How are algebraic, combinatorial, and geometric structures unified within the Brick Space model?
* See the previous post.
* See the previous post.
Related reading from http://m759.net/wordpress/?s=Hot+Wife —
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From Tom McCarthy's review of The Maniac , a novel about 1940s social life at Los Alamos — "The mathematician Martin Davis’s wife, Lydia, storms out of a Trinity dinner party, condemning the men’s failure to fully take on board the consequences of their atom splitting. Besides sharing her name with our own age’s great translator of Blanchot and Proust, this Lydia Davis is a textile artist — a hanging detail that points back toward the novel’s many looms and weavings. For the Greeks, the fates spinning the threads of human lives were female (as Conrad knew, recasting them as Belgian secretaries in 'Heart of Darkness'). So was Theseus’ wool-ball navigator, Ariadne. And so, too, was the Ithacan ur-weaver Penelope, whose perpetual making and unraveling of her tapestry beat Gödel to an incompleteness theory by thousands of years. 'Text,' by the way, means something woven, from which we get 'textile.' It might just be that Penelope was not only testing her own version of the ontological limit, but also embedding it — in absent form, a hole — within the weft and warp of what we would eventually call the novel." |
Deadline yesterday reporting a January 28 death —
". . . 'remembered as a legend of Hollywood publicity,
one who helped define the role . . . .'"
This journal on January 28 —
Earlier in this journal . . .
From the star of Swimming with Sharks today . . .
260128-Branding-agency-featuring-Kiernan_Shipka-navel-pic.jpg
260126-Innie-Outie-Kiernan-Shipka-LA-Times-detail.jpg
Earlier . . .
260128-Crary_Art-Philadelphia_Dawn-photo-date-
synchronology-check-Maltese_Parrot.jpg —
Also on May 23, 2024 . . .

* From a post of June 25, 2008 —
Trevanian (and Kurt Weill) fans may
enjoy a variation on this theme.
260127-Elliott_Smith-memorial-mural-in-LA.jpg
In 2004 —
See as well this journal on the above upload date, 7 August 2010 —
* Related material — The black shirt above, and Elliott Smith.
From https://genius.com/artists/Elliott-smith —
" Smith is best-known for 'Miss Misery', his Oscar-nominated
contribution to the Good Will Hunting soundtrack, and XO’s
addictive and gorgeous family tension meditation, 'Waltz #2'."
Related reading for cultural historians — Quantum Lumps.
"The date chosen to celebrate World Logic Day, 14 January, corresponds to
the date of death of Kurt Gödel and the date of birth of Alfred Tarski,.
two of the most prominent logicians of the twentieth century." — Wikipedia
Some "presentatonal symbols" — The Gray Lotus.
94 sources
The Cullinane diamond theorem establishes a profound link between visual 4×4 patterns and advanced mathematical structures such as finite geometry and group theory. By analyzing symmetry-preserving transformations within a group of 322,560 permutations, the research connects simple grid designs to the complex projective space PG(3,2). These geometric frameworks further extend into coding theory, illustrating how "diamond" rings are isomorphic to matrix algebras over the finite field GF(4). The sources also highlight the Miracle Octad Generator (MOG) and its relationship to the Mathieu group M24, which is essential for understanding error-correcting codes and sphere packings. Beyond pure mathematics, the text explores "brick space" and the Klein correspondence, suggesting that these abstract symmetries represent a "universal" order found in both art and nature. Ultimately, this body of work bridges the gap between ancient motifs and modern computational science through the elegant study of structural invariants.
————————————————————————————.
. These physical or graphic arrangements serve as a concrete visualization (or model) of highly abstract mathematical structures.
. The geometry of is linked to the concept of the 8-set
. The model, developed as a brick space, is analogous to the "line diagrams" construction of found in the Cullinane diamond theorem
. The 35 combinatorial structures arising from permutations of diamond tiles (related to the bricks) are isomorphic to the 35 lines of . These 35 structures also mirror the 35 square patterns within the original MOG
. This same framework connects to the 105 partitions of an 8-set into four 2-sets, which are essentially the same as the 105 lines contained in the Klein quadric
. This provides a geometric dictionary for abstract algebraic combinatorics
. The algebraic rules of dictate the structure of the affine and projective spaces represented by the bricks
. The closure property of lines in is represented algebraically by the fact that the three line diagrams corresponding to a pattern always sum to zero ()
. This group, which explains the pervasive symmetry found in all -images, is the automorphism group of the underlying finite geometry, establishing the algebraic control over the system's geometric and combinatorial properties
. This links the geometry to the machinery of abstract algebra
, linking the visual patterns and the underlying geometries to the abstract world of sporadic simple groups, coding theory (Golay code), and lattice theory (Leech lattice)
. The entire structure is unified by powerful isomorphisms relating these three mathematical domains through finite geometries like and
Grok showed admirable persistence and depth in its research, but failed
to completely understand what is meant by "coordinatization of a 3×3 array."
It did, however, indicate a related concept in a suggested further prompt.
For the example that suggested the original prompt, see The Coxeter Aleph.
For James Joyce, courtesy of Guillermo del Toro.
I prefer the flower window illustrated here
on December 29, 2025 —

A non-Hollywood approach to symmetry . . .

This suggests a review . . .
Meanwhile, at Davos . . .

For the bricks of the title, see other posts tagged Brick Space.
For some cubes* and flowers, see below.
Combining features of the above two images, one might picture the 24
cells of the 4×6 array underlying the Curtis Miracle Octad Generator
(MOG) as each containing an eightfold cube, pictured as above with seven
of its subcubes showing and an eighth subcube hidden behind them.
The seven visible subcubes may be colored, as in the Curtis image of
the Klein map, with seven distinct colors… corresponding to the seven
edge-colors used in the Curtis-Klein map. Each of the seven visible
subcubes in a cell may also be labeled, on its visible faces, with a symbol
denoting one of the 24 points of the projective line over GF(23), just as the
faces in the Curtis-Klein map are labeled. The hidden subcube in each cell
may be regarded as also so labeled, by the MOG label of the cell's position.
There is then enough information in the array's eightfold cubes' colors and
labels to construct the seven generating permutations of M24 described by
Curtis, and the 24 array cells may be regarded as now containing 24 distinct
entities — which perhaps might be called "octoids."
Those desiring a more decorative approach may replace the 24 labeled cubes
with 24 labeled "flowers." Each flower — like the map's symmetric seven
"petals" and the central "infinity heptagon" they surround — forms an octad.
Related Illustrations . . .
* See as well posts tagged Mathieu Cube . . .
Related material —
The 56 triangles of the eightfold cube . . .
Image from Christmas Day 2005.
Post last revised: December 30, 2025 @ 21:30 E.S.T.
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