How are algebraic, combinatorial, and geometric structures unified within the Brick Space model?
A screenshot of a Gemini 3 summary . . .
Wednesday, December 17, 2025
Tuesday, August 26, 2025
Using AI for Search:
Combinatorial Partitions as Projective Lines …
Within the Klein Quadric
Some backstory: yesterday's post "Using AI: Search vs. Chat."
Vide a PDF of the complete Grok report —
In its five-and-a-half-minute research and reasoning process
Grok was able to reference a post from this weblog, but it missed
the correct answer to the prompt — Cullinane's "four-color
decomposition theorem" in the following weblog image:
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Combinatorial Partitions as Projective Lines …
Within the Klein Quadric
Combinatorial Partitions as Projective Lines …
Within the Klein Quadric
Thursday, August 14, 2025
The Klein Quadric and 105 Lines
Earlier Log24 posts tagged 105 Partitions suggest a look at . . .
Version 4 of the above paper is at https://arxiv.org/abs/2105.13798.
See also this journal on the Version 2 date — April 9, 2022 —
a post titled Academic Rhetoric on visual diagrams in mathematics.
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Sunday, February 9, 2025
An AI Overview of Klein Quadric and MOG
See also a more accurate AI report from January 9, 2025 —
Grok 2 on the MOG and the Klein Correspondence:
HTML version, with corrections, of the above 9 January Grok 2 report —
|
Grok 2: Klein Correspondence and MOG, 9 Jan. 2025 . . . 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 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:
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. * Correction: Should be "a correspondence between points in a five-dimensional projective space and lines in a three-dimensional projective space." |
Update of ca. 9 AM ET Monday, Feb. 10, 2024 —
Neither AI report above mentions the Cullinane model of the five-
dimensional projective space PG(5,2) as a brick space — a space
whose points are the 2×4 bricks used in thte MOG. This is
understandable, as the notion of using bricks to model both PG(5,2)
and PG(3,2) has appeared so far only in this journal. See an
illustration from New Year's Eve . . . Dec. 31, 2024 —
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Sunday, January 5, 2025
Friday, July 5, 2024
De Bruyn on the Klein Quadric
— De Bruyn, Bart. “Quadratic Sets on the Klein Quadric.”
JOURNAL OF COMBINATORIAL THEORY SERIES A,
vol. 190, 2022, doi:10.1016/j.jcta.2022.105635.
Related material —
Log24 on Wednesday, July 3, 2024: "The Nutshell Miracle" . . .
In particular, within that post, my own 2019 "nutshell" diagram of PG(5,2):
PG(5,2)
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Wednesday, December 11, 2019
Klein Quadric
The architecture of the recent post
Geometry of 6 and 8 is in part
a reference to the Klein quadric.
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Monday, February 23, 2026
Decomposition Theory
For some group actions on simpler decompositions — in finite spaces — of
point-sets at the vertices of n-dimensional cubes into point-sets at the vertices
of the cubes' n-2-dimensional subcubes . . . See the Feb. 13, 2026, post
Cube Space as well as the post below from the date of Daverman's death —
Another finite-geometry decomposition result that can be applied to the
representation, by 8-set-four-colorings, of lines in the Klein quadric —
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Friday, February 6, 2026
Lander in Artspace
Evolution of an image . . .
( Not to be confused with The Tin Man’s Hat. ) |
From the monograph preprint Diamond Theory (1976) —
(See pages 2 and 3 of the monograph.)
The above theorem underlies a revised anatomy of the Fano plane . . .
The fundamental theorem, expounded further in a 2001 web page, also
underlies the "seventh seal" derived from Peter J. Cameron's 1976 book
Parallelisms of Complete Designs — a representation of the 105 lines of the
Klein Quadric in PG(5,2) as the 105 partitions of an 8-set into four 2-sets.
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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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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 . . .
- in Aitchison's March 9, 2018, talk (slides 32-34), and
- in this journal on July 25, 2008, and later.
Image from Christmas Day 2005.
Post last revised: December 30, 2025 @ 21:30 E.S.T.
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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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Tuesday, December 2, 2025
Today’s NotebookLM “Diamond Theory” Summary
|
Diamond Theory by NotebookLM 92 sources This collection of texts examines the profound mathematical unity connecting finite geometry, group theory, and visual combinatorics, centered largely on the projective space PG(3,2) and the associated Affine Group AGL(4,2). The geometry is often modeled using structures like the 4×4 array or "Brick Space," where the action of the group AGL(4,2) (order 322,560) explains the symmetries of abstract diamond patterns. Central to this framework are classical structures like Conwell's Heptads and the Klein Quadric, which are shown to be crucial in partitioning spaces like PG(5,2) and constructing spreads used in coding theory. The material extensively links these geometric models, including the Miracle Octad Generator (MOG), to the exceptional symmetries of the Mathieu group M24 through stabilizer subgroups. Furthermore, these abstract concepts find applications in diverse fields, providing geometric insights into Mutually Orthogonal Latin Squares (MOLS), algebraic ring structures, and analogies within quantum physics related to qubit observables. The overarching theme demonstrates how symmetry, whether in abstract geometric configurations or visual quilt designs, is rooted in the deep logic of finite algebraic structure. |
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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
Friday, November 14, 2025
A Moduli Space
On the Miracle Octad Generator of R. T. Curtis —
|
December 2025 Notices of the American Mathematical Society Jarod Alper, "Evolution of Stacks and Moduli" — "By a moduli space, we mean a geometric space whose points are in 'natural' bijection (more on what we mean by 'natural' in a moment) with isomorphism classes of your favorite mathematical objects, for example, Riemann surfaces or vector bundles on a fixed space. A moduli space is a solution to the classification problem: it packages all of the data of the geometric objects into a single space, a mathematical catalogue where any object can be located by selecting the corresponding point." |
Analogous notions:
Klein Space and Klein Quadric in this journal.
The Source:
Related art from a Log24 post of July 1, 2018 —
Greg Egan’s animated image of the Klein quartic —
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Friday, November 7, 2025
Wednesday, October 8, 2025
Cube-Brick Columns
This post was suggested by yesterday's update to
the "Analogy Between Analogies" post of October 6.
The reason for the above columns . . .
The action of S8 on the rows of an 8-row 3-column matrix
000
001
010
011
100
101
110
111
is intimately connected, via the 30 labelings of a Fano plane
and via the Klein quadric in PG(5, 2), with the action of a
group of order 322,560 on the 16 squares of a 4×4 array.
See Conwell, 1910 [1] and the Log24 tag 105 partitions.
1. Conwell, George M. “The 3-Space PG(3, 2) and Its Group.”
Annals of Mathematics, vol. 11, no. 2, 1910, pp. 60–76.
JSTOR, https://doi.org/10.2307/1967582.
For those who prefer narratives to mathematics: The Cubes.
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Monday, October 6, 2025
Analogy Between Analogies
Consider . . .
A. The nontrivial analogy between the two parts of the well-known natural
15+15 partition of the 30 labelings of the Fano plane PG(2, 2)
B. The nontrivial analogy between the two parts of the well-known natural
15+15 partition of the 30 planes of the Klein quadric in PG(5, 2)
Are A and B nontrivially analogous? If so, how?
Update of 6:58 PM EDT Oct. 7 . . .
Hint:
Use as labels for PG(2, 2) points the seven nonzero vectors in the
3-space over GF(2), expressed as 001, 010, 011, 100, 101, 110, 111.
Then form three seven-digit vectors by taking the first, second, and third
digit in each 3-digit vector. View these seven-digit vectors as points of
the Klein quadric in PG(5, 2).
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Sunday, September 28, 2025
For Day 28 of September 2025: Fundamental Structures
http://m759.net/wordpress/?s="Ron+Shaw" —
The Klein quadric as background for
the Miracle Octad Generator of R. T. Curtis —
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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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Friday, June 27, 2025
Epistemic Art:
Plato at the Checkout
Geometry related to the famed
"divided line" of Plato's Republic —
Analogous "dividing line" . . .
Other geometry of interest . . .
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Plato at the Checkout
Plato at the Checkout
Tuesday, June 17, 2025
Lines
Related geometry . . .
"Drop me a line" — Request attributed to Emma Stone.
Meditation on the dropped line —
Analogous "dividing line" . . .
Related vocabulary: Stoicheia.
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Friday, April 11, 2025
Brick Space: The Yellow Brick* Road Now and in 2014
Now —
… and in 2014 —
* "Brick" is a term coined by R. T. Curtis that denotes any of the three
4-row 2-column arrays that form his 4-row 6-column Miracle Octad Generator.
Comments Off on Brick Space: The Yellow Brick* Road Now and in 2014
Tuesday, February 18, 2025
Grok 3 on the Log24 post of last New Year’s Eve
The Miracle Octad Generator (MOG) of R. T. Curtis —
|
Grok 3 beta, 12:56 AM ET Feb. 18, 2025 — The diagram you've provided from the webpage titled "The Yellow Brick Road to the Miracle Octad Generator; with Conwell's Heptads" illustrates the concept of "brick space" within the context of projective geometry, specifically PG(5,2), which is the 5-dimensional projective space over the field with 2 elements (GF(2)). Here's a detailed breakdown based on the image: Diagram Explanation:
Components of the Diagram:
|
For some remarks by Grok 3 beta made before it saw the
"yellow brick" diagram, see a PDF of prompts and responses.
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Friday, February 7, 2025
Tuesday, February 4, 2025
Diagrams
"The most powerful diagram in mathematics" —
The YouTube lecturer is not referring to the Fano plane diagram cited
in the AI Overview below, but to a much more sophisticated figure,
the Miracle Octad Generator (MOG) of R. T. Curtis.
Some context —
A rearrangement of the Miracle Octad Generator —
The diagram below may be less powerful , but it illustrates a result that,
although less miraculous , is perhaps more historically significant —
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Tuesday, January 14, 2025
Proofs
A phrase by Aitchison at Hiroshima . . .
"The proof of the above is a relabelling of the Klein quartic . . . ."
Related art — A relabelling of the Klein quadric by Curtis bricks:
Update of 12:26 PM EST Wednesday, January 15, 2025 —
Here is a large (17.5 MB) PDF file containing all posts touching upon
the concept underlying the above illustration — the Klein correspondence.
(A PDF reader such as Foxit is recommended for such large files.)
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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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Saturday, January 4, 2025
For a Contemporary Faustus
A contemporary minimalist composer whose work resembles that of
Thomas Mann's Doctor Faustus reportedly died at 85 Tuesday
in Paris on New Year's Eve (December 31, 2024). The phrase
"mathematical clarity" in his obituary in today's New York Times
suggests a synchronology check —
Compare and contrast.
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Tuesday, December 31, 2024
The Yellow Brick Road to the
Miracle Octad Generator, with Conwell’s Heptads
The Klein quadric as background for the Miracle Octad Generator of R. T. Curtis —
See also Saniga on heptads in this journal.
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Miracle Octad Generator, with Conwell’s Heptads
Miracle Octad Generator, with Conwell’s Heptads