# Log24

## Thursday, August 9, 2018

### True Grids

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

From a search in this journal for "True Grid,"
a fanciful description of  the 3×3 grid —

"This is the garden of Apollo,
the field of Reason…."
John Outram, architect

A fanciful instance of the 4×2 grid in
a scene from the film "The Master" —

A fanciful novel referring to the number 8,
and a not -so-fanciful reference:

Illustrated above are Katherine Neville's novel The Eight  and the
"knight" coordinatization of the 4×2 grid from a page on the exceptional
isomorphism between PSL(3,2) (alias GL(3,2)) and PSL(2,7) — groups
of, respectively, degree 7 and degree 8.

Literature related to the above remarks on grids:

Ross Douthat's New York Times  column yesterday purported, following
a 1946 poem by Auden, to contrast students of the humanities with
technocrats by saying that the former follow Hermes, the latter Apollo.

I doubt that Apollo would agree.

## Saturday, January 8, 2011

### True Grid (continued)

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

"Rosetta Stone" as a Metaphor
in Mathematical Narratives

For some backgound, see Mathematics and Narrative from 2005.

Yesterday's posts on mathematics and narrative discussed some properties
of the 3×3 grid (also known as the ninefold square ).

For some other properties, see (at the college-undergraduate, or MAA, level)–
Ezra Brown, 2001, "Magic Squares, Finite Planes, and Points of Inflection on Elliptic Curves."

His conclusion:

When you are done, you will be able to arrange the points into [a] 3×3 magic square,
which resembles the one in the book [5] I was reading on elliptic curves….

This result ties together threads from finite geometry, recreational mathematics,
combinatorics, calculus, algebra, and number theory. Quite a feat!

5. Viktor Prasolov and Yuri Solvyev, Elliptic Functions and Elliptic Integrals ,
American Mathematical Society, 1997.

Brown fails to give an important clue to the historical background of this topic —
the word Hessian . (See, however, this word in the book on elliptic functions that he cites.)

Investigation of this word yields a related essay at the graduate-student, or AMS, level–
Igor Dolgachev and Michela Artebani, 2009, "The Hesse Pencil of Plane Cubic Curves ."

From the Dolgachev-Artebani introduction–

In this paper we discuss some old and new results about the widely known Hesse
configuration
of 9 points and 12 lines in the projective plane P2(k ): each point lies
on 4 lines and each line contains 3 points, giving an abstract configuration (123, 94).

PlanetMath.org on the Hesse configuration

A picture of the Hesse configuration–

(See Visualizing GL(2,p), a note from 1985).

Related notes from this journal —

From last November —

 m759 @ 10:12 PM From the December 2010 American Mathematical Society Notices— Related material from this  journal— Consolation Prize (August 19, 2010)

From 2006 —

 Sunday December 10, 2006    m759 @ 9:00 PM “Function defined form, expressed in a pure geometry that the eye could easily grasp in its entirety.” – J. G. Ballard on Modernism (The Guardian , March 20, 2006) “The greatest obstacle to discovery is not ignorance – it is the illusion of knowledge.” — Daniel J. Boorstin, Librarian of Congress, quoted in Beyond Geometry

Also from 2006 —

 Sunday November 26, 2006   m759 @ 7:26 AM Rosalind Krauss in "Grids," 1979: "If we open any tract– Plastic Art and Pure Plastic Art  or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter.  They are talking about Being or Mind or Spirit.  From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete. Or, to take a more up-to-date example…." "He was looking at the nine engravings and at the circle, checking strange correspondences between them." – The Club Dumas ,1993 "And it's whispered that soon if we all call the tune Then the piper will lead us to reason." – Robert Plant ,1971 The nine engravings of The Club Dumas (filmed as "The Ninth Gate") are perhaps more an example of the concrete than of the universal. An example of the universal*– or, according to Krauss, a "staircase" to the universal– is the ninefold square: "This is the garden of Apollo, the field of Reason…." – John Outram, architect     For more on the field of reason, see Log24, Oct. 9, 2006. A reasonable set of "strange correspondences" in the garden of Apollo has been provided by Ezra Brown in a mathematical essay (pdf). Unreason is, of course, more popular. * The ninefold square is perhaps a "concrete universal" in the sense of Hegel: "Two determinations found in all philosophy are the concretion of the Idea and the presence of the spirit in the same; my content must at the same time be something concrete, present. This concrete was termed Reason, and for it the more noble of those men contended with the greatest enthusiasm and warmth. Thought was raised like a standard among the nations, liberty of conviction and of conscience in me. They said to mankind, 'In this sign thou shalt conquer,' for they had before their eyes what had been done in the name of the cross alone, what had been made a matter of faith and law and religion– they saw how the sign of the cross had been degraded." – Hegel, Lectures on the History of Philosophy ,    "Idea of a Concrete Universal Unity" "For every kind of vampire, there is a kind of cross." – Thomas Pynchon

And from last October —

Friday, October 8, 2010

m759 @ 12:00 PM

Starting Out in the Evening
… and Finishing Up at Noon

This post was suggested by last evening's post on mathematics and narrative and by Michiko Kakutani on Vargas Llosa in this morning's New York Times .

 Above: Frank Langella in "Starting Out in the Evening" Right: Johnny Depp in "The Ninth Gate"

"One must proceed cautiously, for this road— of truth and falsehood in the realm of fiction— is riddled with traps and any enticing oasis is usually a mirage."

– "Is Fiction the Art of Lying?"* by Mario Vargas Llosa,
New York Times  essay of October 7, 1984

* The Web version's title has a misprint—

## Wednesday, January 5, 2011

### True Grid Example

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

See today's earlier posts Ode and True Grid (continued) and, in the latter's
context of tic-tac-toe war games —  Balance, from Halloween 2005 —

“An asymmetrical balance is sought since it possesses more movement.
This is achieved by the imaginary plotting of the character
upon a nine-fold square, invented by some ingenious writer of the Tang dynasty.
If the square were divided in half or in four, the result would be symmetrical,
but the nine-fold square permits balanced asymmetry."

Paraphrase of a passage in Chiang Yee's Chinese Calligraphy

### True Grid (continued)

Filed under: General — m759 @ 7:59 PM

For material related to the Dec. 29, 2010, posts Toy Stories and True Grid

See Theological Points of Tron: Legacy, by the anonymous blogger hilbertthm90.

An excerpt —

Another story told in the movie is how this girl was about to be destroyed by the evil guys only to find “the father” standing over her after she blacked out and he “saved her”. The father’s disk contains all the information about everything in the grid, so in a sense the father is omniscient. Clearly the father is a stand in for the Biblical God since he creates a whole universe and is omniscient (and by some of the actions he performs in the movie, he seems omnipotent as well). I don’t think the Christian parallels are all in my head.

There is some sketchy theology that occurs that might be in my head, though. The “real world” outside of the grid seems to be a symbol for heaven. If this is true, it presents a very interesting theological point.

Compare and contrast —

Woody and Buzz  from Christmas Day, 2010 —

Update of 8:55 PM — Related material from today's  Ghost Light weblog

See also today's New York midday lottery number, 596, as it appears in a search
for "legacy code" in this journal. It is China's code for its first nuclear test.

Today's evening New York number is less alarming — 401, suggesting the date
4/01 — i.e., April Fools' Day.

## Wednesday, December 29, 2010

### True Grid

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

Part I: True

Bulletin of the American Mathematical Society , October 2002, page 563

“…  the study of symmetries of patterns led to… finite geometries….”

– David W. Henderson, Cornell University

This statement may be misleading, if not (see Part II below) actually false. In truth, finite geometries appear to have first arisen from Fano's research on axiom systems. See The Axioms of Projective Geometry  by Alfred North Whitehead, Cambridge University Press, 1906, page 13.

Part II: Grid

For the story of how symmetries of patterns later did  lead to finite geometries, see the diamond theorem.

## Friday, December 29, 2017

### On Becket’s Day

Filed under: General — m759 @ 1:22 PM

For those who prefer Becket to Beckett
See a Log24 search for True Grid.

Update of 1:37 PM —

## Friday, February 14, 2014

### Ein Satz

Filed under: General — m759 @ 9:00 PM

Continued from Game Boy (August 6, 2006) —

See True Grid (Jan. 8, 2011) and Snow Queen.

## Wednesday, August 17, 2011

### Marginal Remarks

Filed under: General — m759 @ 11:17 AM

Today's Google Doodle is in honor of Fermat's birthday—

"I have discovered a truly marvelous proof of this theorem,
which this doodle is too small to contain."

Another marginal remark, from a link target in last night's "Ein Kampf"—

"We are talking about the spatial and temporal phenomenon of language,
not about some non-spatial, non-temporal chimera [Note in margin:
Only it is possible to be interested in a phenomenon in a variety of ways]."

— Ludwig Wittgenstein, Philosophical Investigations  (1953),  Section 108

Related material on spatial and temporal phenomena—

A Dec. 29, 2010, comment to a Dec. 26 weblog post on
"Unexpected Connections in Mathematics"—

Connoisseurs of synchronicities  in the phenomena of language may note that
these December dates mark the feasts of St. Stephen and St. Thomas Becket.

From the feast of the latter, two Log24 posts: Toy Stories and True Grid.

Those less enchanted by pop math than Google may prefer to observe
two other birthdays today— those of Robert De Niro and of Sean Penn:

## Sunday, April 17, 2011

### Sunday School

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

Apollo and the Tricksters

From The Story of N (Oct. 15, 2010)—

Roberta Smith on what she calls "endgame art"—

"Fear of form above all means fear of compression— of an artistic focus that condenses experiences, ideas and feelings into something whole, committed and visually comprehensible."

Margaret Atwood on tricksters and art—

"If it’s a seamless whole you want, pray to Apollo."

Here is some related material In memory of CIA officer Clare Edward Petty, who died at 90 on March 18—

A review of a sort of storyteller's MacGuffin — the 3×3 grid. This is, in Smith's terms, an "artistic focus" that appears  to be visually comprehensible but is not as simple as it seems.

The Hesse configuration can serve as more than a sort of Dan Brown MacGuffin. As a post of January 14th notes, it can (rather fancifullly) illustrate the soul—

" … I feel I understand
Existence, or at least a minute part
Of my existence, only through my art,
In terms of combinational delight…."

## Friday, January 14, 2011

### Ironic Butterfly

Filed under: General,Geometry — Tags: , — m759 @ 11:07 AM

David Brooks’s column today quotes Niebuhr. From the same source—
Reinhold Niebuhr, The Irony of American History

Chapter 8: The Significance of Irony

Any interpretation of historical patterns and configurations raises the question whether the patterns, which the observer discerns, are “objectively” true or are imposed upon the vast stuff of history by his imagination. History might be likened to the confusion of spots on the cards used by psychiatrists in a Rorschach test. The patient is asked to report what he sees in these spots; and he may claim to find the outlines of an elephant, butterfly or frog. The psychiatrist draws conclusions from these judgments about the state of the patient’s imagination rather than about the actual configuration of spots on the card. Are historical patterns equally subjective?
….
The Biblical view of human nature and destiny moves within the framework of irony with remarkable consistency. Adam and Eve are expelled from the Garden of Eden because the first pair allowed “the serpent” to insinuate that, if only they would defy the limits which God had set even for his most unique creature, man, they would be like God. All subsequent human actions are infected with a pretentious denial of human limits. But the actions of those who are particularly wise or mighty or righteous fall under special condemnation. The builders of the Tower of Babel are scattered by a confusion of tongues because they sought to build a tower which would reach into the heavens.

Niebuhr’s ironic butterfly may be seen in the context of last
Tuesday’s post Shining and of last Saturday’s noon post True Grid

The “butterfly” in the above picture is a diagram showing the 12 lines* of the Hesse configuration from True Grid.

It is also a reference to James Hillman’s classical image (see Shining) of the psyche, or soul, as a butterfly.

Fanciful, yes, but this is in exact accordance with Hillman’s remarks on the soul (as opposed to the spirit— see Tuesday evening’s post).

The 12-line butterfly figure may be viewed as related to the discussions of archetypes and universals in Hillman’s Re-Visioning Psychology  and in Charles Williams’s The Place of the Lion . It is a figure intended here to suggest philosophy, not entertainment.

Niebuhr and Williams, if not the more secular Hillman, might agree that those who value entertainment above all else may look forward to a future in Hell (or, if they are lucky, Purgatory). Perhaps such a future might include a medley of Bob Lind’s “Elusive Butterfly” and Iron Butterfly’s “In-a-Gadda-da-Vida.”

* Three horizontal, three vertical, two diagonal, and four arc-shaped.

## Monday, January 10, 2011

### Big Time*

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

(True Grid continued)

"They're gonna put me in the movies,
They're gonna make a big star out of me…"

“Thus the whirligig of time brings in his revenges.”
Twelfth Night ,
Act V, Sc. I  [text]