Log24

Saturday, May 23, 2020

Eightfold Geometry: A Surface Code “Unit Cell”

Filed under: General — Tags: , — m759 @ 1:50 AM

A unit cell in 'a lattice geometry for a surface code'

The resemblance to the eightfold cube  is, of course,
completely coincidental.

Some background from the literature —

Wednesday, April 28, 2010

Eightfold Geometry

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

Image-- The 35 partitions of an 8-set into two 4-sets

Image-- Analysis of structure of the 35 partitions of an 8-set into two 4-sets

Image-- Miracle Octad Generator of R.T. Curtis

Related web pages:

Miracle Octad Generator,
Generating the Octad Generator,
Geometry of the 4×4 Square

Related folklore:

“It is commonly known that there is a bijection between the 35 unordered triples of a 7-set [i.e., the 35 partitions of an 8-set into two 4-sets] and the 35 lines of PG(3,2) such that lines intersect if and only if the corresponding triples have exactly one element in common.” –“Generalized Polygons and Semipartial Geometries,” by F. De Clerck, J. A. Thas, and H. Van Maldeghem, April 1996 minicourse, example 5 on page 6

The Miracle Octad Generator may be regarded as illustrating the folklore.

Update of August 20, 2010–

For facts rather than folklore about the above bijection, see The Moore Correspondence.

Wednesday, December 11, 2019

Miracle Octad Generator Structure

Filed under: General — Tags: , — m759 @ 11:44 PM

Miracle Octad Generator — Analysis of Structure

(Adapted from Eightfold Geometry, a note of April 28, 2010.
See also the recent post Geometry of 6 and 8.)

Sunday, December 8, 2019

Geometry of 6 and 8

Filed under: General — Tags: , — m759 @ 4:03 AM

Just as
the finite space PG(3,2) is
the geometry of the 6-set, so is
the finite space PG(5,2)
the geometry of the 8-set.*

Selah.

* Consider, for the 6-set, the 32
(16, modulo complementation)
0-, 2-, 4-, and 6-subsets,
and, for the 8-set, the 128
(64, modulo complementation)
0-, 2-, 4-, 6-, and 8-subsets.

Update of 11:02 AM ET the same day:

See also Eightfold Geometry, a note from 2010.

Thursday, October 22, 2015

Objective Quality

Filed under: General,Geometry — Tags: — m759 @ 2:26 AM

Software writer Richard P. Gabriel describes some work of design
philosopher Christopher Alexander in the 1960's at Harvard:

A more interesting account of these 35 structures:

"It is commonly known that there is a bijection between
the 35 unordered triples of a 7-set [i.e., the 35 partitions
of an 8-set into two 4-sets] and the 35 lines of PG(3,2)
such that lines intersect if and only if the corresponding
triples have exactly one element in common."
— "Generalized Polygons and Semipartial Geometries,"
by F. De Clerck, J. A. Thas, and H. Van Maldeghem,
April 1996 minicourse, example 5 on page 6.

For some context, see Eightfold Geometry by Steven H. Cullinane.

Friday, May 3, 2013

Structure

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

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.”

Friday, April 6, 2012

Spectral Theory

Filed under: General,Geometry — Tags: , , , — m759 @ 8:00 AM

“And we may see the meadow in December,
icy white and crystalline” — Johnny Mercer

“At another end of the sexual confusion spectrum….”

IMAGE- Frank Langella and Liam Neeson in 'Unknown'

The devil likes metamorphoses.

Saturday, August 6, 2011

Correspondences

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

Comme de longs échos qui de loin se confondent
Dans une ténébreuse et profonde unité….

— Baudelaire, “Correspondances

From “A Four-Color Theorem”

http://www.log24.com/log/pix11B/110806-Four_Color_Correspondence.gif

Figure 1

Note that this illustrates a natural correspondence
between

(A) the seven highly symmetrical four-colorings
of the 4×2 array at the left of Fig. 1, and

(B) the seven points of the smallest
projective plane at the right of Fig. 1.

To see the correspondence, add, in binary
fashion, the pairs of projective points from the
“points” section that correspond to like-colored
squares in a four-coloring from the left of Fig. 1.
(The correspondence can, of course, be described
in terms of cosets rather than of colorings.)

A different correspondence between these 7 four-coloring
structures and these 7 projective-line structures appears in
a structural analysis of the Miracle Octad Generator
(MOG) of R.T. Curtis—

http://www.log24.com/log/pix11B/110806-Analysis_of_Structure.gif

Figure 2

Here the correspondence between the 7 four-coloring structures (left section) and the 7 projective-line structures (center section) is less obvious, but more fruitful.  It yields, as shown, all of the 35 partitions of an 8-element set  (an 8-set ) into two 4-sets. The 7 four-colorings in Fig. 2 also appear in the 35 4×4 parts of the MOG that correspond, in a way indicated by Fig. 2, to the 35 8-set paritions. This larger correspondence— of 35 4×2 arrays with 35 4×4 arrays— is  the MOG, at least as it was originally defined. See The MOG, Generating the Octad Generator, and Eightfold Geometry

For some applications of the Curtis MOG, see
(for instance) Griess’s Twelve Sporadic Groups .

Tuesday, June 14, 2011

Another Opening

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

NY Lottery this evening: 3-digit 444, 4-digit 0519.

444:

"… of our history … and of our destructive paths.
We are beginning to sense the need to restore
the sacred feminine." She paused. "You
mentioned you are writing a manuscript about
the symbols of the sacred feminine, are you not?"
"I …"

519 (or 5/19):

http://www.log24.com/log/pix11A/110519-PhaneSense.jpg

Related material— "Eightfold Geometry" + Spider in this journal.

For this afternoon's NY numbers— 511 and 9891— see
511 in the "Going Up" post of July 12, 2007, as well as
Ben Brantley's recent suggestion of Paris Hilton as a
matinee attraction and her 9891 photo on the Web.

Friday, April 15, 2011

Spider Notes

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

http://www.log24.com/log/pix11/110415-Symm-axes.jpg

Some connotations of the word "eightfold" —

IMAGE- Google search for 'eightfold geometry,' April 15, 2011

See also Damnation Morning and today's New York Times

A Final Bow for Julie Taymor's 'Spider-Man' Vision.

Tuesday, August 10, 2010

A Problem

Filed under: General,Geometry — m759 @ 7:35 PM

From Telegraph.co.uk (published: 5:56 PM BST 10 Aug 2010), a note on British-born Canadian journalist Bruce Garvey, who died at 70 on August 1—

In 1970, while reporting on the Apollo 13 mission at Nasa Mission Control for the Toronto Star, he was one of only two journalists— alongside Richard Killian of the Daily Express— to hear the famous message: "Houston we've had a problem."

See also Log24 posts of 10 AM and noon today.

The latter post poses the problem "You're dead. Now what?"

Again, as in this morning's post, applying Jungian synchronicity—

A check of this journal on the date of Garvey's death yields a link to 4/28's "Eightfold Geometry."

That post deals with a piece of rather esoteric mathematical folklore. Those who prefer easier problems may follow the ongoing struggles of Julie Taymor with "Spider-Man: Turn Off the Dark."

The problems of death, geometry, and Taymor meet in "Spider Woman" (April 29) and "Memorial for Galois" (May 31).

Tuesday, August 3, 2010

The Graduate

Filed under: General,Geometry — m759 @ 6:13 PM

IMAGE-- Robert F. Boyle, production designer for Hitchcock, died Sunday at 100

"The space in which a film takes place"—

See Eightfold Geometry, linked to here on the date of Boyle's death.

Monday, May 31, 2010

Memorial for Galois

Filed under: General,Geometry — m759 @ 7:16 PM

… and for Louise Bourgeois

Image-- Louise Bourgeois, sculptor of giant spiders, dies at 98

"The épateurs  were as boring as the bourgeois,
two halves of one dreariness."

— D. H. Lawrence, The Plumed Serpent

Image-- Google 5/31/2010 search for 'eightfold geometry' yields page on mother goddess as spider figure, also pages on some actual geometry

Friday, May 14, 2010

Competing MOG Definitions

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

A recently created Wikipedia article says that  “The Miracle Octad Generator [MOG] is an array of coordinates, arranged in four rows and six columns, capable of describing any point in 24-dimensional space….” (Clearly any  array with 24 parts is so capable.) The article ignores the fact that the MOG, as defined by R.T. Curtis in 1976, is not  an array of coordinates, but rather a picture of a correspondence between two sets, each containing 35 structures. (As a later commentator has remarked, this correspondence is a well-known one that preserves a certain incidence property. See Eightfold Geometry.)

From the 1976 paper defining the MOG—

“There is a correspondence between the two systems of 35 groups, which is illustrated in Fig. 4 (the MOG or Miracle Octad Generator).” —R.T. Curtis, “A New Combinatorial Approach to M24,” Mathematical Proceedings of the Cambridge Philosophical Society  (1976), 79: 25-42

http://www.log24.com/log/pix10A/100514-Curtis1976MOG.jpg

Curtis’s 1976 Fig. 4. (The MOG.)

The Wikipedia article, like a similar article at PlanetMath, is based on a different definition, from a book first published in 1988—

http://www.log24.com/log/pix10A/100514-SpherePack.jpg

I have not seen the 1973 Curtis paper, so I do not know whether it uses the 35-sets correspondence definition or the 6×4 array definition. The remarks of Conway and Sloane on page 312 of the 1998 edition of their book about “Curtis’s original way of finding octads in the MOG [Cur2]” indicate that the correspondence definition was the one Curtis used in 1973—

http://www.log24.com/log/pix10A/100514-ConwaySloaneMOG.jpg

Here the picture of  “the 35 standard sextets of the MOG”
is very like (modulo a reflection) Curtis’s 1976 picture
of the MOG as a correspondence between two 35-sets.

A later paper by Curtis does  use the array definition. See “Further Elementary Techniques Using the Miracle Octad Generator,” Proceedings of the Edinburgh Mathematical Society  (1989) 32, 345-353.

The array definition is better suited to Conway’s use of his hexacode  to describe octads, but it obscures the close connection of the MOG with finite geometry. That connection, apparent in the phrases “vector space structure in the standard square” and “parallel 2-spaces” (Conway and Sloane, third ed., p. 312, illustrated above), was not discussed in the 1976 Curtis paper.  See my own page on the MOG at finitegeometry.org.

Thursday, April 29, 2010

Spider Woman

Filed under: General,Geometry — m759 @ 11:32 AM

Mathematics and Narrative
(continued from April 26 and 28):

The Web

Image-- Google search for 'eightfold geometry'-- top result-- the Goddess as Spider Woman

See also

Leiber's Big Time, Spider Woman, and The Eight.

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