This is the first colored version of
the Diamond Theory cover
that I have done since 1976.
Monday, July 3, 2023
Latin Squares
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Tuesday, January 21, 2025
The Cullinane Diamond Theorem
and the Miracle Octad Generator
|
The Cullinane Diamond Theorem Document created on Jan. 21. 2025, by Google’s Gemini Advanced 1.5 Pro with Deep Research, in response to the following prompt:
“Research how the Cullinane diamond theorem and The Cullinane diamond theorem and the Miracle Octad Generator (MOG) are two seemingly disparate mathematical concepts that find a surprising connection through the realm of finite projective geometry. This report delves into the relationship between these two concepts, exploring their definitions, properties, and historical context to illuminate their interconnectedness. Cullinane Diamond Theorem The Cullinane diamond theorem, developed by Steven H. Cullinane, is a mathematical concept that explores the symmetrical properties of certain geometric patterns. It focuses on a 4×4 square pattern with alternating colors arranged in a diamond shape, referred to as the "diamond figure".4 The theorem states that every permutation of the 16 tiles within this diamond figure, generated by mixing rows, columns, and quadrants, will result in a pattern with some form of ordinary or color-interchange symmetry.5 This "invariance of symmetry" is a remarkable property that highlights the inherent order within seemingly random arrangements.3 The theorem has deep roots in group theory, with the group of permutations (G) being isomorphic to the affine group A on the linear 4-space over the finite field GF(2).6 This group has an order of 322,560 and plays a crucial role in understanding the symmetry of both the diamond-theorem figures and the square patterns of the MOG.5 A key result used in the proof of the theorem states that every 4-coloring (i.e., every map into a 4-set) can be expressed as a sum of three 2-colorings.1 Interestingly, the Cullinane diamond theorem can be extended to higher dimensions. For instance, extending the action of A to a 4x4x4 array yields a way of generating the 1.3 trillion transformations natural to the 64 hexagrams of the I Ching, an ancient Chinese divination text.1 This connection suggests potential applications of the theorem in diverse fields beyond geometry. Another interesting concept that arises from the Cullinane diamond theorem is that of "diamond rings." These rings are algebraic structures generated by the G-images of the diamond figure, and they are isomorphic to rings of matrices over GF(4).5 This algebraic formulation provides a deeper understanding of the symmetry properties explored by the theorem. Miracle Octad Generator The Miracle Octad Generator (MOG), conceived by R.T. Curtis, is a mathematical tool used to study the symmetries and properties of the Mathieu group M24, a sporadic simple group of significant importance in group theory and coding theory.7 It is a 4×6 array of combinations that can describe any point in 24-dimensional space.7 More precisely, each element in the array represents a combination of coordinates or symbols that contribute to defining a point in this 24-dimensional space. Properties The MOG preserves all the symmetries and maximal subgroups of M24, including the monad, duad, triad, octad, octern, sextet, trio, and duum groups.7 It can be used to visualize partitions of the 24 points, which is important for characterizing these maximal subgroups of M24.8 One of the key applications of the MOG lies in its ability to quickly verify codewords of the binary Golay code, a highly efficient error-correcting code used in various communication systems.7 Each element in the MOG can store a '1' or a '0', and by analyzing the counts (number of '1's) in each column and the top row, one can determine if a set of 24 coordinates forms a valid codeword.7 The MOG achieves this by projecting the 24 Golay code onto a code called the hexacode.8 The MOG is also closely related to the concept of octads and sextets. Any 5 elements of the MOG lie in a unique octad, which is a set of 8 elements.9 A sextet is a set of 6 tetrads (sets of 4 elements) where the union of any two tetrads forms an octad.9 Every tetrad lies in a unique sextet. Furthermore, the MOG is a pairing of the 35 partitions of an 8-set into two 4-sets with the 35 partitions of AG(4,2) (the affine 4-space over GF(2)) into 4 affine planes.10 This pairing preserves specific incidence properties, which refer to the relationships between points, lines, and planes in a geometric space. It's worth noting that there are two competing definitions of the MOG. Curtis originally defined it as a pairing of two 35-member sets.11 However, many sources now define it as a rectangular array based on Conway's hexacode.11 Finally, the MOG helps in understanding the octad stabilizer, a subgroup of M24 that leaves an octad invariant as a set.10 This subgroup is isomorphic to the automorphism group of the affine 4-space over GF(2), highlighting the deep connection between the MOG and finite geometry. Relationship Between the Cullinane Diamond Theorem and the Miracle Octad Generator The relationship between the Cullinane diamond theorem and the MOG emerges from their shared foundation in finite projective geometry. The 35 square patterns within the MOG have an underlying structure based on finite projective geometry.5 These patterns, composed of two-color diagonally-divided square tiles, exhibit surprising symmetry properties that can be explained using the Cullinane diamond theorem.5 A crucial link between the Cullinane diamond theorem and the MOG is the fact that the same symmetry group, of order 322,560, underlies both concepts.5 This shared symmetry group highlights a deep connection between the seemingly different patterns studied by these two concepts. The line diagrams used to visualize the Cullinane diamond theorem also play a crucial role in understanding the symmetry of the square patterns in the MOG.5 These line diagrams, based on the three partitions of the four-set of square tiles, correspond to the 35 lines in the 3-dimensional projective space over GF(2).5 In essence, the Cullinane diamond theorem provides a way to understand and visualize the symmetry properties inherent in the MOG. Furthermore, the underlying geometry of the 4×4 patterns in the Cullinane diamond theorem is closely related to the MOG, which is used in the construction of the Steiner system S(5,8,24).5 This connection extends to the Leech lattice, a highly symmetrical mathematical structure in 24 dimensions, which Walter Feit described as a "blown up version of S(5,8,24)".5 The Leech lattice is a dense sphere-packing in 24 dimensions with remarkable symmetry properties and connections to various areas of mathematics and physics. Interestingly, the Cullinane diamond theorem also sheds light on the relationship between orthogonality of Latin squares and skewness of lines in a finite projective 3-space.12 Latin squares are square arrays filled with symbols, and orthogonality between two Latin squares means that when they are superimposed, each possible pair of symbols appears exactly once. Skewness of lines in projective geometry refers to lines that do not intersect. The Cullinane diamond theorem helps establish a connection between these seemingly unrelated concepts. Another interesting connection is to Beutelspacher's model of the 15 points of PG(3,2).1 This model provides a way to visualize the points of this projective space, and it relates to the Cullinane diamond theorem and the MOG through their shared foundation in finite projective geometry. Applications The relationship between the Cullinane diamond theorem and the MOG has potential applications in various fields, including coding theory and cryptography. The MOG's ability to verify codewords of the binary Golay code is particularly relevant in coding theory, where efficient error-correcting codes are essential for reliable data transmission.7 The Cullinane diamond theorem, with its focus on symmetry and permutations, could potentially contribute to the development of new coding schemes or cryptographic algorithms. For example, the theorem's insights into the structure of finite projective spaces could be used to design codes with specific properties or to analyze the security of existing cryptographic systems. Beyond coding theory, the Cullinane diamond theorem has applications in visualizing various mathematical concepts. For instance, it provides a way to visualize Walsh functions, which are a set of orthogonal functions used in signal processing and other areas of mathematics.1 The theorem also helps visualize the finite projective plane PG(2,4), a fundamental object in finite geometry with connections to various combinatorial structures.12 Additionally, the theorem has connections to the outer automorphisms of S6, the symmetric group on 6 letters, which are transformations that preserve the group structure but are not inner automorphisms.12 Visualizations Visualizations play a crucial role in understanding the connection between the Cullinane diamond theorem and the MOG. The 35 square patterns within the original (1976) MOG, as defined by R. T. Curtis, provide a visual representation of this connection.5 These patterns, along with the line diagrams used to illustrate the Cullinane diamond theorem, offer a tangible way to grasp the interplay between these concepts.5 Here are some examples of visualizations that illustrate this connection:
Historical Connections Historically, the development of the Cullinane diamond theorem and the MOG stemmed from explorations in finite projective geometry and group theory.5 While the exact historical connections between their development remain unclear, both concepts emerged from a desire to understand the symmetries and properties of mathematical structures in finite spaces. Conclusion The Cullinane diamond theorem and the Miracle Octad Generator, while seemingly distinct, are intricately linked through the principles of finite projective geometry. The theorem's focus on symmetry invariance and the MOG's ability to capture the symmetries of the Mathieu group M24 highlight their shared foundation. This connection extends to higher-dimensional structures like the Leech lattice and has potential applications in coding theory, cryptography, and the visualization of various mathematical concepts. The relationship between these two concepts exemplifies the interconnectedness of mathematical ideas. Often, seemingly unrelated areas of mathematics converge in unexpected ways, revealing hidden symmetries and relationships. The Cullinane diamond theorem and the MOG provide a compelling example of this phenomenon, demonstrating the power of finite projective geometry in unifying and illuminating diverse mathematical structures. Works cited 1. Diamond Theory: Symmetry in Binary Spaces, accessed January 21, 2025, https://m759.tripod.com/theory/dtheory.html 2. The Diamond Theorem in Finite Geometry, accessed January 21, 2025, http://finitegeometry.org/sc/16/dtheorem.html 3. finitegeometry.org, accessed January 21, 2025, http://finitegeometry.org/sc/gen/dtcas.html#:~:text=Cullinane,have%20some%20sort%20of%20symmetry. 4. Speak, Memory « Log24 – Home page of m759.net, a domain used by Steven H. Cullinane for a WordPress weblog., accessed January 21, 2025, http://m759.net/wordpress/?p=112809 5. Cullinane diamond theorem – Encyclopedia of Mathematics, accessed January 21, 2025, https://encyclopediaofmath.org/wiki/Cullinane_diamond_theorem 6. What is the Cullinane diamond theorem? – Log24, accessed January 21, 2025, http://log24.com/log24/240303-You.com-Cullinane_Diamond_Theorem-research-paper.pdf 7. Miracle Octad Generator – Wikipedia, accessed January 21, 2025, https://en.wikipedia.org/wiki/Miracle_Octad_Generator 8. Mathieu groups, the Golay code and Curtis' Miracle Octad Generator, accessed January 21, 2025, https://vrs.amsi.org.au/wp-content/uploads/sites/6/2014/09/UWA-Kelly.pdf 9. MOG (Miracle Octad Generator) – Stanford, accessed January 21, 2025, http://xenon.stanford.edu/~hwatheod/mog/mog.html 10. The Miracle Octad Generator (MOG) of R T. Curtis – Elements of Finite Geometry, accessed January 21, 2025, http://finitegeometry.org/sc/24/MOG.html 11. Competing Definitions of the Miracle Octad Generator – Elements of Finite Geometry, accessed January 21, 2025, http://finitegeometry.org/sc/24/mogdefs.html 12. Diamond Theory: Symmetry in Binary Spaces – Elements of Finite Geometry, accessed January 21, 2025, http://finitegeometry.org/sc/gen/dth/DiamondTheory.html 13. arxiv.org, accessed January 21, 2025, https://arxiv.org/abs/1308.1075 |
View this post as a standalone web page at
http://log24.com/log25/DTandMOG.html.
and as a PDF at
http://log24.com/log25/DTandMOG.pdf.
For a more elementary introduction to the MOG, see a YouTube video,
"The Most Powerful Diagram in Mathematics."
For a PDF of the video's metadata and comments, click here.
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and the Miracle Octad Generator
and the Miracle Octad Generator
Friday, February 16, 2024
The Sixteen
Prompt:
Write a poem, in strict sonnet form, about the Cullinane diamond theorem.
Response (slightly tweaked for grammar):
Within a grid of sixteen squares reside
A diamond's gleams where symmetries do guide.
Though tiles may shift, a pattern shall remain,
By unseen lines of ordered form sustained.
Rows mix and columns twist in playful dance,
Quadrants align with geometric chance.
No matter how the elements may turn,
A hidden balance always will return.
Where Latin squares in secret do align,
And finite fields their subtle powers combine,
An ideal forms, with brilliance all its own,
A ring of light where hidden truths are shown.
Though chaos swirls, an order does descend,
The Cullinane's bright theorem will transcend.
________________________________________________________________
As for "a hidden balance" … See Oct. 10, 2022.
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Tuesday, January 30, 2024
Thursday, December 28, 2023
Basque Country Art Book
Book description at Amazon.com, translated by Google —
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Las matemáticas como herramienta
Mathematics as a tool by Raúl Ibáñez Torres Kindle edition in Spanish, 2023 Although the relationship between mathematics and art can be traced back to ancient times, mainly in geometric and technical aspects, it is with the arrival of the avant-garde and abstract art at the beginning of the 20th century that mathematics takes on greater and different relevance: as a source of inspiration and as a tool for artistic creation. Let us think, for example, of the importance of the fourth dimension for avant-garde movements or, starting with Kandisnky and later Max Bill and concrete art, the vindication of mathematical thinking in artistic creation. An idea that would have a fundamental influence on currents such as constructivism, minimalism, the fluxus movement, conceptual art, systematic art or optical art, among others. Following this approach, this book analyzes, through a variety of examples and activities, how mathematics is present in contemporary art as a creative tool. And it does so through five branches and the study of some of its mathematical topics: geometry (the Pythagorean theorem), topology (the Moebius strip), algebra (algebraic groups and matrices), combinatorics (permutations and combinations) and recreational mathematics (magic and Latin squares). |
From the book ("Cullinane Diamond Theorem" heading and picture of
book's cover added) —
Publisher: Los Libros de La Catarata (October 24, 2023)
Author: Raúl Ibáñez Torres, customarily known as Raúl Ibáñez
(Ibáñez does not mention Cullinane as the author of the above theorem
in his book (except indirectly, quoting Josefine Lyche), but he did credit
him fully in an earlier article, "The Truchet Tiles and the Diamond Puzzle"
(translation by Google).)
About Ibáñez (translated from Amazon.com by Google):
Mathematician, professor of Geometry at the University of the Basque Country
and scientific disseminator. He is part of the Chair of Scientific Culture of the
UPV/EHU and its blog Cuaderno de Cultura Cientifica. He has been a scriptwriter
and presenter of the program “Una de Mates” on the television program Órbita Laika.
He has collaborated since 2005 on the programs Graffiti and La mechanica del caracol
on Radio Euskadi. He has also been a collaborator and co-writer of the documentary
Hilos de tiempo (2020) about the artist Esther Ferrer. For 20 years he directed the
DivulgaMAT portal, Virtual Center for the Dissemination of Mathematics, and was a
member of the dissemination commission of the Royal Spanish Mathematical Society.
Author of several books, including The Secrets of Multiplication (2019) and
The Great Family of Numbers (2021), in the collection Miradas Matemáticas (Catarata).
He has received the V José María Savirón Prize for Scientific Dissemination
(national modality, 2010) and the COSCE Prize for the Dissemination of Science (2011).
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Saturday, December 9, 2023
How Many Magic Beans, Jack?
Two notes from this journal on December 2, 2023 —
A flashback for The November Man —
As for " my own 'diamond theorem' " —
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Question — "What is mathematician Steven H. Cullinane known for?" From the Copilot response (shorn of footnotes) — "Steven H. Cullinane is known for his work in finite geometry, particularly his Diamond Theorem. This theorem explains the surprising symmetry properties of some simple graphic designs, such as those found in quilts. It also provides a model for the 15 points, 35 lines, and 15 hyperplanes of the finite projective space PG (3,2). Cullinane’s work underlies the structure of the 35 square patterns in R. T. Curtis’s Miracle Octad Generator. His theorem has applications to sporadic simple groups, the connection between orthogonal Latin squares and projective spreads, and the symmetry of Walsh functions." |
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Wednesday, December 6, 2023
“This, This!” *
Monday, July 3, 2023
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Also on July 3, 2023 —
* See Parul Sehgal, "What We Learn from the Lives of Critics."
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Saturday, December 2, 2023
My Work According to Copilot
Following yesterday's encounter with the latest version of Pi,
here are some more formal (and more informative) remarks
from Windows Copilot today —
|
Question — "What is mathematician Steven H. Cullinane known for?" From the Copilot response (shorn of footnotes) — "Steven H. Cullinane is known for his work in finite geometry, particularly his Diamond Theorem. This theorem explains the surprising symmetry properties of some simple graphic designs, such as those found in quilts. It also provides a model for the 15 points, 35 lines, and 15 hyperplanes of the finite projective space PG (3,2). Cullinane’s work underlies the structure of the 35 square patterns in R. T. Curtis’s Miracle Octad Generator. His theorem has applications to sporadic simple groups, the connection between orthogonal Latin squares and projective spreads, and the symmetry of Walsh functions." |
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Saturday, March 26, 2022
Box Geometry: Space, Group, Art (Work in Progress)
| Name Tag | .Space | .Group | .Art |
|---|---|---|---|
| Box4 |
2×2 square representing the four-point finite affine geometry AG(2,2). (Box4.space) |
S4 = AGL(2,2) (Box4.group) |
(Box4.art) |
| Box6 |
3×2 (3-row, 2-column) rectangular array representing the elements of an arbitrary 6-set. |
S6 | |
| Box8 | 2x2x2 cube or 4×2 (4-row, 2-column) array. | S8 or A8 or AGL(3,2) of order 1344, or GL(3,2) of order 168 | |
| Box9 | The 3×3 square. | AGL(2,3) or GL(2,3) | |
| Box12 | The 12 edges of a cube, or a 4×3 array for picturing the actions of the Mathieu group M12. | Symmetries of the cube or elements of the group M12 | |
| Box13 | The 13 symmetry axes of the cube. | Symmetries of the cube. | |
| Box15 |
The 15 points of PG(3,2), the projective geometry of 3 dimensions over the 2-element Galois field. |
Collineations of PG(3,2) | |
| Box16 |
The 16 points of AG(4,2), the affine geometry of 4 dimensions over the 2-element Galois field. |
AGL(4,2), the affine group of |
|
| Box20 | The configuration representing Desargues's theorem. | ||
| Box21 | The 21 points and 21 lines of PG(2,4). | ||
| Box24 | The 24 points of the Steiner system S(5, 8, 24). | ||
| Box25 | A 5×5 array representing PG(2,5). | ||
| Box27 |
The 3-dimensional Galois affine space over the 3-element Galois field GF(3). |
||
| Box28 | The 28 bitangents of a plane quartic curve. | ||
| Box32 |
Pair of 4×4 arrays representing orthogonal Latin squares. |
Used to represent elements of AGL(4,2) |
|
| Box35 |
A 5-row-by-7-column array representing the 35 lines in the finite projective space PG(3,2) |
PGL(3,2), order 20,160 | |
| Box36 | Eurler's 36-officer problem. | ||
| Box45 | The 45 Pascal points of the Pascal configuration. | ||
| Box48 | The 48 elements of the group AGL(2,3). | AGL(2,3). | |
| Box56 |
The 56 three-sets within an 8-set or |
||
| Box60 | The Klein configuration. | ||
| Box64 | Solomon's cube. |
— Steven H. Cullinane, March 26-27, 2022
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Monday, April 29, 2019
The Hustvedt Array
For Harlan Kane
"This time-defying preservation of selves,
this dream of plenitude without loss,
is like a snow globe from heaven,
a vision of Eden before the expulsion."
— Judith Shulevitz on Siri Hustvedt in
The New York Times Sunday Book Review
of March 31, 2019, under the headline
"The Time of Her Life."
Edenic-plenitude-related material —
"Self-Blazon… of Edenic Plenitude"
(The Issuu text is taken from Speaking about Godard , by Kaja Silverman
and Harun Farocki, New York University Press, 1998, page 34.)
Preservation-of-selves-related material —
Other Latin squares (from October 2018) —
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Friday, November 30, 2018
Latin-Square Structure
Continued from March 13, 2011 —
"…as we saw, there are two different Latin squares of order 4…."
— Peter J. Cameron, "The Shrikhande Graph," August 26, 2010
Cameron counts Latin squares as the same if they are isotopic .
Some further context for Cameron's remark—
A new website illustrates a different approach to Latin squares of order 4 —
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Thursday, October 19, 2017
Design Grammar***
The elementary shapes at the top of the figure below mirror
the looking-glass property of the classical Lo Shu square.
The nine shapes at top left* and their looking-glass reflection
illustrate the looking-glass reflection relating two orthogonal
Latin squares over the three digits of modulo-three arithmetic.
Combining these two orthogonal Latin squares,** we have a
representation in base three of the numbers from 0 to 8.
Adding 1 to each of these numbers yields the Lo Shu square.
* The array at top left is from the cover of
Wonder Years:
Werkplaats Typografie 1998-2008.
** A well-known construction.
*** For other instances of what might be
called "design grammar" in combinatorics,
see a slide presentation by Robin Wilson.
No reference to the work of Chomsky is
intended.
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Tuesday, October 17, 2017
Plan 9 Continues
See also Holy Field in this journal.
Some related mathematics —
Analysis of the Lo Shu structure —
Structure of the 3×3 magic square:
4 9 2
3 5 7 decreased by 1 is …
8 1 6
3 8 1
2 4 6
7 0 5
In base 3 —
10 22 01
02 11 20
21 00 12
As orthogonal Latin squares
(a well-known construction) —
1 2 0 0 2 1
0 1 2 2 1 0
2 0 1 1 0 2 .
— Steven H. Cullinane,
October 17, 2017
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Monday, October 16, 2017
Highway 61 Revisited
"God said to Abraham …." — Bob Dylan, "Highway 61 Revisited"
Related material —
-
"Inner Truth" in this journal
-
"On Linguistic Creation," June 25, 1999
-
"Ultra Super Magic Squares of 5×5" at
http://mathsforeurope.digibel.be/magic.htm .
See as well Charles Small, Harvard '64,
"Magic Squares over Fields" —
— and Conway-Norton-Ryba in this journal.
Some remarks on an order-five magic square over GF(52):
on the numbers 0 to 24:
22 5 18 1 14
3 11 24 7 15
9 17 0 13 21
10 23 6 19 2
16 4 12 20 8
Base-5:
42 10 33 01 24
03 21 44 12 30
14 32 00 23 41
20 43 11 34 02
31 04 22 40 13
Regarding the above digits as representing
elements of the vector 2-space over GF(5)
(or the vector 1-space over GF(52)) …
All vector row sums = (0, 0) (or 0, over GF(52)).
All vector column sums = same.
Above array as two
orthogonal Latin squares:
4 1 3 0 2 2 0 3 1 4
0 2 4 1 3 3 1 4 2 0
1 3 0 2 4 4 2 0 3 1
2 4 1 3 0 0 3 1 4 2
3 0 2 4 1 1 4 2 0 3
— Steven H. Cullinane,
October 16, 2017
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Sunday, June 8, 2014
Vide
"The relevance of a geometric theorem is determined by what the theorem
tells us about space, and not by the eventual difficulty of the proof."
— Gian-Carlo Rota discussing the theorem of Desargues
What space tells us about the theorem :
In the simplest case of a projective space (as opposed to a plane ),
there are 15 points and 35 lines: 15 Göpel lines and 20 Rosenhain lines.*
The theorem of Desargues in this simplest case is essentially a symmetry
within the set of 20 Rosenhain lines. The symmetry, a reflection
about the main diagonal in the square model of this space, interchanges
10 horizontally oriented (row-based) lines with 10 corresponding
vertically oriented (column-based) lines.
Vide Classical Geometry in Light of Galois Geometry.
* Update of June 9: For a more traditional nomenclature, see (for instance)
R. Shaw, 1995. The "simplest case" link above was added to point out that
the two types of lines named are derived from a natural symplectic polarity
in the space. The square model of the space, apparently first described in
notes written in October and December, 1978, makes this polarity clearly visible:
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Saturday, February 1, 2014
The Delft Version
My webpage "The Order-4 Latin Squares" has a rival—
"Latin squares of order 4: Enumeration of the
24 different 4×4 Latin squares. Symmetry and
other features."
The author — Yp de Haan, a professor emeritus of
materials science at Delft University of Technology —
The main difference between de Haan's approach and my own
is my use of the four-color decomposition theorem, a result that
I discovered in 1976. This would, had de Haan known it, have
added depth to his "symmetry and other features" remarks.
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Tuesday, July 9, 2013
Vril Chick
Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo—
Compare to an image of Vril muse Maria Orsitsch.
From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —
|
Josefine Lyche
Keywords (to help place my artwork in the (See also the original catalog page.) |
Clearly most of this (the non-highlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.
For some background, see (for instance)
Conspiracy Theories and Secret Societies for Dummies .
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Thursday, August 2, 2012
Logos
(Continued from December 26th, 2011)
Some material at math.stackexchange.com related to
yesterday evening's post on Elementary Finite Geometry—
Questions on this topic have recently been
discussed at Affine plane of order 4? and at
Turning affine planes into projective planes.
(For a better discussion of the affine plane of order 4,
see Affine Planes and Mutually Orthogonal Latin Squares
at the website of William Cherowitzo, professor at UC Denver.)
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Friday, August 19, 2011
Geezer Puzzle
An RSS item today—
 Diamond squares Fri Aug 19, 2011 05:36 [EDT] from Peter Cameron
If you like Latin squares and such things, take a look at Diamond Geezer’s post for today: a pair of orthogonal Latin squares with two disjoint common transversals, and some entries given (if you do the harder puzzle).
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The post referred to—
"This Jack, joke, poor potsherd, ' patch, matchwood, immortal diamond,
Is immortal diamond." —Gerard Manley Hopkins, Society of Jesus
Those now celebrating the Catholic Church's "World Youth" week in Madrid
may prefer a related puzzle for younger and nimbler minds:
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Sunday, March 13, 2011
The Counter
"…as we saw, there are two different Latin squares of order 4…."
— Peter J. Cameron, "The Shrikhande Graph," August 26, 2010
Cameron counts Latin squares as the same if they are isotopic .
Some further context for Cameron's remark—
Cover Illustration Number 1 (1976):
Cover Illustration Number 2 (1991):
The Shrikhande Graph
______________________________________________________________________________
This post was prompted by two remarks…
1. In a different weblog, also on August 26, 2010—
The Accidental Mathematician— "The Girl Who Played with Fermat's Theorem."
"The worst thing about the series is the mathematical interludes in The Girl Who Played With Fire….
Salander is fascinated by a theorem on perfect numbers—
one can verify it for as many numbers as one wishes, and it never fails!—
and then advances through 'Archimedes, Newton, Martin Gardner,*
and a dozen other classical mathematicians,' all the way to Fermat’s last theorem."
2. "The fact that the pattern retains its symmetry when you permute the rows and columns
is very well known to combinatorial theorists who work with matrices."
[My italics; note resemblance to the Brualdi-Ryser title above.]
–Martin Gardner in 1976 on the diamond theorem
* Compare Eric Temple Bell (as quoted at the MacTutor history of mathematics site)—
"Archimedes, Newton, and Gauss, these three, are in a class by themselves
among the great mathematicians, and it is not for ordinary mortals
to attempt to range them in order of merit."
This is from the chapter on Gauss in Men of Mathematics .
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Friday, March 11, 2011
Table Talk
The following was suggested by a link within this evening's earlier Kane site link.
Peter J. Cameron's weblog on August 26, 2010—
A Latin square of order n is a
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|
Some related literary remarks—
Proginoskes and Latin Squares.
See also "It was a perfectly ordinary night at Christ's high table…."
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Saturday, February 5, 2011
Cover Art
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Saturday, January 22, 2011
High School Squares*
The following is from the weblog of a high school mathematics teacher—
This is related to the structure of the figure on the cover of the 1976 monograph Diamond Theory—
Each small square pattern on the cover is a Latin square,
with elements that are geometric figures rather than letters or numerals.
All order-four Latin squares are represented.
For a deeper look at the structure of such squares, let the high-school
chart above be labeled with the letters A through X, and apply the
four-color decomposition theorem. The result is 24 structural diagrams—
Some of the squares are structurally congruent under the group of 8 symmetries of the square.
This can be seen in the following regrouping—
(Image corrected on Jan. 25, 2011– "seven" replaced "eight.")
* Retitled "The Order-4 (i.e., 4×4) Latin Squares" in the copy at finitegeometry.org/sc.
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Thursday, August 26, 2010
Home from Home continued
Or— Childhood's Rear End
This post was suggested by…
- Today's New York Times—
"For many artists Electric Lady has become a home away from home…. For Jimmy Page the personal imprimaturs of Hendrix and Mr. Kramer made all the difference when Led Zeppelin mixed parts of 'Houses of the Holy' there in 1972." - The album cover pictures for "Houses of the Holy"
- Boleskine House, home to Aleister Crowley and (occasionally) to Jimmy Page.
Related material:
The Zeppelin album cover, featuring rear views of nude children, was shot at the Giant's Causeway.
From a page at led-zeppelin.org—
See also Richard Rorty on Heidegger—
Safranski, the author of ''Schopenhauer and the Wild Years of Philosophy,'' never steps back and pronounces judgment on Heidegger, but something can be inferred from the German title of his book: ''Ein Meister aus Deutschland'' (''A Master From Germany''). Heidegger was, undeniably, a master, and was very German indeed. But Safranski's spine-chilling allusion is to Paul Celan's best-known poem, ''Death Fugue.'' In Michael Hamburger's translation, its last lines are:
death is a master from Germany his eyes are blue
he strikes you with leaden bullets his aim is true
a man lives in the house your golden hair Margarete
he sets his pack on us he grants us a grave in the air
he plays with the serpents and daydreams death is a master from Germany
your golden hair Margarete
your ashen hair Shulamith.
No one familiar with Heidegger's work can read Celan's poem without recalling Heidegger's famous dictum: ''Language is the house of Being. In its home man dwells.'' Nobody who makes this association can reread the poem without having the images of Hitler and Heidegger — two men who played with serpents and daydreamed — blend into each other. Heidegger's books will be read for centuries to come, but the smell of smoke from the crematories — the ''grave in the air'' — will linger on their pages.
Heidegger is the antithesis of the sort of philosopher (John Stuart Mill, William James, Isaiah Berlin) who assumes that nothing ultimately matters except human happiness. For him, human suffering is irrelevant: philosophy is far above such banalities. He saw the history of the West not in terms of increasing freedom or of decreasing misery, but as a poem. ''Being's poem,'' he once wrote, ''just begun, is man.''
For Heidegger, history is a sequence of ''words of Being'' — the words of the great philosophers who gave successive historical epochs their self-image, and thereby built successive ''houses of Being.'' The history of the West, which Heidegger also called the history of Being, is a narrative of the changes in human beings' image of themselves, their sense of what ultimately matters. The philosopher's task, he said, is to ''preserve the force of the most elementary words'' — to prevent the words of the great, houses-of-Being-building thinkers of the past from being banalized.
Related musical meditations—
Shine On (Saturday, April 21, 2007), Shine On, Part II, and Built (Sunday, April 22, 2007).
Related pictorial meditations—
The Giant's Causeway at Peter J. Cameron's weblog
and the cover illustration for Diamond Theory (1976)—
The connection between these two images is the following from Cameron's weblog today—
… as we saw, there are two different Latin squares of order 4;
one, but not the other, can be extended to a complete set
of 3 MOLS [mutually orthogonal Latin squares].
The underlying structures of the square pictures in the Diamond Theory cover are those of the two different Latin squares of order 4 mentioned by Cameron.
Connection with childhood—
The children's book A Wind in the Door, by Madeleine L'Engle. See math16.com. L'Engle's fantasies about children differ from those of Arthur C. Clarke and Led Zeppelin.
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Tuesday, October 3, 2006
Tuesday October 3, 2006
Serious
"I don't think the 'diamond theorem' is anything serious, so I started with blitzing that."
— Charles Matthews at Wikipedia, Oct. 2, 2006
"The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas."
— G. H. Hardy, A Mathematician's Apology
Matthews yesterday deleted references to the diamond theorem and related material in the following Wikipedia articles:
Affine group
Reflection group
Symmetry in mathematics
Incidence structure
Invariant (mathematics)
Symmetry
Finite geometry
Group action
History of geometry
This would appear to be a fairly large complex of mathematical ideas.
See also the following "large complex" cited, following the above words of Hardy, in Diamond Theory:
Affine geometry, affine planes, affine spaces, automorphisms, binary codes, block designs, classical groups, codes, coding theory, collineations, combinatorial, combinatorics, conjugacy classes, the Conwell correspondence, correlations, design theory, duads, duality, error correcting codes, exceptional groups, finite fields, finite geometry, finite groups, finite rings, Galois fields, generalized quadrangles, generators, geometry, GF(2), GF(4), the (24,12) Golay code, group actions, group theory, Hadamard matrices, hypercube, hyperplanes, hyperspace, incidence structures, invariance, Karnaugh maps, Kirkman's schoolgirl problem, Latin squares, Leech lattice, linear groups, linear spaces, linear transformations, Mathieu groups, matrix theory, Meno, Miracle Octad Generator, MOG, multiply transitive groups, octads, the octahedral group, orthogonal arrays, outer automorphisms, parallelisms, partial geometries, permutation groups, PG(3,2), polarities, Polya-Burnside theorem, projective geometry, projective planes, projective spaces, projectivities, Reed-Muller codes, the relativity problem, Singer cycle, skew lines, sporadic simple groups, Steiner systems, symmetric, symmetry, symplectic, synthemes, synthematic, tesseract, transvections, Walsh functions, Witt designs.
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Monday, January 9, 2006
Monday January 9, 2006
Cornerstone
“In 1782, the Swiss mathematician Leonhard Euler posed a problem whose mathematical content at the time seemed about as much as that of a parlor puzzle. 178 years passed before a complete solution was found; not only did it inspire a wealth of mathematics, it is now a cornerstone of modern design theory.”
— Dean G. Hoffman, Auburn U.,
July 2001 Rutgers talk
Diagrams from Dieter Betten’s 1983 proof
of the nonexistence of two orthogonal
6×6 Latin squares (i.e., a proof
of Tarry’s 1900 theorem solving
Euler’s 1782 problem of the 36 officers):
Compare with the partitions into
two 8-sets of the 4×4 Latin squares
discussed in my 1978 note (pdf).
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Tuesday, August 16, 2005
Tuesday August 16, 2005
Narrative and Latin Squares
From The Independent, 15 August 2005:
“Millions of people now enjoy Sudoku puzzles. Forget the pseudo-Japanese baloney: sudoku grids are a version of the Latin Square created by the great Swiss mathematician Leonhard Euler in the late 18th century.”
The Independent was discussing the conference on “Mathematics and Narrative” at Mykonos in July.
From the Wikipedia article on Latin squares:
“The popular Sudoku puzzles are a special case of Latin squares; any solution to a Sudoku puzzle is a Latin square. Sudoku imposes the additional restriction that 3×3 subgroups must also contain the digits 1–9 (in the standard version).
The Diamond 16 Puzzle illustrates a generalized concept of Latin-square orthogonality: that of “orthogonal squares” (Diamond Theory, 1976) or “orthogonal matrices”– orthogonal, that is, in a combinatorial, not a linear-algebra sense (A. E. Brouwer, 1991).”
This last paragraph, added to Wikipedia on Aug. 14, may or may not survive the critics there.
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Wednesday, May 4, 2005
Wednesday May 4, 2005
The Fano Plane
Revisualized:
Revisualized:
or, The Eightfold Cube
Here is the usual model of the seven points and seven lines (including the circle) of the smallest finite projective plane (the Fano plane):
Every permutation of the plane's points that preserves collinearity is a symmetry of the plane. The group of symmetries of the Fano plane is of order 168 and is isomorphic to the group PSL(2,7) = PSL(3,2) = GL(3,2). (See Cameron on linear groups (pdf).)
The above model indicates with great clarity six symmetries of the plane– those it shares with the equilateral triangle. It does not, however, indicate where the other 162 symmetries come from.
Shown below is a new model of this same projective plane, using partitions of cubes to represent points:
The cubes' partitioning planes are added in binary (1+1=0) fashion. Three partitioned cubes are collinear if and only if their partitioning planes' binary sum equals zero.
The second model is useful because it lets us generate naturally all 168 symmetries of the Fano plane by splitting a cube into a set of four parallel 1x1x2 slices in the three ways possible, then arbitrarily permuting the slices in each of the three sets of four. See examples below.
For a proof that such permutations generate the 168 symmetries, see Binary Coordinate Systems.
(Note that this procedure, if regarded as acting on the set of eight individual subcubes of each cube in the diagram, actually generates a group of 168*8 = 1,344 permutations. But the group's action on the diagram's seven partitions of the subcubes yields only 168 distinct results. This illustrates the difference between affine and projective spaces over the binary field GF(2). In a related 2x2x2 cubic model of the affine 3-space over GF(2) whose "points" are individual subcubes, the group of eight translations is generated by interchanges of parallel 2x2x1 cube-slices. This is clearly a subgroup of the group generated by permuting 1x1x2 cube-slices. Such translations in the affine 3-space have no effect on the projective plane, since they leave each of the plane model's seven partitions– the "points" of the plane– invariant.)
To view the cubes model in a wider context, see Galois Geometry, Block Designs, and Finite-Geometry Models.
For another application of the points-as-partitions technique, see Latin-Square Geometry: Orthogonal Latin Squares as Skew Lines.
For more on the plane's symmetry group in another guise, see John Baez on Klein's Quartic Curve and the online book The Eightfold Way. For more on the mathematics of cubic models, see Solomon's Cube.
For a large downloadable folder with many other related web pages, see Notes on Finite Geometry.
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Saturday, August 28, 2004
Saturday August 28, 2004
History of Mathematics
On the history of the relationship between orthogonality (in the Latin-square sense) and skewness (in the projective-space sense)–
See the newly updated
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Tuesday, January 20, 2004
Tuesday January 20, 2004
Screenshot
A search on “vult decipi” at about
3:40 AM today yielded the following, from
http://www.sacklunch.net/Latin/P/
populusvultdecipidecipiatur.html
The ad for “Geometry of Latin Squares,”
my own. is in direct competition with
“Jesus Loves You.”
Good luck, Latin squares.
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Friday, March 21, 2003
Friday March 21, 2003
ART WARS:
Readings for Bach's Birthday
Larry J. Solomon:
Symmetry as a Compositional Determinant,
Chapter VIII: New Transformations
In Solomon's work, a sequence of notes is represented as a set of positions within a Latin square:
Transformations of the Latin square correspond to transformations of the musical notes. For related material, see The Glass Bead Game, by Hermann Hesse, and Charles Cameron's sites on the Game.
Steven H. Cullinane:
Dorothy Sayers:
"The function of imaginative speech is not to prove, but to create–to discover new similarities, and to arrange them to form new entities, to build new self-consistent worlds out of the universe of undifferentiated mind-stuff." (Christian Letters to a Post-Christian World, Grand Rapids: Eerdmans, 1969, p. xiii)
— Quoted by Timothy A. Smith, "Intentionality and Meaningfulness in Bach's Cyclical Works"
Edward Sapir:
"…linguistics has also that profoundly serene and satisfying quality which inheres in mathematics and in music and which may be described as the creation out of simple elements of a self-contained universe of forms. Linguistics has neither the sweep nor the instrumental power of mathematics, nor has it the universal aesthetic appeal of music. But under its crabbed, technical, appearance there lies hidden the same classical spirit, the same freedom in restraint, which animates mathematics and music at their purest."
— "The Grammarian and his Language,"
American Mercury 1:149-155, 1924
Comments Off on Friday March 21, 2003
Saturday, July 20, 2002
Saturday July 20, 2002
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Initial Xanga entry. Updated Nov. 18, 2006.