Log24

Saturday, February 27, 2010

Cubist Geometries

Filed under: Uncategorized — Tags: , — m759 @ 2:01 PM

"The cube has…13 axes of symmetry:
  6 C2 (axes joining midpoints of opposite edges),
4 C3 (space diagonals), and
3C4 (axes joining opposite face centroids)."
–Wolfram MathWorld article on the cube

These 13 symmetry axes can be used to illustrate the interplay between Euclidean and Galois geometry in a cubic model of the 13-point Galois plane.

The geometer's 3×3×3 cube–
27 separate subcubes unconnected
by any Rubik-like mechanism–

The 3x3x3 geometer's cube, with coordinates

The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
  subcubes in the (Galois) 3×3×3 cube–

The 13 symmetry axes of the cube

A closely related structure–
the finite projective plane
with 13 points and 13 lines–

Oxley's 2004 drawing of the 13-point projective plane

A later version of the 13-point plane
by Ed Pegg Jr.–

Ed Pegg Jr.'s 2007 drawing of the 13-point projective plane

A group action on the 3×3×3 cube
as illustrated by a Wolfram program
by Ed Pegg Jr. (undated, but closely
related to a March 26, 1985 note
by Steven H. Cullinane)–

Ed Pegg Jr.'s program at Wolfram demonstrating concepts of a 1985 note by Cullinane

The above images tell a story of sorts.
The moral of the story–

Galois projective geometries can be viewed
in the context of the larger affine geometries
from which they are derived.

The standard definition of points in a Galois projective plane is that they are lines through the (arbitrarily chosen) origin in a corresponding affine 3-space converted to a vector 3-space.

If we choose the origin as the center cube in coordinatizing the 3×3×3 cube (See Weyl's relativity problem ), then the cube's 13 axes of symmetry can, if the other 26 cubes have properly (Weyl's "objectively") chosen coordinates, illustrate nicely the 13 projective points derived from the 27 affine points in the cube model.

The 13 lines of the resulting Galois projective plane may be derived from Euclidean planes  through the cube's center point that are perpendicular to the cube's 13 Euclidean symmetry axes.

The above standard definition of points in a Galois projective plane may of course also be used in a simpler structure– the eightfold cube.

(The eightfold cube also allows a less standard way to picture projective points that is related to the symmetries of "diamond" patterns formed by group actions on graphic designs.)

See also Ed Pegg Jr. on finite geometry on May 30, 2006
at the Mathematical Association of America.

Thursday, December 29, 2005

Thursday December 29, 2005

Filed under: Uncategorized — Tags: — m759 @ 3:31 PM

Parallel Lines
Meet at Infinity

 

From Log24,
Dec. 16, 2005:

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From today's
New York Times,
a man who died
(like Charlie Chaplin
and W. C. Fields)
on Christmas Day:

The image “http://www.log24.com/log/pix05B/051229-DawsonClip.jpg” cannot be displayed, because it contains errors.

 

From Log24, Dec. 6, 2002,
Santa Versus the Volcano:

Well if you want to ride
you gotta ride it like you find it.
Get your ticket at the station
of the Rock Island Line.

Lonnie Donegan   
(d. Nov. 3, 2002)

and others

The Rock Island Line's namesake depot 
in Rock Island, Illinois

Sunday, November 16, 2003

Sunday November 16, 2003

Filed under: Uncategorized — Tags: , — m759 @ 7:59 PM

Russell Crowe as Santa's Helper

From The Age, Nov. 17, 2003:

"Russell Crowe's period naval epic has been relegated to second place at the US box office by an elf raised by Santa's helpers at the North Pole."

From A Midsummer Night's Dream:

"The lunatic,¹ the lover,² and the poet³
  Are of imagination all compact."

1

2

3

In acceping a British Film Award for his work in A Beautiful Mind, Crowe said that

"Richard Harris, one of the finest of this profession, recently brought to my attention the verse of Patrick Kavanagh:

'To be a poet and not know the trade,
To be a lover and repel all women,
Twin ironies by which
    great saints are made,
The agonising
    pincer jaws of heaven.' "

A theological image both more pleasant and more in keeping with the mathematical background of A Beautiful Mind is the following:

This picture, from a site titled Strange and Complex, illustrates a one-to-one correspondence between the points of the complex plane and all the points of the sphere except for the North Pole.

To complete the correspondence (to, in Shakespeare's words, make the sphere's image "all compact"), we may adjoin a "point at infinity" to the plane — the image, under the revised correspondence, of the North Pole.

For related poetry, see Stevens's "A Primitive Like an Orb."

For more on the point at infinity, see the conclusion of Midsummer Eve's Dream.

For Crowe's role as Santa's helper, consider how he has helped make known the poetry of Patrick Kavanagh, and see Kavanagh's "Advent":

O after Christmas we'll have
    no need to go searching….

… Christ comes with a January flower.

i.e. Christ Mass… as, for instance, performed by the six Jesuits who were murdered in El Salvador on this date in 1989.
  

Thursday, June 26, 2003

Thursday June 26, 2003

Filed under: Uncategorized — Tags: , — m759 @ 5:00 AM

ART WARS:
Art at the Vanishing Point

From the web page Art Wars:

"For more on the 'vanishing point,'
or 'point at infinity,' see
Midsummer Eve's Dream."

On Midsummer Eve, June 23, 2003, minimalist artist Fred Sandback killed himself.

Sandback is discussed in The Dia Generation, an April 6, 2003, New York Times Magazine article that is itself discussed at the Art Wars page.

Sandback, who majored in philosophy at Yale, once said that

"Fact and illusion are equivalents."

Two other references that may be relevant:

The Medium is
the Rear View Mirror
,

which deals with McLuhan's book Through the Vanishing Point, and a work I cited on Midsummer Eve  …

Chapter 5 of Through the Looking Glass:

" 'What is it you want to buy?' the Sheep said at last, looking up for a moment from her knitting.

'I don't quite know yet,' Alice said very gently.  'I should like to look all round me first, if I might.'

'You may look in front of you, and on both sides, if you like,' said the Sheep; 'but you ca'n't look all round you — unless you've got eyes at the back of your head.'

But these, as it happened, Alice had not got: so she contented herself with turning round, looking at the shelves as she came to them.

The shop seemed to be full of all manner of curious things — but the oddest part of it all was that, whenever she looked hard at any shelf, to make out exactly what it had on it, that particular shelf was always quite, empty, though the others round it were crowded as full as they could hold.

'Things flow about so here!' she said at last in a plaintive tone…."

 "When Alice went
     through the vanishing point
"
 

Monday, March 10, 2003

Monday March 10, 2003

Filed under: Uncategorized — Tags: , — m759 @ 5:45 AM

ART WARS:

Art at the Vanishing Point

Two readings from The New York Times Book Review of Sunday,

March 9,

2003 are relevant to our recurring "art wars" theme.  The essay on Dante by Judith Shulevitz on page 31 recalls his "point at which all times are present."  (See my March 7 entry.)  On page 12 there is a review of a novel about the alleged "high culture" of the New York art world.  The novel is centered on Leo Hertzberg, a fictional Columbia University art historian.  From Janet Burroway's review of What I Loved, by Siri Hustvedt:

"…the 'zeros' who inhabit the book… dramatize its speculations about the self…. the spectator who is 'the true vanishing point, the pinprick in the canvas.'''

Here is a canvas by Richard McGuire for April Fools' Day 1995, illustrating such a spectator.

For more on the "vanishing point," or "point at infinity," see

"Midsummer Eve's Dream."

Connoisseurs of ArtSpeak may appreciate Burroway's summary of Hustvedt's prose: "…her real canvas is philosophical, and here she explores the nature of identity in a structure of crystalline complexity."

For another "structure of crystalline
complexity," see my March 6 entry,

"Geometry for Jews."

For a more honest account of the
New York art scene, see Tom Wolfe's
 
The Painted Word.
 

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