Click image for a search.

* Parent company of Westworld.
  See also Delos  in this journal.

For further background to this morning's post Plan 9 Continues,
see posts tagged Sacred Space —

 .

See posts tagged Clooney Omega in this journal.

“… the object sets up a kind of 
 frame or space or field 
 within which there can be epiphany.”

— Charles Taylor

A frame or space or field —

Related material —

Star Wars (January 11, 2014),

The Lyche Gate Asterisk , from 10:31 AM ET on May 22, 2010,
the date of Martin Gardner's death —

— and the March 2014 issue of the
Notices of the American Mathematical Society  —

See as well Epiphany 2014 (Jan. 6) in this journal and the
March Notices  on the Shaw prize —

"Established under the auspices of Run Run Shaw
in November 2002, the prize is managed and
administered by the Shaw Prize Foundation
based in Hong Kong." 

Grid from a post linked to in yesterday's 24 Hour DeLillo—

A Study in Art Education

For an example of this grid as slow art , consider the following—

"One can show that the binary tetrahedral group
is isomorphic to the special linear group SL(2,3)—
the group of all 2×2 matrices over the finite field F₃
with unit determinant." —Wikipedia

As John Baez has noted, these two groups have the same structure as the geometric 24-cell.

For the connection of the grid to the groups and the 24-cell, see Visualizing GL(2,p).

Related material—

The 3×3 grid has been called a symbol of Apollo (Greek god of reason and of the sun).

"This is where we sat through his hushed hour,
a torchlit sky, the closeness of hills barely visible
at high white noon." — Don DeLillo, Point Omega

A reader comments on yesterday afternoon's New York Times
"The Stone" column by Justin E.H. Smith—

"I did indeed appreciate Mr. Smith’s essay.
And I’m curious as to what future contributions of his,
to the Stoner series, that we can look forward to."

From August 24, 2010—

Happy day 23 of Mental Health Month.

Leading today's New York Times  obituaries —

— is that of Nassos Daphnis, a painter of geometric abstractions
who in 1995 had an exhibition at a Leo Castelli gallery
titled "Energies in Outer Space." (See pictures here.)

Daphnis died, according to the Times, on November 23.
See Art Object, a post in this journal on that date—

There is more than one way
to look at a cube.

Some context— this morning's previous post (Apollo's 13,
on the geometry of the 3×3×3 cube), yesterday's noon post
featuring the 3×3 square grid (said to be a symbol of Apollo),

and, for connoisseurs of the Ed Wood school of cinematic art,
a search in this journal for the phrase "Plan 9."

You can't make this stuff up.

From Doonesbury today—

"What a magnificent load"

From this journal (September 20, 2009)—

The Logic of Dreams

From A Beautiful Mind–

“How could you,” began Mackey, “how could you, a mathematician, a man devoted to reason and logical proof…how could you believe that extraterrestrials are sending you messages? How could you believe that you are being recruited by aliens from outer space to save the world? How could you…?”

Nash looked up at last and fixed Mackey with an unblinking stare as cool and dispassionate as that of any bird or snake. “Because,” Nash said slowly in his soft, reasonable southern drawl, as if talking to himself, “the ideas I had about supernatural beings came to me the same way that my mathematical ideas did. So I took them seriously.”

Ideas:

These numbers may, in the mad way so well portrayed by Sylvia Nasar in the above book, be regarded as telling a story… a story that should, of course, not be taken too seriously.

Friday’s New York numbers (midday 214, evening 711) suggest the dates 2/14 and 7/11.  Clicking on these dates will lead the reader to Log24 entries featuring, among others, T. S. Eliot and Stephen King– two authors not unacquainted with the bizarre logic of dreams.

A link in the 7/11 entry leads to a remark of Noel Gray on Plato’s Meno and “graphic austerity as the tool to bring to the surface, literally and figuratively, the inherent presence of geometry in the mind of the slave.”

Also Friday: an example of graphic austerity– indeed, Gray graphic austerity– in Log24:

This illustration refers to chess rather than to geometry, and to the mind of an addict rather than to that of a slave, but chess and geometry, like addiction and slavery, are not unrelated.

Friday’s Pennsylvania numbers, midday 429 and evening 038, suggest that the story includes, appropriately enough in view of the above Beautiful Mind excerpt, Mackey himself.  The midday number suggests the date 4/29, which at Log24 leads to an entry in memory of Mackey.

(Related material: the Harvard Gazette of April 6, 2006, “Mathematician George W. Mackey, 90: Obituary“–  “A memorial service will be held at Harvard’s Memorial Church on April 29 at 2 p.m.“)

Friday’s Pennsylvania evening number 038 tells two other parts of the story involving Mackey…

As Mackey himself might hope, the number may be regarded as a reference to the 38 impressive pages of Varadarajan’s “Mackey Memorial Lecture” (pdf).

More in the spirit of Nash, 38 may also be taken as a reference to Harvard’s old postal address, Cambridge 38, and to the year, 1938, that Mackey entered graduate study at Harvard, having completed his undergraduate studies at what is now Rice University.

Returning to the concept of graphic austerity, we may further simplify the already abstract chessboard figure above to obtain an illustration that has been called both “the field of reason” and “the Garden of Apollo” by an architect, John Outram, discussing his work at Mackey’s undergraduate alma mater:

Let us hope that Mackey,
a devotee of reason,
is now enjoying the company
of Apollo rather than that of
Tom O’Bedlam:

For John Nash on his birthday:

I know more than Apollo,
For oft when he lies sleeping
I see the stars at mortal wars
In the wounded welkin weeping.

— Tom O’Bedlam’s Song

“Something beautiful fills the mind yet invites the search for something beyond itself, something larger or something of the same scale with which it needs to be brought into relation. Beauty, according to its critics, causes us to gape and suspend all thought. This complaint is manifestly true: Odysseus does stand marveling before the palm; Odysseus is similarly incapacitated in front of Nausicaa; and Odysseus will soon, in Book 7, stand ‘gazing,’ in much the same way, at the season-immune orchards of King Alcinous, the pears, apples, and figs that bud on one branch while ripening on another, so that never during the cycling year do they cease to be in flower and in fruit. But simultaneously what is beautiful prompts the mind to move chronologically back in the search for precedents and parallels, to move forward into new acts of creation, to move conceptually over, to bring things into relation, and does all this with a kind of urgency as though one’s life depended on it.”

The above symbol of Apollo suggests, in accordance with Scarry’s remarks, larger structures.   Two obvious structures are the affine 4-space over GF(3), with 81 points, and the affine plane over GF(3²), also with 81 points.  Less obvious are some related projective structures.  Joseph Malkevitch has discussed the standard method of constructing GF(3²) and the affine plane over that field, with 81 points, then constructing the related Desarguesian projective plane of order 9, with 9² + 9 + 1 = 91 points and 91 lines.  There are other, non-Desarguesian, projective planes of order 9.  See Visualizing GL(2,p), which discusses a spreadset construction of the non-Desarguesian translation plane of order 9.  This plane may be viewed as illustrating deeper properties of the 3×3 array shown above. To view the plane in a wider context, see The Non-Desarguesian Translation Plane of Order 9 and a paper on Affine and Projective Planes (pdf). (Click to enlarge the excerpt beow).

See also Miniquaternion Geometry: The Four Projective Planes of Order 9 (pdf), by Katie Gorder (Dec. 5, 2003), and a book she cites:

Miniquaternion geometry: An introduction to the study of projective planes, by T. G. Room and P. B. Kirkpatrick. Cambridge Tracts in Mathematics and Mathematical Physics, No. 60. Cambridge University Press, London, 1971. viii+176 pp.

For “miniquaternions” of a different sort, see my entry on Visible Mathematics for Hamilton’s birthday last year: