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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Thursday, October 10, 2013
Klein Correspondence
(Continued from June 2, 2013)
John Bamberg continues his previous post on this subject.
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Saturday, April 4, 2026
Geometry of the Diamond Theorem
and the Mathieu Group
by NotebookLM on April 4, 2026
|
Based on the provided sources, the connection between the Cullinane Diamond Theorem and the Mathieu group M24 is established through group theory, finite geometry, and a combinatorial tool known as the Miracle Octad Generator (MOG).
1. Group Theoretic Connection:
The Octad Stabilizer The most direct link lies in the symmetry groups governing both structures.
2. The Miracle Octad Generator (MOG)
The "Miracle Octad Generator," developed by R.T. Curtis to study M24 and the binary Golay code, serves as the bridge between the visual patterns of the Diamond Theorem and the abstract algebra of the Mathieu group.
3. Geometric Unification
The connection is further unified by finite geometries that underlie both systems:
In summary, the group that shuffles the Diamond Theorem's 4×4 tiles is physically the same mathematical object (up to isomorphism) as the subgroup of M24 that stabilizes an octad, and the visual patterns produced are the geometric building blocks of the Miracle Octad Generator used to construct M24.
|
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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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 . |
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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 . . .
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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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, 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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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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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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Monday, December 23, 2024
A Projective-Space Home for the Miracle Octad Generator
The natural geometric setting for the "bricks" in the Miracle Octad Generator
(MOG) of Robert T. Curtis is PG(5,2), the projective 5-space over GF(2).
The Klein correspondence mirrors the 35 lines of PG(3,2) — and hence, via the
graphic approach below, the 35 "heavy bricks" of the MOG that match those
lines — in PG(5,2), where the bricks may be studied with geometric methods,
as an alternative to Curtis's original MOG combinatorial construction methods.
The construction below of a PG(5,2) brick space is analogous to the
"line diagrams" construction of a PG(3,2) in Cullinane's diamond theorem.
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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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Thursday, December 19, 2024
Different Angles
"Drawing the same face from different angles sounds fun,
but let me tell you – it’s not. It’s not fun at all. It’s HARD!!"
— Loisvb on Instagram, Dec. 18, 2024
Likewise for PG(5,2).
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.
-coordinates.gif)
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Tuesday, August 27, 2024
For Rubik Worshippers
The above is six-dimensional as an affine space, but only five-dimensional
as a projective space . . . the space PG(5, 2).
As the domain of the smallest model of the Klein correspondence and the
Klein quadric, PG (5,2) is not without mathematical importance.
See Chess Bricks and Ovid.group.
This post was suggested by the date July 6, 2024 in a Warren, PA obituary
and by that date in this journal.
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Wednesday, July 3, 2024
The Nutshell Miracle
From a search in this journal for nocciolo —
From a search in this journal for PG(5,2) —
From a search in this journal for Curtis MOG —
Shown above is a rearranged version of the
Miracle Octad Generator (MOG) of R. T. Curtis
("A new combinatorial approach to M24,"
Math. Proc. Camb. Phil. Soc., 79 (1976), 25-42.)
From a search in this journal for Klein Correspondence —
The picture of PG(5,2) above as an expanded nocciolo
shows that the Miracle Octad Generator illustrates
the Klein correspondence.
Update of 10:33 PM ET Friday, July 5, 2024 —
See the July 5 post "De Bruyn on the Klein Quadric."
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Saturday, September 3, 2022
1984 Revisited
Note the three quadruplets of parallel edges in the 1984 figure above.
The above Gates article appeared earlier, in the June 2010 issue of
Physics World , with bigger illustrations. For instance —
Exercise: Describe, without seeing the rest of the article,
the rule used for connecting the balls above.
Wikipedia offers a much clearer picture of a (non-adinkra) tesseract —
And then, more simply, there is the Galois tesseract —
For parts of my own world in June 2010, see this journal for that month.
The above Galois tesseract appears there as follows:
See also the Klein correspondence in a paper from 1968
in yesterday's 2:54 PM ET post.
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Tuesday, June 21, 2022
For the Church of Synchronology*
See also this journal on the above Cambridge U. Press date.
"There are many places one can read about twistors
and the mathematics that underlies them. One that
I can especially recommend is the book Twistor Geometry
and Field Theory, by Ward and Wells."
— Peter Woit, "Not Even Wrong" weblog post, March 6, 2020.
* A fictional entity. See Synchronology in this journal.
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Wednesday, February 9, 2022
8!
Conwell, 1910 —
(In modern notation, Conwell is showing that the complete
projective group of collineations and dualities of the finite
3-space PG (3,2) is of order 8 factorial, i.e. "8!" —
In other words, that any permutation of eight things may be
regarded as a geometric transformation of PG (3,2).)
Later discussion of this same "Klein correspondence"
between Conwell's 3-space and 5-space . . .
A somewhat simpler toy model —
Related fiction — "The Bulk Beings" of the film "Interstellar."
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Sunday, October 11, 2020
Saniga on Einstein
See “Einstein on Acid” by Stephen Battersby
(New Scientist , Vol. 180, issue 2426 — 20 Dec. 2003, 40-43).
That 2003 article is about some speculations of Metod Saniga.
“Saniga is not a professional mystic or
a peddler of drugs, he is an astrophysicist
at the Slovak Academy of Sciences in Bratislava.
It seems unlikely that studying stars led him to
such a way-out view of space and time. Has he
undergone a drug-induced epiphany, or a period
of mental instability? ‘No, no, no,’ Saniga says,
‘I am a perfectly sane person.'”
Some more recent and much less speculative remarks by Saniga
are related to the Klein correspondence —
arXiv.org > math > arXiv:1409.5691:
Mathematics > Combinatorics
[Submitted on 17 Sep 2014]
The Complement of Binary Klein Quadric
as a Combinatorial Grassmannian
By Metod Saniga
“Given a hyperbolic quadric of PG(5,2), there are 28 points
off this quadric and 56 lines skew to it. It is shown that the
(286,563)-configuration formed by these points and lines
is isomorphic to the combinatorial Grassmannian of type
G2(8). It is also pointed out that a set of seven points of
G2(8) whose labels share a mark corresponds to a
Conwell heptad of PG(5,2). Gradual removal of Conwell
heptads from the (286,563)-configuration yields a nested
sequence of binomial configurations identical with part of
that found to be associated with Cayley-Dickson algebras
(arXiv:1405.6888).”
Related entertainment —
See Log24 on the date, 17 Sept. 2014, of Saniga’s Klein-quadric article:
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Wednesday, September 16, 2020
Critical Invisibility
Synchronology check —
This journal on the above dates —
8 January 2019 (“For the Church of Synchronology“)
and 24 April 2019 (“Critical Visibility“).
Related mathematics: Klein Correspondence posts.
Related entertainment: “The Bulk Beings.”
The above Physical Review remarks were found in a search
for a purely mathematical concept —
“35 points” + projective space . . .
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Wednesday, May 25, 2016
Kummer and Dirac
From "Projective Geometry and PT-Symmetric Dirac Hamiltonian,"
Y. Jack Ng and H. van Dam,
Physics Letters B , Volume 673, Issue 3,
23 March 2009, Pages 237–239
(http://arxiv.org/abs/0901.2579v2, last revised Feb. 20, 2009)
" Studies of spin-½ theories in the framework of projective geometry
have been undertaken before. See, e.g., Ref. [4]. 1 "
" 1 These papers are rather mathematical and technical.
The authors of the first two papers discuss the Dirac equation
in terms of the Plucker-Klein correspondence between lines of
a three-dimensional projective space and points of a quadric
in a five-dimensional projective space. The last paper shows
that the Dirac equation bears a certain relation to Kummer’s
surface, viz., the structure of the Dirac ring of matrices is
related to that of Kummer’s 166 configuration . . . ."
[4]
O. Veblen
Proc. Natl. Acad. Sci. USA , 19 (1933), p. 503
Full Text via CrossRef
E.M. Bruins
Proc. Nederl. Akad. Wetensch. , 52 (1949), p. 1135
F.C. Taylor Jr., Master thesis, University of North Carolina
at Chapel Hill (1968), unpublished
A remark of my own on the structure of Kummer’s 166 configuration . . . .
See as well yesterday morning's post.
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Friday, March 21, 2014
Three Constructions of the Miracle Octad Generator
See also a Log24 post on this subject from Dec. 14, 2013,
especially (scroll down) the update of March 9, 2014.
Related material on the Turyn-Curtis construction
from the University of Cambridge —
— Slide by "Dr. Parker" — Apparently Richard A. Parker —
Lecture 4, "Discovering M24," in slides for lectures 1-8 from lectures
at Cambridge in 2010-2011 on "Sporadic and Related Groups."
See also the Parker lectures of 2012-2013 on the same topic.
A third construction of Curtis's 35 4×6 1976 MOG arrays would use
Cullinane's analysis of the 4×4 subarrays' affine and projective structure,
and point out the fact that Conwell's 1910 correspondence of the 35
4+4-partitions of an 8-set with the 35 lines of the projective 3-space
over the 2-element field, PG(3, 2), is essentially the same correspondence
as that constituting Curtis's 1976 MOG.
See The Diamond Theorem, Finite Relativity, Galois Space,
Generating the Octad Generator, and The Klein Correspondence.
Update of March 22-March 23 —
Adding together as (0,1)-matrices over GF(2) the black parts (black
squares as 1's, all other squares as 0's) of the 35 4×6 arrays of the 1976
Curtis MOG would then reveal* the symmetric role played in octads
by what Curtis called the heavy brick , and so reveal also the action of
S3 on the three Curtis bricks that leaves invariant the set of all 759
octads of the S(5, 8, 24) constructed from the 35 MOG arrays. For more
details of this "by-hand" construction, see Geometry of the 4×4 Square.
For the mathematical properties of the S(5, 8, 24), it is convenient to
have a separate construction (such as Turyn's), not by hand, of the
extended binary Golay code. See the Brouwer preprint quoted above.
* "Then a miracle occurs," as in the classic 1977 Sidney Harris cartoon.
Illustration of array addition from March 23 —
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Saturday, March 8, 2014
Conwell Heptads in Eastern Europe
"Charting the Real Four-Qubit Pauli Group
via Ovoids of a Hyperbolic Quadric of PG(7,2),"
by Metod Saniga, Péter Lévay and Petr Pracna,
arXiv:1202.2973v2 [math-ph] 26 Jun 2012 —
P. 4— "It was found that Q +(5,2) (the Klein quadric)
has, up to isomorphism, a unique one — also known,
after its discoverer, as a Conwell heptad [18].
The set of 28 points lying off Q +(5,2) comprises
eight such heptads, any two having exactly one
point in common."
P. 11— "This split reminds us of a similar split of
63 points of PG(5,2) into 35/28 points lying on/off
a Klein quadric Q +(5,2)."
[18] G. M. Conwell, Ann. Math. 11 (1910) 60–76
A similar split occurs in yesterday's Kummer Varieties post.
See the 63 = 28 + 35 vectors of R8 discussed there.
For more about Conwell heptads, see The Klein Correspondence,
Penrose Space-Time, and a Finite Model.
For my own remarks on the date of the above arXiv paper
by Saniga et. al., click on the image below —
Walter Gropius
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