Tuesday, May 14, 2024
Monday, June 26, 2023
Monday, February 10, 2025
Brick Space: Points with Parts
This post's "Points with Parts" title may serve as an introduction to
what has been called "the most powerful diagram in mathematics" —
the "Miracle Octad Generator" (MOG) of Robert T. Curtis.
Curtis himself has apparently not written on the geometric background
of his diagram — the finite projective spaces PG(5,2) and PG(3,2), of
five and of three dimensions over the two-element Galois field GF(2).
The component parts of the MOG diagram, the 2×4 Curtis "bricks,"
may be regarded* as forming both PG(5,2) and PG(3,2) . . .
Pace Euclid, points with parts. For more on the MOG's geometric
background, see the Klein correspondence in the previous post.
For a simpler example of "points with parts, see
http://m759.net/wordpress/?s=200229.
* Use the notions of Galois (XOR, or "symmetric-difference") addition
of even subsets, and such addition "modulo complementation," to
decrease the number of dimensions of the spaces involved.
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Saturday, December 21, 2024
Coordinatizing Brick Space
Exercise: The eight-part diagrams in the graphic "brick space"
model of PG(5,2) below need to be suitably labeled with six-part
GF(2) coordinates to help illustrate the Klein correspondence that
underlies the large Mathieu group M24.
.gif)
A possible approach: The lines separating dark squares from light
(i.e., blue from white or yellow) in the figure above may be added
in XOR fashion (as if they were diamond theorem line diagrams)
to form a six dimensional vector space, which, after a suitable basis
is chosen, may be represented by six-tuples of 0's and 1's.
Related reading —
log24.com/log24/241221-'Brick Space « Log24' – m759.net.pdf .
This is a large (15.1 MB) file. The Foxit PDF reader is recommended.
The PDF is from a search for Brick Space in this journal.
Some context: http://m759.net/wordpress/?s=Weyl+Coordinatization.
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Friday, December 29, 2023
A Wrinkle in Logic
Also on March 8, 2018 —
This post was suggested by the word "entanglement' in the previous post.
See as well "Galois (Xor) addition."
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Sunday, July 7, 2019
Schoolgirl Problem
Anonymous remarks on the schoolgirl problem at Wikipedia —
"This solution has a geometric interpretation in connection with
Galois geometry and PG(3,2). Take a tetrahedron and label its
vertices as 0001, 0010, 0100 and 1000. Label its six edge centers
as the XOR of the vertices of that edge. Label the four face centers
as the XOR of the three vertices of that face, and the body center
gets the label 1111. Then the 35 triads of the XOR solution correspond
exactly to the 35 lines of PG(3,2). Each day corresponds to a spread
and each week to a packing."
See also Polster + Tetrahedron in this journal.
There is a different "geometric interpretation in connection with
Galois geometry and PG(3,2)" that uses a square model rather
than a tetrahedral model. The square model of PG(3,2) last
appeared in the schoolgirl-problem article on Feb. 11, 2017, just
before a revision that removed it.
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Sunday, October 25, 2015
Celtic Cross
The above illustrations are
from posts tagged
"Universe of Discourse."
Happy birthday to Évariste Galois, who may
prefer a mathematical, not religious,
interpretation of the above Celtic cross.
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Tuesday, February 14, 2012
The Ninth Configuration
The showmanship of Nicki Minaj at Sunday's
Grammy Awards suggested the above title,
that of a novel by the author of The Exorcist .
The Ninth Configuration —
The ninth* in a list of configurations—
"There is a (2d-1)d configuration
known as the Cox configuration."
— MathWorld article on "Configuration"
For further details on the Cox 326 configuration's Levi graph,
a model of the 64 vertices of the six-dimensional hypercube γ6 ,
see Coxeter, "Self-Dual Configurations and Regular Graphs,"
Bull. Amer. Math. Soc. Vol. 56, pages 413-455, 1950.
This contains a discussion of Kummer's 166 as it
relates to γ6 , another form of the 4×4×4 Galois cube.
See also Solomon's Cube.
* Or tenth, if the fleeting reference to 113 configurations is counted as the seventh—
and then the ninth would be a 153 and some related material would be Inscapes.
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Saturday, August 27, 2011
Cosmic Cube*
Prequel — (Click to enlarge)
Background —
See also Rubik in this journal.
* For the title, see Groups Acting.
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