Log24

These stories suggest a meditation by Kierkegaard
and some related remarks from July 2, 2009.

"The final ingredient of the design conception
was its most important and its most dangerous…."

— The Cambridge Companion to Grand Opera

Starring Michael Douglas:

Related material: 

Happy Birthday, Mary Shelley (August 30, 2003)

Disney image:

"But you cannot un-ring that bell." — Matt Damon

"Classy." — Emma Watson

… from yesterday— Bling Ring and Church Logic.

Related narratives— Get Quotes (source of image below)

as well as Helprin's Doors and Trickster.

This post is continued from "Church Narrative," a Log24 post of November 17, 2010.

In memory of Ray Manzarek, remarks from a different weblog on that same date:

Click images for details.

Seth and Stefon, eat your hearts out.

Related material: Diamond Girl and the following ad
for Eliza Doolittle Day:
 

Best vs. Bester

The previous post ended with a reference mentioning Rosenhain.

For a recent application of Rosenhain's work, see
Desargues via Rosenhain (April 1, 2013).

From the next day, April 2, 2013:

"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:

See, too, in the Conway-Sloane book, the Galois tesseract … 
and, in this journal, Geometry for Jews and The Deceivers , by Bester.

From an arXiv preprint submitted July 18, 2011,
and last revised on March 11, 2013 (version 4):

"By our construction, this vector space is the dual
of our hypercube F₂⁴ built on I \ O₉. The vector space
structure of the latter, to our knowledge, is first
mentioned by Curtis in [Cur89]. Hence altogether
our proposition 2.3.4 gives a novel geometric
meaning in terms of Kummer geometry to the known
vector space structure on I \ O₉."

[Cur89] reference:
 R. T. Curtis, "Further elementary techniques using
the miracle octad generator," Proc. Edinburgh
Math. Soc.  32 (1989), 345-353 (received on
July 20, 1987).

— Anne Taormina and Katrin Wendland,
    "The overarching finite symmetry group of Kummer
      surfaces in the Mathieu group M ₂₄ ,"
     arXiv.org > hep-th > arXiv:1107.3834

"First mentioned by Curtis…."

No. I claim that to the best of my knowledge, the 
vector space structure was first mentioned by me,
Steven H. Cullinane, in an AMS abstract submitted
in October 1978, some nine years before the
Curtis article.

See Notes on Finite Geometry for some background.

See in particular The Galois Tesseract.

For the relationship of the 1978 abstract to Kummer
geometry, see Rosenhain and Göpel Tetrads in PG(3,2).

A followup to last midnight's Black Hole Revisited .

See also Victor Turner on liminality, together with Paul Goodman
on public squares, in a post of May 8, 2007. 

Related material: Midnight in Dostoevsky (St. Andrew's Day, 2009).

Backstory: The two previous Log24 posts
Raiders of the Lost Aleph (May 14) and
The Crying of Bucharest (May 15).

The following sequence of images was suggested by
Peter Woit's May 16 post "One Ring to Rule Them All."

Also from Devil's Night 2008:

From the May 16 Nobel Symposium talk discussed in
Woit's "One Ring to Rule Them All":

Related material:

All Souls' Day at the Still Point (Nov. 2, 2003) and

Frodo and the Oxford Murders (Oct. 13, 2011).

From a reported interview with Terry Gilliam:

Question: "You said in an interview that The Zero Theorem
was very influenced by Bucharest. In what way?"

Some related material from Log24:

See also a Google search from this evening—

— and the beginning of the Zero Theorem  screenplay —

FADE IN:
A rim of swirling light appears, circling helplessly into a
gigantic BLACK HOLE. It spirals inward…

See Coxeter + Aleph in this journal.

Epigraph to "The Aleph," a 1945 story by Borges:

"O God! I could be bounded in a nutshell,
and count myself a King of infinite space…"
– Hamlet, II, 2

Continued.

The order-3 affine plane:

Detail from the video in the previous post:

For other permutations of points in the
order-3 affine plane—

See Quaternions in an Affine Galois Plane
and Group Actions, 1984-2009.

See, too, the Mathematics and Narrative post 
from April 28, 2013, and last night's
For Indiana Spielberg.

(Continued from December 30, 2012)

"And let us finally, then, observe the
parallel progress of the formations of thought
across the species of psychical onomatopoeia
of the primitives, and elementary symmetries
and contrasts, to the ideas of substances,
to metaphors, the faltering beginnings of logic,
formalisms, entities, metaphysical existences."

— Paul Valéry, Introduction to the Method of
    Leonardo da Vinci

But first, a word from our sponsor…

Brought to you by two uploads, each from Sept. 11, 2012—

Symmetry and Hierarchy and the above VINCI Genius commercial.

Shem and Shaun present… NewsArse!

Part I:  Camp Germania

Part II:  NewsArse

From Uncle Walt.

For the late Bob Stewart:

"She was a panelist on many game shows, including
'What’s My Line?' and 'The Hollywood Squares.'"

Translation Studies Continued:

See Cameron's Kernel and…

 (Click image for some background.)

For a pundit of pugilism:  Plan 9 continues.

"She was a panelist on many game shows, including
'What’s My Line?' and 'The Hollywood Squares.'*
These appearances had a fitting symmetry:
It was as a game-show contestant that Dr. Brothers
had received her first television exposure."

— Margalit Fox in this evening's online New York Times

* A language game for Hofstadter: click on "Seeing As"
   in today's noon post.

"On Seeing A's and Seeing As" — Hofstadter

"Man redet dann vom Kern der Dinge." — Heidegger

"So we moved, and they, in a formal pattern"

— Four Quartets

See also yesterday's "Multi-Levels to Keep All Happy"
and past posts that mention Multispeech.

Western Washington University in Bellingham maintains a
website to benefit secondary-school math: MathNEXUS.

The MathNEXUS "website of the week" on April 14, 2013,
was the Diamond 16 Puzzle and its related webpages.

Click on the above image for the April 14 webpage.

The above images are from a prequel (March 29, 2013)
to 'Nauts  (March 26, 2006.)

See also Spider Mother,  Gamer Post,  and Spider Tale.

The title is from today's noon post, Core.

It also appears, quoted from Popovič, in Susan Bassnett's
Translation Studies  (third edition, Routledge, 2002)—

"It is an established fact in Translation Studies that if a dozen
translators tackle the same poem, they will produce a dozen
different versions. And yet somewhere in those dozen versions there
will be what Popovič calls the ‘invariant core’ of the original poem.
This invariant core, he claims, is represented by stable, basic and
constant semantic elements in the text, whose existence can be
proved by experimental semantic condensation. Transformations, or
variants, are those changes which do not modify the core of meaning
but influence the expressive form. In short, the invariant can be
defined as that which exists in common between all existing
translations of a single work. So the invariant is part of a dynamic
relationship and should not be confused with speculative arguments
about the ‘nature’, the ‘spirit’ or ‘soul’ of the text; the ‘indefinable
quality’ that translators are rarely supposed to be able to capture."

"A writer hopes to leave behind a work no one forgets…."

— Song sung on NBC's Smash  tonight

Fulsere vere candidi mihi soles….

— André Weil, The Apprenticeship of a Mathematician

nam unguentum dabo, quod meae puellae
donarunt Veneres Cupidinesque….

— Catullus, quoted in Bassnett's Translation Studies

Vale puella, iam Catullus obdurat.

Promotional description of a new book:

"Like Gödel, Escher, Bach  before it, Surfaces and Essences  will profoundly enrich our understanding of our own minds. By plunging the reader into an extraordinary variety of colorful situations involving language, thought, and memory, by revealing bit by bit the constantly churning cognitive mechanisms normally completely hidden from view, and by discovering in them one central, invariant core— the incessant, unconscious quest for strong analogical links to past experiences— this book puts forth a radical and deeply surprising new vision of the act of thinking."

"Like Gödel, Escher, Bach  before it…."

Or like Metamagical Themas . 

Rubik core:

Swarthmore Cube Project, 2008

Non- Rubik cores:

Related material: The Eightfold Cube and…

"A core component in the construction
is a 3-dimensional vector space V  over F₂ ."

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

From Amazon.com:

Book Description
"Publication Date: April 23, 2013
James Blish called him the “finest conscious artist
science fiction ever produced.” Kurt Vonnegut based
the famous character Kilgore Trout on him. And such
luminaries as Harlan Ellison, Stephen King, and Octavia
Butler have hailed him as a mentor. Theodore Sturgeon
was both a popular favorite and a writer’s writer…."

A writers' writer's writer:

"A writer's writer, Mrs. Le Guin 
brings reality itself to the proving ground."

— Theodore Sturgeon

* Shakespeare's birthday, April 23

A page with the above title has been created at
the Encyclopedia of Mathematics.

How long it will stay there remains to be seen.

"Why history?
Well, the essence of history  is story ,
and a good story is an end in itself."

— Barry Mazur, "History of Mathematics  as a tool,"
    February 17, 2013

See also today's previous post.

(Continued)

Click a course description below for some related material.

See also Strike That Pose and Gone to China.

Or:  Ready When You Are, C. B.

Dennis McLellan, Special to The Los Angeles Times , May 7, 2013:

Born in Los Angeles on June 29, 1920, Harryhausen was 13
when he saw "King Kong" during its run at Grauman's Chinese
Theater in Hollywood.

"I haven't been the same since," he is repeatedly quoted
as saying over the years.

"I came out of the theater awestruck," Harryhausen elaborated
in a 1999 interview with the Chicago Tribune. "It was such a
totally different, unusual film. The story line led you from the
mundane world into the most outrageous fantasy that's ever
been put on the screen."

See Damascus Gate.

For Jerusalem Day

Strange Correspondences :

"There are interesting correspondences between
Jewish Kabbala, Torah, and Talmud, and
Chinese Buddhism and Taoism…."

— Tony Smith

See also Chinese Checkers in this journal.

Structure vs. Character continued…

Related vocabulary:

Nick Tosches on the German word "Quell "

and Heidegger on Hölderlin.

* The title is from Heidegger.

(Continued)

From yesterday's post "Structure"—

A meditation on today's New York Times  obituaries:

"Gone to China."

"… Reality is not a given whole. An understanding of this,
a respect for the contingent, is essential to imagination
as opposed to fantasy. Our sense of form, which is an
aspect of our desire for consolation, can be a danger to
our sense of reality as a rich receding background.
Against the consolations of form, the clean crystalline
work, the simplified fantasy-myth, we must pit the
destructive power of the now so unfashionable naturalistic
idea of character.

Real people are destructive of myth, contingency is
destructive of fantasy and opens the way for imagination.
Think of the Russians, those great masters of the contingent.
Too much contingency of course may turn art into journalism.
But since reality is incomplete, art must not be too much
afraid of incompleteness. Literature must always represent a
battle between real people and images; and what it requires
now is a much stronger and more complex conception of the
former."

— Iris Murdoch, January 1961, "Against Dryness"

See also the recent posts Structure and Character.

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."

For the Church of St. Frank:

The phrase "Church of St. Frank" was coined in 1995 by
a Harvard professor sneering at literary critic Frank Kermode.
(See a related Log24 note from 1995.)

Now that Frank Kermode is gone, perhaps the phrase suits Frank Langella.

Above: Langella at Cannes with fellow actors from
Wall Street: Money Never Sleeps . He also starred in
the film version of Starting Out in the Evening  (quoted above).

Some related reflections on character:

Diamond Speech (this journal, July 3, 2012) and
Robert Diamond's Next Life in today's online New York Times .

(Continued)

"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.

Harvard Gazette , March 4th, 2013:

"Winfrey will speak on May 30 during Commencement day’s
Afternoon Exercises, which serve as the annual meeting of
the Harvard Alumni Association. The exercises will take place
in the Tercentenary Theatre of Harvard Yard,
between Memorial Church and Widener Library."

On the 1977 Octavia Butler novel Mind of My Mind :

"The first chapter in a history that Butler has already taken up
at a much later stage in Patternmaster  (1976).
Mind of My Mind  begins with Doro, a ruthless mutant
as old as the pyramids who has spent the last 4,000 years
trying to breed a race in his own image. The culminating
experiment is his daughter Mary. But, to Doro's astonishment,
Mary's first instinct on attaining her full powers is to begin
building a mental community— a Pattern— out of the
wretched thousands of Doro's half-telepathic failures
and partial successes. Despite some ragged moments,
Butler is clearly on to a promising vein— something like
Zenna Henderson's 'People' stories without their
saccharine silliness. There's a lot of intrinsic energy in the
Pattern idea, and one wants to see where this erratic, gifted
storyteller will pick it up next."

— Kirkus Reviews , Vol. XLV, No. 8 (1977), p. 453.

See this journal on Butler's dies natalis , the feast of St. Matthias, 2006.

Those who prefer Eastern approaches to religion may consult
Robert Thurman and his daughter Uma.

"Oprah, Uma. Uma, Oprah." — David Letterman

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:

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.

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

… 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.

Related mathematics and narrative —

Talkin' 'bout my generation.

"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.

Anne Taormina on Mathieu Moonshine —

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

"Well in North Carolina…" — George Jones

For those averse to white lightning —

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

See also Title and 24 Hour Psycho.

(Continued)

For Times Square Church…
Click image for a video.

The Oslo Version and The Lyche Omega

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

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 .

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:

Click to enlarge.

Click to enlarge.

Click for story.

"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