Today the World Wide Web turns 20.
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See also Galois Memorial and Correspondences.
Saturday, August 6, 2011
Happy Web Day
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Correspondences
Comme de longs échos qui de loin se confondent
Dans une ténébreuse et profonde unité….
— Baudelaire, “Correspondances ”
From “A Four-Color Theorem”—
Figure 1
Note that this illustrates a natural correspondence
between
(A) the seven highly symmetrical four-colorings
of the 4×2 array at the left of Fig. 1, and
(B) the seven points of the smallest
projective plane at the right of Fig. 1.
To see the correspondence, add, in binary
fashion, the pairs of projective points from the
“points” section that correspond to like-colored
squares in a four-coloring from the left of Fig. 1.
(The correspondence can, of course, be described
in terms of cosets rather than of colorings.)
A different correspondence between these 7 four-coloring
structures and these 7 projective-line structures appears in
a structural analysis of the Miracle Octad Generator
(MOG) of R.T. Curtis—
Figure 2
| Here the correspondence between the 7 four-coloring structures (left section) and the 7 projective-line structures (center section) is less obvious, but more fruitful. It yields, as shown, all of the 35 partitions of an 8-element set (an 8-set ) into two 4-sets. The 7 four-colorings in Fig. 2 also appear in the 35 4×4 parts of the MOG that correspond, in a way indicated by Fig. 2, to the 35 8-set paritions. This larger correspondence— of 35 4×2 arrays with 35 4×4 arrays— is the MOG, at least as it was originally defined. See The MOG, Generating the Octad Generator, and Eightfold Geometry.
For some applications of the Curtis MOG, see |
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Thursday, August 4, 2011
Midnight in Oslo
For Norway's Niels Henrik Abel (1802-1829)
on his birthday, August Fifth
(6 PM Aug. 4, Eastern Time, is 12 AM Aug. 5 in Oslo.)
Plato's Diamond
The above version by Peter Pesic is from Chapter I of his book Abel's Proof , titled "The Scandal of the Irrational." Plato's diamond also occurs in a much later mathematical story that might be called "The Scandal of the Noncontinuous." The story—
Paradigms"These passages suggest that the Form is a character or set of characters common to a number of things, i.e. the feature in reality which corresponds to a general word. But Plato also uses language which suggests not only that the forms exist separately (χωριστά ) from all the particulars, but also that each form is a peculiarly accurate or good particular of its own kind, i.e. the standard particular of the kind in question or the model (παράδειγμα ) [i.e. paradigm ] to which other particulars approximate…. … Both in the Republic and in the Sophist there is a strong suggestion that correct thinking is following out the connexions between Forms. The model is mathematical thinking, e.g. the proof given in the Meno that the square on the diagonal is double the original square in area." – William and Martha Kneale, The Development of Logic , Oxford University Press paperback, 1985 Plato's paradigm in the Meno—
Changed paradigm in the diamond theorem (2×2 case) —
Aspects of the paradigm change— Monochrome figures to Areas to Continuous transformations to Euclidean geometry to Euclidean quantities to The 24 patterns resulting from the paradigm change—
Each pattern has some ordinary or color-interchange symmetry. This is the 2×2 case of a more general result. The patterns become more interesting in the 4×4 case. For their relationship to finite geometry and finite fields, see the diamond theorem. |
Related material: Plato's Diamond by Oslo artist Josefine Lyche.
“Plato’s Ghost evokes Yeats’s lament that any claim to worldly perfection inevitably is proven wrong by the philosopher’s ghost….”
— Princeton University Press on Plato’s Ghost: The Modernist Transformation of Mathematics (by Jeremy Gray, September 2008)
"Remember me to her."
— Closing words of the Algis Budrys novel Rogue Moon .
Background— Some posts in this journal related to Abel or to random thoughts from his birthday.
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Saturday, July 30, 2011
Groups and Symmetry
"… the best way to understand a group is to
see it as the group of symmetries of something."
— John Baez, p. 239, Bulletin (New Series) of the
American Mathematical Society , Vol. 42, No. 2,
April 2005, book review on pp. 229–243
electronically published on January 26, 2005
"Imagine yourself as a gem cutter,
turning around this diamond…."
— Ibid ., p. 240
See also related material from Log24.
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Thursday, July 28, 2011
Noon
"Anomalies must be expected along the conceptual frontier
between the temporal and the eternal."
– The Death of Adam , by Marilynne Robinson (1998, 2005),
essay on Marguerite de Navarre
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Magical Realism Revisited
The magical part— Synchronicity—
See Roger Cohen in this journal on January 15, 2009 and, on the
same date, Jesse Jarnow on Bob Dylan in The Jewish Daily Forward .
The realism part— Cohen's "smart power" and IQ tests involving pattern blocks.
The above quilt pattern software (both versions) is by Jarnow's father Al.
For a realistic approach to such patterns, see Blockheads in this journal.
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Monday, July 25, 2011
The Game
Virginia Heffernan in Sunday's online New York Times—
"… In the past, information on paper was something to read. Bricks and mortar were a place to be. But, since the first appearance of the Web in 1990, we have come to accept that information in pixels is something to read— and also a place to be . That familiar and yet still jaw-dropping metaphor takes energy to maintain. The odd shared sense that there’s three-dimensionality and immersion and real-world consequences on the Web as in no book or board game— that’s the Web’s sine qua non. Hence, cyberspace . And 'being on' the Internet….
… The dominant social networks are fantasy games built around rigged avatars, outright fictions and a silent— and often unconscious— agreement among players that the game and its somewhat creaky conceits influence the real world…."
— "The Confidence Game at Google+"
"It's just another manic Monday
I wish it was Sunday
'Cause that's my funday"
— The Bangles
"Accentuate the Positive"
— Clint Eastwood, soundtrack album
for "Midnight in the Garden of Good and Evil"
This journal on All Saints' Day, Sunday, November 1, 2009—
|
Suggested by the New York State lottery numbers on All Hallows’ Eve [2009]— 430 (mid-day) and 168 (evening)… From 430 as a date, 4/30— Beyond Grief and Nothing: A Reading of Don DeLillo , by Joseph Dewey, University of South Carolina Press, 2006, page 123: “It is as if DeLillo himself had moved to an endgame….” For such an endgame, see yesterday’s link to a Mira Sorvino drama. The number 168 suggested by the Halloween lottery deals with the properties of space itself and requires a more detailed exegesis… For the full picture, consider the Log24 entries of Feb. 16-28 this year, esp. the entries of Feb. 27 and the phrase they suggest— Flores, flores para los muertos. |
See also Pearly Gates of Cyberspace in this journal.
For flores para los muertos , see today's Times .
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Friday, July 15, 2011
Spider Women
"Of course, the aesthetic program
of cultural modernism
has long been summed up
by the maxim épater la bourgeoisie."
— The New York Times
Sunday Book Review, July 17
Examples:
"This Extreme and Difficult Sense of Spectacular Representation:
Antonin Artaud's Ontology of 'Live'," by Deborah Levitt
of the New School (See the noon post of July 13), as well as…
and, from mathematician Ellen Gethner's home page—
See also Sunday Dinner, A Link for Sunrise, and Inside CBS News.
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Monday, July 11, 2011
And/Or Problem
"It was the simultaneous emergence
and mutual determination
of probability and logic
that von Neumann found intriguing
and not at all well understood."
Context:
Update of 7 AM ET July 12, 2011—
Freeman Dyson on John von Neumann's
Sept. 2, 1954, address to the International
Congress of Mathematicians on
"Unsolved Problems in Mathematics"—
…."The hall was packed with
mathematicians, all expecting to hear a brilliant
lecture worthy of such a historic occasion. The
lecture was a huge disappointment. Von Neumann
had probably agreed several years earlier to give
a lecture about unsolved problems and had then
forgotten about it. Being busy with many other
things, he had neglected to prepare the lecture.
Then, at the last moment, when he remembered
that he had to travel to Amsterdam and say something
about mathematics, he pulled an old lecture
from the 1930s out of a drawer and dusted it off.
The lecture was about rings of operators, a subject
that was new and fashionable in the 1930s. Nothing
about unsolved problems. Nothing about the
future."
— Notices of the American Mathematical Society ,
February 2009, page 220
For a different account, see Giovanni Valente's
2009 PhD thesis from the University of Maryland,
Chapter 2, "John von Neumann's Mathematical
'Utopia' in Quantum Theory"—
"During his lecture von Neumann discussed operator theory and its con-
nections with quantum mechanics and noncommutative probability theory,
pinpointing a number of unsolved problems. In his view geometry was so tied
to logic that he ultimately outlined a logical interpretation of quantum prob-
abilities. The core idea of his program is that probability is invariant under
the symmetries of the logical structure of the theory. This is tantamount to
a formal calculus in which logic and probability arise simultaneously. The
problem that exercised von Neumann then was to construct a geometrical
characterization of the whole theory of logic, probability and quantum me-
chanics, which could be derived from a suitable set of axioms…. As he
himself finally admitted, he never managed to set down the sought-after
axiomatic formulation in a way that he felt satisfactory."
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Accentuate the Positive
An image that may be viewed as
a cube with a “+“ on each face—
The eightfold cube
Underlying structure
For the Pope and others on St. Benedict’s Day
who prefer narrative to mathematics—
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Sunday, July 10, 2011
Wittgenstein’s Diamond
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. |
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Saturday, July 9, 2011
Gleaming the Cube (continued)
The New York Times has a skateboarder obit with a URL date of July 9.
Here is an earlier version from the LA Times—
By Keith Thursby, Los Angeles Times
Chris Cahill, one of the original Dogtown Z-Boys
who brought seismic changes to skateboarding
with their style and attitude, has died. He was 54.
Cahill was found June 24 at his Los Angeles home,
said Larry Dietz of the Los Angeles County
coroner's office. A cause of death has not been
determined and tests are ongoing, Dietz said.
Related material from Midsummer Day, June 24, the day Cahill was found dead—
The Gleaming and The Cube.
An illustration from the latter—
The above was adapted from a 1996 cover—
Vintage Books, July 1996. Cover: Evan Gaffney.
For the significance of the flames,
see PyrE in the book. For the significance
of the cube in the altered cover, see
The 2×2×2 Cube and The Diamond Archetype.
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Wednesday, July 6, 2011
Nordstrom-Robinson Automorphisms
A 2008 statement on the order of the automorphism group of the Nordstrom-Robinson code—
"The Nordstrom-Robinson code has an unusually large group of automorphisms (of order 8! = 40,320) and is optimal in many respects. It can be found inside the binary Golay code."
— Jürgen Bierbrauer and Jessica Fridrich, preprint of "Constructing Good Covering Codes for Applications in Steganography," Transactions on Data Hiding and Multimedia Security III, Springer Lecture Notes in Computer Science, 2008, Volume 4920/2008, 1-22
A statement by Bierbrauer from 2004 has an error that doubles the above figure—
The automorphism group of the binary Golay code G is the simple Mathieu group M24 of order
— Jürgen Bierbrauer, "Nordstrom-Robinson Code and A7-Geometry," preprint dated April 14, 2004, published in Finite Fields and Their Applications , Volume 13, Issue 1, January 2007, Pages 158-170
The error is corrected (though not detected) later in the same 2004 paper—
In fact the symmetry group of the octacode is a semidirect product of an elementary abelian group of order 16 and the simple group GL(3, 2) of order 168. This constitutes a large automorphism group (of order 2688), but the automorphism group of NR is larger yet as we saw earlier (order 40,320).
For some background, see a well-known construction of the code from the Miracle Octad Generator of R.T. Curtis—
For some context, see the group of order 322,560 in Geometry of the 4×4 Square.
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Friday, July 1, 2011
Symmetry Review
Popular novelist Dan Brown is to speak at Chautauqua Institution on August 1.
This suggests a review of some figures discussed here in a note on Brown from February 20, 2004—
Related material: Notes from Nov. 5, 1981, and from Dec. 24, 1981.
For the lower figure in context, see the diamond theorem.
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Wednesday, June 29, 2011
The Shattered Mind
For St. Peter's Day
"For Stevens, the poem 'makes meanings of the rock.'
In the mind, 'its barrenness becomes a thousand things/
And so exists no more.' In fact, in a peculiar irony
that only a poet with Stevens's particular notion
of the imagination's function could develop,
the rock becomes the mind itself, shattered
into such diamond-faceted brilliance
that it encompasses all possibilities for human thought…."
—A discussion of Stevens's late poem "The Rock" (1954)
in Wallace Stevens: A World of Transforming Shapes,
by Alan D. Perlis, Bucknell University Press, 1976, p. 120
Related material on transforming shapes:
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Monday, June 27, 2011
Galois Cube Revisited
The 3×3×3 Galois Cube
This cube, unlike Rubik's, is a
purely mathematical structure.
Its properties may be compared
with those of the order-2 Galois
cube (of eight subcubes, or
elements ) and the order-4 Galois
cube (of 64 elements). The
order-3 cube (of 27 elements)
lacks, because it is based on
an odd prime, the remarkable
symmetry properties of its smaller
and larger cube neighbors.
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Sunday, June 26, 2011
Sunday Dinner
From "Sunday Dinner" in this journal—
"'If Jesus were to visit us, it would have been
the Sunday dinner he would have insisted on
being a part of, not the worship service at the church.'"
—Judith Shulevitz at The New York Times
on Sunday, July 18, 2010
Some table topics—
Today's midday New York Lottery numbers were 027 and 7002.
The former suggests a Galois cube, the latter a course syllabus—
CSC 7002
Graduate Computer Security (Spring 2011)
University of Colorado at Denver
Department of Computer Science
An item from that syllabus:
| Six | 22 February 2011 | DES | History of DES; Encryption process; Decryption; Expander function; S-boxes and their output; Key; the function f that takes the modified key and part of the text as input; mulitple Rounds of DES; Present-day lack of Security in DES, which led to the new Encryption Standard, namely AES. Warmup for AES: the mathematics of Fields: Galois Fields, particularly the one of order 256 and its relation to the irreducible polynomial x^8 + x^4 + x^3 + x + 1 with coefficients from the field Z_2. |
Related material: A novel, PopCo , was required reading for the course.
Discuss a different novel by the same author—
Discuss the author herself, Scarlett Thomas.
Background for the discussion—
Derrida in this journal versus Charles Williams in this journal.
Related topics from the above syllabus date—
Metaphor and Gestell and Quadrat.
Some context— Midsummer Eve's Dream.
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Paradigms Lost
Continued from March 10, 2011 — A post that says
"If Galois geometry is thought of as a paradigm shift
from Euclidean geometry, both… the Kuhn cover
and the nine-point affine plane may be viewed…
as illustrating the shift."
Yesterday's posts The Fano Entity and Theology for Antichristmas,
together with this morning's New York Times obituaries (below)—
—suggest a Sunday School review from last year's
Devil's Night (October 30-31, 2010)—
|
Sunday, October 31, 2010 ART WARS – m759 @ 2:00 AM … There is a Cave – Paradise Lost , by John Milton
|
See also Ash Wednesday Surprise and Geometry for Jews.
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Saturday, June 25, 2011
Theology for Antichristmas
“… the formula ‘Three Hypostases in one Ousia ‘
came to be everywhere accepted as an epitome
of the orthodox doctrine of the Holy Trinity.
This consensus, however, was not achieved
without some confusion….” —Wikipedia
Ousia
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The Fano Entity
The New York Times at 9 PM ET June 23, 2011—
ROBERT FANO: I’m trying to think briefly how to put it.
GINO FANO: "On the Fundamental Postulates"—
"E la prova di questo si ha precisamente nel fatto che si è potuto costruire (o, dirò meglio immaginare) un ente per cui sono verificati tutti i postulati precedenti…."
"The proof of this is precisely the fact that you could build (or, to say it better, imagine) an entity by which are verified all previous assumptions…."
Also from the Times article quoted above…
"… like working on a cathedral. We laid our bricks and knew that others might later replace them with better bricks. We believed in the cause even if we didn’t completely understand the implications.”
— Tom Van Vleck
Some art that is related, if only by a shared metaphor, to Van Vleck's cathedral—
The art is also related to the mathematics of Gino Fano.
For an explanation of this relationship (implicit in the above note from 1984),
see "The Fano plane revisualized—or: the eIghtfold cube."
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Friday, June 24, 2011
For Stephen King Fans
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Wednesday, June 22, 2011
ART WARS continued
This evening's New York Times obituaries—
A work of art suggested by the first and third items above—
I prefer a work of art that is structurally similar—
and is related to a picture, Portrait of O, from October 1, 1983—
For a recent unexpected Web appearance of Portrait of O,
aee Abracadabra from the midnight of June 18-19.
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Tuesday, June 21, 2011
The Longest Day Continues
The AND Publishing weblog page referred to in
a Sunday post has been changed to reflect the
source— my finite-geometry website— of pages
copied and altered by London artist Steve Richards
that are a large part of his contribution to the
AND Publishing Piracy Project.
The new version is as follows—
Note, however, that the cover page is a figure titled
by Richards "metalibrarianship" that has nothing
whatever to do with the concepts in the pages he copied
from my site, finitegeometry.org/sc.
Other pages within Richards's contribution to the
Piracy Project are similarly completely unrelated to
the content of my own site, which deals with geometry.
The image on the cover page also appears, it turns out,
on a website called intertwining.org.
At that site, it occurs in the following resume item:
The links in the resume item do not work,
but some background is available at a page titled
"Circularity, Practicality and Philosophy of Librarianship, or
The Making of 'The Nitecki Trilogy'" by Joanne Twining.
Other images in Richards's contribution to the Piracy Project also occur
in Twining's webpage "Dimensional Advances for Information Architecture."
I never heard of Twining or Nitecki before I encountered Richards's
Piracy Project contribution, and I do not wish to be associated
again in any way with Twining, with Nitecki, or with Richards.
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Piracy Project
Recent piracy of my work as part of a London art project suggests the following.
From http://www.trussel.com/rls/rlsgb1.htm
The 2011 Long John Silver Award for academic piracy
goes to ….
Hermann Weyl, for the remark on objectivity and invariance
in his classic work Symmetry that skillfully pirated
the much earlier work of philosopher Ernst Cassirer.
And the 2011 Parrot Award for adept academic idea-lifting
goes to …
Richard Evan Schwartz of Brown University, for his
use, without citation, of Cullinane’s work illustrating
Weyl’s “relativity problem” in a finite-geometry context.
For further details, click on the above names.
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Monday, June 20, 2011
The Search for Invariants
The title of a recent contribution to a London art-related "Piracy Project" begins with the phrase "The Search for Invariants."
A search for that phrase elsewhere yields a notable 1944* paper by Ernst Cassirer, "The Concept of Group and the Theory of Perception."
Page 20: "It is a process of objectification, the characteristic nature
and tendency of which finds expression in the formation of invariants."
Cassirer's concepts seem related to Weyl's famous remark that
“Objectivity means invariance with respect to the group of automorphisms.”
—Symmetry (Princeton University Press, 1952, page 132)
See also this journal on June 23, 2010— "Group Theory and Philosophy"— as well as some Math Forum remarks on Cassirer and Weyl.
Update of 6 to 7:50 PM June 20, 2011—
Weyl's 1952 remark seems to echo remarks in 1910 and 1921 by Cassirer.
See Cassirer in 1910 and 1921 on Objectivity.
Another source on Cassirer, invariance, and objectivity—
The conclusion of Maja Lovrenov's
"The Role of Invariance in Cassirer’s Interpretation of the Theory of Relativity"—
"… physical theories prove to be theories of invariants
with regard to certain groups of transformations and
it is exactly the invariance that secures the objectivity
of a physical theory."
— SYNTHESIS PHILOSOPHICA 42 (2/2006), pp. 233–241
A search in Weyl's Symmetry for any reference to Ernst Cassirer yields no results.
* Published in French in 1938.
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Sunday, June 19, 2011
The London Piracy Project
My work has been pirated by an artist in London.
An organization there, AND Publishing, sponsors what it calls
"The Piracy Project." The artist's piracy was a contribution
to the project.
The above material now reflects the following update:
|
UPDATE of June 21, 2011, 10:00 PM ET: The organization's weblog (a post for 19th June) In this weblog, changes have been made to correct my "AND Publishing is not sponsored by the art school. |
As this post originally stated…
The web pages from the site finitegeometry.org/sc that
the artist, Steve Richards, copied as part of his contribution to
the AND Publishing Piracy Project have had the author's name,
Steven H. Cullinane, and the date of composition systematically removed.
See a sample (jpg, 2.1 MB).
Here is some background on Richards.
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Abracadabra (continued)
Yesterday's post Ad Meld featured Harry Potter (succeeding in business),
a 4×6 array from a video of the song "Abracadabra," and a link to a post
with some background on the 4×6 Miracle Octad Generator of R.T. Curtis.
A search tonight for related material on the Web yielded…
Weblog post by Steve Richards titled "The Search for Invariants:
The Diamond Theory of Truth, the Miracle Octad Generator
and Metalibrarianship." The artwork is by Steven H. Cullinane.
Richards has omitted Cullinane's name and retitled the artwork.
The author of the post is an artist who seems to be interested in the occult.
His post continues with photos of pages, some from my own work (as above), some not.
My own work does not deal with the occult, but some enthusiasts of "sacred geometry" may imagine otherwise.
The artist's post concludes with the following (note also the beginning of the preceding post)—
"The Struggle of the Magicians" is a 1914 ballet by Gurdjieff. Perhaps it would interest Harry.
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Wednesday, June 15, 2011
A for Anastasios
The title was suggested by this evening's 4-digit NY lottery number.
"… the rhetoric might be a bit over the top."
According to Amazon.com, 2198 (i.e., 2/1/98) was the publication
date of Geometry of Vector Sheaves , Volume I, by Anastasios Mallios.
Related material—
The question of S.S. Chern quoted here June 10: —
"What is Geometry?"— and the remark by Stevens that
accompanied the quotation—
"Reality is the beginning not the end,
Naked Alpha, not the hierophant Omega,
of dense investiture, with luminous vassals."
— Wallace Stevens,
“An Ordinary Evening in New Haven” VI
The work of Mallios in pure mathematics cited above seems
quite respectable (unlike his later remarks on physics).
His Vector Sheaves appears to be trying to explore new territory;
hence the relevance of Stevens's "Alpha." See also the phrase
"A-Invariance" in an undated preprint by Mallios*.
For the evening 3-digit number, 533, see a Stevens poem—
This meditation by Stevens is related to the female form of Mallios's Christian name.
As for the afternoon numbers, see "62" in The Beauty Test (May 23, 2007), Geometry and Death, and "9181" as the date 9/1/81.
* Later published in International Journal of Theoretical Physics , Vol. 47, No. 7, cover date 2008-07-01
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Tuesday, June 14, 2011
Another Opening
NY Lottery this evening: 3-digit 444, 4-digit 0519.
444:
"… of our history … and of our destructive paths.
We are beginning to sense the need to restore
the sacred feminine." She paused. "You
mentioned you are writing a manuscript about
the symbols of the sacred feminine, are you not?"
"I …"
Related material— "Eightfold Geometry" + Spider in this journal.
For this afternoon's NY numbers— 511 and 9891— see
511 in the "Going Up" post of July 12, 2007, as well as
Ben Brantley's recent suggestion of Paris Hilton as a
matinee attraction and her 9891 photo on the Web.
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Thursday, June 9, 2011
Page 679
Good question. See also
Chern died on the evening of Friday, Dec. 3, 2004 (Chinese time).
From the morning of that day (also Chinese time)—
i.e. , the evening of the preceding day here— some poetry.
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Upshot
Suggested by this afternoon’s NY Lottery number, 541—
Related material: Finite Relativity and The Schwartz Notes.
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Wednesday, June 8, 2011
For Saint Peter
… and Arthur Koestler
|
The theme of the January 2010 issue of the
Notices of the American Mathematical Society was “Mathematics and the Arts.”
Related material:
|
See also two posts from the day Peter Jennings died—
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Monday, June 6, 2011
Tree of Life — Jewish Version
Today's midday NY Lottery number was 753, the number of a significant page in Gravity's Rainbow .
An excerpt from that page ((Penguin Classics paperback, June 1, 1995)—
"… the Abyss had crept intolerably close, only an accident away…."
Midrash— See Ben Stein in this journal.
But seriously… See "Geometry and Death" in this journal.
See also PlanetMath.org on the Hesse configuration—
A picture of the Hesse configuration—
.
Some context— A Study in Art Education.
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Sunday, June 5, 2011
Edifice Complex
"Total grandeur of a total edifice,
Chosen by an inquisitor of structures
For himself. He stops upon this threshold,
As if the design of all his words takes form
And frame from thinking and is realized."
— Wallace Stevens, "To an Old Philosopher in Rome"
The following edifice may be lacking in grandeur,
and its properties as a configuration were known long
before I stumbled across a description of it… still…
"What we do may be small, but it has
a certain character of permanence…."
— G.H. Hardy, A Mathematician's Apology
The Kummer 166 Configuration
as seen by Kantor in 1969— (pdf, 2.5 MB)
For some background, see Configurations and Squares.
For some quite different geometry of the 4×4 square that is
original with me, see a page with that title. (The geometry's
importance depends in part on its connection with the
Miracle Octad Generator (MOG) of R.T. Curtis. I of course
had nothing to do with the MOG's discovery, but I do claim credit
for discovering some geometric properties of the 4×4 square
that constitutes two-thirds of the MOG as originally defined .)
Related material— The Schwartz Notes of June 1.
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Wednesday, June 1, 2011
The Schwartz Notes
A Google search today for material on the Web that puts the diamond theorem
in context yielded a satisfyingly complete list. (See the first 21 results.)
(Customization based on signed-out search activity was disabled.)
The same search limited to results from only the past month yielded,
in addition, the following—
This turns out to be a document by one Richard Evan Schwartz,
Chancellor’s Professor of Mathematics at Brown University.
Pages 12-14 of the document, which is untitled, undated, and
unsigned, discuss the finite-geometry background of the R.T.
Curtis Miracle Octad Generator (MOG) . As today’s earlier search indicates,
this is closely related to the diamond theorem. The section relating
the geometry to the MOG is titled “The MOG and Projective Space.”
It does not mention my own work.
See Schwartz’s page 12, page 13, and page 14.
Compare to the web pages from today’s earlier search.
There are no references at the end of the Schwartz document,
but there is this at the beginning—
These are some notes on error correcting codes. Two good sources for
this material are
• From Error Correcting Codes through Sphere Packings to Simple Groups ,
by Thomas Thompson.
• Sphere Packings, Lattices, and Simple Groups by J. H. Conway and N.
Sloane
Planet Math (on the internet) also some information.
It seems clear that these inadequate remarks by Schwartz on his sources
can and should be expanded.
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Saturday, May 28, 2011
Savage Detectives
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From Savage Logic— Sunday, March 15, 2009 5:24 PM The Origin of Change A note on the figure
"Two things of opposite natures seem to depend — Wallace Stevens, |
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Thursday, May 26, 2011
Prime Cubes
The title refers not to numbers of the form p 3, p prime, but to geometric cubes with p 3 subcubes.
Such cubes are natural models for the finite vector spaces acted upon by general linear groups viewed as permutation groups of degree (not order ) p 3.
For the case p =2, see The Eightfold Cube.
For the case p =3, see the "External links" section of the Nov. 30, 2009, version of Wikipedia article "General Linear Group." (That is the version just prior to the Dec. 14, 2009, revision by anonymous user "Greenfernglade.")
For symmetries of group actions for larger primes, see the related 1985 remark* on two -dimensional linear groups—
"Actions of GL(2,p ) on a p ×p coordinate-array
have the same sorts of symmetries,
where p is any odd prime."
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Tuesday, May 24, 2011
Noncontinuous (or Non-Continuous) Groups
The web page has been updated.
An example, the action of the Mathieu group M24
on the Miracle Octad Generator of R.T. Curtis,
was added, with an illustration from a book cover—
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Monday, May 23, 2011
The Stoner Series
A reader comments on yesterday afternoon's New York Times
"The Stone" column by Justin E.H. Smith—
"I did indeed appreciate Mr. Smith’s essay.
And I’m curious as to what future contributions of his,
to the Stoner series, that we can look forward to."
From August 24, 2010—
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Happy day 23 of Mental Health Month.
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Sunday, May 22, 2011
Return of the Stone
The New York Times philosophy column "The Stone" has returned—
"There will certainly always be a place for epistemology,
or the theory of knowledge. But in order for a theory of
knowledge to tell us much, it needs to draw on examples
of knowledge of something or other." — Justin E.H. Smith
Amen.
Examples: Quine on geometry and Quine on universals.
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Asterisk*
A year ago today—
Art Space
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Pictorial version |
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| “Space: what you damn well have to see.” – James Joyce, Ulysses |
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* See Vonnegut.
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Thursday, May 19, 2011
The Aleph, the Lottery, and the Eightfold Way
Three links with a Borges flavor—
Related material
The 236 in yesterday evening's NY lottery may be
viewed as the 236 in March 18's Defining Configurations.
For some background, see Configurations and Squares.
A new illustration for that topic—
This shows a reconcilation of the triples described by Sloane
in Defining Configurations with the square geometric
arrangement described by Coxeter in the Aleph link above.
Note that the 56 from yesterday's midday NY lottery
describes the triples that appear both in the Eightfold Way
link above and also in a possible source for
the eight triples of Sloane's 83 configuration—
The geometric square arrangement discussed in the Aleph link
above appears in a different, but still rather Borgesian, context
in yesterday morning's Minimalist Icon.
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Wednesday, May 18, 2011
Minimalist Icon
The source of the mysterious generic
3×3 favicon with one green cell —
— has been identified.
For minimalists, here is a purer 3×3 matrix favicon—
This may, if one likes, be viewed as the "nothing"
present at the Creation. See Jim Holt on physics.
See also Visualizing GL(2,p), Coxeter and the Aleph, and Ayn Sof.
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Tuesday, May 17, 2011
Anomalies
Perfect Symmetry (Oct. 2008) and Perfect Symmetry single (Dec. 2008)—
Related science…
Heinz Pagels in Perfect Symmetry (paperback, 1985), p. xvii—
The penultimate chapter of this third part of the book—
as far as speculation is concerned— describes some
recent mathematical models for the very origin of the
universe—how the fabric of space, time and matter can
be created out of absolutely nothing. What could have more
perfect symmetry than absolute nothingness? For the first
time in history, scientists have constructed mathematical
models that account for the very creation of the universe
out of nothing.
On Grand Unified Theories (GUT's) of physics (ibid., 284)—
In spite of the fact that GUTs leave deep puzzles unsolved,
they have gone a long way toward unifying the various
quantum particles. For example, many people are disturbed
by the large numbers of gluons, quarks and leptons. Part of
the appeal of the GUT idea is that this proliferation of
quantum particles is really superficial and that all the gluons
as well at the quarks and leptons may be simply viewed as
components of a few fundamental unifying fields. Under the
GUT symmetry operation these field components transform
into one another. The reason quantum particles appear to
have different properties in nature is that the unifying
symmetry is broken. The various gluons, quarks and leptons
are analogous to the facets of a cut diamond, which appear
differently according to the way the diamond is held but in
fact are all manifestations of the same underlying object.
Related art— Puzzle and Particles…
The Diamond 16 Puzzle (compare with Keane art above)—
—and The Standard Model of particle theory—
The fact that both the puzzle and the particles appear
within a 4×4 array is of course completely coincidental.
See also a more literary approach— "The Still Point and the Wheel"—
"Anomalies must be expected along the conceptual frontier between the temporal and the eternal."
— The Death of Adam , by Marilynne Robinson, Houghton Mifflin, 1998, essay on Marguerite de Navarre
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Friday, May 13, 2011
Apollo’s 13
Continued … See related previous posts.
Those who prefer narrative to mathematics
may consult Wikipedia on The Cosmic Cube.
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Tuesday, May 10, 2011
Groups Acting
The LA Times on last weekend's film "Thor"—
"… the film… attempts to bridge director Kenneth Branagh's high-minded Shakespearean intentions with Marvel Entertainment's bottom-line-oriented need to crank out entertainment product."
Those averse to Nordic religion may contemplate a different approach to entertainment (such as Taymor's recent approach to Spider-Man).
A high-minded— if not Shakespearean— non-Nordic approach to groups acting—
"What was wrong? I had taken almost four semesters of algebra in college. I had read every page of Herstein, tried every exercise. Somehow, a message had been lost on me. Groups act . The elements of a group do not have to just sit there, abstract and implacable; they can do things, they can 'produce changes.' In particular, groups arise naturally as the symmetries of a set with structure. And if a group is given abstractly, such as the fundamental group of a simplical complex or a presentation in terms of generators and relators, then it might be a good idea to find something for the group to act on, such as the universal covering space or a graph."
— Thomas W. Tucker, review of Lyndon's Groups and Geometry in The American Mathematical Monthly , Vol. 94, No. 4 (April 1987), pp. 392-394
"Groups act "… For some examples, see
- The 2×2×2 Cube,
- The Diamond 16 Puzzle,
- The Diamond Theorem, and
- Finite Geometry of the Square and Cube.
Related entertainment—
High-minded— Many Dimensions—
Not so high-minded— The Cosmic Cube—
One way of blending high and low—
The high-minded Charles Williams tells a story
in his novel Many Dimensions about a cosmically
significant cube inscribed with the Tetragrammaton—
the name, in Hebrew, of God.
The following figure can be interpreted as
the Hebrew letter Aleph inscribed in a 3×3 square—
The above illustration is from undated software by Ed Pegg Jr.
For mathematical background, see a 1985 note, "Visualizing GL(2,p)."
For entertainment purposes, that note can be generalized from square to cube
(as Pegg does with his "GL(3,3)" software button).
For the Nordic-averse, some background on the Hebrew connection—
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Sunday, May 8, 2011
Matrix
From Thomas Mann, "Schopenhauer," 1938, in Essays of Three Decades , translated by H. T. Lowe-Porter, Alfred A. Knopf, 1947, pp. 372-410—
Page 372: THE PLEASURE we take in a metaphysical system, the gratification purveyed by the intellectual organization of the world into a closely reasoned, complete, and balanced structure of thought, is always of a pre-eminently aesthetic kind. It flows from the same source as the joy, the high and ever happy satisfaction we get from art, with its power to shape and order its material, to sort out life's manifold confusions so as to give us a clear and general view.
Truth and beauty must always be referred the one to the other. Each by itself, without the support given by the other, remains a very fluctuating value. Beauty that has not truth on its side and cannot have reference to it, does not live in it and through it, would be an empty chimera— and "What is truth?"
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Friday, May 6, 2011
Theme and Variations
— Richard Powers, "The Perpetual Calendar," from The Gold Bug Variations , 1991
See also, from last All Hallows' Eve, "Diamond Theorem in Norway."
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Thursday, May 5, 2011
On Art and Magic
Two Blocks Short of a Design:
A sequel to this morning’s post on Douglas Hofstadter
Photo of Hofstadter by Mike McGrath taken May 13, 2006 |
Related material — See Lyche’s “Theme and Variations” in this journal
and Hofstadter’s “Variations on a Theme as the Essence of Imagination”
— Scientific American October 1982
A quotation from a 1985 book by Hofstadter—
“… we need to entice people with the beauties of clarity, simplicity, precision,
elegance, balance, symmetry, and so on.
Those artistic qualities… are the things that I have tried to explore and even
to celebrate in Metamagical Themas . (It is not for nothing that the word
‘magic’ appears inside the title!)”
The artistic qualities Hofstadter lists are best sought in mathematics, not in magic.
An example from Wikipedia —
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Mathematics
The Fano plane block design |
Magic
The Deathly Hallows symbol— |
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Beyond Forgetfulness
From this journal on July 23, 2007—
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It is not enough to cover the rock with leaves. We must be cured of it by a cure of the ground Or a cure of ourselves, that is equal to a cure
Of the ground, a cure beyond forgetfulness.
And if we ate the incipient colorings – Wallace Stevens, "The Rock" |
This quotation from Stevens (Harvard class of 1901) was posted here on when Daniel Radcliffe (i.e., Harry Potter) turned 18 in July 2007.
Other material from that post suggests it is time for a review of magic at Harvard.
On September 9, 2007, President Faust of Harvard
"encouraged the incoming class to explore Harvard’s many opportunities.
'Think of it as a treasure room of hidden objects Harry discovers at Hogwarts,' Faust said."
That class is now about to graduate.
It is not clear what "hidden objects" it will take from four years in the Harvard treasure room.
Perhaps the following from a book published in 1985 will help…
The March 8, 2011, Harvard Crimson illustrates a central topic of Metamagical Themas , the Rubik's Cube—
Hofstadter in 1985 offered a similar picture—
Hofstadter asks in his Metamagical introduction, "How can both Rubik's Cube and nuclear Armageddon be discussed at equal length in one book by one author?"
For a different approach to such a discussion, see Paradigms Lost, a post made here a few hours before the March 11, 2011, Japanese earthquake, tsunami, and nuclear disaster—
Whether Paradigms Lost is beyond forgetfulness is open to question.
Perhaps a later post, in the lighthearted spirit of Faust, will help. See April 20th's "Ready When You Are, C.B."
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Sunday, May 1, 2011
Delos
The late translator Helen Lane in Translation Review , Vol. 5, 1980—
"Among the awards, I submit, should be one for the entire oeuvre of a lifetime "senior" translator— and one for the best first translation…. Similar organization, cooperation, and fund-finding for a first-rate replacement for the sorely missed Delos ."
This leads to one of the founders of Delos , the late Donald Carne-Ross, who died on January 9, 2010.
For one meditation on the date January 9, see Bridal Birthday (last Thursday).
Another meditation, from the date of Carne-Ross's death—
Saturday, January 9, 2010
Positional Meaning"The positional meaning of a symbol derives from its relationship to other symbols in a totality, a Gestalt, whose elements acquire their significance from the system as a whole." – Victor Turner, The Forest of Symbols , Ithaca, NY, Cornell University Press, 1967, p. 51, quoted by Beth Barrie in "Victor Turner." To everything, turn, turn, turn …
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See also Delos in this journal.
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Saturday, April 30, 2011
Crimson Walpurgisnacht
Part I — Unity and Multiplicity
(Continued from The Talented and Galois Cube)
Part II — "A feeling, an angel, the moon, and Italy"—
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Thursday, April 28, 2011
26 Today
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Wednesday, April 27, 2011
Block That Metaphor–
A Note on Galois Geometry
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Simple groups as the (Click image to enlarge.)
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Points, lines, etc., as the
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Related material —
(Click images for some background.)
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Building blocks and
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Building blocks and |
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Tuesday, April 26, 2011
25 Years Ago Today
The above points and hyperplanes underlie the symmetries discussed
in the diamond theorem. See The Oslo Version and related remarks
for a different use in art.
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Unity and Multiplicity
Today's earlier post mentions one approach to the concepts of unity and multiplicity. Here is another.
Unity:
The 3×3×3 Galois Cube
Multiplicity:
One of a group, GL(3,3), of 11,232
natural transformations of the 3×3×3 Cube
See also the earlier 1985 3×3 version by Cullinane.
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Monday, April 25, 2011
Poetry and Physics
One approach to the storied philosophers' stone, that of Jim Dodge in Stone Junction , was sketched in yesterday's Easter post. Dodge described a mystical "spherical diamond." The symmetries of the sphere form what is called in mathematics a Lie group . The "spherical" of Dodge therefore suggests a review of the Lie group E8 in Garrett Lisi's poetic theory of everything.
A check of the Wikipedia article on Lisi's theory yields…
Diamond and E8 at Wikipedia
Related material — E8 as "a diamond with thousands of facets"—
Also from the New Yorker article—
“There’s a dream that underlying the physical universe is some beautiful mathematical structure, and that the job of physics is to discover that,” Smolin told me later. “The dream is in bad shape,” he added. “And it’s a dream that most of us are like recovering alcoholics from.” Lisi’s talk, he said, “was like being offered a drink.”
A simpler theory of everything was offered by Plato. See, in the Timaeus , the Platonic solids—
Figure from this journal on August 19th, 2008.
See also July 19th, 2008.
“It’s all in Plato, all in Plato:
bless me, what do they
teach them at these schools!”
— C. S. Lewis
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Friday, April 22, 2011
Romancing the Hyperspace
For the title, see Palm Sunday.
"There is a pleasantly discursive treatment of
Pontius Pilate's unanswered question 'What is truth?'" — H. S. M. Coxeter, 1987
From this date (April 22) last year—
 Richard J. Trudeau in The Non-Euclidean Revolution , chapter on "Geometry and the Diamond Theory of Truth"– "… Plato and Kant, and most of the philosophers and scientists in the 2200-year interval between them, did share the following general presumptions: (1) Diamonds– informative, certain truths about the world– exist. Presumption (1) is what I referred to earlier as the 'Diamond Theory' of truth. It is far, far older than deductive geometry." Trudeau's book was published in 1987. The non-Euclidean* figures above illustrate concepts from a 1976 monograph, also called "Diamond Theory." Although non-Euclidean,* the theorems of the 1976 "Diamond Theory" are also, in Trudeau's terminology, diamonds. * "Non-Euclidean" here means merely "other than Euclidean." No violation of Euclid's parallel postulate is implied. |
Trudeau comes to reject what he calls the "Diamond Theory" of truth. The trouble with his argument is the phrase "about the world."
Geometry, a part of pure mathematics, is not about the world. See G. H. Hardy, A Mathematician's Apology .
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Romancing the Symmetry
From a story about mathematician Emmy Noether and 1882, the year she was born—
"People were then slowly becoming 'modern'— fortunately they had finally discovered not just that there are no Easter bunnies and Santa Claus, but also that there probably never were women who were led to evil ways by their curiosity and ended up, depending on their level of education, as common witches, as 'wiccans,' or as those particularly mysterious 'benandanti.'"
"… in the Balkans people believe that the souls of the dead rise to heaven in the guise of butterflies."
— "The Fairytale of the Totally Symmetrical Butterfly," by Dietmar Dath, in Intoxicating Heights (Eichborn AG, Frankfurt 2003)
An insect perhaps more appropriate for the afternoon of Good Friday— the fly in the logo of Dath's publisher—
Related material— Holy Saturday of 2004 and Wittgenstein and the Fly Bottle.
(After clicking, scroll down to get past current post.)
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Wednesday, April 20, 2011
Romancing the Cube
It was a dark and stormy night…
— Page 180, Logicomix
“… the class of reflections is larger in some sense over an arbitrary field than over a characteristic zero field.”
– Julia Hartmann and Anne V. Shepler, “Jacobians of Reflection Groups”
For some context, see the small cube in “A Simple Reflection Group of Order 168.”
See also the larger cube in “Many Dimensions” + Whitehead in this journal (scroll down to get past the current post).
That search refers to a work by Whitehead published in 1906, the year at the top of the Logicomix page above—
A related remark on axiomatics that has metaphysical overtones suitable for a dark and stormy night—
“An adequate understanding of mathematical identity requires a missing theory that will account for the relationships between formal systems that describe the same items. At present, such relationships can at best be heuristically described in terms that invoke some notion of an ‘intelligent user standing outside the system.'”
— Gian-Carlo Rota, “Syntax, Semantics, and…” in Indiscrete Thoughts . See also the original 1988 article.
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Tuesday, April 19, 2011
Romancing the Omega
Today's news from Oslo suggests a review—
The circular sculpture in the foreground
is called by the artist "The Omega Point."
This has been described as
"a portal that leads in or out of time and space."
Some related philosophical remarks—
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Sunday, April 17, 2011
Annals of Search
The following has rather mysteriously appeared in a search at Google Scholar for "Steven H. Cullinane."
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[HTML] Romancing the Non-Euclidean Hyperspace |
This turns out to be a link to a search within this weblog. I do not know why Google Scholar attributes the resulting web page to a journal article by "AB Story" or why it drew the title from a post within the search and applied it to the entire list of posts found. I am, however, happy with the result— a Palm Sunday surprise with an eclectic mixture of styles that might please the late Robert de Marrais.
I hope the late George Temple would also be pleased. He appears in "Romancing" as a resident of Quarr Abbey, a Benedictine monastery.
The remarks by Martin Hyland quoted in connection with Temple's work are of particular interest in light of the Pope's Christmas remark on mathematics quoted here yesterday.
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Sunday School
Apollo and the Tricksters
From The Story of N (Oct. 15, 2010)—
Roberta Smith on what she calls "endgame art"—
"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."
Margaret Atwood on tricksters and art—
"If it’s a seamless whole you want, pray to Apollo."
Here is some related material In memory of CIA officer Clare Edward Petty, who died at 90 on March 18—
A review of a sort of storyteller's MacGuffin — the 3×3 grid. This is, in Smith's terms, an "artistic focus" that appears to be visually comprehensible but is not as simple as it seems.
- The Pope in Plato's Cave (Sept. 16, 2006)
- A search for "einsatz" in this journal
- True Grid (Jan. 8, 2011)
- The Hesse configuration revisited (a post initially from the date of Petty's death, but later much reworked)
The Hesse configuration can serve as more than a sort of Dan Brown MacGuffin. As a post of January 14th notes, it can (rather fancifullly) illustrate the soul—
" … I feel I understand
Existence, or at least a minute part
Of my existence, only through my art,
In terms of combinational delight…."
— Vladimir Nabokov, Pale Fire
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Friday, April 15, 2011
Spider Notes
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Friday, April 8, 2011
Concepts of Space
Part I — Roberta Smith in today's New York Times —
"… the argument that painting may ultimately be about
little more than the communication of some quality of
light and space, however abstract or indirect."
– Review of "Rooms With a View" at the Met
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Pictorial version |
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“Space: what you damn well have to see.”
– James Joyce, Ulysses
Part II — Window from A Crooked House
"Teal lifted the blind a few inches. He saw nothing, and raised it a little more—still nothing. Slowly he raised it until the window was fully exposed. They gazed out at—nothing.
Nothing, nothing at all. What color is nothing? Don't be silly! What shape is it? Shape is an attribute of something . It had neither depth nor form. It had not even blackness. It was nothing ."
Part III — Not So Crooked: The Cabinet of Dr. Montessori
An April 5 Wall Street Journal article on Montessori schools, and…
A cabinet from Dr. Montessori's own
explanation of her method
Part IV — Pilate Goes to Kindergarten and The Seven
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Wednesday, April 6, 2011
De Marrais Memorial
In memory of Robert de Marrais, an excerpt from an obituary at Legacy.com—
Robert “Bob” Paul de Marrais died April 4, 2011 in Boston, Mass. One measure of a life is those that grieve our absence. Bob is dearly missed. He is survived by his 92 year old mother Yvette (nee Pétronille) in NY, his brother John A. in NY, his Aunt Mae in NJ; three children Luc, Sylvie, and Nathalie in Mass, and his devoted wife Dali (nee Zangurashvili) from Georgia of the ex-Soviet-Union. Bob was born Nov. 30, 1948, grew up in Cresskill, NJ, made life-long friends during some of his happiest days at MIT in Mass., and did not wander far from there for the rest of his life. He had a lifelong interest in history, his French heritage, music, history of science, and multidimensional algebras. His wife, friends Izzy and Mitch, brother John (and wife Caroline), little nephew Louis J., and two of his own children got to say goodbye. He found the energy to reward us with a smile. Bob has now joined his loving dad Louis J., Uncle Jack, Aunt Ginny, Uncle Gil, et. al.
For some details of de Marrais's life, see a separate biography from Legacy.com.
Related material— A search for "deMarrais" in this journal. (The name often occurs only within links.)
Cached copies of the 5-part "Kaleidoscopes" work by de Marrais referred to in the search can be found here.
A more personal note, from a quotation linked to here on the date of de Marrais's death—
… and who shall ever tell the sorrow of being on this earth,
lying, on quilts, on the grass, in a summer evening, among the sounds of the night.
May God bless my people, my uncle, my aunt, my mother, my good father,
oh, remember them kindly in their time of trouble;
and in the hour of their taking away.
After a little I am taken in and put to bed.
— James Agee, "Knoxville: Summer of 1915"
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Tuesday, March 29, 2011
Diamond Star
From last night's note on finite geometry—
"The (83, 83) Möbius-Kantor configuration here described by Coxeter is of course part of the larger (94, 123) Hesse configuration. Simply add the center point of the 3×3 Galois affine plane and the four lines (1 horizontal, 1 vertical, 2 diagonal) through the center point." An illustration—
This suggests a search for "diamond+star."
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Design Theory
See the new note Configurations and Squares at finitegeometry.org/sc/.
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Sunday, March 27, 2011
A Many-Sided Theory
On this date 106 years ago…
Prefatory note from Hudson's classic Kummer's Quartic Surface ,
Cambridge University Press, 1905—
RONALD WILLIAM HENRY TURNBULL HUDSON would have
been twenty-nine years old in July of this year; educated at
St Paul's School, London, and at St John's College, Cambridge,
he obtained the highest honours in the public examinations of the
University, in 1898, 1899, 1900; was elected a Fellow of St John's
College in 1900; became a Lecturer in Mathematics at University
College, Liverpool, in 1902; was D.Sc. in the University of London
in 1903; and died, as the result of a fall while climbing in Wales,
in the early autumn of 1904….
A many-sided theory such as that of this volume is
generally to be won only by the work of many lives;
one who held so firmly the faith that the time is well spent
could ill be spared.
— H. F. Baker, 27 March 1905
For some more recent remarks related to the theory, see
Defining Configurations and its updates, March 20-27, 2011.
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Friday, March 18, 2011
Defining Configurations*
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:
- DEFINITION: A combinatorial configuration of type (n_3) consists of an (abstract) set of n points together with a set of n triples of points, called lines, such that each point belongs to 3 lines and each line contains 3 points.
- EXAMPLE: The unique (8_3) configuration consists of the triples 125, 148, 167, 236, 278, 347, 358, 456.
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.
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Thursday, March 17, 2011
Remarks on Reality
Conclusion of “The Place of Pure Mathematics” —
“Dogmas and philosophies, it would seem, rise and fall. But gradually accumulating throughout the ages, from the earliest dawn of history, there is a body of doctrine, a reasoned insight into the relations of exact ideas, painfully won and often tested. And this remains the main heritage of man; his little beacon of light amidst the solitudes and darknesses of infinite space; or, if you prefer, like the shout of children at play together in the cultivated valleys, which continues from generation to generation.
Yes, and continues for ever! A universe which has the potentiality of becoming thus conscious of itself is not without something of which that which we call memory is but an image. Somewhere, somehow, in ways we dream not of, when you and I have merged again into the illimitable whole, when all that is material has ceased, the faculty in which we now have some share, shall surely endure; the conceptions we now dimly struggle to grasp, the joy we have in the effort, these are but part of a greater whole. Some may fear, and some may hope, that they and theirs shall not endure for ever. But he must have studied Nature in vain who does not see that our spiritual activities are inherent in the mighty process of which we are part; who can doubt of their persistence.
And, on the intellectual side, of all that is best ascertained, and surest, and most definite, of these; of all that is oldest and most universal; of all that is most fundamental and far-reaching, of these activities, Pure Mathematics is the symbol and the sum.”
— From a 1913 address by geometry saint Henry Frederick Baker, who died on this date in 1956
The feast of another saint, Patrick, also falls on 3/17. The date itself is related, if only by chance, to the following remark—
“317 is a prime, not because we think so,
or because our minds are shaped in one way
rather than another, but because it is so,
because mathematical reality is built that way.”
— From a 1940 book by the somewhat less saintly number theorist G. H. Hardy
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Wednesday, March 16, 2011
Time and Chance (continued)
Accidental Time and Space
New York Lottery today— midday 987, evening 522.
Time
The midday 987 may be interpreted as "…nine, eight, seven, …."—
"The countdown as we know it, 10-9-8-u.s.w.,
was invented by Fritz Lang in 1929 for
the Ufa film Die Frau im Mond . He put it into
the launch scene to heighten the suspense.
'It is another of my damned "touches,"' Fritz Lang said."
Space
The evening 522 suggests the date 5/22. From that date last year—
Art Space (2:02 AM EDT)
|
Pictorial version |
|
“Space: what you damn well have to see.”
– James Joyce, Ulysses
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Sunday, March 13, 2011
The Counter
"…as we saw, there are two different Latin squares of order 4…."
— Peter J. Cameron, "The Shrikhande Graph," August 26, 2010
Cameron counts Latin squares as the same if they are isotopic .
Some further context for Cameron's remark—
Cover Illustration Number 1 (1976):
Cover Illustration Number 2 (1991):
The Shrikhande Graph
______________________________________________________________________________
This post was prompted by two remarks…
1. In a different weblog, also on August 26, 2010—
The Accidental Mathematician— "The Girl Who Played with Fermat's Theorem."
"The worst thing about the series is the mathematical interludes in The Girl Who Played With Fire….
Salander is fascinated by a theorem on perfect numbers—
one can verify it for as many numbers as one wishes, and it never fails!—
and then advances through 'Archimedes, Newton, Martin Gardner,*
and a dozen other classical mathematicians,' all the way to Fermat’s last theorem."
2. "The fact that the pattern retains its symmetry when you permute the rows and columns
is very well known to combinatorial theorists who work with matrices."
[My italics; note resemblance to the Brualdi-Ryser title above.]
–Martin Gardner in 1976 on the diamond theorem
* Compare Eric Temple Bell (as quoted at the MacTutor history of mathematics site)—
"Archimedes, Newton, and Gauss, these three, are in a class by themselves
among the great mathematicians, and it is not for ordinary mortals
to attempt to range them in order of merit."
This is from the chapter on Gauss in Men of Mathematics .
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Friday, March 11, 2011
Table Talk
The following was suggested by a link within this evening's earlier Kane site link.
Peter J. Cameron's weblog on August 26, 2010—
A Latin square of order n is a
|
|
Some related literary remarks—
Proginoskes and Latin Squares.
See also "It was a perfectly ordinary night at Christ's high table…."
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Thursday, March 10, 2011
Paradigms Lost
(Continued from February 19)
The cover of the April 1, 1970 second edition of The Structure of Scientific Revolutions , by Thomas S. Kuhn—
This journal on January 19, 2011—
If Galois geometry is thought of as a paradigm shift from Euclidean geometry,
both images above— the Kuhn cover and the nine-point affine plane—
may be viewed, taken together, as illustrating the shift. The nine subcubes
of the Euclidean 3x3x3 cube on the Kuhn cover do not form an affine plane
in the coordinate system of the Galois cube in the second image, but they
at least suggest such a plane. Similarly, transformations of a
non-mathematical object, the 1974 Rubik cube, are not Galois transformations,
but they at least suggest such transformations.
See also today's online Harvard Crimson illustration of problems of translation—
not unrelated to the problems of commensurability discussed by Kuhn.
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Monday, March 7, 2011
Point Taken
Recommended— An essay (part 1 of 5 parts) in today's New York TImes—
|
I don’t want to die in |
"I agree with one of the earlier commenters that this is a piece of fine literary work. And in response to some of those who have wondered 'WHAT IS THE POINT?!' of this essay, I would like to say: Must literature always answer that question for us (and as quickly and efficiently as possible)?"
For an excellent survey of the essay's historical context, see The Stanford Encyclopedia of Philosophy article
"The Incommensurability of Scientific Theories,"
First published Wed., Feb. 25, 2009,
by Eric Oberheim and Paul Hoyningen-Huene.
Related material from this journal—
Paradigms, Paradigms Lost, and a search for "mere geometry." This last includes remarks contrasting Euclid's definition of a point ("that which has no parts") with a later notion useful in finite geometry.
See also (in the spirit of The Abacus Conundrum )…
(Note the Borges epigraph above.)
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Punto
"Time it goes so fast
When you're having fun"
"….mirando il punto 
a cui tutti li tempi son presenti"
– Dante, Paradiso , XVII, 17-18
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Saturday, March 5, 2011
Personal Link
A search for some background on Dmitri Tymoczko, the subject of yesterday's evening entry on music theory, shows that his name and mine once both appeared in the same web page— "This Week's Finds in Mathematical Physics (Week 234)," by John Baez, June 12, 2006 (linked to by the Wikipedia article on transformational music theory).
In that page, Baez speculates on the possibility of a connection between music theory and Mathieu groups and says—
"For a pretty explanation of M24, also try this:
Steven H. Cullinane, Geometry of the 4 × 4 square, http://finitegeometry.org/sc/16/geometry.html."
I know of no connection* between the groups I discussed there and music theory. For some background on Tymoczko's work, see the helpful survey "Exploring Musical Space," by Julian Hook (Science magazine, 7 July 2006).
* Apart, that is, from the tesseract (see Geometry of the 4 × 4 Square) shown by Tymoczko in a 2010 lecture—
This is perhaps "Chopin's tesseract" from section 8.5 of Tymoczko's new book
A Geometry of Music (Oxford University Press, 2011).
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Friday, March 4, 2011
Ageometretos Medeis Eisito*
Your mission, should you choose to accept it…
See also “Mapping Music” from Harvard Magazine , Jan.-Feb. 2007—
“Life inside an orbifold is a non-Euclidean world”
— as well as the cover story “The Shape of Music” from Princeton Alumni Weekly ,
Feb. 9, 2011, and “Bead Game” + music in this journal (click, then scroll down).
Those impressed by the phrase “non-Euclidean” may also enjoy
Non-Euclidean Blocks and Pilate Goes to Kindergarten.
The “Bead Game” + music search above includes, notably, a passage describing a
sort of non-Euclidean abacus in the classic 1943 story “Mimsy Were the Borogoves.”
For a visually related experience, see the video “Chord Geometries Demo: Chopin
on a Mobius Strip” at a music.princeton.edu web page.
* Motto of the American Mathematical Society, said to be also the motto of Plato’s Academy.
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Wednesday, March 2, 2011
Labyrinth of the Line
“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”
Another single-line labyrinth—
Robert A. Wilson on the projective line with 24 points
and its image in the Miracle Octad Generator (MOG)—
Related material —
The remarks of Scott Carnahan at Math Overflow on October 25th, 2010
and the remarks at Log24 on that same date.
A search in the latter for miracle octad is updated below.
This search (here in a customized version) provides some context for the
Benedictine University discussion described here on February 25th and for
the number 759 mentioned rather cryptically in last night’s “Ariadne’s Clue.”
Update of March 3— For some historical background from 1931, see The Mathieu Relativity Problem.
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Monday, February 28, 2011
His Kind of Diamond
In memory of Jane Russell —
H.S.M. Coxeter's classic
Introduction to Geometry (2nd ed.):
Note the resemblance of the central part to
a magical counterpart— the Ojo de Dios
of Mexico's Sierra Madre.
Related material— page 55 of Polly and the Aunt ,
by Mary E. Blatchford.
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Friday, February 25, 2011
Diamond Theorem Exposition
"THE DIAMOND THEOREM AND QUILT PATTERNS
Victoria Blumen, Mathematics, Junior, Benedictine University
Tim Comar, Benedictine University
Mathematics
Secondary Source Research
Let D be a 4 by 4 block quilt shape, where each of the 16 square blocks is consists of [sic ] two triangles, one of which is colored red and the other of which is colored blue. Let G: D -> D_g be a mapping of D that interchanges a pair of columns, rows, or quadrants of D. The diamond theorem states that G(D) = D_g has either ordinary or color-interchange symmetry. In this talk, we will prove the diamond theorem and explore symmetries of quilt patterns of the form G(D)."
Exercise— Correct the above statement of the theorem.
Background— This is from a Google search result at about 10:55 PM ET Feb. 25, 2011—
[DOC] THE DIAMOND THEOREM AND QUILT PATTERNS – acca.elmhurst.edu
File Format: Microsoft Word – 14 hours ago –
Let G: D -> D_g be a mapping of D that interchanges a pair of columns, rows, or quadrants of D. The diamond theorem states that G(D) = D_g has either …
acca.elmhurst.edu/…/victoria_blumen9607_
THE%20DIAMOND%20THEOREM%20AND%20QUILT%20PATTERNS…
The document is from a list of mathematics abstracts for the annual student symposium of the ACCA (Associated Colleges of the Chicago Area) held on April 10, 2010.
Update of Feb. 26— For a related remark quoted here on the date of the student symposium, see Geometry for Generations.
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Monday, February 21, 2011
The Abacus Conundrum*
From Das Glasperlenspiel (Hermann Hesse, 1943) —
“Bastian Perrot… constructed a frame, modeled on a child’s abacus, a frame with several dozen wires on which could be strung glass beads of various sizes, shapes, and colors. The wires corresponded to the lines of the musical staff, the beads to the time values of the notes, and so on. In this way he could represent with beads musical quotations or invented themes, could alter, transpose, and develop them, change them and set them in counterpoint to one another. In technical terms this was a mere plaything, but the pupils liked it.… …what later evolved out of that students’ sport and Perrot’s bead-strung wires bears to this day the name by which it became popularly known, the Glass Bead Game.”
From “Mimsy Were the Borogoves” (Lewis Padgett, 1943)—
…”Paradine looked up. He frowned, staring. What in—
…”Is that an abacus?” he asked. “Let’s see it, please.”
…Somewhat unwillingly Scott brought the gadget across to his father’s chair. Paradine blinked. The “abacus,” unfolded, was more than a foot square, composed of thin, rigid wires that interlocked here and there. On the wires the colored beads were strung. They could be slid back and forth, and from one support to another, even at the points of jointure. But— a pierced bead couldn’t cross interlocking wires—
…So, apparently, they weren’t pierced. Paradine looked closer. Each small sphere had a deep groove running around it, so that it could be revolved and slid along the wire at the same time. Paradine tried to pull one free. It clung as though magnetically. Iron? It looked more like plastic.
…The framework itself— Paradine wasn’t a mathematician. But the angles formed by the wires were vaguely shocking, in their ridiculous lack of Euclidean logic. They were a maze. Perhaps that’s what the gadget was— a puzzle.
…”Where’d you get this?”
…”Uncle Harry gave it to me,” Scott said on the spur of the moment. “Last Sunday, when he came over.” Uncle Harry was out of town, a circumstance Scott well knew. At the age of seven, a boy soon learns that the vagaries of adults follow a certain definite pattern, and that they are fussy about the donors of gifts. Moreover, Uncle Harry would not return for several weeks; the expiration of that period was unimaginable to Scott, or, at least, the fact that his lie would ultimately be discovered meant less to him than the advantages of being allowed to keep the toy.
…Paradine found himself growing slightly confused as he attempted to manipulate the beads. The angles were vaguely illogical. It was like a puzzle. This red bead, if slid along this wire to that junction, should reach there— but it didn’t. A maze, odd, but no doubt instructive. Paradine had a well-founded feeling that he’d have no patience with the thing himself.
…Scott did, however, retiring to a corner and sliding beads around with much fumbling and grunting. The beads did sting, when Scott chose the wrong ones or tried to slide them in the wrong direction. At last he crowed exultantly.
…”I did it, dad!”
…””Eh? What? Let’s see.” The device looked exactly the same to Paradine, but Scott pointed and beamed.
…”I made it disappear.”
…”It’s still there.”
…”That blue bead. It’s gone now.”
…Paradine didn’t believe that, so he merely snorted. Scott puzzled over the framework again. He experimented. This time there were no shocks, even slight. The abacus had showed him the correct method. Now it was up to him to do it on his own. The bizarre angles of the wires seemed a little less confusing now, somehow.
…It was a most instructive toy—
…It worked, Scott thought, rather like the crystal cube.
* Title thanks to Saturday Night Live (Dec. 4-5, 2010).
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Sunday, February 20, 2011
Sermon
From Galleri MGM in Oslo —
Josefine Lyche "Why do we remember the past, but not the future?" "If you want a vision of the future, imagine a boot stamping on a human face – forever." Galleri MGM is pleased to present the second solo exhibition by Josefine Lyche (b.1973). Lyche presents a body of work consisting of sculptures and a wall painting, as well as a series of new paperwork, all using geometrical abstractions and light as a medium. Her work dissolves boundaries between fiction and documentation, depicting how fantasies and dreams collide with and yet help determine the shape of reality. Theme and variations is a term most commonly used in the music genre as a musical form in which the fundamental musical idea, or theme, is repeated in altered form or accompanied in a different manner. The exhibition explores geometric shapes and solids and revisits work of artists like Robert Morris and Ellsworth Kelly, giving it a disco treatment of glitter, neon and gloss. The mathematical, science-fiction and new age references incorporated in the works comments on the ambivalent foretelling of utopian hope and dystopian vision of a near, yet unknown future. The transmission between past and future is shown in the sculpture "The Omega Point" a portal that leads in or out of time and space. … |
A connection to today's earlier post, Sunday School— The Oslo Version, from Friday, May 21, 2010.
Lyche's "Omega Point" portal, together with her last name, suggested three posts from the following Saturday morning— which later proved to be the date of Martin Gardner's death—
Art Space, Through the Lyche Gate and The Lyche Gate Asterisk.
For some further religious remarks, see November 9th, 2010— A Theory of Pure Design.
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Sunday School
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Saturday, February 19, 2011
Brightness at Noon (continued)
From Epiphany Revisited —
A star figure and the Galois quaternion.
The square root of the former is the latter.
… Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte.
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Thursday, February 17, 2011
Paradigms
"These passages suggest that the Form is a character or set of characters
common to a number of things, i.e. the feature in reality which corresponds
to a general word. But Plato also uses language which suggests not only
that the forms exist separately (χωριστά ) from all the particulars, but also
that each form is a peculiarly accurate or good particular of its own kind,
i.e. the standard particular of the kind in question or the model (παράδειγμα )
[i.e. paradigm ] to which other particulars approximate….
… Both in the Republic and in the Sophist there is a strong suggestion
that correct thinking is following out the connexions between Forms.
The model is mathematical thinking, e.g. the proof given in the Meno
that the square on the diagonal is double the original square in area."
— William and Martha Kneale, The Development of Logic,
Oxford University Press paperback, 1985
Plato's paradigm in the Meno —
Changed paradigm in the diamond theorem (2×2 case) —
Aspects of the paradigm change* —
Monochrome figures to
colored figures
Areas to
transformations
Continuous transformations to
non-continuous transformations
Euclidean geometry to
finite geometry
Euclidean quantities to
finite fields
Some pedagogues may find handling all of these
conceptual changes simultaneously somewhat difficult.
* "Paradigm shift " is a phrase that, as John Baez has rightly pointed out,
should be used with caution. The related phrase here was suggested by Plato's
term παράδειγμα above, along with the commentators' specific reference to
the Meno figure that serves as a model. (For "model" in a different sense,
see Burkard Polster.) But note that Baez's own beloved category theory
has been called a paradigm shift.
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The Form, the Pattern
"… Only by the form, the pattern,
Can words or music reach
The stillness…."
— T. S. Eliot,
Four Quartets
For further details, see Time Fold.
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Monday, February 14, 2011
Simplify (continued)
"Plato acknowledges how khora challenges our normal categories
of rational understanding. He suggests that we might best approach it
through a kind of dream consciousness."
—Richard Kearney, quoted here yesterday afternoon
"You make me feel like I'm living a teenage dream."
— Song at last night's Grammy awards
Richard Kiley in "Blackboard Jungle" (1955)
Note the directive on the blackboard.
Quoted here last year on this date—
Alexandre Borovik's Mathematics Under the Microscope (American Mathematical Society, 2010)—
"Once I mentioned to Gelfand that I read his Functions and Graphs ; in response, he rather sceptically asked me what I had learned from the book. He was delighted to hear my answer: 'The general principle of always looking at the simplest possible example.'….
So, let us look at the principle in more detail:
Always test a mathematical theory on the simplest possible example…
This is a banality, of course. Everyone knows it; therefore, almost no one follows it."
Related material— Geometry Simplified and A Simple Reflection Group of Order 168.
"Great indeed is the riddle of the universe.
Beautiful indeed is the source of truth."
– Shing-Tung Yau, Chairman,
Department of Mathematics, Harvard University
"Always keep a diamond in your mind."
– King Solomon at the Paradiso
Image from stoneship.org
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Wednesday, February 9, 2011
An Abstract Window
The sliding window in blue below
Click for the web page shown.
is an example of a more general concept.
Such a sliding window,* if one-dimensional of length n , can be applied to a sequence of 0's and 1's to yield a sequence of n-dimensional vectors. For example— an "m-sequence" (where the "m" stands for "maximum length") of length 63 can be scanned by a length-6 sliding window to yield all possible 6-dimensional binary vectors except (0,0,0,0,0,0).
For details, see A Galois Field—
The image is from Bert Jagers at his page on the Galois field GF(64) that he links to as "A Field of Honor."
For a discussion of the m-sequence shown in circular form above, see Jagers's "Pseudo-Random Sequences from GF(64)." Here is a noncircular version of the length-63 m-sequence described by Jagers (with length scale below)—
100000100001100010100111101000111001001011011101100110101011111
123456789012345678901234567890123456789012345678901234567890123
This m-sequence may be viewed as a condensed version of 63 of the 64 I Ching hexagrams. (See related material in this journal.)
For a more literary approach to the window concept, see The Seventh Symbol (scroll down after clicking).
* Moving windows also appear (in a different way) In image processing, as convolution kernels .
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An Abstract Power
Two characters named “Black” and “White” debate religion and the afterlife in the Cormac McCarthy play “The Sunset Limited.”
The play opened in Chicago in a Steppenwolf Theatre production on May 18, 2006.
A New York Times theater review from All Hallows’ Eve, 2006—
“…there is an abstract power in the mysteriousness of Mr. McCarthy’s
vision’s allowing for a multitude of interpretations.” –Jason Zinoman
The current New Yorker (Feb.14) has a note
by Lillian Ross on the same play— “Two-Man Show: O Death”
Some purely visual black-and-white variations that are less dramatic, but have their own “abstract power”—
A book cover pictured here last November to contrast with
“the sound and fury of the rarified Manhattan art world”—
and a web page with multiple interpretations of the book cover’s pattern—
A synchronicity— The first version of “Symmetry Framed” was done
on May 18, 2006— the day “The Sunset Limited” opened.
Another synchronicity relates the mathematics underlying
such patterns to the Halloween date of the above review.
See “To Announce a Faith,” from October 31, 2006.
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Saturday, February 5, 2011
Zen and the Art of Philosophy
| An Ordinary Evening in New Haven | |
|---|---|
| line 540 (xxx.18): | In which hundreds of eyes, in one mind, see at once. |
The cover art of a 1976 monograph, "Diamond Theory," was described in this morning's post.
As Madeleine L'Engle noted in 1976, the cover art resembles the character Proginoskes in her novel A Wind in the Door.
A search today for Proginoskes yields a description by Brendan Kidwell…
A link at Kidwell's site leads to a weblog by Jeff Atwood, a founder of Stack Overflow, a programmers' question-and-answer site.
(Stack Overflow is said to have inspired the similar site for mathematicians, Math Overflow.)
Yesterday Atwood discussed technical writing.
This suggests a look at Robert M. Pirsig on that subject in his 1974 philosophical novel Zen and the Art of Motorcycle Maintenance.
(See also a document on Pirsig's technical-writing background.)
Pirsig describes his novel as "a sort of Chautauqua."
This, together with the Stevens and Proginoskes quotes above, leads back to the Log24 Feb. 1 post The Search.
An image from that post (click to enlarge)—
Here the apparently fragmented nature of the set of
images imagined as rising above the podium of the
Hall of Philosophy at Chautauqua rather naturally
echoes Stevens's "hundreds of eyes" remark.
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Cover Art
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Thursday, February 3, 2011
Brightness at Noon (continued)
"The predicate of bright origin"
— A phrase of Wallace Stevens from "An Ordinary Evening in New Haven" (1950)
Perhaps the predicate Stevens means is "bright."
If so, an apt illustration can be found on the cover of
the 1943 first edition of Hesse's Glasperlenspiel—
See also Stevens's use of the phrase "heaven-haven" in "Notes" (1942),
the original plan of New Haven, and related scholia in this journal.
… Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte.
– Rubén Darío
An academic work from 2003 discusses Stevens's "Notes" as
"a perfect geometric whole."
Note that "perfect" means "complete, finished, done."
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Wednesday, February 2, 2011
Origin
"the predicate* of bright origin"
— A phrase of Wallace Stevens quoted here yesterday
One origin, noted here on January 25—
This commemorated the death of noted discographer Brian Rust.
Rust appears in today's New York Times obituary index—
Also in today's obituaries: artist Alan Uglow, who reportedly died on January 20.
A link ("Noland") from this journal on that date leads to… a geometric origin.
|
“At the still point, |
|
What Stevens's "predicate" is, I do not know.
Eliot's predicate would seem to be "still."
Related material— The dance from "Pulp Fiction"** illustrated here
on the Feast of St. Michael and All Angels last year.
* Some background for the Hall of Philosophy (yesterday's post)—
"Unity of the Proposition" at Wikipedia and at Oxford University Press.
** A flickr.com page gives examples. (The link is thanks to The Ghost Light).
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Thursday, January 27, 2011
Mathematics and Narrative, continued…
Indiana Jones and the Magical Oracle
Mathematician Ken Ono in the December 2010 American Mathematical Society Notices—
The "dying genius" here is Ramanujan, not Galois. The story now continues at the AMS website—
(Excerpt from Jan. 27 screenshot;
the partitions story has been the top
news item at the site all week.)
From a Jan. 20, 2011, Emory University press release —
"Finite formula found for partition 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. P does not return gruesome numbers with infinitely many decimal places. It's the finite, algebraic formula that we have all been looking for."
For an introduction to the magical oracle, see a preprint, "Bruinier-Ono," at the American Institute of Mathematics website.
Ono also discussed the oracle in a video (see minute 25) recorded Jan. 21 and placed online today.
See as well "Exact formulas for the partition function?" at mathoverflow.net.
A Nov. 29, 2010, remark by Thomas Bloom on that page leads to a 2006 preprint by Ono and Kathrin Bringmann, "An Arithmetic Formula for the Partition Function*," that seems not unrelated to Ono's new "magical oracle" formula—
The Bruinier-Ono paper does not mention the earlier Bringmann-Ono work.
(Both the 2011 Bruinier-Ono paper and the 2006 Bringmann-Ono paper mention their debt to a 2002 work by Zagier— Don Zagier, "Traces of singular moduli," in Motives, Polylogarithms and Hodge theory, Part II (Irvine, CA, 1998), International Press Lecture Series 3 (International Press, Somerville, MA, 2002), pages 211-244.)
Some background for those who prefer mathematics to narrative—
The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series ,
by Ken Ono, American Mathematical Society CBMS Series, 2004.
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Wednesday, January 26, 2011
Early Nothing
(Continued from yesterday)
Today's New York Times obituaries —
From Wes Clark's site Web Noir —
Scenes from "The Set-Up," a 1949 noir classic by Robert Wise—
From Bruce Gordon's obituary in today's New York Times —
"Mr. Gordon appeared on Broadway many times. He was in the original cast of the hit comedy 'Arsenic and Old Lace,' which opened in 1941 and starred . Uncharacteristically, given his later résumé, Mr. Gordon played a policeman." —Margalit Fox
Related material —
(See Savage Solstice in this journal on December 21st, 2010.)
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Tuesday, January 25, 2011
Another Reappearing Number
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Saturday, January 22, 2011
High School Squares*
The following is from the weblog of a high school mathematics teacher—
This is related to the structure of the figure on the cover of the 1976 monograph Diamond Theory—
Each small square pattern on the cover is a Latin square,
with elements that are geometric figures rather than letters or numerals.
All order-four Latin squares are represented.
For a deeper look at the structure of such squares, let the high-school
chart above be labeled with the letters A through X, and apply the
four-color decomposition theorem. The result is 24 structural diagrams—
Some of the squares are structurally congruent under the group of 8 symmetries of the square.
This can be seen in the following regrouping—
(Image corrected on Jan. 25, 2011– "seven" replaced "eight.")
* Retitled "The Order-4 (i.e., 4×4) Latin Squares" in the copy at finitegeometry.org/sc.
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Thursday, January 20, 2011
Brightness at Noon, continued
"One wild rhapsody a fake for another."
– Wallace Stevens, "Arrival at the Waldorf," in Parts of a World (1942)
"Camelot is an illusion.
That doesn't matter, according to Catherine.
Camelot is an artificial construction, a public perception.
The things that matter are closer, deeper, self-generated, unkillable.
You've got to grow up to discover what those things are."
— Dan Zak, Washington Post movie review on Feb. 27, 2009. See also this journal on that date.
See as well a note on symmetry from Christmas Eve, 1981, and Verbum in this journal.
Some philosophical background— Derrida in the Garden.
Some historical background— A Very Private Woman and Noland.
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Wednesday, January 19, 2011
Intermediate Cubism
The following is a new illustration for Cubist Geometries—
(For elementary cubism, see Pilate Goes to Kindergarten and The Eightfold Cube.
For advanced, see Solomon's Cube and Geometry of the I Ching .)
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Friday, January 14, 2011
Ironic Butterfly
David Brooks's column today quotes Niebuhr. From the same source—
Reinhold Niebuhr, The Irony of American History—
Chapter 8: The Significance of Irony
Any interpretation of historical patterns and configurations raises the question whether the patterns, which the observer discerns, are "objectively" true or are imposed upon the vast stuff of history by his imagination. History might be likened to the confusion of spots on the cards used by psychiatrists in a Rorschach test. The patient is asked to report what he sees in these spots; and he may claim to find the outlines of an elephant, butterfly or frog. The psychiatrist draws conclusions from these judgments about the state of the patient’s imagination rather than about the actual configuration of spots on the card. Are historical patterns equally subjective?
….
The Biblical view of human nature and destiny moves within the framework of irony with remarkable consistency. Adam and Eve are expelled from the Garden of Eden because the first pair allowed "the serpent" to insinuate that, if only they would defy the limits which God had set even for his most unique creature, man, they would be like God. All subsequent human actions are infected with a pretentious denial of human limits. But the actions of those who are particularly wise or mighty or righteous fall under special condemnation. The builders of the Tower of Babel are scattered by a confusion of tongues because they sought to build a tower which would reach into the heavens.
Niebuhr's ironic butterfly may be seen in the context of last
Tuesday's post Shining and of last Saturday's noon post True Grid—
The "butterfly" in the above picture is a diagram showing the 12 lines* of the Hesse configuration from True Grid.
It is also a reference to James Hillman's classical image (see Shining) of the psyche, or soul, as a butterfly.
Fanciful, yes, but this is in exact accordance with Hillman's remarks on the soul (as opposed to the spirit— see Tuesday evening's post).
The 12-line butterfly figure may be viewed as related to the discussions of archetypes and universals in Hillman's Re-Visioning Psychology and in Charles Williams's The Place of the Lion . It is a figure intended here to suggest philosophy, not entertainment.
Niebuhr and Williams, if not the more secular Hillman, might agree that those who value entertainment above all else may look forward to a future in Hell (or, if they are lucky, Purgatory). Perhaps such a future might include a medley of Bob Lind's "Elusive Butterfly" and Iron Butterfly's "In-a-Gadda-da-Vida."
* Three horizontal, three vertical, two diagonal, and four arc-shaped.
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Tuesday, January 11, 2011
Soul and Spirit
This morning's post, "Shining," gave James Hillman's 1976 remarks
on the distinction between soul and spirit .
The following images may help illustrate these concepts.
The distinction as illustrated by Jeff Bridges —
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Soul
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Spirit
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The mirror has two faces (at least).
Postscript from a story, "The Zahir," in the Borges manner,
by Mark Jason Dominus (programmer of the quilt designs above)—
"I left that madhouse gratefully."
Dominus is also the author of…
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Saturday, January 8, 2011
True Grid (continued)
"Rosetta Stone" as a Metaphor
in Mathematical Narratives
For some backgound, see Mathematics and Narrative from 2005.
Yesterday's posts on mathematics and narrative discussed some properties
of the 3×3 grid (also known as the ninefold square ).
For some other properties, see (at the college-undergraduate, or MAA, level)–
Ezra Brown, 2001, "Magic Squares, Finite Planes, and Points of Inflection on Elliptic Curves."
His conclusion:
When you are done, you will be able to arrange the points into [a] 3×3 magic square,
which resembles the one in the book [5] I was reading on elliptic curves….
This result ties together threads from finite geometry, recreational mathematics,
combinatorics, calculus, algebra, and number theory. Quite a feat!
5. Viktor Prasolov and Yuri Solvyev, Elliptic Functions and Elliptic Integrals ,
American Mathematical Society, 1997.
Brown fails to give an important clue to the historical background of this topic —
the word Hessian . (See, however, this word in the book on elliptic functions that he cites.)
Investigation of this word yields a related essay at the graduate-student, or AMS, level–
Igor Dolgachev and Michela Artebani, 2009, "The Hesse Pencil of Plane Cubic Curves ."
From the Dolgachev-Artebani introduction–
In this paper we discuss some old and new results about the widely known Hesse
configuration of 9 points and 12 lines in the projective plane P2(k ): each point lies
on 4 lines and each line contains 3 points, giving an abstract configuration (123, 94).
PlanetMath.org on the Hesse configuration—
A picture of the Hesse configuration–
(See Visualizing GL(2,p), a note from 1985).
Related notes from this journal —
From last November —
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From the December 2010 American Mathematical Society Notices—
Related material from this journal— Consolation Prize (August 19, 2010) |
From 2006 —
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Sunday December 10, 2006
“Function defined form, expressed in a pure geometry
– J. G. Ballard on Modernism
“The greatest obstacle to discovery is not ignorance –
— Daniel J. Boorstin, |
Also from 2006 —
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Sunday November 26, 2006
Rosalind Krauss "If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete. Or, to take a more up-to-date example…."
"He was looking at the nine engravings and at the circle,
"And it's whispered that soon if we all call the tune
The nine engravings of The Club Dumas
An example of the universal*– or, according to Krauss,
"This is the garden of Apollo, the field of Reason…."
For more on the field of reason, see
A reasonable set of "strange correspondences" Unreason is, of course, more popular. * The ninefold square is perhaps a "concrete universal" in the sense of Hegel: "Two determinations found in all philosophy are the concretion of the Idea and the presence of the spirit in the same; my content must at the same time be something concrete, present. This concrete was termed Reason, and for it the more noble of those men contended with the greatest enthusiasm and warmth. Thought was raised like a standard among the nations, liberty of conviction and of conscience in me. They said to mankind, 'In this sign thou shalt conquer,' for they had before their eyes what had been done in the name of the cross alone, what had been made a matter of faith and law and religion– they saw how the sign of the cross had been degraded."
– Hegel, Lectures on the History of Philosophy ,
"For every kind of vampire, |
And from last October —
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Friday, October 8, 2010
Starting Out in the Evening This post was suggested by last evening's post on mathematics and narrative and by Michiko Kakutani on Vargas Llosa in this morning's New York Times .
"One must proceed cautiously, for this road— of truth and falsehood in the realm of fiction— is riddled with traps and any enticing oasis is usually a mirage."
– "Is Fiction the Art of Lying?"* by Mario Vargas Llosa,
* The Web version's title has a misprint— |
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Friday, January 7, 2011
Coxeter and the Aleph
In a nutshell —
Epigraph to "The Aleph," a 1945 story by Borges:
O God! I could be bounded in a nutshell,
and count myself a King of infinite space…
— Hamlet, II, 2
The story in book form, 1949
A 2006 biography of geometer H.S.M. Coxeter:
The Aleph (implicit in a 1950 article by Coxeter):
The details:
Related material: Group Actions, 1984-2009.
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Thursday, January 6, 2011
Epiphany Riddle
"Spaces and geometries, those which we perceive,
which we can’t perceive, or which only some of us perceive,
are a recurring theme in Against the Day ."
"大哉大哉 宇宙之谜
美哉美哉 真理之源"
"Great indeed is the riddle of the universe.
Beautiful indeed is the source of truth."
— Shing-Tung Yau, Chairman,
Department of Mathematics, Harvard University
"Always keep a diamond in your mind."
— King Solomon at the Paradiso
Image from stoneship.org
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Wednesday, January 5, 2011
True Grid Example
See today's earlier posts Ode and True Grid (continued) and, in the latter's
context of tic-tac-toe war games — Balance, from Halloween 2005 —
“An asymmetrical balance is sought since it possesses more movement.
This is achieved by the imaginary plotting of the character
upon a nine-fold square, invented by some ingenious writer of the Tang dynasty.
If the square were divided in half or in four, the result would be symmetrical,
but the nine-fold square permits balanced asymmetry."
— Paraphrase of a passage in Chiang Yee's Chinese Calligraphy
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Sunday, January 2, 2011
A Universal Form
Simon Critchley today in the New York Times series "The Stone"—
Philosophy, among other things, is that living activity of critical reflection in a specific context, by which human beings strive to analyze the world in which they find themselves, and to question what passes for common sense or public opinion— what Socrates called doxa— in the particular society in which they live. Philosophy cuts a diagonal through doxa. It does this by raising the most questions of a universal form: “What is X?”
Actually, that's two diagonals. See Kulturkampf at the Times and Geometry of the I Ching .
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Horseness
"Art has to reveal to us ideas, formless spiritual essences."
— A character clearly talking nonsense, from the National Library section of James Joyce's Ulysses
"Unsheathe your dagger definitions. Horseness is the whatness of allhorse."
— A thought of Stephen Dedalus in the same Ulysses section
For a representation of horseness related to Singer's dagger definitions in Saturday evening's post, see Generating the Octad Generator and Art Wars: Geometry as Conceptual Art.
More seriously, Joyce's "horseness" is related to the problem of universals. For an illuminating approach to universals from a psychological point of view, see James Hillman's Re-Visioning Psychology (Harper Collins, 1977). (See particularly pages 154-157.)
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Wednesday, December 29, 2010
True Grid
Part I: True
Bulletin of the American Mathematical Society , October 2002, page 563 —
“… the study of symmetries of patterns led to… finite geometries….”
– David W. Henderson, Cornell University
This statement may be misleading, if not (see Part II below) actually false. In truth, finite geometries appear to have first arisen from Fano's research on axiom systems. See The Axioms of Projective Geometry by Alfred North Whitehead, Cambridge University Press, 1906, page 13.
Part II: Grid
For the story of how symmetries of patterns later did lead to finite geometries, see the diamond theorem.
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Toy Stories
From the Bulletin of the American Mathematical Society , October 2002, p. 563:
“To produce decorations for their weaving, pottery, and other objects, early artists experimented with symmetries and repeating patterns. Later the study of symmetries of patterns led to tilings, group theory, crystallography, finite geometries, and in modern times to security codes and digital picture compactifications. Early artists also explored various methods of representing existing objects and living things. These explorations led to… [among other things] computer-generated movies (for example, Toy Story ).”
– David W. Henderson, Cornell University
THE SOURCE —
From the weblog The Ghost Light on Christmas Day, 2010 —
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Tuesday, December 28, 2010
Insane Symmetry
Continued from yesterday's Church Diamond and from Dec. 17's Fare Thee Well —
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The San Francisco Examiner last year
on New Year's Eve — Entertainment
Discover the modern art of Amish quilts By: Leslie Katz 12/31/09 1:00 AM Arts editor
Quilts made by Amish women in Pennsylvania, Household handicrafts and heirlooms made by American women seen as precursors to modern art is one underlying thesis of “Amish Abstractions: Quilts from the Collection of Faith and Stephen Brown,” a provocative exhibit on view at the de Young Museum through June. Curated by Jill D’Alessandro of the Fine Arts Museums of San Francisco, the show features about 50 full-size and crib quilts made between 1880 and 1940 in Pennsylvania and the Midwest during what experts consider the apex of Amish quilt-making production. Faith and Stephen Brown, Bay Area residents who began collecting quilts in the 1970s after seeing one in a shop window in Chicago and being bowled over by its bold design, say their continued passion for the quilts as art is in part because they’re so reminiscent of paintings by modern masters like Mark Rothko, Josef Albers, Sol LeWitt and Ellsworth Kelly — but the fabric masterpieces came first. “A happy visual coincidence” is how the Browns and D’Alessandro define the connection, pointing to the brilliance in color theory, sophisticated palettes and complex geometry that characterize both the quilts and paintings. “There’s an insane symmetry to these quilts,” says D’Alessandro…. Read more at the San Francisco Examiner . |
The festive nature of the date of the above item, New Year's Eve, suggests Stephen King's
All work and no play makes Jack a dull boy.
and also a (mis)quotation from a photographer's weblog—
"Art, being bartender, is never drunk."
— Quotation from Peter Viereck misattributed to Randall Jarrell in
Art as Bartender and the Golden Gate.
By a different photographer —
See also…
We may imagine the bartender above played by Louis Sullivan.
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Monday, December 27, 2010
Church Diamond
Also known, roughly speaking, as confluence or the Church-Rosser property.
From “NYU Lambda Seminar, Week 2” —
[See also the parent page Seminar in Semantics / Philosophy of Language or:
What Philosophers and Linguists Can Learn From Theoretical Computer Science But Didn’t Know To Ask)]
A computational system is said to be confluent, or to have the Church-Rosser or diamond property, if, whenever there are multiple possible evaluation paths, those that terminate always terminate in the same value. In such a system, the choice of which sub-expressions to evaluate first will only matter if some of them but not others might lead down a non-terminating path.
The untyped lambda calculus is confluent. So long as a computation terminates, it always terminates in the same way. It doesn’t matter which order the sub-expressions are evaluated in.
A computational system is said to be strongly normalizing if every permitted evaluation path is guaranteed to terminate. The untyped lambda calculus is not strongly normalizing: ω ω doesn’t terminate by any evaluation path; and (\x. y) (ω ω) terminates only by some evaluation paths but not by others.
But the untyped lambda calculus enjoys some compensation for this weakness. It’s Turing complete! It can represent any computation we know how to describe. (That’s the cash value of being Turing complete, not the rigorous definition. There is a rigorous definition. However, we don’t know how to rigorously define “any computation we know how to describe.”) And in fact, it’s been proven that you can’t have both. If a computational system is Turing complete, it cannot be strongly normalizing.
There is no connection, apart from the common reference to an elementary geometric shape, between the use of “diamond” in the above Church-Rosser sense and the use of “diamond” in the mathematics of (Cullinane’s) Diamond Theory.
Any attempt to establish such a connection would, it seems, lead quickly into logically dubious territory.
Nevertheless, in the synchronistic spirit of Carl Jung and Arthur Koestler, here are some links to such a territory —
Link One — “Insane Symmetry” (Click image for further details)—
See also the quilt symmetry in this journal on Christmas Day.
Link Two — Divine Symmetry
(George Steiner on the Name in this journal on Dec. 31 last year (“All about Eve“)) —
“The links are direct between the tautology out of the Burning Bush, that ‘I am’ which accords to language the privilege of phrasing the identity of God, on the one hand, and the presumptions of concordance, of equivalence, of translatability, which, though imperfect, empower our dictionaries, our syntax, our rhetoric, on the other. That ‘I am’ has, as it were, at an overwhelming distance, informed all predication. It has spanned the arc between noun and verb, a leap primary to creation and the exercise of creative consciousness in metaphor. Where that fire in the branches has gone out or has been exposed as an optical illusion, the textuality of the world, the agency of the Logos in logic—be it Mosaic, Heraclitean, or Johannine—becomes ‘a dead letter.'”
– George Steiner, Grammars of Creation
(See also, from Hanukkah this year, A Geometric Merkabah and The Dreidel is Cast.)
Link Three – Spanning the Arc —
Part A — Architect Louis Sullivan on “span” (see also Kindergarten at Stonehenge)
Part B — “Span” in category theory at nLab —
Also from nLab — Completing Spans to Diamonds
“It is often interesting whether a given span in some partial ordered set can be completed into a diamond. The property of a collection of spans to consist of spans which are expandable into diamonds is very useful in the theory of rewriting systems and producing normal forms in algebra. There are classical results e.g. Newman’s diamond lemma, Širšov-Bergman’s diamond lemma (Širšov is also sometimes spelled as Shirshov), and Church-Rosser theorem (and the corresponding Church-Rosser confluence property).”
The concepts in this last paragraph may or may not have influenced the diamond theory of Rudolf Kaehr (apparently dating from 2007).
They certainly have nothing to do with the Diamond Theory of Steven H. Cullinane (dating from 1976).
For more on what the above San Francisco art curator is pleased to call “insane symmetry,” see this journal on Christmas Day.
For related philosophical lucubrations (more in the spirit of Kaehr than of Steiner), see the New York Times “The Stone” essay “Span: A Remembrance,” from December 22—
“To understand ourselves well,” [architect Louis] Sullivan writes, “we must arrive first at a simple basis: then build up from it.”
Around 300 BC, Euclid arrived at this: “A point is that which has no part. A line is breadthless length.”
See also the link from Christmas Day to remarks on Euclid and “architectonic” in Mere Geometry.
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Saturday, December 25, 2010
Not-So-Fearful Symmetry
From a Mennonite homeschooling family —
It's a start. For more advanced remarks from the same date, see Mere Geometry.
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Wednesday, December 22, 2010
Reflections
Published on November 10, 2009 —
The above book may be regarded as an ironic answer to a question posed here on that date—
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“Public commentators assumed the air of kindergarten teachers who had to protect their children from thinking certain impermissible and intolerant thoughts.” – David Brooks in the Nov. 10, 2009, New York Times What else is new? |
For related kindergarten thoughts, see Finite Geometry and Physical Space.
For the connection of the kindergarten thoughts to reflections, see A Simple Reflection Group of Order 168.
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Tuesday, December 21, 2010
Savage Solstice
In memory of kaleidoscope enthusiast Cozy Baker, who died at 86, according to Saturday's Washington Post , on October 19th.
This journal on that date — Savage Logic and Savage Logic continued.
See this journal on All Saints' Day 2006 for some background to those posts—
“Savage logic works like a kaleidoscope whose chips can fall into a variety of patterns while remaining unchanged in quantity, form, or color. The number of patterns producible in this way may be large if the chips are numerous and varied enough, but it is not infinite. The patterns consist in the disposition of the chips vis-a-vis one another (that is, they are a function of the relationships among the chips rather than their individual properties considered separately). And their range of possible transformations is strictly determined by the construction of the kaleidoscope, the inner law which governs its operation. And so it is too with savage thought. Both anecdotal and geometric, it builds coherent structures out of ‘the odds and ends left over from psychological or historical process.’
These odds and ends, the chips of the kaleidoscope, are images drawn from myth, ritual, magic, and empirical lore. (How, precisely, they have come into being in the first place is one of the points on which Levi-Strauss is not too explicit, referring to them vaguely as the ‘residue of events… fossil remains of the history of an individual or a society.’) Such images are inevitably embodied in larger structures– in myths, ceremonies, folk taxonomies, and so on– for, as in a kaleidoscope, one always sees the chips distributed in some pattern, however ill-formed or irregular. But, as in a kaleidoscope, they are detachable from these structures and arrangeable into different ones of a similar sort. Quoting Franz Boas that ‘it would seem that mythological worlds have been built up, only to be shattered again, and that new worlds were built from the fragments,’ Levi-Strauss generalizes this permutational view of thinking to savage thought in general.”
– Clifford Geertz, “The Cerebral Savage: the Structural Anthropology of Claude Levi-Strauss,” in Encounter, Vol. 28 No. 4 (April 1967), pp. 25-32.
Related material —
See also "Levi-Strauss" in this journal and "At Play in the Field."
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Sunday, December 19, 2010
Stella
In the spirit of last night's SNL Hanukkah version of "It's a Wonderful Life" —
A Geometric Merkabah for Hanukkah from December 1, and…
"The Homepage of Contemporary Structuralism" (click to enlarge) —
Detail —
Summary —
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Saturday, December 18, 2010
Friday, December 17, 2010
Time and Chance
New York Lottery yesterday, December 16, 2010— midday 282, evening 297.
Suggested by a Jesuit commentary that mentions the midday number —
Page 282 of Encyclopaedia of Religion and Ethics , Volume 2,
edited by James Hastings, John Alexander Selbie, and Louis Herbert Gray,
New York, Charles Scribner's Sons, 1910 —
"Philosophy seeks not absolute first principles, nor yet purely immediate insights,
but the self-mediation of the system of truth, and an insight into this self-mediation.
Axioms, in the language of modern theory, are best defined, neither as certainties
nor as absolutely first principles, but as those principles which are used as the first
in a special theory.
LITERATURE — A complete view of the literature of the problems
regarding axioms is impossible, since the topic is connected with all
the fundamental philosophical issues…. JOSIAH ROYCE"
Suggested by the evening number, 297, and by Amy Adams (see previous post) —
See also a cartoon version of Russell and Whitehead discussing axioms.
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Fare Thee Well
Excerpt from a post of 8 AM May 26, 2006 —
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A Living Church "The man who lives in contact with what he believes to be a living Church is a man always expecting to meet Plato and Shakespeare to-morrow at breakfast." – G. K. Chesterton
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A related scene from the opening of Blake Edwards's "S.O.B." —
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