Sunday, June 18, 2023
Efficient Packing
Friday, December 4, 2020
Here’s to Efficient Packing!
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.
and the Miracle Octad Generator
Thursday, December 19, 2024
Language … Locked and Loaded
See as well other posts tagged with the above search result date.
Saturday, November 23, 2024
The Square Aspect (vs. Stevens’s “Radial* Aspect”)
"New horizons open up." — New York Times, Nov. 21, 1924.
A search for some such 96-square division yields . . .
https://www.atariuptodate.de/img/kidshapes.png —
"Here's to efficient packing."
* See as well radial vs. square aspects in a post on Mackey's classic
Harmonic Analysis as the Exploitation of Symmetry.
Sunday, August 4, 2024
The Bright Elusive Butterfly Chair of Love
From an Instagram story captioned "An Actor Prepares" —
Related logline for "I Am Not Okay with This" fans . . .
Wednesday, January 3, 2024
Saturday, October 21, 2023
Monday, October 16, 2023
Saturday, June 3, 2023
Sunday, December 11, 2022
Friday, October 14, 2022
Recreational Crystallography
"Boron atoms and metal atoms can form a configuration . . . ."
See as well other posts now tagged Death Valley Days.
Monday, December 13, 2021
Pretty
On the song "Different Drum" by the late Michael Nesmith:
"Ronstadt's version flips the gender references
in Nesmith's original lyrics, replacing 'girl' with 'boy'
when describing her lover, but still referring to him
being 'pretty'." — Wikipedia
Wednesday, December 1, 2021
“Show Details” — Art Pallets
Monday, September 13, 2021
Saturday, September 4, 2021
Wednesday, August 4, 2021
“Old men ought to be explorers” — T. S. Eliot
Friday, April 16, 2021
Schoolgirl Problem (Kirkman’s, not Epstein’s)
Friday, December 4, 2020
German Lesson: Untergang
The recent posts "Bunker Bingo" and "Here's to Efficient Packing!"
suggest a review.
Alex Ross in The New Yorker on Dec. 2, 2020, on the German
word "Untergang " —
"The usual translation is 'downfall,' although
the various implications of the word—
literally, “going-under”—are difficult to capture
in English. In some contexts, Untergang simply
means descent: a sunset is a Sonnenuntergang .
Lauren German in a 2005 film —
See as well . . .
Thursday, August 27, 2020
Sunday, July 7, 2019
Schoolgirl Problem
Anonymous remarks on the schoolgirl problem at Wikipedia —
"This solution has a geometric interpretation in connection with
Galois geometry and PG(3,2). Take a tetrahedron and label its
vertices as 0001, 0010, 0100 and 1000. Label its six edge centers
as the XOR of the vertices of that edge. Label the four face centers
as the XOR of the three vertices of that face, and the body center
gets the label 1111. Then the 35 triads of the XOR solution correspond
exactly to the 35 lines of PG(3,2). Each day corresponds to a spread
and each week to a packing."
See also Polster + Tetrahedron in this journal.
There is a different "geometric interpretation in connection with
Galois geometry and PG(3,2)" that uses a square model rather
than a tetrahedral model. The square model of PG(3,2) last
appeared in the schoolgirl-problem article on Feb. 11, 2017, just
before a revision that removed it.
Tuesday, December 25, 2018
Simply
"So to obtain the isomorphism from L2(7) onto L3(2) we simply
— Sphere Packings, Lattices and Groups , |
Compare and contrast —
This post was suggested by a New York Times headline today —
Thursday, November 29, 2018
The Fortune Cookie
The mathematician Chuanming Zong in the previous post
has also written about Aristotle's pyramid scheme —
Monday, May 2, 2016
Subjective Quality
The previous post deals in part with a figure from the 1988 book
Sphere Packings, Lattices and Groups , by J. H. Conway and
N. J. A. Sloane.
Siobhan Roberts recently wrote a book about the first of these
authors, Conway. I just discovered that last fall she also had an
article about the second author, Sloane, published:
"How to Build a Search Engine for Mathematics,"
Nautilus , Oct 22, 2015.
Meanwhile, in this journal …
Log24 on that same date, Oct. 22, 2015 —
Roberts's remarks on Conway and later on Sloane are perhaps
examples of subjective quality, as opposed to the objective quality
sought, if not found, by Alexander, and exemplified by the
above bijection discussed here last October.
Sunday, May 1, 2016
Sunday Appetizer from 1984
Judith Shulevitz in The New York Times
on Sunday, July 18, 2010
(quoted here Aug. 15, 2010) —
“What would an organic Christian Sabbath look like today?”
The 2015 German edition of Beautiful Mathematics ,
a 2011 Mathematical Association of America (MAA) book,
was retitled Mathematische Appetithäppchen —
Mathematical Appetizers . The German edition mentions
the author's source, omitted in the original American edition,
for his section 5.17, "A Group of Operations" (in German,
5.17, "Eine Gruppe von Operationen") —
Mathematische Appetithäppchen: Autor: Erickson, Martin —
"Weitere Informationen zu diesem Themenkreis finden sich |
That source was a document that has been on the Web
since 2002. The document was submitted to the MAA
in 1984 but was rejected. The German edition omits the
document's title, and describes it as merely a source for
"further information on this subject area."
The title of the document, "Binary Coordinate Systems,"
is highly relevant to figure 11.16c on page 312 of a book
published four years after the document was written: the
1988 first edition of Sphere Packings, Lattices and Groups ,
by J. H. Conway and N. J. A. Sloane —
A passage from the 1984 document —
Thursday, March 24, 2016
Friday, January 17, 2014
The 4×4 Relativity Problem
The sixteen-dot square array in yesterday’s noon post suggests
the following remarks.
“This is the relativity problem: to fix objectively a class of
equivalent coordinatizations and to ascertain the group of
transformations S mediating between them.”
— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—
The 1977 matrix Q is echoed in the following from 2002—
A different representation of Cullinane’s 1977 square model of the
16-point affine geometry over the two-element Galois field GF(2)
is supplied by Conway and Sloane in Sphere Packings, Lattices and Groups
(first published in 1988) :
Here a, b, c, d are basis vectors in the vector 4-space over GF(2).
(For a 1979 version of this vector space, see AMS Abstract 79T-A37.)
See also a 2011 publication of the Mathematical Association of America —
Sunday, December 15, 2013
Sermon
Odin's Jewel
Jim Holt, the author of remarks in yesterday's
Saturday evening post—
"It turns out that the Kyoto school of Buddhism
makes Heidegger seem like Rush Limbaugh—
it’s so rarified, I’ve never been able to
understand it at all. I’ve been knocking my head
against it for years."
— Vanity Fair Daily , July 16, 2012
Backstory: Odin + Jewel in this journal.
See also Odin on the Kyoto school —
For another version of Odin's jewel, see Log24
on the date— July 16, 2012— that Holt's Vanity Fair
remarks were published. Scroll to the bottom of the
"Mapping Problem continued" post for an instance of
the Galois tesseract —
Saturday, December 14, 2013
Beautiful Mathematics
The title, which I dislike, is taken from a 2011 publication
of the MAA, also sold by Cambridge University Press.
Some material relevant to the title adjective:
"For those who have learned something of higher mathematics, nothing could be more natural than to use the word 'beautiful' in connection with it. Mathematical beauty, like the beauty of, say, a late Beethoven quartet, arises from a combination of strangeness and inevitability. Simply defined abstractions disclose hidden quirks and complexities. Seemingly unrelated structures turn out to have mysterious correspondences. Uncanny patterns emerge, and they remain uncanny even after being underwritten by the rigor of logic."— Jim Holt, opening of a book review in the Dec. 5, 2013, issue of The New York Review of Books |
Some relevant links—
- Strangeness and inevitability
- Simply defined abstractions
- Hidden quirks and complexities
- Seemingly unrelated structures
- Mysterious correspondences
- Uncanny patterns
- The rigor of logic
- Beethoven quartet
The above list was updated on Jan. 31, 2014, to include the
"Strangeness" and "Hidden quirks" links. See also a post of
Jan. 31, 2014.
Update of March 9, 2014 —
The link "Simply defined abstractions" is to the construction of the Steiner
system S(5, 8, 24) described by R. T. Curtis in his 1976 paper defining the
Miracle Octad Generator. It should be noted that this construction is due
to Richard J. Turyn, in a 1967 Sylvania research report. (See Emily Jennings's
talk of 1 Nov. 2012.) Compare the Curtis construction, written in 1974,
with the Turyn construction of 1967 as described in Sphere Packings, Lattices
and Groups , by J. H. Conway and N. J. A. Sloane (first published in 1988).
Sunday, May 19, 2013
Sermon
Best vs. Bester
The previous post ended with a reference mentioning Rosenhain.
For a recent application of Rosenhain's work, see
Desargues via Rosenhain (April 1, 2013).
From the next day, April 2, 2013:
"The proof of Desargues' theorem of projective geometry
comes as close as a proof can to the Zen ideal.
It can be summarized in two words: 'I see!' "
– Gian-Carlo Rota in Indiscrete Thoughts (1997)
Also in that book, originally from a review in Advances in Mathematics ,
Vol. 84, Number 1, Nov. 1990, p. 136:
See, too, in the Conway-Sloane book, the Galois tesseract …
and, in this journal, Geometry for Jews and The Deceivers , by Bester.
Tuesday, April 2, 2013
Rota in a Nutshell
"The proof of Desargues' theorem of projective geometry
comes as close as a proof can to the Zen ideal.
It can be summarized in two words: 'I see!' "
— Gian-Carlo Rota in Indiscrete Thoughts (1997)
Also in that book, originally from a review in Advances in Mathematics,
Vol. 84, Number 1, Nov. 1990, p. 136:
Related material:
Pascal and the Galois nocciolo ,
Conway and the Galois tesseract,
Gardner and Galois.
See also Rota and Psychoshop.
Saturday, January 5, 2013
Vector Addition in a Finite Field
The finite (i.e., Galois) field GF(16),
according to J. J. Seidel in 1974—
The same field according to Steven H. Cullinane in 1986,
in its guise as the affine 4-space over GF(2)—
The same field, again disguised as an affine 4-space,
according to John H. Conway and N.J.A. Sloane in
Sphere Packings, Lattices, and Groups , first published in 1988—
The above figure by Conway and Sloane summarizes, using
a 4×4 array, the additive vector-space structure of the finite
field GF(16).
This structure embodies what in Euclidean space is called
the parallelogram rule for vector addition—
(Thanks to June Lester for the 3D (uvw) part of the above figure.)
For the transition from this colored Euclidean hypercube
(used above to illustrate the parallelogram rule) to the
4×4 Galois space (illustrated by Cullinane in 1979 and
Conway and Sloane in 1988— or later… I do not have
their book’s first edition), see Diamond Theory in 1937,
Vertex Adjacency in a Tesseract and in a 4×4 Array,
Spaces as Hypercubes, and The Galois Tesseract.
For some related narrative, see tesseract in this journal.
(This post has been added to finitegeometry.org.)
Update of August 9, 2013—
Coordinates for hypercube vertices derived from the
parallelogram rule in four dimensions were better
illustrated by Jürgen Köller in a web page archived in 2002.
Update of August 13, 2013—
The four basis vectors in the 2002 Köller hypercube figure
are also visible at the bottom of the hypercube figure on
page 7 of “Diamond Theory,” excerpts from a 1976 preprint
in Computer Graphics and Art , Vol. 2, No. 1, February 1977.
A predecessor: Coxeter’s 1950 hypercube figure from
“Self-Dual Configurations and Regular Graphs.”
Wednesday, October 31, 2012
The Malfunctioning TARDIS
(Continued from previous TARDIS posts)
Summary: A review of some posts from last August is suggested by the death,
reportedly during the dark hours early on October 30, of artist Lebbeus Woods.
An (initially unauthorized) appearance of his work in the 1995 film
Twelve Monkeys …
… suggests a review of three posts from last August.
Wednesday, August 1, 2012Defining FormContinued from July 29 in memory of filmmaker Chris Marker, See Slides and Chanting†and Where Madness Lies. See also Sherrill Grace on Malcolm Lowry. * Washington Post. Other sources say Marker died on July 30. † These notably occur in Marker's masterpiece |
Wednesday, August 1, 2012Triple FeatureFor related material, see this morning's post Defining Form. |
Sunday, August 12, 2012Doctor WhoOn Robert A. Heinlein's novel Glory Road— "Glory Road (1963) included the foldbox , a hyperdimensional packing case that was bigger inside than outside. It is unclear if Glory Road was influenced by the debut of the science fiction television series Doctor Who on the BBC that same year. In Doctor Who , the main character pilots a time machine called a TARDIS, which is built with technology which makes it 'dimensionally transcendental,' that is, bigger inside than out." — Todd, Tesseract article at exampleproblems.com From the same exampleproblems.com article— "The connection pattern of the tesseract's vertices is the same as that of a 4×4 square array drawn on a torus; each cell (representing a vertex of the tesseract) is adjacent to exactly four other cells. See geometry of the 4×4 square." For further details, see today's new page on vertex adjacency at finitegeometry.org. |
"It was a dark and stormy night."— A Wrinkle in Time
Sunday, August 12, 2012
Doctor Who
On Robert A. Heinlein's novel Glory Road—
"Glory Road (1963) included the foldbox , a hyperdimensional packing case that was bigger inside than outside. It is unclear if Glory Road was influenced by the debut of the science fiction television series Doctor Who on the BBC that same year. In Doctor Who , the main character pilots a time machine called a TARDIS, which is built with technology which makes it 'dimensionally transcendental,' that is, bigger inside than out."
— Todd, Tesseract article at exampleproblems.com
From the same exampleproblems.com article—
"The connection pattern of the tesseract's vertices is the same as that of a 4×4 square array drawn on a torus; each cell (representing a vertex of the tesseract) is adjacent to exactly four other cells. See geometry of the 4×4 square."
For further details, see today's new page on vertex adjacency at finitegeometry.org.
Sunday, July 29, 2012
The Galois Tesseract
The three parts of the figure in today's earlier post "Defining Form"—
— share the same vector-space structure:
0 | c | d | c + d |
a | a + c | a + d | a + c + d |
b | b + c | b + d | b + c + d |
a + b | a + b + c | a + b + d | a + b + c + d |
(This vector-space a b c d diagram is from Chapter 11 of
Sphere Packings, Lattices and Groups , by John Horton
Conway and N. J. A. Sloane, first published by Springer
in 1988.)
The fact that any 4×4 array embodies such a structure was implicit in
the diamond theorem (February 1979). Any 4×4 array, regarded as
a model of the finite geometry AG(4, 2), may be called a Galois tesseract.
(So called because of the Galois geometry involved, and because the
16 cells of a 4×4 array with opposite edges identified have the same
adjacency pattern as the 16 vertices of a tesseract (see, for instance,
Coxeter's 1950 "Self-Dual Configurations and Regular Graphs," figures
5 and 6).)
A 1982 discussion of a more abstract form of AG(4, 2):
Source:
The above 1982 remarks by Brouwer may or may not have influenced
the drawing of the above 1988 Conway-Sloane diagram.
Monday, July 16, 2012
Mapping Problem continued
Another approach to the square-to-triangle
mapping problem (see also previous post)—
For the square model referred to in the above picture, see (for instance)
- Picturing the Smallest Projective 3-Space,
- The Relativity Problem in Finite Geometry, and
- Symmetry of Walsh Functions.
Coordinates for the 16 points in the triangular arrays
of the corresponding affine space may be deduced
from the patterns in the projective-hyperplanes array above.
This should solve the inverse problem of mapping,
in a natural way, the triangular array of 16 points
to the square array of 16 points.
Update of 9:35 AM ET July 16, 2012:
Note that the square model's 15 hyperplanes S
and the triangular model's 15 hyperplanes T —
— share the following vector-space structure —
0 | c | d | c + d |
a | a + c | a + d | a + c + d |
b | b + c | b + d | b + c + d |
a + b | a + b + c | a + b + d | a + b + c + d |
(This vector-space a b c d diagram is from
Chapter 11 of Sphere Packings, Lattices
and Groups , by John Horton Conway and
N. J. A. Sloane, first published by Springer
in 1988.)
Thursday, September 1, 2011
Wednesday, June 1, 2011
The Schwartz Notes
A Google search today for material on the Web that puts the diamond theorem
in context yielded a satisfyingly complete list. (See the first 21 results.)
(Customization based on signed-out search activity was disabled.)
The same search limited to results from only the past month yielded,
in addition, the following—
This turns out to be a document by one Richard Evan Schwartz,
Chancellor’s Professor of Mathematics at Brown University.
Pages 12-14 of the document, which is untitled, undated, and
unsigned, discuss the finite-geometry background of the R.T.
Curtis Miracle Octad Generator (MOG) . As today’s earlier search indicates,
this is closely related to the diamond theorem. The section relating
the geometry to the MOG is titled “The MOG and Projective Space.”
It does not mention my own work.
See Schwartz’s page 12, page 13, and page 14.
Compare to the web pages from today’s earlier search.
There are no references at the end of the Schwartz document,
but there is this at the beginning—
These are some notes on error correcting codes. Two good sources for
this material are
• From Error Correcting Codes through Sphere Packings to Simple Groups ,
by Thomas Thompson.
• Sphere Packings, Lattices, and Simple Groups by J. H. Conway and N.
Sloane
Planet Math (on the internet) also some information.
It seems clear that these inadequate remarks by Schwartz on his sources
can and should be expanded.
Wednesday, October 27, 2010
Review
A mathematical review—
— Gian-Carlo Rota
A science fiction—
— Alfred Bester
Saturday, May 29, 2010
Packed
Significant Passage:
On the Writing Style of Visual Thinkers
"The words are filled with unstated meaning.
They are (the term is Ricoeur's) 'packed'
and need unpacking." —Gerald Grow
From the date of Ricoeur's death,
May 20, 2005—
“Plato’s most significant passage
may be found in Phaedrus 265b…."
With a little effort, cross-referenced." — Opening sentence
|
Mozart's K 265 is variations on the theme
now known as "Twinkle, Twinkle, Little Star."
For darker variations on the Twinkle theme,
see the film "Joshua" and Martin Gardner's
Annotated Alice (Norton, 2000, pp. 73-75).
For darker variations on the asterisk theme,
see Darkness Visible (May 25)
and Vonnegut's Asterisk.
Friday, May 28, 2010
Multispeech for Oxford
Happy Birthday,
Carey Mulligan
Star of "An Education"
In "An Education," Mulligan's character
applies for admission to Oxford.
Today's New York Times:
Education »
Oxford Tradition Comes to This: ‘Death’ (Expound) |
Related material:
Such words arrive on the page like suitcases at the baggage claim: You know there is something in them and they have travelled far, but you cannot tell what the writer means. The words are filled with unstated meaning. They are (the term is Ricoeur's) "packed" and need unpacking. This method of using language, however, is not always a defect; radiantly evocative words have long been the language of myth, mysticism, and love. Also, in earlier centuries, educated readers expected to interpret writing on several different levels at once (e.g., literal, allegorical, moral, and anagogical or spiritual), so that multiple meanings were the norm. This was before the era of clear, expository, fully-explicit prose. Visual thinkers are accustomed to their own kind of interpreting; the very act of visual perception, as Gregory (1966, 1970) and Gombrich (1959) have shown, is interpretive. When oral thinkers leave you to guess at something they have written, it is usually something that would have been obvious had the writing been a conversation. Such is not the case with visual thinkers, even whose spoken words can be mysterious references to visual thoughts invisible to anyone but the thinker. Writing done in this "packed" manner makes more sense when read as poetry than when read as prose. References: Gombrich, E. H. (1959). Art and Illusion: A Study in the Psychology of Pictorial Representation. London: Phaidon. Gregory, R. L. (1966). Eye and Brain: The Psychology of Seeing. New York: McGraw-Hill. Gregory, R. L. (1970). The Intelligent Eye. New York: McGraw-Hill. — "Stacking, Packing, and Enfolding Words," by Gerald Grow in "The Writing Problems of Visual Thinkers" |
Those wishing to emulate Mulligan's
character in "An Education" might,
having read the Times article above,
consult this journal's post of May 17,
"Rolling the Stone."
That post contains the following
image from the Times—
May 17 was, by the way, the day
that R. L. Gregory, author of
The Intelligent Eye, died.
Wednesday, November 30, 2005
Wednesday November 30, 2005
For St. Andrew’s Day
“The miraculous enters…. When we investigate these problems, some fantastic things happen….”
— John H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, preface to first edition (1988)
The remarkable Mathieu group M24, a group of permutations on 24 elements, may be studied by picturing its action on three interchangeable 8-element “octads,” as in the “Miracle Octad Generator” of R. T. Curtis.
A picture of the Miracle Octad Generator, with my comments, is available online.
Related material:
Mathematics and Narrative.
Friday, November 18, 2005
Friday November 18, 2005
a fight for love and…
Wikipedia on the tesseract:
Robert A. Heinlein in Glory Road:
And opened it again.
And kept on opening it– And kept right on unfolding its sides and letting them down until the durn thing was the size of a small moving van and even more packed….
… Anyone who has studied math knows that the inside does not have to be smaller than the outside, in theory…. Rufo’s baggage just carried the principle further.”
Johnny Cash: “And behold, a white horse.”
On The Last Battle, a book in the Narnia series by C. S. Lewis:
Lewis said in “The Weight of Glory”—
On enchantments that need to be broken:
See the description of the Eater of Souls in Glory Road and of Scientism in
- and the C. S. Lewis classic,
That Hideous Strength.
Tuesday, September 14, 2004
Tuesday September 14, 2004
The Square Wheel
Harmonic analysis may be based either on the circular (i.e., trigonometric) functions or on the square (i. e., Walsh) functions. George Mackey's masterly historical survey showed that the discovery of Fourier analysis, based on the circle, was of comparable importance (within mathematics) to the discovery (within general human history) of the wheel. Harmonic analysis based on square
For some observations of Stephen Wolfram on square-wheel analysis, see pp. 573 ff. in Wolfram's magnum opus, A New Kind of Science (Wolfram Media, May 14, 2002). Wolfram's illustration of this topic is closely related, as it happens, to a note on the symmetry of finite-geometry hyperplanes that I wrote in 1986. A web page pointing out this same symmetry in Walsh functions was archived on Oct. 30, 2001.
That web page is significant (as later versions point out) partly because it shows that just as the phrase "the circular functions" is applied to the trigonometric functions, the phrase "the square functions" might well be applied to Walsh
"While the reader may draw many a moral from our tale, I hope that the story is of interest for its own sake. Moreover, I hope that it may inspire others, participants or observers, to preserve the true and complete record of our mathematical times."
— From Error-Correcting Codes
Through Sphere Packings
To Simple Groups,
by Thomas M. Thompson,
Mathematical Association of America, 1983
Friday, May 23, 2003
Friday May 23, 2003
Mental Health Month, Day 23:
The Prime Cut Gospel
On Christmas Day, 1949,
Mary Elizabeth Spacek was born in Texas.
Lee Marvin, Sissy Spacek in “Prime Cut”
Exercises for Mental Health Month:
Read this discussion of the phrase, suggested by Spacek’s date of birth, “God’s gift to men.”
Read this discussion of the phrase “the same yesterday, today, and forever,” suggested by the previous reading.
Read the more interesting of these discussions of the phrase “the eternal in the temporal.”
Read this discussion of eternal, or “necessary,” truths versus other sorts of alleged “truths.”
Read this discussion of unimportant mathematical properties of the prime number 23.
Read these discussions of important properties of 23:
- R. D. Carmichael’s 1937 discussion of the linear fractional group modulo 23 in
Introduction to the Theory of Groups of Finite Order, Ginn, Boston, 1937 (reprinted by Dover in 1956), final chapter, “Tactical Configurations,” and
- Conway’s 1969 discussion of the same group in
J. H. Conway, “Three Lectures on Exceptional Groups,” pp. 215-247 in Finite Simple Groups (Oxford, 1969), edited by M. B. Powell and G. Higman, Academic Press, London, 1971….. Reprinted as Ch. 10 in Sphere Packings, Lattices, and Groups
Read this discussion of what might be called “contingent,” or “literary,” properties of the number 23.
Read also the more interesting of these discussions of the phrase “the 23 enigma.”
Having thus acquired some familiarity with both contingent and necessary properties of 23…
Read this discussion of Aquinas’s third proof of the existence of God.
Note that the classic Spacek film “Prime Cut” was released in 1972, the year that Spacek turned 23:
1949
+ 23 1972 |