For the Church of St. Frank:
See Strange Correspondences and Eightfold Geometry.
Correspondences , by Steven H. Cullinane, August 6, 2011
“The rest is the madness of art.”
Friday, May 3, 2013
Wednesday, May 1, 2013
The Crosswicks Curse
"There is such a thing as a tesseract." —A novel from Crosswicks
Related material from a 1905 graduate of Princeton,
"The 3-Space PG(3,2) and Its Group," is now available
at Internet Archive (1 download thus far).
The 3-space paper is relevant because of the
connection of the group it describes to the
"super, overarching" group of the tesseract.
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Tuesday, April 30, 2013
Projective Analysis
A Nested Sequence of Complete N-points and Their Sections
The complete space 6-point
(6 points in general position in space,
5 lines on each point, and 15 lines, 2 points on each)
has as a section
the large Desargues configuration
(15 points, 4 lines on each, and 20 lines, 3 points on each).
(Veblen and Young, Vol. 1, exercise 11, p. 53)
The large Desargues configuration may in turn be viewed as
the complete space 5-point
(5 points, 4 lines on each, and 10 lines, 2 points on each)
together with its section
the Desargues configuration
(10 points, 3 lines on each, and 10 lines, 3 points on each).
(Veblen and Young, Vol. I, pages 40-42)
The Desargues configuration may in turn be viewed as
the complete space 4-point (tetrahedron)
(4 points, 3 lines on each, and 6 lines, 2 points on each)
together with its section
the complete (plane) 4-side (complete quadrilateral)
(6 points, 2 lines on each, and 4 lines, 3 points on each).
The complete quadrilateral may in turn be viewed as
the complete 3-point (triangle)
(3 points, 2 lines on each, and 3 lines, 2 points on each)
together with its section
the three-point line
(3 points, 1 line on each, and 1 line, 3 points on the line).
The three-point line may in turn be viewed as
the complete 2-point
(2 points, 1 line on each, and 1 line with 2 points on the line)
together with its section
the complete 1-point
(1 point and 0 lines).
Update of May 1: For related material, see the exercises at the end of Ch. II
in Veblen and Young's Projective Geometry, Vol. I (Ginn, 1910). For instance:
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Logline
Found this morning in a search:
A logline is a one-sentence summary of your script.
www.scriptologist.com/Magazine/Tips/Logline/logline.html
It's the short blurb in TV guides that tells you what a movie
is about and helps you decide if you're interested …
The search was suggested by a screenwriting weblog post,
"Loglines: WHAT are you doing?".
What is your story about?
No, seriously, WHAT are you writing about?
Who are the characters? What happens to them?
Where does it take place? What’s the theme?
What’s the style? There are nearly a million
little questions to answer when you set out
to tell a story. But it all starts with one
super, overarching question.
What are you writing about? This is the first
big idea that we pull out of the ether, sometimes
before we even have any characters.
What is your story about?
The screenwriting post was found in an earlier search for
the highlighted phrase.
The screenwriting post was dated December 15, 2009.
What I am doing now is checking for synchronicity.
This weblog on December 15, 2009, had a post
titled A Christmas Carol. That post referred to my 1976
monograph titled Diamond Theory .
I guess the script I'm summarizing right now is about
the heart of that theory, a group of 322,560 permutations
that preserve the symmetry of a family of graphic designs.
For that group in action, see the Diamond 16 Puzzle.
The "super overarching" phrase was used to describe
this same group in a different context:
This is from "Mathieu Moonshine," a webpage by Anne Taormina.
A logline summarizing my approach to that group:
Finite projective geometry explains
the surprising symmetry properties
of some simple graphic designs—
found, for instance, in quilts.
The story thus summarized is perhaps not destined for movie greatness.
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Sunday, April 28, 2013
The Octad Generator
… And the history of geometry —
Desargues, Pascal, Brianchon and Galois
in the light of complete n-points in space.
(Rewritten for clarity at about 10 AM ET April 29, with quote from Dowling added.
Updated with a reference to a Veblen and Young exercise (on p. 53) on April 30.)
Veblen and Young, Projective Geometry, Vol. I ,
Ginn and Company, 1910, page 39:
"The Desargues configuration. A very important configuration
is obtained by taking the plane section of a complete space five-point."
Each of figures 14 and 15 above has 15 points and 20 lines.
The Desargues configuration within each figure is denoted by
10 white points and 10 solid lines, with 3 points on each line and
3 lines on each point. Black points and dashed lines indicate the
complete space five-point and lines connecting it to the plane section
containing the Desargues configuration.
In a 1915 University of Chicago doctoral thesis, Archibald Henderson
used a complete space six -point to construct a configuration of
15 points and 20 lines in the context not of Desargues ' theorem, but
rather of Brianchon 's theorem and of the Pascal hexagram.
Henderson's 1915 configuration is, it turns out, isomorphic to that of
the 15 points and 20 lines in the configuration constructed via a
complete space five -point five years earlier by Veblen and Young.
(See, in Veblen and Young's 1910 Vol. I, exercise 11, page 53:
"A plane section of a 6-point in space can be considered as
3 triangles perspective in pairs from 3 collinear points with
corresponding sides meeting in 3 collinear points." This is the
large Desargues configuration. See Classical Geometry in Light of
Galois Geometry.)
For this large Desargues configuration see April 19.
For Henderson's complete six –point, see The Six-Set (April 23).
That post ends with figures relating the large Desargues configuration
to the Galois geometry PG(3,2) that underlies the Curtis
Miracle Octad Generator and the large Mathieu group M24 —
See also Note on the MOG Correspondence from April 25, 2013.
That correspondence was also discussed in a note 28 years ago, on this date in 1985.
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Mathematics and Narrative (continued)
See Snakes on a Projective Plane by Andrew Spann (Sept. 26, 2006):
Click image for some related posts.
"…what he was trying to get across was not that he was the Soldier of a Power that was fighting across all of time to change history, but simply that we men were creatures with imaginations and it was our highest duty to try to tell what it was really like to live in other times and places and bodies. Once he said to me, 'The growth of consciousness is everything… the seed of awareness sending its roots across space and time. But it can grow in so many ways, spinning its web from mind to mind like the spider or burrowing into the unconscious darkness like the snake. The biggest wars are the wars of thought.' "
— Fritz Leiber, Changewar , page 22
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Saturday, April 27, 2013
Mark and Remark
“Fact and fiction weave in and out of novels like a shell game.” —R.B. Kitaj
Not just novels.
Fact:
The mark preceding A8 in the above denotes the semidirect product.
 |
Symbol from the box-style I Ching (Cullinane, 1/6/89). This is Hexagram 55, “Abundance [Fullness].” |
The mathematical quote, from last evening’s Symmetry, is from Anne Taormina.
The I Ching remark is not.
Another version of Abbondanza —
Fiction:
Found in Translation and the giorno June 22, 2009, here.
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Friday, April 26, 2013
Symmetry
Anne Taormina on Mathieu Moonshine —
This is, of course, the same group (of order 322,560) underlying the Diamond 16 Puzzle.
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Thursday, April 25, 2013
Rosenhain and Göpel Revisited
Some historical background for today's note on the geometry
underlying the Curtis Miracle Octad Generator (MOG):
The above incidence diagram recalls those in today's previous post
on the MOG, which is used to construct the large Mathieu group M24.
For some related material that is more up-to-date, search the Web
for Mathieu + Kummer .
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Note on the MOG Correspondence
In light of the April 23 post "The Six-Set,"
the caption at the bottom of a note of April 26, 1986
seems of interest:
"The R. T. Curtis correspondence between the 35 lines and the
2-subsets and 3-subsets of a 6-set. This underlies M24."
A related note from today:
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Tuesday, April 23, 2013
The Six-Set
The configurations recently discussed in
Classical Geometry in Light of Galois Geometry
are not unrelated to the 27 "Solomon's Seal Lines"
extensively studied in the 19th century.
See, in particular—
The following figures supply the connection of Henderson's six-set
to the Galois geometry previously discussed in "Classical Geometry…"—
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Sunday, April 21, 2013
Abstraction
(Continued from December 31st, 2012)
"Principles before personalities." — AA saying
Art Principles
Part I:
Part II:
Baker's 1922 Principles of Geometry—
Art Personalities
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Friday, April 19, 2013
The Large Desargues Configuration
Desargues' theorem according to a standard textbook:
"If two triangles are perspective from a point
they are perspective from a line."
The converse, from the same book:
"If two triangles are perspective from a line
they are perspective from a point."
Desargues' theorem according to Wikipedia
combines the above statements:
"Two triangles are in perspective axially [i.e., from a line]
if and only if they are in perspective centrally [i.e., from a point]."
A figure often used to illustrate the theorem,
the Desargues configuration , has 10 points and 10 lines,
with 3 points on each line and 3 lines on each point.
A discussion of the "if and only if" version of the theorem
in light of Galois geometry requires a larger configuration—
15 points and 20 lines, with 3 points on each line
and 4 lines on each point.
This large Desargues configuration involves a third triangle,
needed for the proof (though not the statement ) of the
"if and only if" version of the theorem. Labeled simply
"Desargues' Theorem," the large configuration is the
frontispiece to Volume I (Foundations) of Baker's 6-volume
Principles of Geometry .
Point-line incidence in this larger configuration is,
as noted in a post of April 1, 2013, described concisely
by 20 Rosenhain tetrads (defined in 1905 by
R. W. H. T. Hudson in Kummer's Quartic Surface ).
The third triangle, within the larger configuration,
is pictured below.
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Monday, April 15, 2013
M Theory
This morning's previous post pictured the cover
of a book titled "The Mystery of the Quantum World."
That title, together with Peter Woit's post on Hawking
yesterday, suggests a review of the phrase "M -theory."
See remarks on that topic in the October 1998
Notices of the American Mathematical Society :
"The richer theory, which has as limiting cases
the five string theories studied in the last generation,
has come to be called M -theory, where M stands for
magic, mystery, or matrix, according to taste."
— "Magic, Mystery, and Matrix," by Edward Witten
See also, in this journal, a post mentioning Hawking's
vulgarized M -theory book The Grand Design , as well
as a post of January 9, 2012, titled "M Theory."
Of Witten's three alternative meanings for M , I prefer "matrix."
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Sunday, April 14, 2013
Space Itself
From The Cambridge Companion to Wallace Stevens ,
John N. Serio, ed., "Stevens's Late Poetry," by B.J. Leggett,
pp. 62-75, an excerpt from page 70:
Click the above image for further details.
See also Nothingness and "The Rock" in this journal.
Further readings along these lines:
For pure mathematics, rather than theories of the physical world,
see the properties of the cube illustrated on the second (altered)
book cover above.
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Saturday, April 13, 2013
Princeton’s Christopher Robin
The title is that of a talk (see video) given by
George Dyson at a Princeton land preservation trust,
reportedly on March 21, 2013. The talk's subtitle was
"Oswald Veblen and the Six-hundred-acre Woods."
Meanwhile…
Thursday, March 21, 2013
|
Related material for those who prefer narrative
to mathematics:
|
Log24 on June 6, 2006:
The Omen :
|
Related material for those who prefer mathematics
to narrative:
What the Omen narrative above and the mathematics of Veblen
have in common is the number 6. Veblen, who came to
Princeton in 1905 and later helped establish the Institute,
wrote extensively on projective geometry. As the British
geometer H. F. Baker pointed out, 6 is a rather important number
in that discipline. For the connection of 6 to the Göpel tetrads
figure above from March 21, see a note from May 1986.
See also last night's Veblen and Young in Light of Galois.
"There is such a thing as a tesseract." — Madeleine L'Engle
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Veblen and Young in Light of Galois
Another in a series of figures illustrating
Desargues's theorem in light of Galois geometry:
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Friday, April 12, 2013
Midnight in Paris
Surreal requiem for the late Jonathan Winters:
"They 'burn, burn, burn like fabulous yellow roman candles
exploding like spiders across the stars,'
as Jack Kerouac once wrote. It was such a powerful
image that Wal-Mart sells it as a jigsaw puzzle."
— "When the Village Was the Vanguard,"
by Henry Allen, in today's Wall Street Journal
See also Damnation Morning and the picture in
yesterday evening's remarks on art:
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Thursday, April 11, 2013
Naked Art
The New Yorker on Cubism:
"The style wasn’t new, exactly— or even really a style,
in its purest instances— though it would spawn no end
of novelties in art and design. Rather, it stripped naked
certain characteristics of all pictures. Looking at a Cubist
work, you are forced to see how you see. This may be
gruelling, a gymnasium workout for eye and mind.
It pays off in sophistication."
— Online "Culture Desk" weblog, posted today by Peter Schjeldahl
Non-style from 1911:
See also Cube Symmetry Planes in this journal.
A comment at The New Yorker related to Schjeldahl's phrase "stripped naked"—
"Conceptualism is the least seductive modern-art movement."
POSTED 4/11/2013, 3:54:37 PM BY CHRISKELLEY
(The "conceptualism" link was added to the quoted comment.)
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Wednesday, April 10, 2013
Caution: Slow Art
"Of course, DeLillo being DeLillo,
it’s the deeper implications of the piece —
what it reveals about the nature of
film, perception and time — that detain him."
— Geoff Dyer, review of Point Omega
Related material:
A phrase of critic Robert Hughes,
"slow art," in this journal.
A search for that phrase yields the following
figure from a post on DeLillo of Oct. 12, 2011:
The above 3×3 grid is embedded in a
somewhat more sophisticated example
of conceptual art from April 1, 2013:
Update of April 12, 2013
The above key uses labels from the frontispiece
to Baker's 1922 Principles of Geometry, Vol. I ,
that shows a three-triangle version of Desargues's theorem.
A different figure, from a site at National Tsing Hua University,
shows the three triangles of Baker's figure more clearly:
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Tuesday, April 9, 2013
Four Quartets
For the cruelest month
Click for a much larger version of the photo below.
These four Kountry Korn quartets are from the Fox Valleyaires
Men's Barbershop Chorus of Appleton, Wisconsin.
See also the fine arts here on Saturday, April 6, 2013—
The New York Times Magazine cover story
a decade ago, on Sunday, April 6, 2003:
"The artists demanded space
in tune with their aesthetic."
— "The Dia Generation,"
by Michael Kimmelman
Related material:
See Wikipedia for the difference between binary numbers
and binary coordinates from the finite Galois field GF(2).
For some background, see the relativity problem.
See also the chapter on vector spaces in Korn & Korn
(originally published by McGraw-Hill)—
.
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Monday, April 8, 2013
Magic for Jews
A commenter on Saturday's "Seize the Dia" has
suggested a look at the work of one Mark Collins.
Here is such a look (click to enlarge):
I find attempts to associate pure mathematics with the words
"magic" or "mystic" rather nauseating. (H. F. Baker's work
on Pascal's mystic hexagram is no exception; Baker was
stuck with Pascal's obnoxious adjective, but had no truck
with any mystic aspects of the hexagram.)
The remarks above by Clifford Pickover on Collins, Dürer, and
binary representations may interest some non-mathematicians,
who should not be encouraged to waste their time on this topic.
For the mathematics underlying the binary representation of
Dürer's square, see, for instance, my 1984 article "Binary
Coordinate Systems."
Those without the background to understand that article
may enjoy, instead of Pickover's abortive attempts above at
mathematical vulgarization, his impressively awful 2009 novel
Jews in Hyperspace .
Pickover's 2002 book on magic squares was, unfortunately,
published by the formerly reputable Princeton University Press.
Related material from today's Daily Princetonian :
See also Nash + Princeton in this journal.
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Sunday, April 7, 2013
Pascal Inscape
Background: Inscapes and The 2-subsets of a 6-set are the points of a PG(3,2).
Related remarks: Classical Geometry in Light of Galois Geometry.
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Saturday, April 6, 2013
Pascal via Curtis
Click image for some background.
Shown above is a rearranged version of the
Miracle Octad Generator (MOG) of R. T. Curtis
("A new combinatorial approach to M24,"
Math. Proc. Camb. Phil. Soc., 79 (1976), 25-42.)
The 8-subcell rectangles in the left part of the figure may be
viewed as illustrating (if the top left subcell is disregarded)
the thirty-five 3-subsets of a 7-set.
Such a view relates, as the remarks below show, the
MOG's underlying Galois geometry, that of PG(3,2), to
the hexagrammum mysticum of Pascal.
On Danzer's 354 Configuration:
"Combinatorially, Danzer’s configuration can be interpreted
as defined by all 3-sets and all 4-sets that can be formed
by the elements of a 7-element set; each 'point' is represented
by one of the 3-sets, and it is incident with those lines
(represented by 4-sets) that contain the 3-set."
— Branko Grünbaum, "Musings on an Example of Danzer's,"
European Journal of Combinatorics , 29 (2008),
pp. 1910–1918 (online March 11, 2008)
"Danzer's configuration is deeply rooted in
Pascal's Hexagrammum Mysticum ."
— Marko Boben, Gábor Gévay, and Tomaž Pisanski,
"Danzer's Configuration Revisited," arXiv.org, Jan. 6, 2013
For an approach to such configurations that differs from
those of Grünbaum, Boben, Gévay, and Pisanski, see
Classical Geometry in Light of Galois Geometry.
Grünbaum has written little about Galois geometry.
Pisanski has recently touched on the subject;
see Configurations in this journal (Feb. 19, 2013).
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Wednesday, April 3, 2013
Museum Piece
Roberta Smith in 2011 on the American Folk Art Museum (see previous post):
"It could be argued that we need a museum of folk art
the way we need a museum of modern art,
to shine a very strong, undiluted light on
a very important achievement."
Some other aesthetic remarks:
"We have had a gutful of fast art and fast food.
What we need more of is slow art: art that holds time
as a vase holds water: art that grows out of modes
of perception and whose skill and doggedness
make you think and feel; art that isn't merely sensational,
that doesn't get its message across in 10 seconds,
that isn't falsely iconic, that hooks onto something
deep-running in our natures. In a word, art that is
the very opposite of mass media. For no spiritually
authentic art can beat mass media at their own game."
— Robert Hughes, speech of June 2, 2004,
quoted here June 15, 2007.
Perhaps, as well as museums of modern art and of folk art,
we need a Museum of Slow Art.
One possible exhibit, from this journal Monday:
The diagram on the left is from 1922. The 20 small squares at right
that each have 4 subsquares darkened were discussed, in a different
context, in 1905. They were re-illustrated, in a new context
(Galois geometry), in 1986. The "key" square, and the combined
illustration, is from April 1, 2013. For deeper background, see
Classical Geometry in Light of Galois Geometry.
Those who prefer faster art may consult Ten Years After.
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Tuesday, April 2, 2013
Hermite
A sequel to the quotation here March 8 (Pinter Play)
of Joan Aiken's novel The Shadow Guests—
Supposing that one's shadow guests are
Rosenhain and Göpel (see March 18)…
Hans Freudenthal at Encyclopedia.com on Charles Hermite:
"In 1855 Hermite took advantage of Göpel’s and Rosenhain’s work
when he created his transformation theory (see below)."
"One of his invariant theory subjects was the fifth-degree equation,
to which he later applied elliptic functions.
Armed with the theory of invariants, Hermite returned to
Abelian functions. Meanwhile, the badly needed theta functions
of two arguments had been found, and Hermite could apply what
he had learned about quadratic forms to understanding the
transformation of the system of the four periods. Later, Hermite’s
1855 results became basic for the transformation theory of Abelian
functions as well as for Camille Jordan’s theory of 'Abelian' groups.
They also led to Herrnite’s own theory of the fifth-degree equation
and of the modular equations of elliptic functions. It was Hermite’s
merit to use ω rather than Jacobi’s q = eπi ω as an argument and to
prepare the present form of the theory of modular functions.
He again dealt with the number theory applications of his theory,
particularly with class number relations or quadratic forms.
His solution of the fifth-degree equation by elliptic functions
(analogous to that of third-degree equations by trigonometric functions)
was the basic problem of this period."
See also Hermite in The Catholic Encyclopedia.
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Rota in a Nutshell
"The proof of Desargues' theorem of projective geometry
comes as close as a proof can to the Zen ideal.
It can be summarized in two words: 'I see!' "
— Gian-Carlo Rota in Indiscrete Thoughts (1997)
Also in that book, originally from a review in Advances in Mathematics,
Vol. 84, Number 1, Nov. 1990, p. 136:
Related material:
Pascal and the Galois nocciolo ,
Conway and the Galois tesseract,
Gardner and Galois.
See also Rota and Psychoshop.
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Baker on Configurations
The geometry posts of Sunday and Monday have been
placed in finitegeometry.org as
Classical Geometry in Light of Galois Geometry.
Some background:
See Baker, Principles of Geometry , Vol. II, Note I
(pp. 212-218)—
On Certain Elementary Configurations, and
on the Complete Figure for Pappus's Theorem
and Vol. II, Note II (pp. 219-236)—
On the Hexagrammum Mysticum of Pascal.
Monday's elucidation of Baker's Desargues-theorem figure
treats the figure as a 154203 configuration (15 points,
4 lines on each, and 20 lines, 3 points on each).
Such a treatment is by no means new. See Baker's notes
referred to above, and
"The Complete Pascal Figure Graphically Presented,"
a webpage by J. Chris Fisher and Norma Fuller.
What is new in the Monday Desargues post is the graphic
presentation of Baker's frontispiece figure using Galois geometry :
specifically, the diamond theorem square model of PG(3,2).
See also Cremona's kernel, or nocciolo :
Baker on Cremona's approach to Pascal—
"forming, in Cremona's phrase, the nocciolo of the whole."
A related nocciolo :
Click on the nocciolo for some
geometric background.
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Monday, April 1, 2013
Desargues via Rosenhain
Background: Rosenhain and Göpel Tetrads in PG(3,2)
|
Introduction: The Large Desargues Configuration Added by Steven H. Cullinane on Friday, April 19, 2013 Desargues' theorem according to a standard textbook:
"If two triangles are perspective from a point The converse, from the same book:
"If two triangles are perspective from a line
Desargues' theorem according to Wikipedia
"Two triangles are in perspective axially [i.e., from a line]
A figure often used to illustrate the theorem,
A discussion of the "if and only if" version of the theorem
This large Desargues configuration involves a third triangle,
Point-line incidence in this larger configuration is,
The third triangle, within the larger configuration,
|
A connection discovered today (April 1, 2013)—
(Click to enlarge the image below.)
Update of April 18, 2013
Note that Baker's Desargues-theorem figure has three triangles,
ABC, A'B'C', A"B"C", instead of the two triangles that occur in
the statement of the theorem. The third triangle appears in the
course of proving, not just stating, the theorem (or, more precisely,
its converse). See, for instance, a note on a standard textbook for
further details.
(End of April 18, 2013 update.)
Update of April 14, 2013
See Baker's Proof (Edited for the Web) for a detailed explanation
of the above picture of Baker's Desargues-theorem frontispiece.
(End of April 14, 2013 update.)
Update of April 12, 2013
A different figure, from a site at National Tsing Hua University,
shows the three triangles of Baker's figure more clearly:
(End of update of April 12, 2013)
Update of April 13, 2013
Another in a series of figures illustrating
Desargues's theorem in light of Galois geometry:
See also the original Veblen-Young figure in context.
(End of update of April 13, 2013)
Rota's remarks, while perhaps not completely accurate, provide some context
for the above Desargues-Rosenhain connection. For some other context,
see the interplay in this journal between classical and finite geometry, i.e.
between Euclid and Galois.
For the recent context of the above finite-geometry version of Baker's Vol. I
frontispiece, see Sunday evening's finite-geometry version of Baker's Vol. IV
frontispiece, featuring the Göpel, rather than the Rosenhain, tetrads.
For a 1986 illustration of Göpel and Rosenhain tetrads (though not under
those names), see Picturing the Smallest Projective 3-Space.
In summary… the following classical-geometry figures
are closely related to the Galois geometry PG(3,2):
|
Volume I of Baker's Principles has a cover closely related to the Rosenhain tetrads in PG(3,2) |
Volume IV of Baker's Principles has a cover closely related to the Göpel tetrads in PG(3,2) |
|
Foundations (click to enlarge)
|
Higher Geometry (click to enlarge)
|
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Preparation
"First published between 1922 and 1925,
the six-volume Principles of Geometry was
a synthesis of Baker's lecture series on geometry…."
From a different university press, a new logo
can be seen either as six volumes or as
the letter H —
"What is the H for?"
"Preparation."
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Sunday, March 31, 2013
For Baker
Baker, Principles of Geometry, Vol. IV (1925), Title:
Baker, Principles of Geometry, Vol. IV (1925), Frontispiece:
Baker's Vol. IV frontispiece shows "The Figure of fifteen lines
and fifteen points, in space of four dimensions."
Another such figure in a vector space of four dimensions
over the two-element Galois field GF(2):
(Some background grid parts were blanked by an image resizing process.)
Here the "lines" are actually planes in the vector 4-space over GF(2),
but as planes through the origin in that space, they are projective lines .
For some background, see today's previous post and Inscapes.
Update of 9:15 PM March 31—
The following figure relates the above finite-geometry
inscape incidences to those in Baker's frontispiece. Both the inscape
version and that of Baker depict a Cremona-Richmond configuration.
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For Pascal
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Thursday, March 28, 2013
Woman at the Well
For some images related to this rather biblical topic,
see Hillman + Dream in this journal.
“She was dazzled by light and shade, by the confusing
duplication of reflections and of frames. All coming from
too many directions for the mind to take account of.
The various images bounced against each other
until she felt a desperate vertigo….”
Summary image:
“… Margaret Murry, wrapped in an old patchwork quilt, sat on the foot of her bed….”
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Tuesday, March 26, 2013
Blockheads
"It should be emphasized that block models are physical models, the elements of which can be physically manipulated. Their manipulation differs in obvious and fundamental ways from the manipulation of symbols in formal axiomatic systems and in mathematics. For example the transformations described above, in which two linear arrays are joined together to form one array, or a rectangle of blocks is re-assembled into a linear array, are physical transformations not symbolic transformations. …"
— Storrs McCall, Department of Philosophy, McGill University, "The Consistency of Arithmetic"
"It should be emphasized…."
OK:
Storrs McCall at a 2008 philosophy conference .
His blocks talk was at 2:50 PM July 21, 2008.
See also this journal at noon that same day:
Froebel's Third Gift and the Eightfold Cube
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Thursday, March 21, 2013
Geometry of Göpel Tetrads (continued)
An update to Rosenhain and Göpel Tetrads in PG(3,2)
supplies some background from
Notes on Groups and Geometry, 1978-1986,
and from a 2002 AMS Transactions paper.
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Tuesday, March 19, 2013
Mathematics and Narrative (continued)
Angels & Demons meet Hudson Hawk
Dan Brown's four-elements diamond in Angels & Demons :
The Leonardo Crystal from Hudson Hawk :
Mathematics may be used to relate (very loosely)
Dan Brown's fanciful diamond figure to the fanciful
Leonardo Crystal from Hudson Hawk …
-
Compare Brown's fictional Illuminati Diamond to the
nonfictional figures in The Diamond Theorem and
Theme and Variations. -
Compare the fictional Leonardo Crystal to Hudson's
nonfictional desmic system of tetrahedra (above), and
see, in Rosenhain and Göpel Tetrads in PG(3,2), how
the diamond theorem is related to Hudson's work.
For the tetrads ' relationship to tetrahedra , see
Hudson's own book.
"Giving himself a head rub, Hawk bears down on
the three oddly malleable objects. He TANGLES
and BENDS and with a loud SNAP, puts them together,
forming the Crystal from the opening scene."
— A screenplay of Hudson Hawk
Happy birthday to Bruce Willis.
Comments Off on Mathematics and Narrative (continued)
Monday, March 18, 2013
Back to the Present: The Sequel
|
From Tom Stoppard's play "Rosencrantz and Guildenstern Are Dead"
GUIL: Yes, one must think of the future. |
Related material: Quotes from H. F. Baker in posts from March 2011—
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Sunday, March 17, 2013
Back to the Present
The previous post discussed some tesseract–
related mathematics from 1905.
Returning to the present, here is some arXiv activity
in the same area from March 11, 12, and 13, 2013.
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Saturday, March 16, 2013
The Crosswicks Curse
From the prologue to the new Joyce Carol Oates
novel Accursed—
"This journey I undertake with such anticipation
is not one of geographical space but one of Time—
for it is the year 1905 that is my destination.
1905!—the very year of the Curse."
Today's previous post supplied a fanciful link
between the Crosswicks Curse of Oates and
the Crosswicks tesseract of Madeleine L'Engle.
The Crosswicks Curse according to L'Engle
in her classic 1962 novel A Wrinkle in Time —
"There is such a thing as a tesseract."
A tesseract is a 4-dimensional hypercube that
(as pointed out by Coxeter in 1950) may also
be viewed as a 4×4 array (with opposite edges
identified).
Meanwhile, back in 1905…
For more details, see how the Rosenhain and Göpel tetrads occur naturally
in the diamond theorem model of the 35 lines of the 15-point projective
Galois space PG(3,2).
See also Conwell in this journal and George Macfeely Conwell in the
honors list of the Princeton Class of 1905.
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Puzzles
For readers of The Daily Princetonian :
(From a site advertised in the
Princetonian on March 11, 2013)
For readers of The Harvard Crimson :
For some background, see Crimson Easter Egg and the Diamond 16 Puzzle.
For some (very loosely) related narrative, see Crosswicks in this journal
and the Crosswicks Curse in a new novel by Joyce Carol Oates.
"There is such a thing as a tesseract."
— Crosswicks author Madeleine L'Engle
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Wednesday, March 13, 2013
Blackboard Jungle
From a review in the April 2013 issue of
Notices of the American Mathematical Society—
"The author clearly is passionate about mathematics
as an art, as a creative process. In reading this book,
one can easily get the impression that mathematics
instruction should be more like an unfettered journey
into a jungle where an individual can make his or her
own way through that terrain."
From the book under review—
"Every morning you take your machete into the jungle
and explore and make observations, and every day
you fall more in love with the richness and splendor
of the place."
— Lockhart, Paul (2009-04-01). A Mathematician's Lament:
How School Cheats Us Out of Our Most Fascinating and
Imaginative Art Form (p. 92). Bellevue Literary Press.
Kindle Edition.
Related material: Blackboard Jungle in this journal.
See also Galois Space and Solomon's Mines.
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Tuesday, March 12, 2013
Smoke and Mirrors
Sistine Chapel Smoke
Tromso Kunsthall Mirrors
Background for the smoke image:
A remark by Michelangelo in a 2007 post, High Concept.
Background for the mirrors image:
Note the publication date— Mar. 10, 2013.
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Sunday, March 10, 2013
Galois Space
The 16-point affine Galois space:
Further properties of this space:
In Configurations and Squares, see the
discusssion of the Kummer 166 configuration.
Some closely related material:
- Wolfgang Kühnel,
"Minimal Triangulations of Kummer Varieties,"
Abh. Math. Sem. Univ. Hamburg 57, 7-20 (1986).For the first two pages, click here.
- Jonathan Spreer and Wolfgang Kühnel,
"Combinatorial Properties of the K 3 Surface:
Simplicial Blowups and Slicings,"
preprint, 26 pages. (2009/10) (pdf).
(Published in Experimental Math. 20,
issue 2, 201–216 (2011).)
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Thursday, March 7, 2013
Proof Symbol
Today's previous post recalled a post
from ten years before yesterday's date.
The subject of that post was the
Galois tesseract.
Here is a post from ten years before
today's date.
The subject of that post is the Halmos
tombstone:
"The symbol 
is used throughout the entire book
in place of such phrases as 'Q.E.D.' or 'This
completes the proof of the theorem' to signal
the end of a proof."
— Measure Theory (1950)
For exact proportions, click on the tombstone.
For some classic mathematics related
to the proportions, see September 2003.
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Ten Years After
Rock guitarist Alvin Lee, a founder of
the band Ten Years After , died
on March 6, 2013 (Michelangelo's
birthday). In his memory, a figure
from a post Ten Years Before —
Plato's reported motto for his Academy:
"Let no one ignorant of geometry enter."
For visual commentary by an artist ignorant
of geometry, see a work by Sol LeWitt.
For verbal commentary by an art critic ignorant
of geometry, see a review of LeWitt by
Robert Hughes—
"A Beauty Really Bare" (TIME, Feb. 6, 2001).
See also Ten Years Group and Four Gods.
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Wednesday, March 6, 2013
Midnight in Pynchon*
"It is almost as though Pynchon wishes to
repeat the grand gesture of Joyce’s Ulysses…."
— Vladimir Tasic on Pynchon's Against the Day
Related material:
Tasic's Mathematics and the Roots of Postmodern Thought
and Michael Harris's "'Why Mathematics?' You Might Ask"
*See also Occupy Galois Space and Midnight in Dostoevsky.
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Monday, March 4, 2013
Occupy Galois Space
Continued from February 27, the day Joseph Frank died…
"Throughout the 1940s, he published essays
and criticism in literary journals, and one,
'Spatial Form in Modern Literature'—
a discussion of experimental treatments
of space and time by Eliot, Joyce, Proust,
Pound and others— published in
The Sewanee Review in 1945, propelled him
to prominence as a theoretician."
— Bruce Weber in this morning's print copy
of The New York Times (p. A15, NY edition)
That essay is reprinted in a 1991 collection
of Frank's work from Rutgers University Press:
See also Galois Space and Occupy Space in this journal.
Frank was best known as a biographer of Dostoevsky.
A very loosely related reference… in a recent Log24 post,
Freeman Dyson's praise of a book on the history of
mathematics and religion in Russia:
"The intellectual drama will attract readers
who are interested in mystical religion
and the foundations of mathematics.
The personal drama will attract readers
who are interested in a human tragedy
with characters who met their fates with
exceptional courage."
Frank is survived by, among others, his wife, a mathematician.
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Thursday, February 28, 2013
Tuesday, February 26, 2013
Publication
"I’ve had the privilege recently of being a Harvard University
professor, and there I learned one of the greatest of Harvard
jokes. A group of rabbis are on the road to Golgotha and
Jesus is coming by under the cross. The young rabbi bursts
into tears and says, 'Oh, God, the pity of it!' The old rabbi says,
'What is the pity of it?' The young rabbi says, 'Master, Master,
what a teacher he was.'
'Didn’t publish!'
That cold tenure- joke at Harvard contains a deep truth.
Indeed, Jesus and Socrates did not publish."
— George Steiner, 2002 talk at York University
See also Steiner on Galois.
Les Miserables at the Academy Awards
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Thursday, February 21, 2013
Galois Space
The previous post suggests two sayings:
"There is such a thing as a Galois space."
— Adapted from Madeleine L'Engle
"For every kind of vampire, there is a kind of cross."
Illustrations—
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Tuesday, February 19, 2013
Configurations
Yesterday's post Permanence dealt with the cube
as a symmetric model of the finite projective plane
PG(2,3), which has 13 points and 13 lines. The points
and lines of the finite geometry occur in the cube as
the 13 axes of symmetry and the 13 planes through
the center perpendicular to those axes. If the three
axes lying in a plane that cuts the cube in a hexagon
are supplemented by the axis perpendicular to that
plane, each plane is associated with four axes and,
dually, each axis is associated with four planes.
My web page on this topic, Cubist Geometries, was
written on February 27, 2010, and first saved to the
Internet Archive on Oct. 4, 2010.
For a more recent treatment of this topic that makes
exactly the same points as the 2010 page, see p. 218
of Configurations from a Graphical Viewpoint , by
Tomaž Pisanski and Brigitte Servatius, published by
Springer on Sept. 23, 2012 (date from both Google
Books and Amazon.com):
For a similar 1998 treatment of the topic, see Burkard Polster's
A Geometrical Picture Book (Springer, 1998), pp. 103-104.
The Pisanski-Servatius book reinforces my argument of Jan. 13, 2013,
that the 13 planes through the cube's center that are perpendicular
to the 13 axes of symmetry of the cube should be called the cube's
symmetry planes , contradicting the usual use of of that term.
That argument concerns the interplay between Euclidean and
Galois geometry. Pisanski and Servatius (and, in 1998, Polster)
emphasize the Euclidean square and cube as guides* to
describing the structure of a Galois space. My Jan. 13 argument
uses Galois structures as a guide to re-describing those of Euclid .
(For a similar strategy at a much more sophisticated level,
see a recent Harvard Math Table.)
Related material: Remarks on configurations in this journal
during the month that saw publication of the Pisanski-Servatius book.
* Earlier guides: the diamond theorem (1978), similar theorems for
2x2x2 (1984) and 4x4x4 cubes (1983), and Visualizing GL(2,p)
(1985). See also Spaces as Hypercubes (2012).
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Monday, February 18, 2013
Permanence
Inscribed hexagon (1984)
The well-known fact that a regular hexagon
may be inscribed in a cube was the basis
in 1984 for two ways of coloring the faces
of a cube that serve to illustrate some graphic
aspects of embodied Galois geometry—
Inscribed hexagon (2013)
A redefinition of the term "symmetry plane"
also uses the well-known inscription
of a regular hexagon in the cube—
Related material
"Here is another way to present the deep question 1984 raises…."
— "The Quest for Permanent Novelty," by Michael W. Clune,
The Chronicle of Higher Education , Feb. 11, 2013
“What we do may be small, but it has a certain character of permanence.”
— G. H. Hardy, A Mathematician’s Apology
Comments Off on Permanence
Wednesday, February 13, 2013
Form:
Story, Structure, and the Galois Tesseract
Recent Log24 posts have referred to the
"Penrose diamond" and Minkowski space.
The Penrose diamond has nothing whatever
to do with my 1976 monograph "Diamond Theory,"
except for the diamond shape and the connection
of the Penrose diamond to the Klein quadric—
The Klein quadric occurs in the five-dimensional projective space
over a field. If the field is the two-element Galois field GF(2), the
quadric helps explain certain remarkable symmetry properties
of the R. T. Curtis Miracle Octad Generator (MOG), hence of
the large Mathieu group M24. These properties are also
relevant to the 1976 "Diamond Theory" monograph.
For some background on the quadric, see (for instance)…
See also The Klein Correspondence,
Penrose Space-Time, and a Finite Model.
Related material:
|
"… one might crudely distinguish between philosophical – J. M. E. Hyland. "Proof Theory in the Abstract." (pdf) |
Those who prefer story to structure may consult
- today's previous post on the Penrose diamond
- the remarks of Scott Aaronson on August 17, 2012
- the remarks in this journal on that same date
- the geometry of the 4×4 array in the context of M24.
Comments Off on Form:
Midnight in the Garden
(Continued, to mark Tuesday's birthdays of Lincoln and Darwin.)
A British reporter who died at 97 on Tuesday is said to have
"covered the space race in its entirety." In his honor, here
in review are posts containing the phrase Space Race
and, more generally, the two words Galois + Space.
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Monday, February 11, 2013
The Penrose Diamond
Related material:
See also remarks on Penrose linked to in Sacerdotal Jargon.
(For a connection of these remarks to
the Penrose diamond, see April 1, 2012.)
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Sunday, February 10, 2013
The Cleaning
"In 2005 Arthur Jaffe succeeded Sir Michael Atiyah as
Chair of the Board of the Dublin Institute for Advanced Study,
School of Theoretical Physics."
Related material:
Biddies in this journal and…
Detail:
An early version of quaternions.
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Saturday, February 9, 2013
Snow Dance
The Snow White dance from last Nov. 14
features an ad that was originally embedded
in an American Mathematical Society Notices
review describing three books of vulgarized
mathematics. These books all use "great
equations" as a framing device.
This literary strategy leads to a more abstract
snow dance. See the ballet blanc in this journal
on Balanchine's birthday (old style) in 2003.
That dance involves equation (C) below.
Recall that in a unit ring ,
"0" denotes the additive identity,
"1" the multiplicative identity, and "-1" the
additive inverse of the multiplicative identity.
Three classic equations:
(A) 1 + 1 = 2 (Characteristic 0, ordinary arithmetic)
(B) 1 + 1 = 0 (Characteristic 2 arithmetic, in which 2 = 0)
(C) 1 + 1 = -1 (Characteristic 3 arithmetic, in which 2 = -1)
Cases (B) and (C), in which the characteristic is prime,
occur in Galois geometry.
For a more elaborate snow dance, see Master Class.
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Friday, February 8, 2013
Dictum
A review of the life of physicist Arthur Wightman,
who died at 90 on January 13th, 2013. yields
the following.
Wightman at Wikipedia:
"His graduate students include
Arthur Jaffe, Jerrold Marsden, and Alan Sokal."
"I think of Arthur as the spiritual leader
of mathematical physics and his death
really marks the end of an era."
— Arthur Jaffe in News at Princeton , Jan. 30
Marsden at Wikipedia:
"He [Marsden] has laid much of the foundation for
symplectic topology." (Link redirects to symplectic geometry.)
A Wikipedia reference in the symplectic geometry article leads to…
|
THE SYMPLECTIZATION OF SCIENCE:
Mark J. Gotay
James A. Isenberg February 18, 1992 Acknowledgments:
We would like to thank Jerry Marsden and Alan Weinstein Published in: Gazette des Mathématiciens 54, 59-79 (1992). Opening:
"Physics is geometry . This dictum is one of the guiding |
A different account of the dictum:
The strange term Geometrodynamics
is apparently due to Wheeler.
Physics may or may not be geometry, but
geometry is definitely not physics.
For some pure geometry that has no apparent
connection to physics, see this journal
on the date of Wightman's death.
Comments Off on Dictum
Tuesday, February 5, 2013
Arsenal
The previous post discussed some fundamentals of logic.
The name “Boole” in that post naturally suggests the
concept of Boolean algebra . This is not the algebra
needed for Galois geometry . See below.
Some, like Dan Brown, prefer to interpret symbols using
religion, not logic. They may consult Diamond Mandorla,
as well as Blade and Chalice, in this journal.
See also yesterday’s Universe of Discourse.
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Thursday, January 31, 2013
Scholarship in 1961…
… Before Derrida's writings on Plato and on inscription
A remark by the late William Harris:
"Scholarship has many dark ages, and they do not all fall
in the safe confines of remote antiquity."
For more about Harris, see the previous post.
Discussing an approach to solving a geometrical problem
from section 86e of the Meno , Harris wrote that
"… this is a very important element of method and purpose,
one which must be taken with great seriousness and respect.
In fact it is as good an example of the master describing for us
his method as Plato ever gives us. Tricked by the appearance
of brevity and unwilling to follow through Plato's thought on
the road to Euclid, we have garbled or passed over a unique
piece of philosophical information."
Harris, though not a geometer, was an admirable man.
His remark on the Meno method is itself worthy of respect.
In memory of Harris, Plato, and pre-Derrida scholarship, here
are some pages from 1961 on the problem Harris discussed.
A pair of figures from the 1961 pages indicates how one view of the
section 86e problem (at right below) resembles the better-known
demonstration earlier in the Meno of how to construct
a square of area 2 —
Comments Off on Scholarship in 1961…
Wednesday, January 30, 2013
The Problem Problem
Given these choices for a solution ,
what is a suitable problem ?
The problem sketched on Jan. 22 was a joke.
A more serious triangle-circle-square problem:
Introductory commentary from the same source—
See also a description of this problem by the late William Harris,
Harvard '48, Professsor Emeritus of Classics at Middlebury College,
who died on February 22, 2009*—
"… this is a very important element of method and purpose,
one which must be taken with great seriousness and respect.
In fact it is as good an example of the master describing for us
his method as Plato ever gives us. Tricked by the appearance
of brevity and unwilling to follow through Plato's thought on
the road to Euclid, we have garbled or passed over a unique
piece of philosophical information."
The problem itself, from the Perseus site:
[87a] whether a certain area is capable of being inscribed as a triangular space in a given circle: they reply—“I cannot yet tell whether it has that capability; but I think, if I may put it so, that I have a certain helpful hypothesis for the problem, and it is as follows: If this area is such that when you apply it to the given line of the circle you find it falls short by a space similar to that which you have just applied, then I take it you have one consequence, and if it is impossible for it to fall so, then some other. Accordingly I wish to put a hypothesis, before I state our conclusion as regards inscribing this figure [87b] in the circle by saying whether it is impossible or not.” In the same way with regard to our question about virtue, since we do not know either what it is or what kind of thing it may be, we had best make use of a hypothesis in considering whether it can be taught or not, as thus: what kind of thing must virtue be in the class of mental properties, so as to be teachable or not? In the first place, if it is something dissimilar or similar to knowledge, is it taught or not—or, as we were saying just now, remembered? Let us have no disputing about the choice of a name: [87c] is it taught? Or is not this fact plain to everyone—that the one and only thing taught to men is knowledge?
Meno
I agree to that.
Socrates
Then if virtue is a kind of knowledge, clearly it must be taught?
Meno
Certainly.
Socrates
So you see we have made short work of this question—if virtue belongs to one class of things it is teachable, and if to another, it is not.
Meno
To be sure.
For further details, consult (for instance) a 1955 paper at JSTOR.
* See a post from that date in this journal.
See also a remark by Harris:
"Scholarship has many dark ages, and they do not all fall
in the safe confines of remote antiquity."
Comments Off on The Problem Problem
Thursday, January 24, 2013
Moondance
The title was suggested by an ad for a film that opens
at 10 PM EST today: "Hansel & Gretel: Witch Hunters."
Related material: Grimm Day 2012, as well as
Amy Adams in Raiders of the Lost Tesseract
and in a Film School Rejects page today.
See also some Norwegian art in
Trish Mayo's Photostream today and in
Omega Point (Log24, Oct. 15, 2012)—
Monday, October 15, 2012
|
Comments Off on Moondance
Wednesday, January 23, 2013
DNA and a Galois Field
From Ewan Birney's weblog today:
| WEDNESDAY, 23 JANUARY 2013
Using DNA as a digital archive media Today sees the publication in Nature of “Toward practical high-capacity low-maintenance storage of digital information in synthesised DNA,” a paper spearheaded by my colleague Nick Goldman and in which I played a major part, in particular in the germination of the idea. |
Birney appeared in Log24 on Dec. 30, 2012, quoted as follows:
"It is not often anyone will hear the phrase 'Galois field' and 'DNA' together…."
— Birney's weblog on July 3, 2012, "Galois and Sequencing."
Birney's widespread appearance in news articles today about the above Nature publication suggests a review of the "Galois-field"-"DNA" connection.
See, for instance, the following papers:
- Gail Rosen and Jeff Moore. "Investigation of Coding Structure in DNA," IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Hong Kong, April 2003. [pdf]
- Gail Rosen. "Finding Near-Periodic DNA Regions using a Finite-Field Framework," 2nd IEEE Genomic Signal Processing Workshop (GENSIPS), Baltimore, MD, May 2004. [pdf]
- Gail Rosen. "Examining Coding Structure and Redundancy in DNA," IEEE Engineering in Medicine and Biology Magazine, Volume 25, Issue 1, January/February 2006. [pdf]
A Log24 post of Sept. 17, 2012, also mentions the phrases "Galois field" and "DNA" together.
Comments Off on DNA and a Galois Field
Tuesday, January 22, 2013
The Problem Problem
For connoisseurs of psychological tests,
here is an inverse puzzle:
Given these choices for a solution ,
what is a suitable problem ?
There is, of course, no single right answer.
One path to an answer might involve
a British webpage and the recent film Branded.
Max von Sydow in Branded (2012)
(See, too, related remarks on The Queen's Privy Council.)
Comments Off on The Problem Problem
Monday, January 21, 2013
Shining Forth
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Sunday, January 20, 2013
In the Details
Comments Off on In the Details
Friday, January 18, 2013
Solomon’s Rep-tiles
"Rep-tiles Revisited," by Viorel Nitica, in MASS Selecta: Teaching and Learning Advanced Undergraduate Mathematics , American Mathematical Society,
"The goal of this note is to take a new look at some of the most amazing objects discovered in recreational mathematics. These objects, having the curious property of making larger copies of themselves, were introduced in 1962 by Solomon W. Golomb [2], and soon afterwards were popularized by Martin Gardner [3] in Scientific American…."
2. S. W. Golomb: "Replicating Figures in the Plane," Mathematical Gazette 48, 1964, 403-412
3. M. Gardner: "On 'Rep-tiles,' Polygons That Can Make Larger and Smaller Copies of Themselves," Scientific American 208, 1963, 154-157
Two such "amazing objects"—
|
Triangle |
Square |
For a different approach to the replicating properties of these objects, see the square-triangle theorem.
For related earlier material citing Golomb, see Not Quite Obvious (July 8, 2012; scroll down to see the update of July 15.).
Golomb's 1964 Gazette article may now be purchased at JSTOR for $14.
Comments Off on Solomon’s Rep-tiles
Thursday, January 17, 2013
Brazil Revisited
Yesterday's post Treasure Hunt, on a Brazilian weblog,
suggests a review of Brazil in this journal. The post
most relevant to yesterday's remarks is from
August 15, 2003, with a link, now broken, to the work
of Brazilian artist Nicole Sigaud* that also uses the
four half-square tiles used in 1704 by Sebastien Truchet
and somewhat later by myself in Diamond Theory
(see a 1977 version).
A more recent link that works:
|
http://vismath9.tripod.com/sigaud/e-index.html ANACOM PROJECT
APPLICATIONS
© 1997 – 2002 Nicole Sigaud |
* Sigaud shares the interests of her fellow Brazilian
whose weblog was the subject of yesterday's
Treasure Hunt.—
"For many years I have dedicated myself to the study
of medieval magic, demonology, Kabbalah, Astrology,
Alchemy, Tarot and divination in general."
— Nicole Sigaud (translated by Google) in a self-profile:
http://www.recantodasletras.com.br/autor.php?id=78359.
I do not share the interest of these authors in such matters,
except as they are reflected in the works of authors like
Charles Williams and Umberto Eco.
Comments Off on Brazil Revisited
Wednesday, January 16, 2013
Treasure Hunt
The Mathematical Association of America (MAA)
newsmagazine Focus for December 2012/January 2013:
The Babylonian tablet on the cover illustrates the
"Mathematical Treasures" article.
A search for related material yields a Babylonian tablet
reproduced in a Brazilian weblog on July 4, 2012:
In that weblog on the same day, July 4, 2012,
another post quotes at length my Diamond Theory page,
starting with the following image from that page—
That Brazilian post recommends use of geometry together
with Tarot and astrology. I do not concur with this
recommendation, but still appreciate the mention.
Comments Off on Treasure Hunt
Medals
National…
International…
Click medal for some background. The medal may be regarded
as illustrating the 16-point Galois space. (See previous post.)
Related material: Jews in Hyperspace.
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Space Race
Japanese character
for "field"
This morning's leading
New York Times obituaries—
For other remarks on space, see
Galois + Space in this journal.
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Sunday, January 13, 2013
Spinning in Infinity
A note for day 13 of 2013
How the cube's 13 symmetry planes*
are related to the finite projective plane
of order 3, with 13 points and 13 lines—
For some background, see Cubist Geometries.
* This is not the standard terminology. Most sources count
only the 9 planes fixed pointwise under reflections as
"symmetry planes." This of course obscures the connection
with finite geometry.
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Wednesday, January 9, 2013
Bad Idea
For the 2013 Joint Mathematics Meetings in San Diego,
which start today, a cartoon by Andrew Spann—
(Click for larger image.)
Related remarks:
This journal on the Feast of Epiphany, 2013—
"The Tesseract is where it belongs: out of our reach."
— The Avengers' Nick Fury, played by Samuel L. Jackson
"You never know what could happen.
If you have Sam, you’re going to be cool."
— The late David R. Ellis, film director
If anyone in San Diego cares about the relationship
of Spann's plane to Fury's Tesseract, he or she may
consult Finite Geometry of the Square and Cube.
Comments Off on Bad Idea
Saturday, January 5, 2013
Vector Addition in a Finite Field
The finite (i.e., Galois) field GF(16),
according to J. J. Seidel in 1974—
The same field according to Steven H. Cullinane in 1986,
in its guise as the affine 4-space over GF(2)—
The same field, again disguised as an affine 4-space,
according to John H. Conway and N.J.A. Sloane in
Sphere Packings, Lattices, and Groups , first published in 1988—
The above figure by Conway and Sloane summarizes, using
a 4×4 array, the additive vector-space structure of the finite
field GF(16).
This structure embodies what in Euclidean space is called
the parallelogram rule for vector addition—
(Thanks to June Lester for the 3D (uvw) part of the above figure.)
For the transition from this colored Euclidean hypercube
(used above to illustrate the parallelogram rule) to the
4×4 Galois space (illustrated by Cullinane in 1979 and
Conway and Sloane in 1988— or later… I do not have
their book’s first edition), see Diamond Theory in 1937,
Vertex Adjacency in a Tesseract and in a 4×4 Array,
Spaces as Hypercubes, and The Galois Tesseract.
For some related narrative, see tesseract in this journal.
(This post has been added to finitegeometry.org.)
Update of August 9, 2013—
Coordinates for hypercube vertices derived from the
parallelogram rule in four dimensions were better
illustrated by Jürgen Köller in a web page archived in 2002.
Update of August 13, 2013—
The four basis vectors in the 2002 Köller hypercube figure
are also visible at the bottom of the hypercube figure on
page 7 of “Diamond Theory,” excerpts from a 1976 preprint
in Computer Graphics and Art , Vol. 2, No. 1, February 1977.
A predecessor: Coxeter’s 1950 hypercube figure from
“Self-Dual Configurations and Regular Graphs.”
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Thursday, January 3, 2013
Two Poems and Some Images
From an obituary of singer Patti Page, who died on New Year's Day—
"Clara Ann Fowler was born Nov. 8, 1927, in Claremore, Okla., and grew up in Tulsa. She was one of 11 children and was raised during the Great Depression by a father who worked for the railroad.
She told the Times that her family often did not have enough money to buy shoes. To save on electricity bills, the Fowlers listened to only a few select radio programs. Among them was 'Grand Ole Opry.'"
See also two poems by Wallace Stevens and some images related to yesterday's Log24 post.
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Wednesday, January 2, 2013
PlanetMath link
Update of May 27, 2013:
The post below is now outdated. See
http://planetmath.org/cullinanediamondtheorem .
__________________________________________________________________
The brief note on the diamond theorem at PlanetMath
disappeared some time ago. Here is a link to its
current URL: http://planetmath.org/?op=getobj;from=lec;id=49.
Update of 3 PM ET Jan. 2, 2013—
Another item recovered from Internet storage:
Click on the Monthly page for some background.
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Tuesday, January 1, 2013
The Simplest Situation
Thanks to a Harvard math major for the following V. I. Arnold quote
in a weblog post yesterday titled "Abstraction and Generality"—
"… the author has attempted to adhere to the principle of
minimal generality, according to which every idea should first
be clearly understood in the simplest situation;*
only then can the method developed be extended to
more complicated cases.
— Vladimir I. Arnold, Lectures on Partial Differential Equations
(Russian edition 1997; English translation 2004),
Preface to the second Russian edition
Thanks also to the math major for his closing post today.
* For instance… Natalie Angier's New Year's meditation
on a Buddha Field—
"… the multiverse as envisioned in Tibetan Buddhism,
'a vast system of 1059 [sic ; corrected to 10^59 on Jan. 3]
universes, that together are called a Buddha Field,' said
Jonathan C. Gold, who studies Buddhist philosophy at
Princeton."
— versus a search in this journal for "Japanese character" that yields…
Japanese character
for "field"
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Monday, December 31, 2012
Holiday Philosophy
Obituaries for New Year's Eve—
A link from Christmas Day—
Easter meditations—
See also…
- Valéry's reference to "a broken phrase by Pascal,"
- The shattered rock of a Wallace Stevens critic, and
- Pascal's diamond.
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Sunday, December 30, 2012
The Kick
George Steiner, Real Presences , first published in 1989—
The inception of critical thought, of a philosophic anthropology,
is contained in the archaic Greek definition of man as a
'language-animal'….
Richard Powers, The Gold Bug Variations , first published in 1991—
Botkin, whatever her gifts as a conversationist, is almost as old
as the rediscovery of Mendel. The other extreme in age,
Joe Lovering, beat a time-honored path out of pure math
into muddy population statistics. Ressler has seen the guy
potting about in the lab, although exactly what the excitable kid
does is anybody's guess. He looks decidedly gumfooted holding
any equipment more corporeal than a chi-square. Stuart takes
him to the Y for lunch, part of a court-your-resources campaign.
He has the sub, Levering the congealed mac and cheese.
Hardly are they seated when Joe whips out a napkin and begins
sketching proofs. He argues that the genetic code, as an
algorithmic formal system, is subject to Gödel's Incompleteness
Theorem. "That would mean the symbolic language of the code
can't be both consistent and complete. Wouldn't that be a kick
in the head?"
Kid talk, competitive showing off, intellectual fantasy.
But Ressler knows what Joe is driving at. He's toyed with similar
ideas, cast in less abstruse terms. We are the by-product of the
mechanism in there. So it must be more ingenious than us.
Anything complex enough to create consciousness may be too
complex for consciousness to understand. Yet the ultimate paradox
is Lovering, crouched over his table napkin, using proofs to
demonstrate proof's limits. Lovering laughs off recursion and takes
up another tack: the key is to find some formal symmetry folded
in this four-base chaos. Stuart distrusts this approach even more.
He picks up the tab for their two untouched lunches, thanking
Lovering politely for the insight.
Edith Piaf—
See last midnight's post and Theme and Variations.
"The key is to find some formal symmetry…."
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Not Often
"It is not often anyone will hear the phrase 'Galois field' and 'DNA' together…."
— Ewan Birney at his weblog on July 3, 2012.
Try a Google search. (And see such a search as of Dec. 30, 2012.)
See also "Context Part III" in a Log24 post of Sept. 17, 2012.
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Saturday, December 29, 2012
Mapping Problem
A mapping problem posed (informally) in 1985
and solved 27 years later, in 2012:
See also Finite Relativity and Finite Relativity: The Triangular Version.
(A note for fans of the recent film Looper (see previous post)—
Hunter S. Thompson in this journal on February 22, 2005 …
Hunter S. Thompson, photos from The New York Times
… and on March 3, 2009.)
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Friday, December 28, 2012
Cube Koan
|
From Don DeLillo's novel Point Omega — I knew what he was, or what he was supposed to be, a defense intellectual, without the usual credentials, and when I used the term it made him tense his jaw with a proud longing for the early weeks and months, before he began to understand that he was occupying an empty seat. "There were times when no map existed to match the reality we were trying to create." "What reality?" "This is something we do with every eyeblink. Human perception is a saga of created reality. But we were devising entities beyond the agreed-upon limits of recognition or interpretation. Lying is necessary. The state has to lie. There is no lie in war or in preparation for war that can't be defended. We went beyond this. We tried to create new realities overnight, careful sets of words that resemble advertising slogans in memorability and repeatability. These were words that would yield pictures eventually and then become three-dimensional. The reality stands, it walks, it squats. Except when it doesn't." He didn't smoke but his voice had a sandlike texture, maybe just raspy with age, sometimes slipping inward, becoming nearly inaudible. We sat for some time. He was slouched in the middle of the sofa, looking off toward some point in a high corner of the room. He had scotch and water in a coffee mug secured to his midsection. Finally he said, "Haiku." I nodded thoughtfully, idiotically, a slow series of gestures meant to indicate that I understood completely. "Haiku means nothing beyond what it is. A pond in summer, a leaf in the wind. It's human consciousness located in nature. It's the answer to everything in a set number of lines, a prescribed syllable count. I wanted a haiku war," he said. "I wanted a war in three lines. This was not a matter of force levels or logistics. What I wanted was a set of ideas linked to transient things. This is the soul of haiku. Bare everything to plain sight. See what's there. Things in war are transient. See what's there and then be prepared to watch it disappear." |
What's there—
This view of a die's faces 3, 6, and 5, in counter-
clockwise order (see previous post) suggests a way
of labeling the eight corners of a die (or cube):
123, 135, 142, 154, 246, 263, 365, 456.
Here opposite faces of the die sum to 7, and the
three faces meeting at each corner are listed
in counter-clockwise order. (This corresponds
to a labeling of one of MacMahon's* 30 colored cubes.)
A similar vertex-labeling may be used in describing
the automorphisms of the order-8 quaternion group.
For a more literary approach to quaternions, see
Pynchon's novel Against the Day .
* From Peter J. Cameron's weblog:
"The big name associated with this is Major MacMahon,
an associate of Hardy, Littlewood and Ramanujan,
of whom Robert Kanigel said,
His expertise lay in combinatorics, a sort of
glorified dice-throwing, and in it he had made
contributions original enough to be named
a Fellow of the Royal Society.
Glorified dice-throwing, indeed…"
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Thursday, December 27, 2012
Object Lesson
Yesterday's post on the current Museum of Modern Art exhibition
"Inventing Abstraction: 1910-1925" suggests a renewed look at
abstraction and a fundamental building block: the cube.
From a recent Harvard University Press philosophical treatise on symmetry—
The treatise corrects Nozick's error of not crediting Weyl's 1952 remarks
on objectivity and symmetry, but repeats Weyl's error of not crediting
Cassirer's extensive 1910 (and later) remarks on this subject.
For greater depth see Cassirer's 1910 passage on Vorstellung :
This of course echoes Schopenhauer, as do discussions of "Will and Idea" in this journal.
For the relationship of all this to MoMA and abstraction, see Cube Space and Inside the White Cube.
"The sacramental nature of the space becomes clear…." — Brian O'Doherty
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Monday, December 24, 2012
Post-Mortem for Quincy
Raiders of the Lost Trunk, or:
Stars in the Attic
See also A Glass for Klugman :
Context: Poetry and Truth, Eternal Recreation,
Solid Symmetry, and Stevens's Rock.
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All Over Again
"… the movement of analogy
begins all over once again."
See A Reappearing Number in this journal.
Illustrations:
Figure 1 —
Background: MOG in this journal.
Figure 2 —
Background —
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Eternal Recreation
Memories, Dreams, Reflections
by C. G. Jung
Recorded and edited By Aniela Jaffé, translated from the German
by Richard and Clara Winston, Vintage Books edition of April 1989
From pages 195-196:
"Only gradually did I discover what the mandala really is:
'Formation, Transformation, Eternal Mind's eternal recreation.'*
And that is the self, the wholeness of the personality, which if all
goes well is harmonious, but which cannot tolerate self-deceptions."
* Faust , Part Two, trans. by Philip Wayne (Harmondsworth,
England, Penguin Books Ltd., 1959), p. 79. The original:
… Gestaltung, Umgestaltung,
Des ewigen Sinnes ewige Unterhaltung….
Jung's "Formation, Transformation" quote is from the realm of
the Mothers (Faust , Part Two, Act 1, Scene 5: A Dark Gallery).
The speaker is Mephistopheles.
See also Prof. Bruce J. MacLennan on this realm
in a Web page from his Spring 2005 seminar on Faust:
"In alchemical terms, F is descending into the dark, formless
primary matter from which all things are born. Psychologically
he is descending into the deepest regions of the
collective unconscious, to the source of life and all creation.
Mater (mother), matrix (womb, generative substance), and matter
all come from the same root. This is Faust's next encounter with
the feminine, but it's obviously of a very different kind than his
relationship with Gretchen."
The phrase "Gestaltung, Umgestaltung " suggests a more mathematical
approach to the Unterhaltung . Hence…
Part I: Mothers
"The ultimate, deep symbol of motherhood raised to
the universal and the cosmic, of the birth, sending forth,
death, and return of all things in an eternal cycle,
is expressed in the Mothers, the matrices of all forms,
at the timeless, placeless originating womb or hearth
where chaos is transmuted into cosmos and whence
the forms of creation issue forth into the world of
place and time."
— Harold Stein Jantz, The Mothers in Faust:
The Myth of Time and Creativity ,
Johns Hopkins Press, 1969, page 37
Part II: Matrices
Part III: Spaces and Hypercubes
Click image for some background.
Part IV: Forms
Forms from the I Ching :
Click image for some background.
Forms from Diamond Theory :
Click image for some background.
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Saturday, December 22, 2012
Web Links:
Spidey Goes to Church
More realistically…
- "Nick Bostrom … is a Swedish philosopher at
St. Cross College, University of Oxford…." - "The early location of St Cross was on a site in
St Cross Road, immediately south of St Cross Church." - "The church building is located on St Cross Road
just south of Holywell Manor." - "Balliol College has had a presence in the area since
the purchase by Benjamin Jowett, the Master, in the 1870s
of the open area which is the Balliol sports ground
'The Master's Field.' " - Leaving Wikipedia, we find a Balliol field at Log24:
- A different view of the same field, from 1950—
. - A view from 1974, thanks to J. J. Seidel —
- Yesterday's Analogies.
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Friday, December 21, 2012
Analogies*
The Moore correspondence may be regarded
as an analogy between the 35 partitions of an
8-set into two 4-sets and the 35 lines in the
finite projective space PG(3,2).
Closely related to the Moore correspondence
is a correspondence (or analogy) between the
15 2-subsets of a 6-set and the 15 points of PG(3,2).
An analogy between the two above analogies
is supplied by the exceptional outer automorphism of S6.
See…
The 2-subsets of a 6-set are the points of a PG(3,2),
Picturing outer automorphisms of S6, and
A linear complex related to M24.
(Background: Inscapes, Inscapes III: PG(2,4) from PG(3,2),
and Picturing the smallest projective 3-space.)
* For some context, see Analogies and
"Smallest Perfect Universe" in this journal.
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Thursday, December 20, 2012
Riddled
See reference AR in "Binary Coordinate Systems."
See also Prewar Berlin in this journal.
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Wednesday, December 19, 2012
Identity Shining Forth
From William M. Kantor's 1978 review of Peter J. Cameron's
1976 book Parallelisms of Complete Designs—
"There are three ways an area of mathematics
can be surveyed: by a vast, comprehensive treatise;
by a monograph on a small corner of the field; or by
a monograph on a cross section."
An area of mathematics—
A small corner of the field—
A cross section—
The area— Four.
The corner— Identity.
The cross section— Window.
The three ways— December 8 ten years ago.
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Tuesday, December 18, 2012
Monkey Grammar
For a modern Adam and Eve—
W. Tecumseh Fitch and Gesche Westphal Fitch,
editors of a new four-volume collection titled
Language Evolution (Feb. 2, 2012, $1,360)—
Related material—
|
"At the point of convergence
by Octavio Paz, translated by |
The "play of mirrors" link above is my own.
Click on W. Tecumseh Fitch for links to some
examples of mirror-play in graphic design—
from, say, my own work in a version of 1977, not from
the Fitches' related work published online last June—
See also Log24 posts from the publication date
of the Fitches' Language Evolution—
Happy birthday to the late Alfred Bester.
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Thursday, December 13, 2012
Interplay
Review:
Cubic models of finite geometries
display an interplay between
Euclidean and Galois geometry.
Related literary remarks: Congregated Light.
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Vibrations
… Or: A Funny Thing Happened
on the Way to the Embedding
This journal on the morning of Saturday, Dec. 8, 2012:
Marilyn Monroe and her music coach in 1954,
from last night's online New York Times :
" 'We were very close to making love; I don’t remember
the stage we were at, but I would say half-dressed,'
Mr. Schaefer recalled. He added: 'And all of a sudden
for some reason, Marilyn got these vibrations, and
we went over to the window….' " more »
"Mr. Schaefer died on Saturday at 87 at his home in
Fort Lauderdale, Fla. ….
He [had] coached Monroe through 'Diamonds Are
a Girl’s Best Friend,' her signature number in the
1953 movie 'Gentlemen Prefer Blondes' (he arranged
the music as well)…."
Perhaps on Saturday she returned the favor.
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Wednesday, December 12, 2012
Chromatic Plenitude
(Continued from 2 PM ET Tuesday)
“… the object sets up a kind of frame or space or field
within which there can be epiphany.”
— Charles Taylor, "Epiphanies of Modernism,"
Chapter 24 of Sources of the Self
(Cambridge U. Press, 1989, p. 477)
"The absolute consonance is a state of chromatic plenitude."
"… the nearest precedent might be found in Becky Sharp .
The opening of the Duchess of Richmond's ball,
with its organization of strong contrasts and
display of chromatic plenitude, presents a schema…."
— Scott Higgins, Harnessing the Technicolor Rainbow:
Color Design in The 1930s , University of Texas Press,
2007, page 142
Note the pattern on the dance floor.
(Click for wider image.)
"At the still point…" — Four Quartets
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Tuesday, December 11, 2012
Plenitude
In memory of Charles Rosen:
Related material:
The Magic Square in Doctor Faustus (October 10th, 2012)
Elementary Finite Geometry (August 1st, 2012)
The Space of Horizons (August 7th, 2012)
Chromatic Plenitude (Rosen on Schoenberg)
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Monday, December 10, 2012
Review of Leonardo Article
Review of an often-cited Leonardo article that is
now available for purchase online…
|
The Tiling Patterns of Sebastien Truchet Authors: Cyril Stanley Smith and Pauline Boucher
Source: Leonardo , Vol. 20, No. 4, Published by: The MIT Press Stable URL: http://www.jstor.org/stable/1578535 . |
Smith and Boucher give a well-illustrated account of
the early history of Truchet tiles, but their further remarks
on the mathematics underlying patterns made with
these tiles (see the diamond theorem* of 1976) are
worthless.
For instance…
Excerpt from pages 383-384—
"A detailed analysis of Truchet's
patterns touches upon the most fundamental
questions of the relation between
mathematical formalism and the structure
of the material world. Separations
between regions differing in density
require that nothing be as important as
something and that large and small cells of
both must coexist. The aggregation of
unitary choice of directional distinction
at interfaces lies at the root of all being
and becoming."
* This result is about Truchet-tile patterns, but the
underlying mathematics was first discovered by
investigating superimposed patterns of half-circles .
See Half-Circle Patterns at finitegeometry.org.
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Sunday, December 9, 2012
Eve’s Menorah
"Now the serpent was more subtle
than any beast of the field…."
— Genesis 3:1
"“The serpent’s eyes shine
As he wraps around the vine….”
– Don Henley
"Nine is a vine."
— Folk rhyme
Click images for some background.
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Adam in Eden
"… we have taken the first steps
in decoding the uniquely human
fascination with visual patterns…."
— W. Tecumseh Fitch et al. , July 2012
Fitch cites the following as a reference:
Washburn and Crowe discuss symmetries in general, but
not the Galois geometry underlying patterns like some of
those shown on their book's cover.
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Deep Structure
The concept of "deep structure," once a popular meme,
has long been abandoned by Chomskians.
It still applies, however, to the 1976 mathematics, diamond theory ,
underlying the formal patterns discussed in a Royal Society paper
this year.
A review of deep structure, from the Wikipedia article Cartesian linguistics—
|
[Numbers in parentheses refer to pages in the original 1966 Harper edition of Chomsky's book Cartesian Linguistics .] Deep structure vs. surface structure "Pursuing the fundamental distinction between body and mind, Cartesian linguistics characteristically assumes that language has two aspects" (32). These are namely the sound/character of a linguistic sign and its significance (32). Semantic interpretation or phonetic interpretation may not be identical in Cartesian linguistics (32). Deep structures are often only represented in the mind (a mirror of thought), as opposed to surface structures, which are not. Deep structures vary less between languages than surface structures. For instance, the transformational operations to derive surface forms of Latin and French may obscure common features of their deep structures (39). Chomsky proposes, "In many respects, it seems to me quite accurate, then, to regard the theory of transformational generative grammar, as it is developing in current work, as essentially a modern and more explicit version of the Port-Royal theory" (39). Summary of Port Royal Grammar The Port Royal Grammar is an often cited reference in Cartesian Linguistics and is considered by Chomsky to be a more than suitable example of Cartesian linguistic philosophy. "A sentence has an inner mental aspect (a deep structure that conveys its meaning) and an outer, physical aspect as a sound sequence"***** This theory of deep and surface structures, developed in Port Royal linguistics, meets the formal requirements of language theory. Chomsky describes it in modern terms as "a base system that generates deep structures and a transformational system that maps these into surface structures", essentially a form of transformational grammar akin to modern studies (42). |
The corresponding concepts from diamond theory are…
"Deep structure"— The line diagrams indicating the underlying
structure of varying patterns
"A base system that generates deep structures"—
Group actions on square arrays… for instance, on the 4×4 square
"A transformational system"— The decomposition theorem
that maps deep structure into surface structure (and vice-versa)
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Saturday, December 8, 2012
It’s 10 PM
Do you know where the mushrooms are?
Above: Image from Log24 on Dec. 4th, 2012, at 4:23 PM ET.
See also… on that date at that time …
The American College of Neuropsychopharmacology… (click to enlarge)—
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Defining the Contest…
… Chomsky vs. Santa
From a New Yorker weblog yesterday—
"Happy Birthday, Noam Chomsky." by Gary Marcus—
"… two titans facing off, with Chomsky, as ever,
defining the contest"
"Chomsky sees himself, correctly, as continuing
a conversation that goes back to Plato, especially
the Meno dialogue, in which a slave boy is
revealed by Socrates to know truths about
geometry that he hadn’t realized he knew."
See Meno Diamond in this journal. For instance, from
the Feast of Saint Nicholas (Dec. 6th) this year—
The Meno Embedding
For related truths about geometry, see the diamond theorem.
For a related contest of language theory vs. geometry,
see pattern theory (Sept. 11, 16, and 17, 2012).
See esp. the Sept. 11 post, on a Royal Society paper from July 2012
claiming that
"With the results presented here, we have taken the first steps
in decoding the uniquely human fascination with visual patterns,
what Gombrich* termed our ‘sense of order.’ "
The sorts of patterns discussed in the 2012 paper —
"First steps"? The mathematics underlying such patterns
was presented 35 years earlier, in Diamond Theory.
* See Gombrich-Douat in this journal.
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Friday, December 7, 2012
The Embedding
( Continued from December 6th — and from Duelle here and in Pynchon )
Part I
The Galois Embedding
Part II
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Wednesday, December 5, 2012
Arte Programmata*
The 1976 monograph "Diamond Theory" was an example
of "programmed art" in the sense established by, for
instance, Karl Gerstner. The images were produced
according to strict rules, and were in this sense
"programmed," but were drawn by hand.
Now an actual computer program has been written,
based on the Diamond Theory excerpts published
in the Feb. 1977 issue of Computer Graphics and Art
(Vol. 2, No. 1, pp. 5-7), that produces copies of some of
these images (and a few malformed images not in
Diamond Theory).
See Isaac Gierard's program at GitHub—
https://github.com/matthewepler/ReCode_Project/
blob/dda7b23c5ad505340b468d9bd707fd284e6c48bf/
isaac_gierard/StevenHCullinane_DiamondTheory/
StevenHCullinane_DiamondTheory.pde
As the suffix indicates, this program is in the
Processing Development Environment language.
It produces the following sketch:
The rationale for selecting and arranging these particular images is not clear,
and some of the images suffer from defects (exercise: which ones?), but the
overall effect of the sketch is pleasing.
For some background for the program, see The ReCode Project.
It is good to learn that the Processing language is well-adapted to making the
images in such sketches. The overall structure of the sketch gives, however,
no clue to the underlying theory in "Diamond Theory."
For some related remarks, see Theory (Sept. 30, 2012).
* For the title, see Darko Fritz, "Notions of the Program in 1960s Art."
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Tuesday, December 4, 2012
McKenna Theory
A 1976 monograph:
A 2012 mixtape cover:
A new "Diamond Theory" image found on the Web
today links my work to the "Stoned Ape Theory"
of human evolution due to Terence McKenna.
This link is via a picture, apparently copied from deviantart.com,
of two apes contemplating some psychedelic mushrooms.
The picture is titled "Stoned Ape Theory." The mushrooms in
the picture are apparently taken from an image at DrugNet.net:
Actually, the mathematical work called "diamond theory"
has nothing whatever to do with psychedelic experiences,
although some of the illustrations may appeal to McKenna fans.
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Monday, December 3, 2012
The Revisiting
Alan Cowell in the The New York Times ,
October 21, 2006—
"Mr. Pinter played the role of Krapp,
a 69-year-old man revisiting
a tape recording he had made at 39…."
See also a weblog post by a 69-year-old man
revisiting a drawing he had made at 39.
The revisiting:
On Guy Fawkes Day 2011,
a return to Guy Fawkes day 2005—
Contrapuntal Themes in a Shadowland.
The drawing:
A clearer version, from 1981, of the central object below —
For commentary on the original 1981 drawing, see
Diamond-Faceted: Transformations of the Rock.
(A link in that page to "an earlier note from 1981"
leads to remarks from exactly thirty years before
the 2011 post, made on another Guy Fawkes Day.)
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Thursday, November 29, 2012
Lines of Symbols
C. P. Snow on G. H. Hardy, in Snow's foreword to A Mathematician's Apology—
"One morning early in 1913, he found, among the letters on his breakfast table, a large untidy envelope decorated with Indian stamps. When he opened it, he found sheets of paper by no means fresh, on which, in a non-English holograph, were line after line of symbols. Hardy glanced at them without enthusiasm. He was by this time, at the age of thirty-six, a world famous mathematician: and world famous mathematicians, he had already discovered, are unusually exposed to cranks. He was accustomed to receiving manuscripts from strangers, proving the prophetic wisdom of the Great Pyramid, the revelations of the Elders of Zion, or the cryptograms that Bacon has inserted in the plays of the so-called Shakespeare."
Some related material (click to enlarge)—
The author links to, but does not name, the source of the above
"line after line of symbols." It is "Visualizing GL(2,p)." See that webpage
for some less esoteric background.
See also the two Wikipedia articles Finite geometry and Hesse configuration
and an image they share—
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Conceptual Art
Quotes from the Bremen site
http://dada.compart-bremen.de/ —
" 'compArt | center of excellence digital art' is a project
at the University of Bremen, Germany. It is dedicated
to research and development in computing, design,
and teaching. It is supported by Rudolf Augstein Stiftung,
the University of Bremen, and Karin und Uwe Hollweg Stiftung."
See also Stiftung in this journal.
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Wednesday, November 28, 2012
Diamond Theory
A pdf of a 1977 three-page article with this title
has been added at finitegeometry.org/sc.
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Tuesday, November 27, 2012
Counterexample
The non-Coxeter simple reflection group of order 168
is a counterexample to the statement that
"Every finite reflection group is a Coxeter group."
The counterexample is based on a definition of "reflection group"
that includes reflections defined over finite fields.
Today I came across a 1911 paper that discusses the counterexample.
Of course, Coxeter groups were undefined in 1911, but the paper, by
Howard H. Mitchell, discusses the simple order-168 group as a reflection group .
(Naturally, Mitchell's definition of "reflection" and his statement that
"The discussion of the binary groups
applies also to the case p = 2."
should be approached with care.)
A review of this topic might be appropriate for Jessica Fintzen's 2012 fall tutorial at Harvard
on reflection groups and Coxeter groups. The syllabus for the tutorial states that
"finite Coxeter groups correspond precisely to finite reflection groups." This statement
is based on Fintzen's definition of "reflection group"—
"Reflection groups are— as their name indicates—
groups generated by reflections across
hyperplanes of Rn which contain the origin."
For some background, see William Kantor's 1981 paper "Generation of Linear Groups"
(quoted at the finitegeometry.org page on the simple order-168 counterexample).
Kantor discusses Mitchell's work in some detail, but does not mention the
simple order-168 group explicitly.
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Sunday, November 25, 2012
Grim Joke
"It's a grim joke." — Amy Adams in "The Master"
When Irish Eyes Are Smiling…
Click diagram for some background from 3/17.
See, too, some background on Amy Adams and on Leap Day.
For related Harvard humor, see Venn Diagram.
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Saturday, November 24, 2012
Will and Representation*
Robert A. Wilson, in an inaugural lecture in April 2008—
Representation theory
A group always arises in nature as the symmetry group of some object, and group
theory in large part consists of studying in detail the symmetry group of some
object, in order to throw light on the structure of the object itself (which in some
sense is the “real” object of study).
But if you look carefully at how groups are used in other areas such as physics
and chemistry, you will see that the real power of the method comes from turning
the whole procedure round: instead of starting from an object and abstracting
its group of symmetries, we start from a group and ask for all possible objects
that it can be the symmetry group of .
This is essentially what we call Representation theory . We think of it as taking a
group, and representing it concretely in terms of a symmetrical object.
Now imagine what you can do if you combine the two processes: we start with a
symmetrical object, and find its group of symmetries. We now look this group up
in a work of reference, such as our big red book (The ATLAS of Finite Groups),
and find out about all (well, perhaps not all) other objects that have the same
group as their group of symmetries.
We now have lots of objects all looking completely different, but all with the same
symmetry group. By translating from the first object to the group, and then to
the second object, we can use everything we know about the first object to tell
us things about the second, and vice versa.
As Poincaré said,
Mathematicians do not study objects, but relations between objects.
Thus they are free to replace some objects by others, so long as the
relations remain unchanged.
Fano plane transformed to eightfold cube,
and partitions of the latter as points of the former:
* For the "Will" part, see the PyrE link at Talk Amongst Yourselves.
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Reappearing All Over Again
For the title, see the phrase "reappearing number" in this journal.
Some related mathematics—
the Greek labyrinth of Borges, as well as…
.
Note that "0" here stands for "23," while ∞ corresponds to today's date.
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Thursday, November 22, 2012
Finite Relativity
(Continued from 1986)
|
S. H. Cullinane This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.
— H. Weyl, The Classical Groups , In finite geometry "points" are often defined as ordered n-tuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such n-tuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S. This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame. The frame: A 4×4 array. The invariant structure: The following set of 15 partitions of the frame into two 8-sets.
A representative coordinatization:
0000 0001 0010 0011
The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2). |
S. H. Cullinane This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.
— H. Weyl, The Classical Groups , In finite geometry "points" are often defined as ordered n-tuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such n-tuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S. This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame. The frame: An array of 16 congruent equilateral subtriangles that make up a larger equilateral triangle. The invariant structure: The following set of 15 partitions of the frame into two 8-sets.
The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2). |
For some background on the triangular version,
see the Square-Triangle Theorem,
noting particularly the linked-to coordinatization picture.
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Monday, November 19, 2012
Poetry and Truth
From today's noon post—
"In all his poems with all their enchantments
for the poet himself, there is the final enchantment
that they are true. The significance of the poetic act
then is that it is evidence. It is instance and illustration.
It is an illumination of a surface,
the movement of a self in the rock.
Above all it is a new engagement with life.
It is that miracle to which the true faith of the poet
attaches itself."
— Wallace Stevens at Bard College, March 30, 1951
Stevens also said at Bard that
"When Joan of Arc said:
Have no fear: what I do, I do by command.
My brothers of Paradise tell me what I have to do.
these words were the words of an hallucination.
No matter what her brothers of Paradise drove her to do,
what she did was never a poetic act of faith in reality
because it could not be."
There are those who would dispute this.
Some related material:
"Ageometretos me eisito."—
"Let no one ignorant of geometry enter."—
Said to be a saying of Plato, part of the
seal of the American Mathematical Society—
A poetic approach to geometry—
"A surface" and "the rock," from All Saints' Day, 2012—
— and from 1981—
Some mathematical background for poets in Purgatory—
"… the Klein correspondence underlies Conwell's discussion
of eight heptads. These play an important role in another
correspondence, illustrated in the Miracle Octad Generator
of R. T. Curtis, that may be used to picture actions
of the large Mathieu group M24."
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Sunday, November 18, 2012
Sermon
Happy birthday to…
Today's sermon, by Marie-Louise von Franz—
For more on the modern physicist analyzed by von Franz,
see The Innermost Kernel , by Suzanne Gieser.
Another modern physicist, Niels Bohr, died
on this date in 1962…
|
The circle above is marked with a version For the square, see the diamond theorem. "Two things of opposite natures seem to depend — Wallace Stevens, |
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Thursday, November 15, 2012
A Passage to India
On a mathematician who died on All Souls' Day 2012—
"… he enthusiastically shared with us the many stories
of Indian epics like Mahabharata." — Online tribute
This suggests a pictorial review incorporating some
images from past Log24 posts.
Best Exotic Ananga Ranga
Click images for some background.
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Wednesday, November 14, 2012
Group Actions
The December 2012 Notices of the American
Mathematical Society has an ad on page 1564
(in a review of two books on vulgarized mathematics)
for three workshops next year on “Low-dimensional
Topology, Geometry, and Dynamics”—
(Only the top part of the ad is shown; for further details
see an ICERM page.)
(ICERM stands for Institute for Computational
and Experimental Research in Mathematics.)
The ICERM logo displays seven subcubes of
a 2x2x2 eight-cube array with one cube missing—
The logo, apparently a stylized image of the architecture
of the Providence building housing ICERM, is not unlike
a picture of Froebel’s Third Gift—
© 2005 The Institute for Figuring
Photo by Norman Brosterman from the Inventing Kindergarten
exhibit at The Institute for Figuring (co-founded by Margaret Wertheim)
The eighth cube, missing in the ICERM logo and detached in the
Froebel Cubes photo, may be regarded as representing the origin
(0,0,0) in a coordinatized version of the 2x2x2 array—
in other words the cube invariant under linear , as opposed to
more general affine , permutations of the cubes in the array.
These cubes are not without relevance to the workshops’ topics—
low-dimensional exotic geometric structures, group theory, and dynamics.
See The Eightfold Cube, A Simple Reflection Group of Order 168, and
The Quaternion Group Acting on an Eightfold Cube.
Those who insist on vulgarizing their mathematics may regard linear
and affine group actions on the eight cubes as the dance of
Snow White (representing (0,0,0)) and the Seven Dwarfs—
.
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Saturday, November 10, 2012
Descartes Field of Dreams
(A prequel to Galois Field of Dreams)
The opening of Descartes' Dream ,
by Philip J. Davis and Reuben Hersh—
"The modern world,
our world of triumphant rationality,
began on November 10, 1619,
with a revelation and a nightmare."
For a revelation, see Battlefield Geometry.
For a nightmare, see Joyce's Nightmare.
Some later work of Descartes—
From "What Descartes knew of mathematics in 1628,"
by David Rabouin, CNRS-Univ. Paris Diderot,
Historia Mathematica , Volume 37, Issue 3,
Contexts, emergence and issues of Cartesian geometry,
August 2010, pages 428–459 —
Fig. 5. How to represent the difference between white, blue, and red
according to Rule XII [from Descartes, 1701, p. 34].
The 4×4 array of Descartes appears also in the Battlefield Geometry posts.
For its relevance to Galois's field of dreams, see (for instance) block designs.
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