The resemblance to the eightfold cube is, of course,
completely coincidental.
Some background from the literature —
Saturday, May 23, 2020
Eightfold Geometry: A Surface Code “Unit Cell”
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Wednesday, April 28, 2010
Eightfold Geometry
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.
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Wednesday, December 11, 2019
Miracle Octad Generator Structure
(Adapted from Eightfold Geometry, a note of April 28, 2010.
See also the recent post Geometry of 6 and 8.)
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Sunday, December 8, 2019
Geometry of 6 and 8
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.
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Thursday, October 22, 2015
Objective Quality
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.
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Friday, May 3, 2013
Structure
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.”
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Friday, April 6, 2012
Spectral Theory
“And we may see the meadow in December,
icy white and crystalline” — Johnny Mercer
“At another end of the sexual confusion spectrum….”
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Saturday, August 6, 2011
Correspondences
Comme de longs échos qui de loin se confondent
Dans une ténébreuse et profonde unité….
— Baudelaire, “Correspondances ”
From “A Four-Color Theorem”—
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—
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 |
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Tuesday, June 14, 2011
Another Opening
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 …"
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.
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Friday, April 15, 2011
Spider Notes
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Tuesday, August 10, 2010
A Problem
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).
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Tuesday, August 3, 2010
The Graduate
"The space in which a film takes place"—
See Eightfold Geometry, linked to here on the date of Boyle's death.
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Monday, May 31, 2010
Memorial for Galois
… and for Louise Bourgeois
"The épateurs were as boring as the bourgeois,
two halves of one dreariness."
— D. H. Lawrence, The Plumed Serpent
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Friday, May 14, 2010
Competing MOG Definitions
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
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—
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—
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.
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Thursday, April 29, 2010
Spider Woman
Mathematics and Narrative
(continued from April 26 and 28):
The Web
See also
Leiber's Big Time, Spider Woman, and The Eight.
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