Log24

Saturday, February 7, 2026

A Sequel to Lowell Space* — Klein Space

Filed under: General — Tags: — m759 @ 4:52 am

* See the previous post.

Wednesday, February 4, 2026

Floor Show

Filed under: General — Tags: , , , — m759 @ 9:34 am

http://www.log24.com/log/pix10B/101027-LangerSymbolicLogic.jpg

Related reading from http://m759.net/wordpress/?s=Hot+Wife

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."

Tuesday, February 3, 2026

Surreal Memorial

Filed under: General — Tags: — m759 @ 6:38 am

260203-Surreal-Three-Dog-Memorial.jpg

Saturday, January 31, 2026

Times That Try Men’s Souls:
Another Opening, Another Show

Filed under: General — Tags: , — m759 @ 11:34 am

Synchronology check — Nov. 21, 2018

SAS Drama:  Sharkbait And Showgirls

Filed under: General — Tags: — m759 @ 8:54 am

Friday, January 30, 2026

Where Entertainment Is God:
Definition

Filed under: General — Tags: , , — m759 @ 6:26 am

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 . . .

Wednesday, January 28, 2026

For Your Consideration:  California Navel-Gaze . . .
Art by Marcela Nowak, Performance by Kiernan Shipka

Filed under: General — Tags: — m759 @ 1:47 pm

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 . . .

Surreal Art vs. Photo Realism

Filed under: General — Tags: — m759 @ 11:41 am

260128-Crary_Art-Philadelphia_Dawn-photo-date-
synchronology-check-Maltese_Parrot.jpg —

Also on May 23, 2024 . . .

Annals of Popular Culture:
PopCult for Quine*

Filed under: General — Tags: , — m759 @ 9:27 am

* From a post of June 25, 2008

From A Logical Point of View,  Harvard U. Press, 1980, p. 72

From A Logical Point of View,  Harvard U. Press, 1980, p. 73

Trevanian (and Kurt Weill) fans may
enjoy a variation on this theme.

Tuesday, January 27, 2026

Annals of Cultural Appropriation:
Waymark for Erewhon

Filed under: General — Tags: , , , — m759 @ 7:09 am

260127-Elliott_Smith-memorial-mural-in-LA.jpg

In 2004 —

See as well this  journal on the above upload date, 7 August 2010 —

The Matrix ReloadedCamp InceptionRift Designs .

Monday, January 26, 2026

For Your Consideration:
Multispeech Illustrated*

Filed under: General — Tags: , , , — m759 @ 9:28 pm

* 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'."

Saturday, January 24, 2026

A Polish Author’s Object Lesson

Filed under: General — Tags: — m759 @ 3:59 am

From a search in this  journal for "Abracadabra" . . .

Thursday, January 22, 2026

Sundance Tools Help

Filed under: General — Tags: , , — m759 @ 3:26 pm

Who’s Afraid of the Bloomsbury Group?

Filed under: General — Tags: , — m759 @ 4:02 am

Related reading for cultural historians — Quantum Lumps.

Sunday, January 18, 2026

Carpet Rider

Filed under: General — Tags: , , — m759 @ 12:03 pm

Wednesday, January 14, 2026

Today Is World Logic Day 2026

Filed under: General — Tags: , — m759 @ 12:34 pm

"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.

Diamond Theory
by NotebookLM

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.

 
Thursday, November 20, 2025

How are algebraic, combinatorial, and geometric structures unified within the Brick Space model?

The unification of algebraic, combinatorial, and geometric structures within the Brick Space model is achieved through its role as a graphic, isomorphic representation of key finite geometries, particularly the projective spaces PG(5,2) and PG(3,2), and their fundamental relationship via the Klein correspondence.

————————————————————————————.

The term "Brick Space" originates from the arrays, or "bricks," coined by R. T. Curtis as component parts of his Miracle Octad Generator (MOG)

. These physical or graphic arrangements serve as a concrete visualization (or model) of highly abstract mathematical structures.

1. Geometric Unification
The Brick Space model provides a geometric foundation for combinatorial objects through direct visualization of finite geometries defined over the two-element field, GF(2)
.
Modeling Projective Space: The Brick Space model is the natural geometric setting for the MOG's components, which may be regarded as forming both the 5-dimensional projective space and the 3-dimensional projective space
.
Visualizing PG(5,2) and the Klein Quadric: The model visually represents the space, which has 63 points, by partitioning these points into 28 points lying off the Klein quadric and 35 points lying on the Klein quadric

. The geometry of is linked to the concept of the 8-set

.
The Klein Correspondence: The geometric relationship between and is formalized by the Klein correspondence

. The model, developed as a brick space, is analogous to the "line diagrams" construction of found in the Cullinane diamond theorem

.
Lines and Points: The Brick Space implicitly models how the 35 lines of correspond to the 35 points on the Klein quadric in
.
2. Combinatorial Unification
Combinatorial structures, primarily partitions of sets and block designs, are mapped directly onto geometric entities within the Brick Space framework
.
Partitions and Lines: The central combinatorial equivalence involves mapping partitions of sets to geometric objects

. 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

.
Partitions of the 8-Set: The lines correspond to the 35 partitions of an 8-set into two 4-sets

. 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

.
Latin Squares: The combinatorial property of orthogonality of Latin squares (specifically, order-4 Latin squares) is unified geometrically by corresponding exactly to the property of skewness of lines (non-intersecting lines) in

. This provides a geometric dictionary for abstract algebraic combinatorics

.
3. Algebraic Unification
Algebraic structures, rooted in finite field theory and group theory, govern the behavior and symmetries observed in the combinatorial and geometric models
.
Binary Field : The foundation of the geometry is the Galois field of two elements, (binary arithmetic: )

. The algebraic rules of dictate the structure of the affine and projective spaces represented by the bricks

.
Binary Addition (XOR): The line diagrams used to formalize the patterns (related to the bricks) can be "added" via component-wise binary addition (XOR)

. 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 ()

.
Group Theory and Symmetry: The massive permutation group (of order 322,560) that transforms the visual patterns is isomorphic to the affine group

. 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

.
Ring Theory: The patterns can be endowed with algebraic structure to generate an ideal within a ring of symmetric patterns, leading to an infinite family of "diamond rings" isomorphic to rings of matrices over

. This links the geometry to the machinery of abstract algebra

.
M24 Connection: The affine group is identified as the octad stabilizer subgroup of the Mathieu group

, 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 Brick Space model, therefore, functions as a visual tapestry where graphic arrangements (geometry/visualization) encode set partitions (combinatorics) that behave according to the rules of binary algebra ( and affine groups)

. The entire structure is unified by powerful isomorphisms relating these three mathematical domains through finite geometries like and

Ex Fano Apollinis
Grok Benchmark

Filed under: General — Tags: , , — m759 @ 12:23 am

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.

Tuesday, January 13, 2026

Abstract Six

Filed under: General — Tags: , , — m759 @ 4:34 pm

Princess Mandorla

Filed under: General — Tags: — m759 @ 2:18 am

Ex Fano Apollinis

Filed under: General — Tags: , — m759 @ 2:03 am

Sunday, January 11, 2026

History as It Is Writ

Filed under: General — Tags: , , , , — m759 @ 10:48 pm

For James Joyce, courtesy of Guillermo del Toro.

I prefer the flower window illustrated here 
on December 29, 2025 —

“Tantamount”

Filed under: General — Tags: — m759 @ 10:02 pm

    See as well . . .

https://m759.net/wordpress/?s=Backwards .

Hollywood’s Search for . . .

Filed under: General — Tags: , , — m759 @ 1:20 pm

A non-Hollywood approach to symmetry . . .

Gypsy Rose Lily

Filed under: General — Tags: — m759 @ 3:34 am

Netflix Today: Expanding the Limits

Filed under: General — Tags: , , — m759 @ 1:41 am

https://m759.net/wordpress/?tag=harvard-munch

Saturday, January 10, 2026

But Seriously . . .

Filed under: General — Tags: , — m759 @ 3:03 am

This suggests a review . . .

Meanwhile, at Davos . . .

Thursday, January 8, 2026

Epiphany Dramarama

Filed under: General — Tags: — m759 @ 3:22 pm

Related art —

Epiphany 2014 piece on TV miniseries 'Spoils of Babylon'

 

Saturday, January 3, 2026

The Plugin

Filed under: General — Tags: , — m759 @ 5:04 pm

Chorus: ♫ “This Old Man Came Rolling Home

Filed under: General — Tags: , , — m759 @ 10:34 am

Backstory:

 

Monday, December 29, 2025

Octad Art — Bricks, Cubes, Flowers

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 . . .

The Eightfold Cube: The Beauty of Klein's Simple Group

   Image from Christmas Day 2005.

Post last revised:  December 30, 2025 @ 21:30 E.S.T.

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