How are algebraic, combinatorial, and geometric structures unified within the Brick Space model?
Illustration using Cullinane's four-color decomposition theorem —
" … fare forward, voyager . . . ." — T. S. Eliot
Friday, September 26, 2025
On the Klein Correspondence in Finite Geometry
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Thursday, May 22, 2025
Four-Color Decomposition and the Klein Correspondence
Ron Shaw in "Configurations of planes in PG(5,2)" . . .
"There are some rather weird things happening here."
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Tuesday, May 6, 2025
Grok 3 “Deeper Search” on
the Klein Correspondence and the MOG*
https://x.com/i/grok/share/6S3wugra7IfdMQGn0AvDjEC5V .
|
Conclusion "In conclusion, the Klein correspondence and the MOG are intricately linked through Conwell's correspondence, which aligns partitions of an 8-set with lines in PG(3,2), forming the backbone of the MOG's construction. This relationship enables the MOG to effectively study the Mathieu group M24 and related structures, bridging geometric and combinatorial mathematics. The detailed exploration reveals the depth of this connection, highlighting its significance in advanced mathematical research as of May 6, 2025." — Grok 3 "Deeper Search" |
* The "Miracle Octad Generator" of R. T. Curtis.
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the Klein Correspondence and the MOG*
the Klein Correspondence and the MOG*
Thursday, February 20, 2025
Coloring the Klein Correspondence
A Google search for "four color decomposition" yields an AI Overview —
My "four-color decomposition" theorem supplies some background
for last New Year's Eve's post on the Klein Correspondence.
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Thursday, January 9, 2025
Grok 2 on the MOG and the Klein Correspondence
Related illustration —
|
— Transcription —
Grok 2 on Klein correspondence and MOG — 9 Jan. 2025 Prompt: How is the Klein correspondence related to the Miracle Octad Generator? Grok 2 response (with citations and links removed): The Klein correspondence is closely related to the Miracle Octad Generator (MOG) through the mathematical structure they both engage with, particularly in the context of finite geometry and combinatorics. Klein Correspondence: This refers to a mathematical framework that establishes a correspondence between points and lines in a projective space. Specifically, in the context of the MOG, it relates to the geometry of PG(3,2), the three-dimensional projective space over the field with two elements. The Klein correspondence, in this context, essentially connects the combinatorial aspects of partitions of an 8-set to the geometric structure of lines in PG(3,2). Miracle Octad Generator (MOG): Introduced by R.T. Curtis, the MOG is a tool used in the study of the Mathieu group M24, the binary Golay code, and related structures like the Steiner system S(5,8,24). It consists of a 4×6 array where the positions are used to construct octads (sets of 8 elements) from given sets of 5 elements, reflecting the structure of the Steiner system. Relation Between Klein Correspondence and MOG: Combinatorial and Geometric Link: The MOG uses the Klein correspondence to map partitions of an 8-set (into two 4-sets) to 4 parallel affine planes within PG(3,2). This mapping is natural and symmetrical, which is crucial for the MOG's functionality. The 35 partitions of an 8-set correspond to the 35 lines in PG(3,2), and this correspondence helps in understanding how to generate and visualize octads within the MOG framework. Structural Insight: The Klein correspondence provides a way to see how the MOG's method of generating octads can be understood geometrically. This insight is vital for exploring the symmetry and subgroup structure of M24, as the MOG's layout preserves these symmetries. Thus, the Klein correspondence serves as a foundational link that allows the MOG to operate by translating combinatorial problems into geometric ones, facilitating the study of complex mathematical structures like the Mathieu groups and Steiner systems. |
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Monday, June 20, 2022
Thursday, June 9, 2022
Correspondence* Club
The new URL "inscape.club" forwards to …
http://m759.net/wordpress/?s=Inscape .
* For the "correspondences" of the above title, see …
http://m759.net/wordpress/?s=Correspondences+Ninth .
"He was looking at the nine engravings and at the circle,
checking strange correspondences between them."
– The Club Dumas , 1993
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Friday, February 7, 2020
Correspondences
The 15 2-subsets of a 6-set correspond to the 15 points of PG(3,2).
(Cullinane, 1986*)
The 35 3-subsets of a 7-set correspond to the 35 lines of PG(3,2).
(Conwell, 1910)
The 56 3-subsets of an 8-set correspond to the 56 spreads of PG(3,2).
(Seidel, 1970)
Each correspondence above may have been investigated earlier than
indicated by the above dates , which are the earliest I know of.
See also Correspondences in this journal.
* The above 1986 construction of PG(3,2) from a 6-set also appeared
in the work of other authors in 1994 and 2002 . . .
-
Gonzalez-Dorrego, Maria R. (Maria del Rosario),
(16,6) Configurations and Geometry of Kummer Surfaces in P3.
American Mathematical Society, Providence, RI, 1994. -
Dolgachev, Igor, and Keum, JongHae,
"Birational Automorphisms of Quartic Hessian Surfaces."
Trans. Amer. Math. Soc. 354 (2002), 3031-3057.
Addendum at 5:09 PM suggested by an obituary today for Stephen Joyce:
See as well the word correspondences in
"James Joyce and the Hermetic Tradition," by William York Tindall
(Journal of the History of Ideas , Jan. 1954).
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Saturday, August 6, 2016
Mystic Correspondence:
The Cube and the Hexagram
The above illustration, by the late Harvey D. Heinz,
shows a magic cube* and a corresponding magic
hexagram, or Star of David, with the six cube faces
mapped to the six hexagram lines and the twelve
cube edges mapped to the twelve hexagram points.
The eight cube vertices correspond to eight triangles
in the hexagram (six small and two large).
Exercise: Is this noteworthy mapping** of faces to lines,
edges to points, and vertices to triangles an isolated
phenomenon, or can it be viewed in a larger context?
* See the discussion at magic-squares.net of
"perimeter-magic cubes"
** Apparently derived from the Cube + Hexagon figure
discussed here in various earlier posts. See also
"Diamonds and Whirls," a note from 1984.
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Sunday, July 17, 2016
Symmetries and Correspondences
Continued from July 14, 2016 —
Symmetries and Correspondences in 1879 —
|
Cyparissos Stephanos Sur les systèmes desmiques de trois tétraèdres Bulletin des sciences mathématiques et astronomiques 2e série, Tome 3, No. 1 (1879), pp. 424-456. <http://www.numdam.org/item?id=BSMA_1879_2_3_1_424_1> © Gauthier-Villars, 1879, tous droits réservés. |
Symmetries and Correspondences in 1905 —
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Thursday, July 14, 2016
Symmetries and Correspondences
The title is that of a large-scale British research project
in mathematics. On a more modest scale …
"Hanks + Cube" in this journal —
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Wednesday, January 6, 2016
Correspondences
The above passage is from a Dec. 19, 2015, post,
Nunc Stans , on the death of New York Philharmonic
music director emeritus Kurt Masur.
See also a Log24 search for the word "Correspondences."
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Monday, October 6, 2014
Mysterious Correspondences
(Continued from Beautiful Mathematics, Dec. 14, 2013)
“Seemingly unrelated structures turn out to have
mysterious correspondences.” — Jim Holt, opening
paragraph of a book review in the Dec. 5, 2013, issue
of The New York Review of Books
One such correspondence:
For bibliographic information and further details, see
the March 9, 2014, update to “Beautiful Mathematics.”
See as well posts from that same March 9 now tagged “Story Creep.”
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Thursday, October 10, 2013
Klein Correspondence
(Continued from June 2, 2013)
John Bamberg continues his previous post on this subject.
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Tuesday, May 7, 2013
Strange Correspondences
For Jerusalem Day
"There are interesting correspondences between
Jewish Kabbala, Torah, and Talmud, and
Chinese Buddhism and Taoism…."
See also Chinese Checkers in this journal.
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Thursday, April 25, 2013
Note on the MOG Correspondence
In light of the April 23 post "The Six-Set,"
the caption at the bottom of a note of April 26, 1986
seems of interest:
"The R. T. Curtis correspondence between the 35 lines and the
2-subsets and 3-subsets of a 6-set. This underlies M24."
A related note from today:
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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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Friday, August 20, 2010
The Moore Correspondence
There is a remarkable correspondence between the 35 partitions of an eight-element set H into two four-element sets and the 35 partitions of the affine 4-space L over GF(2) into four parallel four-point planes. Under this correspondence, two of the H-partitions have a common refinement into 2-sets if and only if the same is true of the corresponding L-partitions (Peter J. Cameron, Parallelisms of Complete Designs, Cambridge U. Press, 1976, p. 60). The correspondence underlies the isomorphism* of the group A8 with the projective general linear group PGL(4,2) and plays an important role in the structure of the large Mathieu group M24.
A 1954 paper by W.L. Edge suggests the correspondence should be named after E.H. Moore. Hence the title of this note.
Edge says that
It is natural to ask what, if any, are the 8 objects which undergo
permutation. This question was discussed at length by Moore…**.
But, while there is no thought either of controverting Moore's claim to
have answered it or of disputing his priority, the question is primarily
a geometrical one….
Excerpts from the Edge paper—
Excerpts from the Moore paper—
Pages 432, 433, 434, and 435, as well as the section mentioned above by Edge— pp. 438 and 439
* J.W.P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Oxford U. Press, 1985, p. 72
** Edge cited "E.H. Moore, Math. Annalen, 51 (1899), 417-44." A more complete citation from "The Scientific Work of Eliakim Hastings Moore," by G.A. Bliss, Bull. Amer. Math. Soc. Volume 40, Number 7 (1934), 501-514— E.H. Moore, "Concerning the General Equations of the Seventh and Eighth Degrees," Annalen, vol. 51 (1899), pp. 417-444.
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Wednesday, January 14, 2026
Today Is World Logic Day 2026
"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.
How does the diamond theorem bridge visual tile patterns and abstract group theory?
In what ways does PG(3,2) serve as a foundation for Diamond Theory?
How do symmetry and automorphism groups link 4×4 arrays to the Mathieu group?
__________________________________________________________________
Thursday, November 20, 2025
————————————————————————————.
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
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Friday, December 5, 2025
Today’s “Diamond Theory” NotebookLM Summary
|
Diamond Theory by NotebookLM 92 sources The collected sources discuss the intricate confluence of finite geometry and abstract combinatorics, focusing heavily on the smallest three-dimensional projective space, PG(3,2), which acts as the geometric model for structures derived from the 6-set and 8-set. A primary focus is the Cullinane Diamond Theorem and the visual representation of abstract symmetries using 4×4 arrays, whose enormous automorphism group, the Affine group AGL(4,2), relates combinatorial design to geometric transformations. These connections are formalized using the Miracle Octad Generator (MOG) and the Klein Correspondence, which map partitions of an 8-set onto geometric objects like the lines of PG(3,2) and the points of the Klein quadric in PG(5,2). Furthermore, this framework bridges pure mathematics to applied fields, establishing relationships between geometric concepts like Conwell's Heptads and spreads (line partitions) and applications in algebraic ring theory, error-correcting codes, and the study of the sporadic simple group M24. Ultimately, the sources highlight how the symmetry inherent in these designs offers essential geometric insight into complex algebraic and combinatorial problems. |
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Wednesday, December 3, 2025
Today’s Diamond Theory Summary from NotebookLM
|
Diamond Theory by NotebookLM 92 sources The sources detail the profound mathematical correspondences linking visual, combinatorial, and abstract algebraic structures, primarily focusing on the finite projective space PG(3,2) and the affine group AGL(4,2). A central component is the Cullinane diamond theorem, which uses highly symmetric 4×4 grid patterns to model the AGL(4,2) transformation group, whose large order of 322,560 governs the symmetry of the arrangements. These geometric models are tied directly to deep combinatorial structures, specifically the Miracle Octad Generator (MOG) and the sporadic simple group Mathieu group M24, offering a unified framework for understanding octads and partitions like Conwell's Heptads. Further discussion establishes how geometric entities such as spreads, packings, and the Klein correspondence provide solutions for classic problems like the "schoolgirl problem" and inform contemporary areas like error-correcting codes and the classification of group orbits. This interplay extends even to physics, connecting the geometries to quantum space-time and two-qubit observables, demonstrating how abstract finite geometry underlies sophisticated concepts across various scientific and artistic disciplines. |
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Monday, December 1, 2025
“Diamond Theory” at NotebookLM Today
NotebookLM — Dec. 1, 2025 — "A Unifying Framework"
|
Diamond Theory by NotebookLM 92 sources These documents comprehensively examine the tight relationships among abstract algebra, combinatorics, and finite geometry, primarily through the lens of the projective spaces PG(3,2) and PG(5,2). A central focus is the Cullinane Diamond Theory, which utilizes highly symmetric 4×4 arrays over the Galois field GF(2) to model affine space, whose transformation group is the extensive Affine Group AGL(4,2). This visual and geometric structure connects directly to classic combinatorial objects, such as Conwell's Heptads and Dye's Partitions, which define specific sets and orbits within the geometry. Critically, these underlying symmetries are fundamental to the operation of the Miracle Octad Generator (MOG), establishing a direct link between the patterns of the 4×4 arrays and the intricate structure of the sporadic simple Mathieu group M24. The established correspondence provides crucial geometric interpretations for abstract algebraic problems, including the classification of reguli and the construction of Mutually Orthogonal Latin Squares. Collectively, the material demonstrates a unifying mathematical framework that spans theoretical geometry, error-correcting codes, quantum information science, and visual art. |
Comments Off on “Diamond Theory” at NotebookLM Today
Thursday, November 20, 2025
On Brick Space
A question suggested by NotebookLM in the "Diamond Theory" notebook,
and its answer today by NotebookLM —
|
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 . |
Comments Off on On Brick Space
Thursday, November 6, 2025
On Middlemarch: “The Patterns Are Out There!”
|
The Hunt for the World’s Oldest Story From thunder gods to serpent slayers, scholars are reconstructing myths that vanished millennia ago. How much further can we go—and what might we find? By Manvir Singh in The New Yorker
October 13, 2025 The Reverend Edward Casaubon is Eliot’s grand study in futility: an aging, self-important, faintly ridiculous clergyman who has dedicated his life to an audacious quest. Casaubon is convinced that every mythic system is a decayed remnant of a single original revelation—a claim he plans to substantiate in his magnum opus, “The Key to All Mythologies.” He means to chart the world’s myths, trace their similarities, and produce a codex that, as Eliot puts it, would make “the vast field of mythical constructions . . . intelligible, nay, luminous with the reflected light of correspondences.” The ill-fated project founders between the unruly diversity of cultural traditions and the fantasy of a single source, between the expanse of his material and the impossibility of ever mastering it, between the need for theory and the distortions it introduces. These failures are deepened by Casaubon’s limitations—his pedantic love of minutiae (he “dreams footnotes”) and his refusal to engage with scholarship in languages he doesn’t know (if only he’d learned German).
Casaubon’s quest stands as both an indictment of overreach and a warning about the senselessness of such sweeping comparisons. But is this entirely fair? The patterns are out there. Floods, tricksters, battles with monsters, creation and apocalypse—sometimes the resemblances are uncanny. |
"Before time began . . ." — Optimus Prime
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Sunday, November 2, 2025
The Mystery of Brick Space . . .
. . . is the title of a NotebookLM video now on YouTube —
See https://youtu.be/6zUKg4dNEbM .
|
The current NotebookLM summary for the Diamond Theory notebook, the source of the above video — "These sources comprehensively explore the deep connections between finite geometry, particularly the projective spaces PG(3,2) and PG(5,2) over GF(2), and various topics in combinatorics, group theory, and coding theory. Central to this discussion are the Miracle Octad Generator (MOG) and the Cullinane Diamond Theorem, which model highly symmetric structures like the affine group AGL(4,2) and the sporadic Mathieu group M24 using geometric figures such as 4×4 arrays or 'brick space.' The geometry of PG(3,2), described as the 'smallest perfect universe,' is shown to be crucial, relating to concepts like Conwell's Heptads, Klein correspondence, spreads, and mutually orthogonal Latin squares (MOLS), which also have applications in error-correcting codes and quantum information theory involving n-qubits. Ultimately, these texts demonstrate how abstract mathematical symmetry is intrinsically linked across algebra, geometry, and visual art, often leveraging automorphism groups to reveal structural invariants." |
Comments Off on The Mystery of Brick Space . . .
Thursday, October 16, 2025
Story Space: “The Greek Letter”
For this year's winner of the Nobel prize for literature,
a Hungarian enthusiast of run-on sentences whose "bible"
is said to be the classic novel about Cuernavaca by
Malcolm Lowry Under the Volcano . . .
As season three of "The Diplomat" unfolds, I visit my
memory garden to recall the time I sat with Judge Flick in his
chambers at my hometown courthouse to get a reference for
my application to Harvard and noticed on his desk a copy of
E. B. White's "little book" on prose style which, along with
a library book by Norbert Wiener, may have influenced my
mentioning to the judge the rather strange word "cybernetics,"
derived from the term for the steersman of the ship of Odysseus
who was lost at sea in Homer's epic tale.
"Wiener" of course is another term for a resident of Vienna.
And so, returning to much more recent memories — from
yesterday — of the long strange journey that has been my life . . .
Two references from a much less subjective and much more
objective tale that might amuse the late Hermann Weyl —
Related picture from a cartoon graveyard —
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Wednesday, October 15, 2025
Sextet Space
“Perhaps the philosophically most relevant feature of modern science
is the emergence of abstract symbolic structures as the hard core
of objectivity behind— as Eddington puts it— the colorful tale of
the subjective storyteller mind.”
— Hermann Weyl, Philosophy of Mathematics and
Natural Science , Princeton, 1949, p. 237
Melissa C. Wong, illustration for "Atlas to the Text,"
by Nicholas T. Rinehart:
The above fanciful illustration pictures 6*9=54 colored squares on the six
faces of a 3x3x3 cube.
Compare and contrast the Aitchison labeling, not unlike the one above,
of 6*4=24 unit squares (or, equivalently, 24 pips at the squares' centers)
on a 2x2x2 cube.
Now consider how the 8-square "brick" of R. T. Curtis may be colored with
four colors using the 105 ways to partition its eight squares into four 2-sets.
By analogy, the 24 squares on a cube's surface, as above, afford a cubical
space for applying six colors to the sextet partitions (into six 4-sets) of Curtis's
Miracle Octad Generator (MOG), using Aitchson's cubical model (with, of course,
the parts to be moved being pips or squares rather than cuboctahedron edges).
The 4-coloring of Curtis bricks is useful in picturing the Klein correspondence.
Are there similar uses of cube 6-colorings? Or 4-colorings? (Group actions on
a 6-set are of considerable combinatorial and algebraic interest because of
the exceptional outer automorphism of S6.)
For a colored presentation of sextet space modeled with a rectangle,
as in the Curtis MOG, see . . .
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Tuesday, October 14, 2025
Annals of AI Research: The Psi Mapping
Clicking "Explain the ψ mapping" led to Grok attempting to retrieve
a 1992 paper by Baartmans et al. from sciencedirect.com, which has
a captcha to block robot access. This led to . . .
For non-robots, the relevant sciencedirect.com page is . . .
Alphonse Baartmans, Walter Wallis, Joseph Yucas,
A geometric construction of the Steiner system S(4, 7, 23),
Discrete Mathematics, Volume 102, Issue 2, 1992, Pages 177-186,
ISSN 0012-365X,
https://doi.org/10.1016/0012-365X(92)90052-H.
(https://www.sciencedirect.com/science/article/pii/0012365X9290052H)
Abstract: The Steiner system S(4, 7, 23) is constructed from the geometry of PG(3, 2).
Some background from Google's AI Overview —
My own illustration of "a line lying entirely within the quadric" —
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Saturday, September 27, 2025
Four-Color Monolith
Those who find Kubrick's black 2001 monolith too dark
may prefer a more colorful image, taken from yesterday's
post on the Klein correspondence —
|
When?
Going to dark bed there was a square round Where?
— Ulysses , conclusion of Chapter 17. |
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Monday, September 22, 2025
Patterns: “Perceived Coherence”
From a Log24 post of Oct. 22, 2015 —
Software writer Richard P. Gabriel describes some work of design
philosopher Christopher Alexander in the 1960s at Harvard:
The above 35 strips are, it turns out, isomorphic to
the 35 points of of the Klein quadric over GF(2).
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