An alternative narrative:
Plan 9 Continues (Log24, Dec. 11, 2024).
Wednesday, March 5, 2025
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.
Comments Off on The Cullinane Diamond Theorem
and the Miracle Octad Generator
and the Miracle Octad Generator
Thursday, November 21, 2024
Six-Set Geometry and Witt Designs
From other posts tagged Six-Set —
The six-set also underlies the 21-point projective plane PG(2,4) —
PG(2,4) may be used to construct S(5,8,24), also known as
the large Witt design. Some related research . . .
On four codes with automorphism group PΣL(3,4)
and pseudo-embeddings of the large Witt designs,
by Bart De Bruyn (UGent) and Mou Gao (UGent),
(2020) DESIGNS CODES AND CRYPTOGRAPHY. 88(2), pp. 429-452.
Some background reading —
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Saturday, August 26, 2023
For the Church of Synchronology
Art Blocks in the previous post —
"… making accessibility and IRL viewership a core component" . . .
From this journal on the above art date — April 6, 2021 —
Tuesday, April 6, 2021
|
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Monday, August 21, 2023
Art Song: “… And They Call the Wind Pariah”
Update …
"Here are 3 brands building on the blockchain…"
♫ "Way out west they have a name for rain and wind and fire …."
https://www.classic-country-song-lyrics.com/ theycallthewindmariahlyricschords.html They Call The Wind Mariah Recorded by Jim Ed Brown written by Alan J. Lerner and Frederick Loewe C Am C Am Way out west they have a name for rain and wind and fire C Am F G7 C The rain is Tess the fire's Joe and they call the wind Mariah Am C Am Mariah blows the stars around and sets the clouds a-flying C Am F G7 C Mariah makes the mountains sound like folks up there were dying Am Em Mariah Mariah F G7 C They call the wind Mariah Am C Am Before I knew Mariah's name or heard her wail and whining C Am F G7 C I had a gal and she had me and the sun was always shining Am C Am And then one day I left that gal I left her far behind me C Am F G7 C And now I'm lost I'm so darn lost not even God can find me Am Em Mariah Mariah F G7 C They call the wind Mariah Am C Am Out here they’ve got a name for rain wind and fire only C Am F G7 C And when you're lost and all alone there ain't no word for lonely Am C Am Well I'm a lost and lonely man without a star to guide me C Am F G7 C Mariah blow my love to me I need her here beside me Am Em Mariah Mariah F G7 C They call the wind Mariah |
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Friday, July 28, 2023
To Phrase a Coin
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Thursday, June 15, 2023
Michaelmas 2019
Transcribed from a PDF:
|
Received September 29, 2019, accepted October 15, 2019, Digital Object Identifier 10.1109/ACCESS.2019.2949310
A Method for Determining
ZIYU WANG1 , XIAO ZENG1 , JINZHAO WU2,3, AND
1Big Data Research Center, School of Computer Science
2Guangxi Key Laboratory of Hybrid Computation and
3School of Computer and Electronic Information,
Corresponding authors:
This work was supported in part by the National Natural Science Foundation
ABSTRACT
INDEX TERMS Logic synthesis, Boolean functions, |
Meanwhile . . .
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Friday, November 13, 2020
Raiders of the Lost Dorm Room
“That really is, really, I think, the Island of the Misfit Toys at that point.
You have crossed the Rubicon, you jumped on the crazy train and
you’re headed into the cliffs that guard the flat earth at that time, brother,”
said Rep. Denver Riggleman, a Republican congressman from Virginia,
in an interview."
— Jon Ward, political correspondent, Yahoo News , Nov. 12, 2020
The instinct for heaven had its counterpart:
The instinct for earth, for New Haven, for his room,
The gay tournamonde as of a single world
In which he is and as and is are one.
— Wallace Stevens, "An Ordinary Evening in New Haven"
Related material for comedians —
See as well Sallows in this journal.
“There exists a considerable literature
devoted to the Lo shu , much of it infected
with the kind of crypto-mystic twaddle
met with in Feng Shui.”
— Lee C. F. Sallows, Geometric Magic Squares ,
Dover Publications, 2013, page 121
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Tuesday, January 22, 2019
Iconic Remotivation
From an obituary in yesterday's online New York Times —
Photo credit: Gabriella Angotti-Jones/The New York Times
This journal in the time frame of events leading to the obituary —
On "Wakean Cryptogenetics" —
"… Joyce now disposes of a complex machine thanks to which
any linguistic item culled when reading a book, a magazine,
overhearing a conversation, meditating upon a dream, can find
an actor who will underwrite it . . . ."
— P. 81 in Joyce upon the Void by Jean-Michel Rabaté,
Palgrave-Macmillan, 1991.
An Actor —
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Thursday, August 4, 2016
The Soltan Diagnosis
Professor Soltan, a fan of James Joyce, would do well
to apply her diagnostic powers to Finnegans Wake ,
a word salad if ever there was one.
Related recommended reading:
-
Publications list of Peter Mahon, sessional lecturer
at the University of British Columbia, Vancouver - Cryptomorphisms in Log24
-
A search in Soltan's weblog for why
"people hate professors."
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Sunday, November 15, 2015
The Diamond and the Cube
Anyone who clicked on the Dirac search at the end of
the previous post, "Dirac's Diamond," may wonder why the
"Solomon's Cube" post of 11 AM Sunday, March 1, 2009,
appeared in the Dirac search results, since there is no
apparent mention of Dirac in that Sunday post.
<!– See also "a linear transformation of V6… which preserves
the Klein quadric; in this way we arrive at the isomorphism of
Sym(8) withthe full orthogonal group O+(6; 2)." in "The
Classification of Flats in PG(9,2) which are External to the
Grassmannian G1,4,2 Authors: Shaw, Ron;
 Maks, Johannes; Gordon, Neil; Source: Designs,
Codes and Cryptography, Volume 34, Numbers 2-3, February
2005 , pp. 203-227; Publisher: Springer.  For more details,
see "Finite Geometry, Dirac Groups and the Table of Real
Clifford Algebras," by R. Shaw (U. of Hull), pp. 59-99 in
Clifford Algebras and Spinor Structures, by By Albert
Crumeyrolle, Rafał Abłamowicz, Pertti Lounesto,
published by Springer, 1995. –>
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Monday, August 31, 2015
Nightmare for Wes Craven
Adapted from posts tagged Cryptomorphisms
in this journal:
"Hear it not, Craven, for it is a knell
That summons thee to heaven, or to hell. “
Max von Sydow in Branded (2012)
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Thesis, Antithesis, Synthesis
Thesis and antithesis at last night's
MTV video music awards:
A geometric synthesis —
Related material —
The Wrench and the Nut (Aug. 24) and Cryptomorphisms.
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Sunday, August 9, 2015
Cryptomorphisms
Backstory: Other posts tagged “Cryptomorphisms,”
and the word itself in Wikipedia.
Compare and contrast:
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Sunday, November 9, 2014
Twaddle
“There exists a considerable literature
devoted to the Lo shu , much of it infected
with the kind of crypto-mystic twaddle
met with in Feng Shui.”
— Lee C. F. Sallows, Geometric Magic Squares ,
Dover Publications, 2013, page 121
Cf. Raiders of the Lost Theorem, Oct. 13, 2014.
See also tonight’s previous post and
“Feng Shui” in this journal.
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Thursday, November 29, 2012
Lines of Symbols
C. P. Snow on G. H. Hardy, in Snow's foreword to A Mathematician's Apology—
"One morning early in 1913, he found, among the letters on his breakfast table, a large untidy envelope decorated with Indian stamps. When he opened it, he found sheets of paper by no means fresh, on which, in a non-English holograph, were line after line of symbols. Hardy glanced at them without enthusiasm. He was by this time, at the age of thirty-six, a world famous mathematician: and world famous mathematicians, he had already discovered, are unusually exposed to cranks. He was accustomed to receiving manuscripts from strangers, proving the prophetic wisdom of the Great Pyramid, the revelations of the Elders of Zion, or the cryptograms that Bacon has inserted in the plays of the so-called Shakespeare."
Some related material (click to enlarge)—
The author links to, but does not name, the source of the above
"line after line of symbols." It is "Visualizing GL(2,p)." See that webpage
for some less esoteric background.
See also the two Wikipedia articles Finite geometry and Hesse configuration
and an image they share—
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Monday, November 5, 2012
Sitting Specially
Some webpages at finitegeometry.org discuss
group actions on Sylvester’s duads and synthemes.
Those pages are based on the square model of
PG(3,2) described in the 1980’s by Steven H. Cullinane.
A rival tetrahedral model of PG(3,2) was described
in the 1990’s by Burkard Polster.
Polster’s tetrahedral model appears, notably, in
a Mathematics Magazine article from April 2009—
Click for a pdf of the article.
Related material:
“The Religion of Cubism” (May 9, 2003) and “Art and Lies”
(Nov. 16, 2008).
This post was suggested by following the link in yesterday’s
Sunday School post to High White Noon, and the link from
there to A Study in Art Education, which mentions the date of
Rudolf Arnheim‘s death, June 9, 2007. This journal
on that date—
The Fink-Guy article was announced in a Mathematical
Association of America newsletter dated April 15, 2009.
Those who prefer narrative to mathematics may consult
a Log24 post from a few days earlier, “Where Entertainment is God”
(April 12, 2009), and, for some backstory, The Judas Seat
(February 16, 2007).
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Tuesday, July 10, 2012
Space Cadets
From this journal on June 19, 2012—
Walter Gropius on space—
"Was ist Raum, wie können wir ihn
erfassen und gestalten?"
The Theory and
Organization of the
Bauhaus (1923)
A book published on the same date—
June 19, 2012:
"… what Chalmers called the convergence of coincidence—
a force majeure of unrelated events that shaped one's life,
that perhaps defined the concept of life itself.
He believed in the power of that force."
— The Cryptos Conundrum , by Chase Brandon
See also Chase Brandon in Sunday's Huffington Post .
"I wrote another book."
— Robert De Niro as Harlan Kane
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Sunday, May 17, 2009
Sunday May 17, 2009
Design Theory
Laura A. Smit, Calvin College, "Towards an Aesthetic Teleology: Romantic Love, Imagination and the Beautiful in the Thought of Simone Weil and Charles Williams"–
"My work is motivated by a hope that there may be a way to recapture the ancient and medieval vision of both Beauty and purpose in a way which is relevant to our own century. I even dare to hope that the two ideas may be related, that Beauty is actually part of the meaning and purpose of life."
Hans Ludwig de Vries, "On Orthogonal Resolutions of the Classical Steiner Quadruple System SQS(16)," Designs, Codes and Cryptography Vol. 48, No. 3 (Sept. 2008) 287-292 (DOI 10.1007/s10623-008-9207-5)–
"The Reverend T. P. Kirkman knew in 1862 that there exists a group of degree 16 and order 322560 with a normal, elementary abelian, subgroup of order 16 [1, p. 108]. Frobenius identified this group in 1904 as a subgroup of the Mathieu group M24 [4, p. 570]…."
1. Biggs N.L., "T. P. Kirkman, Mathematician," Bulletin of the London Mathematical Society 13, 97–120 (1981).
4. Frobenius G., "Über die Charaktere der mehrfach transitiven Gruppen," Sitzungsber. Königl. Preuss. Akad. Wiss. zu Berlin, 558–571 (1904). Reprinted in Frobenius, Gesammelte Abhandlungen III (J.-P. Serre, editor), pp. 335–348. Springer, Berlin (1968).
Olli Pottonen, "Classification of Steiner Quadruple Systems" (Master's thesis, Helsinki, 2005)–
"The concept of group actions is very useful in the study of isomorphisms of combinatorial structures."
"Simplify, simplify."
— Thoreau
"Beauty is bound up
with symmetry."
— Weyl
Pottonen's thesis is
dated Nov. 16, 2005.
For some remarks on
images and theology,
see Log24 on that date.
Click on the above image
for some further details.
Comments Off on Sunday May 17, 2009
Sunday, March 1, 2009
Sunday March 1, 2009
Solomon's Cube
continued
"There is a book… called A Fellow of Trinity, one of series dealing with what is supposed to be Cambridge college life…. There are two heroes, a primary hero called Flowers, who is almost wholly good, and a secondary hero, a much weaker vessel, called Brown. Flowers and Brown find many dangers in university life, but the worst is a gambling saloon in Chesterton run by the Misses Bellenden, two fascinating but extremely wicked young ladies. Flowers survives all these troubles, is Second Wrangler and Senior Classic, and succeeds automatically to a Fellowship (as I suppose he would have done then). Brown succumbs, ruins his parents, takes to drink, is saved from delirium tremens during a thunderstorm only by the prayers of the Junior Dean, has much difficulty in obtaining even an Ordinary Degree, and ultimately becomes a missionary. The friendship is not shattered by these unhappy events, and Flowers's thoughts stray to Brown, with affectionate pity, as he drinks port and eats walnuts for the first time in Senior Combination Room."
— G. H. Hardy, A Mathematician's Apology
"The Solomon Key is the working title of an unreleased novel in progress by American author Dan Brown. The Solomon Key will be the third book involving the character of the Harvard professor Robert Langdon, of which the first two were Angels & Demons (2000) and The Da Vinci Code (2003)." — Wikipedia
"One has O+(6) ≅ S8, the symmetric group of order 8! …."
— "Siegel Modular Forms and Finite Symplectic Groups," by Francesco Dalla Piazza and Bert van Geemen, May 5, 2008, preprint.
"The complete projective group of collineations and dualities of the [projective] 3-space is shown to be of order [in modern notation] 8! …. To every transformation of the 3-space there corresponds a transformation of the [projective] 5-space. In the 5-space, there are determined 8 sets of 7 points each, 'heptads' …."
— George M. Conwell, "The 3-space PG(3, 2) and Its Group," The Annals of Mathematics, Second Series, Vol. 11, No. 2 (Jan., 1910), pp. 60-76
"It must be remarked that these 8 heptads are the key to an elegant proof…."
— Philippe Cara, "RWPRI Geometries for the Alternating Group A8," in Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference (July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis, James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97
Comments Off on Sunday March 1, 2009
Saturday, December 22, 2007
Saturday December 22, 2007
"It has been said that the unexamined life isn't worth living. Nachman wasn't against examining his life, but then what was a life? ….
… As for 'a life,' it was what you read about in newspaper obituaries. He didn't need one. He would return to California and think only about mathematics."
— Leonard Michaels, "Cryptology"
Related material:
Today's online
New York Times
obituaries
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Sunday, August 12, 2007
Sunday August 12, 2007
In the context of quantum information theory, the following structure seems to be of interest–
"… the full two-by-two matrix ring with entries in GF(2), M2(GF(2))– the unique simple non-commutative ring of order 16 featuring six units (invertible elements) and ten zero-divisors."
— "Geometry of Two-Qubits," by Metod Saniga (pdf, 17 pp.), Jan. 25, 2007
This ring is another way of looking at the 16 elements of the affine space A4(GF(2)) over the 2-element field. (Arrange the four coordinates of each element– 1's and 0's– into a square instead of a straight line, and regard the resulting squares as matrices.) (For more on A4(GF(2)), see Finite Relativity and related notes at Finite Geometry of the Square and Cube.) Using the above ring, Saniga constructs a system of 35 objects (not unlike the 35 lines of the finite geometry PG(3,2)) that he calls a "projective line" over the ring. This system of 35 objects has a subconfiguration isomorphic to the (2,2) generalized quadrangle W2 (which occurs naturally as a subconfiguration of PG(3,2)– see Inscapes.)
Saniga concludes:
"We have demonstrated that the basic properties of a system of two interacting spin-1/2 particles are uniquely embodied in the (sub)geometry of a particular projective line, found to be equivalent to the generalized quadrangle of order two. As such systems are the simplest ones exhibiting phenomena like quantum entanglement and quantum non-locality and play, therefore, a crucial role in numerous applications like quantum cryptography, quantum coding, quantum cloning/teleportation and/or quantum computing to mention the most salient ones, our discovery thus
- not only offers a principally new geometrically-underlined insight into their intrinsic nature,
- but also gives their applications a wholly new perspective
- and opens up rather unexpected vistas for an algebraic geometrical modelling of their higher-dimensional counterparts."
is not without relevance to
the physics of quantum theory.
the physics of quantum theory.
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Thursday, June 14, 2007
Thursday June 14, 2007
"Kurt Waldheim, the former United Nations Secretary General and President of Austria whose hidden ties to Nazi organizations and war crimes was [sic] exposed late in his career, died today at his home in Vienna. He was 88." —The New York Times this afternoon
Related material:
From a story by
Leonard Michaels
linked to on
Aaron Sorkin's
birthday, June 9:
"Induction and analogy, in which he was highly gifted, were critical to mathematical intelligence.
It has been said that the unexamined life isn't worth living. Nachman wasn't against examining his life, but then what was a life? ….
… As for 'a life,' it was what you read about in newspaper obituaries. He didn't need one. He would return to California and think only about mathematics."
Mathematics:
2. A quotation from an anonymous
Internet user signed
"George Polya"–
"Steven Cullinane is a Liar."
Internet user signed
"George Polya"–
"Steven Cullinane is a Liar."
3. L'Affaire Dharwadker continues
(May 31, 2007)
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Saturday, June 9, 2007
Saturday, April 5, 2003
Saturday April 5, 2003
Art Wars:
Mathematics and the
Emperor's New Art
From Maureen Dowd's New York Times column of June 9, 2002:
"The shape of the government is not as important as the policy of the government. If he makes the policy aggressive and pre-emptive, the president can conduct the war on terror from the National Gallery of Art."
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NY Times, April 5, 2003: |
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Meanwhile, at the Washington Post, another example of great determination and strength of character:
Donald Coxeter Dies: Leader in Geometry
By Martin Weil
Washington Post Staff Writer
Saturday, April 5, 2003
"Donald Coxeter, 96, a mathematician who was one of the 20th century's foremost specialists in geometry and a man of great determination and strength of character as well, died March 31 at his home in Toronto."
From another Coxeter obituary:
In the Second World War, Coxeter was asked by the American government to work in Washington as a code-breaker. He accepted, but then backed out, partly because of his pacifist views and partly for aesthetic reasons: "The work didn't really appeal to me," he explained; "it was a different sort of mathematics."
For a differing account of how geometry is related to code-breaking, see the "Singer 7-cycle" link in yesterday's entry, "The Eight," of 3:33 PM. This leads to a site titled
An Introduction to the
Applications of Geometry in Cryptography.
"Now I have precisely the right instrument, at precisely the right moment of history, in exactly the right place."
— "Patton,"
the film
Added Sunday, April 6, 2003, 3:17 PM:
The New York Times Magazine of April 6
continues this Art Wars theme.
(Cover typography revised)
The military nature of our Art Wars theme appears in the Times's choice of words for its cover headline: "The Greatest Generation." (This headline appears in the paper, but not the Internet, version.)
Some remarks in today's Times Magazine article seem especially relevant to my journal entry for Michelangelo's birthday, March 6.
"…Conceptualism — suddenly art could be nothing more than an idea….
LeWitt moved between his syntax of geometric sculptures and mental propositions for images: concepts he wrote on paper that could be realized by him or someone else or not at all. Physical things are perishable. Ideas need not be."
— Michael Kimmelman, chief art critic of the New York Times, April 6, 2003
Compare this with a mathematician's aesthetics:
"A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas."
— G. H. Hardy, A Mathematician's Apology (1940), reprinted 1969, Cambridge U. Press, p. 84
It seems clear from these two quotations that the real conceptual art is mathematics and that Kimmelman is peddling the emperor's new clothes.