
Some related resources from Malcolm Lowry—
"…his eyes ranged the Consul's books disposed quite neatly… on high shelves around the walls: Dogme et Ritual de la Haute Magie , Serpent and Siva Worship in Central America , there were two long shelves of this, together with the rusty leather bindings and frayed edges of the numerous cabbalistic and alchemical books, though some of them looked fairly new, like the Goetia of the Lemegaton of Solomon the King , probably they were treasures, but the rest were a heterogeneous collection…."
— Under the Volcano , Chapter VI
— and from Matilde Marcolli—
Seven books on analytical psychology
See also Marcolli in this morning's previous post, The Garden Path.
For the relevance of alchemy to form, see Alchemy in this journal.
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The Garden Path
"Not all those who have sought to decode the symbolism of the Tarot pack
have been occultists; some have been serious scholars…."
— Michael Dummett, The Game of Tarot , Ch. 20
“Eliot by his own admission took the ‘still point of the turning world’
in Burnt Norton from the Fool in Williams’s The Greater Trumps .”
— Humphrey Carpenter, The Inklings , Ballantine Books, 1981, p. 106
From a talk on April 16, 2010, in Cuernavaca—
Related material—
See also The Martial Art of Giving Talks.
(Thanks to Lieven Le Bruyn for his Twelfth Night post on this topic.)
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Thursday, January 5, 2012
Crown Archetype
"Publisher: Crown Archetype (November 1, 2011)"
— Amazon.com on Piper Laurie's new self-portrait
See also last year's For All Hallows Day and today's previous post.
Some context: "God's Girlfriend" in this journal
and "Shouts & Murmurs" in The New Yorker
of January 9th, 2012—
Life is like a box of chocolates.
ILLUSTRATION: Maximilian Bode
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