Log24

Tuesday, April 30, 2013

Projective Analysis

Filed under: General,Geometry — m759 @ 8:00 PM

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).

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:

Logline

Filed under: General,Geometry — Tags: — m759 @ 9:29 AM

Found this morning in a search:

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?“.

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.

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.

Monday, April 29, 2013

Seal

Filed under: General — m759 @ 10:10 AM

For Poetry Month:

See posts containing
the above image.

“The theory of poetry, that is to say,
the total of the theories of poetry,
often seems to become in time
a mystical theology or, more simply,
a mystique."

Wallace Stevens, The Necessary Angel

Sunday, April 28, 2013

Filed under: General,Geometry — Tags: , , — m759 @ 11:00 PM

… 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 —

That correspondence was also discussed in a note 28 years ago, on this date in 1985.

Nominee

Filed under: General,Geometry — m759 @ 6:40 PM

Related mathematics and narrative —

Talkin' 'bout my generation.

Slow Art

Filed under: General — Tags: — m759 @ 12:00 PM

and posts of the past 24 hours.

Mathematics and Narrative (continued)

Filed under: General,Geometry — Tags: , — m759 @ 11:30 AM

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

Red October’s Sermon

Filed under: General — Tags: — m759 @ 10:30 AM

For the Harvard Arts Weekend:

"Grids, You Say?" by Josefine Lyche, with
Lyche's quotation from Rosalind Krauss in October
(Vol. 9, Summer 1979) —

"For every kind of vampire, there is a kind of cross." — Gravity's Rainbow

C’mon Baby…

Filed under: General — Tags: — m759 @ 3:13 AM

 Let's do the twist. The image at left is from a poster for a film released on March 28, 2003. See this journal on that date.

A phrase from yesterday's noon post:

Sinking the Magic 8-Ball .

A scene from the above film is related to this phrase.
Another image from the film poster:

A review of the film:

"The final 'twist' seems to negate the entire story,

Such a joke:

“Words and numbers are of equal value,
for, in the cloak of knowledge,
one is warp and the other woof.”

— The princesses Rhyme and Reason
in The Phantom Tollbooth

"A core component in the construction
is a 3-dimensional vector space over F."

—  Page 29 of "A twist in the M24 moonshine story,"
by Anne Taormina and Katrin Wendland.
(Submitted to the arXiv on 13 Mar 2013.)

The number of points in such a space is, of course, 8.

Saturday, April 27, 2013

For Bright Star*

Filed under: General — Tags: — m759 @ 9:00 PM

* See Title.

Elevation of the Host

Filed under: General — Tags: — m759 @ 7:59 PM

The title is a reference to a scheduled SNL.

Related material:

Cooper Union Borg,  Master Class,  and

White Noon

Filed under: General — Tags: — m759 @ 12:00 PM

(Continued from 24 hours ago)

Sinking the Magic 8-Ball

Mark and Remark

Filed under: General,Geometry — m759 @ 11:00 AM

“Fact and fiction weave in and out of novels like a shell game.” —R.B. Kitaj

Not just novels.

Fact:

The mark preceding A 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, 2009here.

Friday, April 26, 2013

Master Class

Filed under: General — Tags: , — m759 @ 8:00 PM

Philip Seymour Hoffman in "Doubt"—

and in "A Late Quartet"—

Symmetry

Filed under: General,Geometry — Tags: , — m759 @ 7:00 PM

This is, of course, the same group (of order 322,560) underlying the Diamond 16 Puzzle.

The Cruelest Month continues…

Filed under: General — m759 @ 5:24 PM

"Well in North Carolina…" — George Jones

For those averse to white lightning —

A link in yesterday 's 5:24 PM post yields moonshine.

High White

Filed under: General — Tags: , — m759 @ 12:00 PM

For Times Square Church
Click image for a video.

Review

Filed under: General — Tags: — m759 @ 11:00 AM

Those who prefer more traditional art
may consult The Portal Project.

Thursday, April 25, 2013

Rosenhain and Göpel Revisited

Filed under: General,Geometry — Tags: , — m759 @ 5:24 PM

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 .

Note on the MOG Correspondence

Filed under: General,Geometry — Tags: , — m759 @ 4:15 PM

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:

Wednesday, April 24, 2013

Art Wars for Odin’s Day

Filed under: General — m759 @ 9:25 PM

"By groping toward the light we are made to realize how deep the darkness is around us."

— Arthur Koestler, The Call Girls: A Tragi-Comedy , Random House, 1973, page 118

Character

Filed under: General — m759 @ 3:14 PM

"He said, 'I wrote a piece of code
that they just can’t seem to do without.'
He was a symbolic logician.
That was his career…."

She said, "It's a grim joke."

Title

Filed under: General — m759 @ 1:06 PM

Google search result at 1 PM ET April 24, 2013:

New York Stage and Film 2013 Musicals – EPA – Playbill
www.playbill.com/jobs/find/job_detail/51922.html

Casting: Howie Cherpakov
Music by Edie Brickell and Steve Martin
Lyrics by Edie Brickell Book by Steve Martin…

The musical is set in North Carolina.

From Howie Cherpakov:

From North Carolina:

Archibald Henderson monument, Chestnut Hill Cemetery, Salisbury, NC

Henderson died in 1963 on the Feast of St. Nicholas.
Related material: Santa vs. the Obelisk.

Tuesday, April 23, 2013

Say When

Filed under: General — Tags: — m759 @ 2:01 PM

Beneath the word "When" above, there appears
the date of a journal post— "July 27, 2012."

A check of synchronicity for this  journal on that date
yields two posts related to this morning's remarks.

The Six-Set

Filed under: General,Geometry — Tags: , — m759 @ 3:00 AM

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…"—

Sunday, April 21, 2013

The Grandmother Ship

Filed under: General — Tags: — m759 @ 9:00 PM

"E.L. Konigsburg— the author of one of my favorite
childhood books, the brilliantly quirky mystery
From The Mixed-Up Files of Mrs. Basil E. Frankweiler
died April 19 at the age of 83."

From other mixed-up files:

Detail:

Abstraction

Filed under: General,Geometry — Tags: — m759 @ 2:00 PM

(Continued from December 31st, 2012)

"Principles before personalities." — AA saying

Art Principles

Part I:

Part II:

Baker's 1922 Principles of Geometry

Art Personalities

Stoppard Update

Filed under: General — Tags: — m759 @ 12:30 PM

Meanwhile

Related material:

A sermon by the man named today the new President of Princeton.
The sermon is from October 7, 2012. See also Log24 on that date.

Sermon

Filed under: General — Tags: — m759 @ 11:01 AM

"There is  such a thing as a figure in four dimensions."

Friday, April 19, 2013

The Large Desargues Configuration

Filed under: General,Geometry — Tags: — m759 @ 9:25 AM

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.

Monday, April 15, 2013

M Theory

Filed under: General,Geometry — m759 @ 9:25 AM

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 "-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 -theory, where M  stands for
magic, mystery, or matrix, according to taste."

— "Magic, Mystery, and Matrix," by Edward Witten

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."

Suspense

Filed under: General — m759 @ 8:25 AM

"The suspension, as in solid space,
The suspending hand withdrawn, would be
An invisible gesture."

— Wallace Stevens

Part I… An image from this journal yesterday morning

Part II… A Google Doodle image  from this  morning—

Sunday, April 14, 2013

Raffiniert

Filed under: General — m759 @ 11:00 AM

Three interpretations:

Space Itself

Filed under: General,Geometry — Tags: — m759 @ 9:25 AM

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.

For pure mathematics, rather than theories of the physical world,
see the properties of the cube illustrated on the second (altered
book cover above.

Saturday, April 13, 2013

Occupy Space

Filed under: General — m759 @ 8:28 PM

(Continued from Seize the Dia,  April 6)

Two chess games by Fischer, against two brothers—

1956: "In this game, Fischer (playing Black) demonstrates
noteworthy innovation and improvisation." — Wikipedia

1963: "Fischer [playing Black] had engineered a brilliantly
disguised trap for him and … he had fallen into it." — NY Times

Some context:  The Crosswicks Curse.

Narrative

Filed under: General — m759 @ 6:16 PM

"… both marveled at early Ingmar Bergman movies."

One of the friends' "humor was inspired by
surrealist painters and Franz Kafka."

"Most of Marvel's fictional characters operate in
a single reality known as the Marvel Universe…."

Related material:  The Cosmic Cube.

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

m759 @ 7:00 PM

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.

Related material for those who prefer narrative
to mathematics:

 The Omen : Now we are …

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.

"There is  such a thing as a tesseract." — Madeleine L'Engle

Veblen and Young in Light of Galois

Filed under: General,Geometry — m759 @ 1:00 AM

Another in a series of figures illustrating
Desargues's theorem in light of Galois geometry:

Friday, April 12, 2013

An Education

Filed under: General — m759 @ 7:59 PM

For little Colva The Mother Ship :

.  .  .  .

For more light, see "Merton College" + Cameron
in this journal, as well as

An Education

Leonardo DiCaprio and Carey Mulligan in Baz Luhrmann's
new version of The Great Gatsby :

We're going to Disney World!

(For a more up-to-date version of little Colva,
see Primitive Groups and Maximal Subgroups.)

Midnight in Paris

Filed under: General,Geometry — m759 @ 6:00 PM

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

yesterday evening's remarks on art:

R.I.P.

Filed under: General — m759 @ 1:14 PM

Thursday, April 11, 2013

Naked Art

Filed under: General,Geometry — m759 @ 9:48 PM

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:

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

Dante Prize (continued)

Filed under: General — m759 @ 3:00 PM

For the title, see posts of March 25, 2013.

For little Colva  a tune from November 2005

and a New York Times  review.

Wednesday, April 10, 2013

Caution: Slow Art

Filed under: General,Geometry — Tags: , — m759 @ 9:00 PM

"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:

Art Wars (continued)

Filed under: General — Tags: — m759 @ 5:01 PM

This Way to the Egress:

Click images for some background.

A Text (continued)

Filed under: General — Tags: — m759 @ 9:00 AM

This journal on July 2, 2007:

(Click for more of the post)

A text:

Related material from July 3, 2007:

(Click for a clearer image of the quiz below.)

For answers to the quiz, see Jonathan Langdale.

For a deeper look at Achebe, see the following quote
in the context of last night's post on Hitchcock

— as well as Time + Eternity + Cloth in this journal.

Tuesday, April 9, 2013

Kountry Korn Kandy

Filed under: General — m759 @ 9:00 PM

For the first two words of the title,
see the previous post.

For the third word, see a review of the recent film "Hitchcock"
about the director and Janet Leigh during the filming of "Psycho"—

Hopkins' Hitchcock more or less eats out of Janet's hand
when she feeds him candy corn during a drive together
(the reference is to the candy Norman Bates is devouring
when he's interviewed by Martin Balsam's detective).

A story that demands the blended talents of Hitchcock and of
Mel Brooks to do it justice:

DeLillo's novel Point Omega . The review is titled,
without any other reference to L'Engle's classic tale
of the same name, "A Wrinkle in Time."

Related material: The Crosswicks Curse.

Four Quartets

Filed under: General,Geometry — m759 @ 5:10 PM

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.

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.

.

Filed under: General — Tags: — m759 @ 3:06 PM

Click images for some background.

Monday, April 8, 2013

Talkin’ ’Bout My Generation

Filed under: General — m759 @ 7:00 PM

A theme from the Mother Ship:

"When you wish upon a star…"

A related song, from last night at the
MGM Grand Garden Arena in Las Vegas:

"Highway Don't Care."

Concession to Sentiment…

Filed under: General — m759 @ 2:10 PM

The following may serve as an artistic sequel
to a post of April 3, "Mother Ship Wannabe."

Magic for Jews

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

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
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,

Related material from today's Daily Princetonian :

Star Wars

Filed under: General — m759 @ 11:01 AM

See searches in this journal for Balliol and for Star Quality.

Related material:

Above: A Google image search for Göpel tetrads  today. Click to enlarge.

Sunday, April 7, 2013

Pascal Inscape

Filed under: General,Geometry — m759 @ 1:00 PM

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.

Saturday, April 6, 2013

Seize the Dia

Filed under: General — m759 @ 7:00 PM

On this journal:

"he seems to repeat stuff compulsively punctuated with citing others and berating them for note taken nor credit given of his precedence .. but like i said, he more than makes up for that, dredging up and dusting off his all time faves like a super expensive store keeper who moves a piece only once a decade"

— "poetpiet" on Feb. 23, 2013

This suggests moving a piece linked to here
(in an update; scroll down) a decade ago.

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:  Occupy Space in this journal.

Pascal via Curtis

Filed under: General,Geometry — Tags: , — m759 @ 9:17 AM

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

Grünbaum has written little about Galois geometry.
Pisanski has recently touched on the subject;
see Configurations in this journal (Feb. 19, 2013).

Friday, April 5, 2013

Reflections (continued)

Filed under: General — m759 @ 4:30 PM

From a search for Mirror-Play in this journal:

That search was suggested by a much lengthier
search, for Core, that itself was suggested
by yesterday's post (on Katherine Neville's birthday)
titled A Philosopher's Stone.

See, too, Tom Hanks (shown above as symbologist
Robert Langdon) in "Lucky Guy" (reviewed by TIME
yesterday), and a related poem:

Prospective purchasers of the poet's work
may consult a press release from LSU Press
dated 4/4/2013. The poet died, apparently*
unlamented by his publisher, on 3/30.

* But only  apparently.

The Crucible

Filed under: General — Tags: — m759 @ 6:12 AM

"Though we had many pieces, we did not have the whole.
It was thirty years before we deciphered the formula.
But we did it at last.

There at night in the darkness of Fourier’s laboratory,
the four of us stood and watched the philosophers’ stone
forming in the crucible."

The Eight , by Katherine Neville
(2008 Ballantine Books mass market edition, p. 640)

A journal post from August 25, 2009:

Image from a different journal earlier that same day, August 25, 2009:

Thirty-year medallion from Alcoholics Anonymous —

Thursday, April 4, 2013

Soul of a Poet

Filed under: General — m759 @ 6:00 PM

"We buy it lock, stock, and shot glass"
— Ebert on Newman's performance in "The Verdict."

A Philosopher’s Stone

Filed under: General — m759 @ 4:00 PM

"Core" (in the original, Kern ) is perhaps
not the best translation of hypokeimenon :

Related material: In this journal, "underlie" and "underlying."

Hammer Down

Filed under: General — m759 @ 7:24 AM

Last Things

Filed under: General — m759 @ 7:00 AM

Session guitarist Hugh McCracken died at 70
last week Some say on Holy Thursday
some say on Good Friday.

"Be very very quiet
Clock everything you see
Little things might matter later
At the start of the end of history"

Wednesday, April 3, 2013

Museum Piece

Filed under: General,Geometry — Tags: — m759 @ 3:01 PM

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.

Mother Ship Wannabe

Filed under: General — m759 @ 6:29 AM

This morning's print edition of The New York Times
on the American Folk Art Museum:

"The museum’s executive director, Anne-Imelda Radice,
six months into the job, says her strategy is not to compete
with the Mets and MoMAs of the world, but to make
the institution a niche destination with a clear specialty.

'In a way, you become the mother ship of that particular
subject,' she said. 'We can be the leader. We’re the place
you check out first.' "

Presumably her lapel button, too small to read, does not say "6."

Tuesday, April 2, 2013

Hermite

Filed under: General,Geometry — Tags: , — m759 @ 7:14 PM

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
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 = eπω 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."

Intercultural Whatever

Filed under: General — Tags: — m759 @ 6:29 PM

University Diaries on March 22:

"An Intercultural Whatever professor at Florida Atlantic University…" (FAU).

An obituary in this afternoon's New York Times :

Related Log24 posts:

Pinter Play (March 8, about FAU) and a post
from the reported day of Gumperz's death.

Rota in a Nutshell

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

"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.

Baker on Configurations

Filed under: General,Geometry — Tags: , — m759 @ 11:11 AM

The geometry posts of Sunday and Monday have been
placed in finitegeometry.org as

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 15420configuration (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).

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.

Monday, April 1, 2013

Desargues via Rosenhain

Filed under: General,Geometry — Tags: , — m759 @ 6:00 PM

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 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 the post of April 1 that follows this introduction, 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.

A connection discovered today (April 1, 2013)—

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:

(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)

Preparation

Filed under: General,Geometry — m759 @ 7:20 AM

"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."