For related material on the geometry of a 2×3 rectangular array —
a domino — see the previous post and also a search in this journal for attic .
Friday, July 21, 2023
The Domino Model: Tricksters in the Attic
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Monday, September 12, 2016
The Kummer Lattice
The previous post quoted Tom Wolfe on Chomsky's use of
the word "array."
An example of particular interest is the 4×4 array
(whether of dots or of unit squares) —
.
Some context for the 4×4 array —
The following definition indicates that the 4×4 array, when
suitably coordinatized, underlies the Kummer lattice .
Further background on the Kummer lattice:
Alice Garbagnati and Alessandra Sarti,
"Kummer Surfaces and K3 surfaces
with $(Z/2Z)^4$ symplectic action."
To appear in Rocky Mountain J. Math. —
The above article is written from the viewpoint of traditional
algebraic geometry. For a less traditional view of the underlying
affine 4-space from finite geometry, see the website
Finite Geometry of the Square and Cube.
Some further context …
"To our knowledge, the relation of the Golay code
to the Kummer lattice … is a new observation."
— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of
Kummer surfaces in the Mathieu group M24 "
As noted earlier, Taormina and Wendland seem not to be aware of
R. W. H. T. Hudson's use of the (uncoordinatized*) 4×4 array in his
1905 book Kummer's Quartic Surface. The array was coordinatized,
i.e. given a "vector space structure," by Cullinane eight years prior to
the cited remarks of Curtis.
* Update of Sept. 14: "Uncoordinatized," but parametrized by 0 and
the 15 two-subsets of a six-set. See the post of Sept. 13.
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Friday, April 19, 2024
Benchmarking Meta.ai
Transcription —
(With one correction … The Meta.ai phrase "the group of 322,560"
was changed to "a group of 322,560" … and one addition — the
acronym MOG was defined. Superscript numerals for missing footnotes
were removed.)
|
You What is the Cullinane diamond theorem? Meta AI The Cullinane diamond theorem states that every image of D (a 4×4 array of two-color diagonally divided square tiles) under the group G (a group of 322,560 permutations of these 16 tiles) has some ordinary or color-interchange symmetry. Here are some key points about the theorem :
|
Tuesday, March 19, 2024
Art for Nihilists
|
"I am serious about my study. I am a distinguished professor of mathematics at Brown University, though I have not for decades concerned myself with arithmetic, calculus, matrices, theorems, Hausdorff spaces, finite lattice representations, or anything else that involves values or numbers or representations of values or numbers or any such somethings, whether they have substance or not. I have spent my career in my little office on George Street in Providence contemplating and searching for nothing. I have not found it."
Everett, Percival. Dr. No: A Novel (p. 6). |
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Thursday, February 8, 2024
Google Bard is now Gemini.
Gemini Prompt:
Write a poem in the style of Wallace Stevens
about the Cullinane diamond theorem.
Response:
Absolutely! Here's a poem inspired by Wallace Stevens
and the Cullinane diamond theorem:
A theorem not of Euclid's line,
But diamonds cut with strange design.
Sixteen squares in shifting light,
A symmetry of black and white.
Where rows and columns intertwine,
A hidden order lies supine.
Permutations twist and turn,
And latent symmetries discern.
Not fixed like stars in ordered rows,
But like reflections, pattern grows.
Projective planes, a finite field,
The diamond's inner world revealed.
The proof, a dance of abstract thought,
Where shifting forms find space untaught.
Not nature's chaos, but design,
A brilliance of the human mind.
A lattice born of Stevens' art,
Where form and thought find counterpart.
Amidst the fragments, patterns gleam,
The Cullinane, a geometer's dream.
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Monday, December 18, 2023
AI Class
|
Bringing Up Bing : A Tutorial for Bing on Monday, December 18, 2023 Prompt: How is the Cullinane Diamond Theorem related to the Curtis Miracle Octad Generator?
Response: Searching for: Generating answers for you… The Cullinane Diamond Theorem and the Curtis Miracle Octad Generator (MOG) are both mathematical concepts that are deeply intertwined with finite projective geometry1,2. The Cullinane Diamond Theorem explains the surprising symmetry properties of some simple graphic designs and underlies the structure of the 35 square patterns in R. T. Curtis’s MOG1,2. The theorem uses a 4×4 array of two-color diagonally-divided square tiles, and considers the group of permutations of these 16 tiles generated by mixing random permutations of rows, columns, and the four 2×2 quadrants1,2. The theorem states that every image of the 4×4 array has some ordinary or color-interchange symmetry1,2. The 35 structures of the 840 images of the 4×4 array are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2), a finite field with two elements1,2. This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets1,2. The MOG is a 4×6 array of combinations describing any point in 24-dimensional space1. It preserves all of the symmetries and maximal subgroups of the Mathieu group M241. The MOG is used in the construction of the Steiner system S(5,8,24), and hence is also related to the Leech lattice2. In summary, the Cullinane Diamond Theorem provides a geometric explanation for the structure and symmetries of the patterns in the MOG1,2. Both theorems are connected through their use of finite projective geometry and their applications to the study of symmetry1,2. Learn more Reply to Bing's Response — Actually, the Cullinane diamond theorem, by itself, does NOT explain the patterns in the MOG. The MOG is a natural correspondence, or pairing, of the 35 diamond-theorem structures with the 35 partitions of an eight-element set into two four-element sets. That correspondence is explained in a 1910 paper by G. M. Conwell, "The 3-Space PG(3,2) and Its Group." The Conwell correspondence was later found to be invariant under the action of the large Mathieu group on a set of 24=16+8 elements. Response to the Above Reply — Generating answers for you… Thank you for the clarification. You’re correct that the Cullinane Diamond Theorem, by itself, does not explain the patterns in the Miracle Octad Generator (MOG). The MOG is indeed a natural correspondence of the 35 structures from the Diamond Theorem with the 35 partitions of an eight-element set into two four-element sets. This correspondence was explained by G. M. Conwell in his 1910 paper “The 3-Space PG(3,2) and Its Group.” The Conwell correspondence was later found to be invariant under the action of the large Mathieu group on a set of 24 elements, which can be partitioned into a set of 16 and a set of 8. I appreciate your input and the opportunity to learn from it. If you have any more questions or need further clarification, feel free to ask!
|
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Wednesday, December 6, 2023
Das Wiener Meisterstück
|
Introduction In the present article, the research work of many years is summarized in an interim report concerning the connection between logic, space, time and matter. The author always had in mind two things, namely 1. The discovery/construction of an interface between matter and mind, and 2. some entry points for the topos view that concern graphs, grade rotations and contravariant involutions in geometric Boolean lattices. In this part of the MiTopos theme I follow the historic approach to mathematical physics and remain with the Clifford algebra of the Minkowski space. It turns out that this interface is a basic morphogenetic structure inherent in both matter and thought. It resides in both oriented spaces and logic, and most surprisingly is closely linked to the symmetries of elementary particle physics.
— "MiTopos: Space Logic I," |
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Saturday, October 21, 2023
Monday, October 16, 2023
A Harlan Kane Rite Aid Special: Chapter 11
From a search in this journal for "Chapter 11" —
Inner structure —
The above three images 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 + |
(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.)
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Wednesday, October 4, 2023
Quantum Dots: “The Thing and I” Continues.
See as well "The Thing and I."
Comments Off on Quantum Dots: “The Thing and I” Continues.
Friday, June 30, 2023
Trickster Fuge (German for Joint)
|
Margaret Atwood on Lewis Hyde's "Trickster is among other things the gatekeeper who opens the door into the next world; those who mistake him for a psychopath never even know such a door exists." (159) What is "the next world"? It might be the Underworld…. The pleasures of fabulation, the charming and playful lie– this line of thought leads Hyde to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the con-artist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation and art all come from the same ancient root, a word meaning "to join," "to fit," and "to make." (254) If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist. Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart. |
"Drop me a line" — Request attributed to Emma Stone
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Friday, October 21, 2022
Dimensional Positioning*
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Wednesday, October 5, 2022
Physicality
"Battles argues that 'the experience of the physicality
of the book is strongest in large libraries,' and stand
among the glass cube at the center of the British Library,
the stacks upon stacks in Harvard’s Widener Library, or
the domed portico of the Library of Congress and tell me
any differently."
— Ed Simon, Binding the Ghost: Theology, Mystery, and
the Transcendence of Literature. Hardcover – April 19, 2022.
… And back to cube:
Related meditation: Beer Summit.
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Tuesday, April 26, 2022
The Nocciolo
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Thursday, January 27, 2022
The Lexicographic Octad Generator
A Lexicographic Basis for the Binary Golay Code:
Brouwer and Guven — "Long ago," in
"The generating rank of the space of short vectors
in the Leech lattice mod 2," by
Andries Brouwer & Cicek Guven,
https://www.win.tue.nl/~aeb/preprints/udim24a.pdf —
"One checks by computer" that this is a basis:
000000000000000011111111
000000000000111100001111
000000000011001100110011
000000000101010101010101
000000001001011001101001
000000110000001101010110
000001010000010101100011
000010010000011000111010
000100010001000101111000
001000010001001000011101
010000010001010001001110
100000010001011100100100
(Copied from the Brouwer-Guven paper)
_________________________________________________
Adlam at Harvard —
"Constructing the Extended Binary Golay Code,"
by Ben Adlam, Harvard University, August 9, 2011,
https://fliphtml5.com/llqx/wppz/basic —
Adlam also asserts, citing a reference, that this same
set of twelve vectors is a basis:
000000000000000011111111
000000000000111100001111
000000000011001100110011
000000000101010101010101
000000001001011001101001
000000110000001101010110
000001010000010101100011
000010010000011000111010
000100010001000101111000
001000010001001000011101
010000010001010001001110
100000010001011100100100
(Copied from the Adlam paper)
__________________________________________________
Sources —
"One checks by computer" —
At http://magma.maths.usyd.edu.au/calc/ —
|
> V24 := VectorSpace(FiniteField(2), 24); > G := sub< V24 | > [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1], > [0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1], > [0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1], > [0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1], > [0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1], > [0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,1,0,1,0,1,0,1,1,0], > [0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,1,1,0,0,0,1,1], > [0,0,0,0,1,0,0,1,0,0,0,0,0,1,1,0,0,0,1,1,1,0,1,0], > [0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,1,1,1,1,0,0,0], > [0,0,1,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,0,1,1,1,0,1], > [0,1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,0,1,1,1,0], > [1,0,0,0,0,0,0,1,0,0,0,1,0,1,1,1,0,0,1,0,0,1,0,0]>; > Dimension(G); 12 |
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Friday, September 17, 2021
Adventures in Mix-and-Match Reality . . .
Continued from September 8 .
The New York Times yesterday —
“Art is another way to try to exercise your imagination
at connecting incongruous things,” Anthony Doerr said.
“It’s a way to say, hey, reader, let’s work together and
practice and train our imagination to connect things
that you don’t readily think of as connected.”
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Tuesday, August 10, 2021
Ex Fano Apollinis
|
Margaret Atwood on Lewis Hyde's "Trickster is among other things the gatekeeper who opens the door into the next world; those who mistake him for a psychopath never even know such a door exists." (159) What is "the next world"? It might be the Underworld…. The pleasures of fabulation, the charming and playful lie– this line of thought leads Hyde to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the con-artist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation and art all come from the same ancient root, a word meaning "to join," "to fit," and "to make." (254) If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist. Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart. |
"As a Chinese jar . . . ."
— Four Quartets
|
Rosalind Krauss "If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete. Or, to take a more up-to-date example…."
"He was looking at the nine engravings and at the circle,
"And it's whispered that soon if we all call the tune
The nine engravings of The Club Dumas
An example of the universal— or, according to Krauss,
"This is the garden of Apollo, |
The "Katz" of the August 7 post Art Angles
is a product of Princeton's
Department of Art and Archaeology.
ART —
.jpg)
ARCHAEOLOGY —
"This pattern is a square divided into nine equal parts.
It has been called the 'Holy Field' division and
was used throughout Chinese history for many
different purposes, most of which were connected
with things religious, political, or philosophical."
– The Magic Square: Cities in Ancient China,
by Alfred Schinz, Edition Axel Menges, 1996, p. 71
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Thursday, June 24, 2021
Persons of Interest
|
Margaret Atwood on Lewis Hyde's "Trickster is among other things the gatekeeper who opens the door into the next world; those who mistake him for a psychopath never even know such a door exists." (159) What is "the next world"? It might be the Underworld…. The pleasures of fabulation, the charming and playful lie– this line of thought leads Hyde to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the con-artist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation and art all come from the same ancient root, a word meaning "to join," "to fit," and "to make." (254) If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist. Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart. |
Punch Line . . .
Wrap Party!
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Sunday, February 21, 2021
Cube Woo
"Before time began, there was the Cube."
— Hassenfeld Brothers merchandising slogan
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Thursday, August 27, 2020
The Complete Extended Binary Golay Code
All 4096 vectors in the code are at . . .
http://neilsloane.com/oadir/oa.4096.12.2.7.txt.
Sloane’s list* contains the 12 generating vectors
listed in 2011 by Adlam —
As noted by Conway in Sphere Packings, Lattices and Groups ,
these 4096 vectors, constructed lexicographically, are exactly
the same vectors produced by using the Conway-Sloane version
of the Curtis Miracle Octad Generator (MOG). Conway says this
lexico-MOG equivalence was first discovered by M. J. T. Guy.
(Of course, any permutation of the 24 columns above produces
a version of the code qua code. But because the lexicographic and
the MOG constructions yield the same result, that result is in
some sense canonical.)
See my post of July 13, 2020 —
The lexicographic Golay code
contains, embedded within it,
the Miracle Octad Generator.
For some related results, Google the twelfth generator:
* Sloane’s list is of the codewords as the rows of an orthogonal array —
See also http://neilsloane.com/oadir/.
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Saturday, May 23, 2020
Eightfold Geometry: A Surface Code “Unit Cell”
The resemblance to the eightfold cube is, of course,
completely coincidental.
Some background from the literature —
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Friday, May 22, 2020
Surface Code News
From a paper cited in the above story:
“Fig. 4 A lattice geometry for a surface code.” —
The above figure suggests a search for “surface code” cube :
Related poetic remarks — “Illumination of a surface.”
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Monday, April 20, 2020
Geometry for Poets
From Atomicity and Quanta by James Jeans,
Cambridge University Press, 1926, pp. 55-56 —
| “So far as we can at present conjecture, the investigation of the structure which produces this atomicity appears to be the big problem in the path of the quantum-theory. To conform to the principle of relativity, the new atomicity must admit of expression in terms of the space-time continuum, although we have seen that it cannot be an atomicity of the continuum itself. It may conceivably be an atomicity of its metric properties, such as determine its curvatures. We may perhaps form a very rude picture of it by imagining the curvature of the continuum in the neighbourhood of an atom not to be of the continuous nature imagined by Weyl, but to occur in finite chunks—a straight piece, then a sudden bend, then another straight bit, and so on. A small bit of the continuum viewed through a five-dimensional microscope might look rather like a cubist picture; and, conversely, perhaps a cubist picture looks rather more like a little fragment of the continuum than like anything else.” |
This is, of course, not the “atomicity” of the previous post.
For examples of that atomicity, a concept of pure geometry
rather than of physics, see …
Faure, C. A., and Frölicher, A., “Fundamental Notions of
Lattice Theory,” in Modern Projective Geometry (2000).
(Mathematics and Its Applications, vol 521. Springer, Dordrecht.)
Related art (a “cubist picture”) —
Juan Gris, Fruit Dish and Carafe , 1914
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Saturday, June 29, 2019
That’s “Merry” … And Quite Contrary
"John Horton Conway is a cross between
Archimedes, Mick Jagger and Salvador Dalí."
— The Guardian paraphrasing Siobhan Roberts,
John Horton Conway and his Leech lattice doodle
in The Guardian . Photo: Hollandse Hoogte/Eyevine.
. . . .
"In junior school, one of Conway’s teachers had nicknamed him 'Mary'.
He was a delicate, effeminate creature. Being Mary made his life
absolute hell until he moved on to secondary school, at Liverpool’s
Holt High School for Boys. Soon after term began, the headmaster
called each boy into his office and asked what he planned to do with
his life. John said he wanted to read mathematics at Cambridge.
Instead of 'Mary' he became known as 'The Prof'. These nicknames
confirmed Conway as a terribly introverted adolescent, painfully aware
of his own suffering." — Siobhan Roberts, loc. cit.
From the previous post —
See as well this journal on the above Guardian date —
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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 —
.gif)
This post was suggested by a New York Times headline today —
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Sunday, December 2, 2018
Symmetry at Hiroshima
A search this morning for articles mentioning the Miracle Octad Generator
of R. T. Curtis within the last year yielded an abstract for two talks given
at Hiroshima on March 8 and 9, 2018 —
|
http://www.math.sci.hiroshima-u.ac.jp/ branched/files/2018/abstract/Aitchison.txt
Iain AITCHISON Title: Construction of highly symmetric Riemann surfaces, related manifolds, and some exceptional objects, I, II Abstract: Since antiquity, some mathematical objects have played a special role, underpinning new mathematics as understanding deepened. Perhaps archetypal are the Platonic polyhedra, subsequently related to Platonic idealism, and the contentious notion of existence of mathematical reality independent of human consciousness. Exceptional or unique objects are often associated with symmetry – manifest or hidden. In topology and geometry, we have natural base points for the moduli spaces of closed genus 2 and 3 surfaces (arising from the 2-fold branched cover of the sphere over the 6 vertices of the octahedron, and Klein's quartic curve, respectively), and Bring's genus 4 curve arises in Klein's description of the solution of polynomial equations of degree greater than 4, as well as in the construction of the Horrocks-Mumford bundle. Poincare's homology 3-sphere, and Kummer's surface in real dimension 4 also play special roles. In other areas: we have the exceptional Lie algebras such as E8; the sporadic finite simple groups; the division algebras: Golay's binary and ternary codes; the Steiner triple systems S(5,6,12) and S(5,8,24); the Leech lattice; the outer automorphisms of the symmetric group S6; the triality map in dimension 8; and so on. We also note such as: the 27 lines on a cubic, the 28 bitangents of a quartic curve, the 120 tritangents of a sextic curve, and so on, related to Galois' exceptional finite groups PSL2(p) (for p= 5,7,11), and various other so-called `Arnol'd Trinities'. Motivated originally by the `Eightfold Way' sculpture at MSRI in Berkeley, we discuss inter-relationships between a selection of these objects, illustrating connections arising via highly symmetric Riemann surface patterns. These are constructed starting with a labeled polygon and an involution on its label set. Necessarily, in two lectures, we will neither delve deeply into, nor describe in full, contexts within which exceptional objects arise. We will, however, give sufficient definition and detail to illustrate essential inter-connectedness of those exceptional objects considered. Our starting point will be simplistic, arising from ancient Greek ideas underlying atomism, and Plato's concepts of space. There will be some overlap with a previous talk on this material, but we will illustrate with some different examples, and from a different philosophical perspective. Some new results arising from this work will also be given, such as an alternative graphic-illustrated MOG (Miracle Octad Generator) for the Steiner system S(5,8,24), and an alternative to Singerman – Jones' genus 70 Riemann surface previously proposed as a completion of an Arnol'd Trinity. Our alternative candidate also completes a Trinity whose two other elements are Thurston's highly symmetric 6- and 8-component links, the latter related by Thurston to Klein's quartic curve. |
See also yesterday morning's post, "Character."
Update: For a followup, see the next Log24 post.
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Tuesday, June 12, 2018
Like Decorations in a Cartoon Graveyard
Two visions of happy neurons:
This post was suggested by a link in today's New York Times —
"Simon Denny, the New Zealand artist whose work incorporates
board games, intervenes by introducing his own pieces into an attic of
the late-18th-century Haus zum Kirschgarten, already filled with
'old historical dollhouses, board games, chess games' and the like …."
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Tuesday, August 16, 2016
Midnight Narrative
The images in the previous post do not lend themselves
to any straightforward narrative. Two portions of the
large image search are, however, suggestive —
Cross and Boolean lattice.
The improvised cross in the second pair of images
is perhaps being wielded to counteract the
Boole of the first pair of images. See the heading
of the webpage that is the source of the lattice
diagram toward which the cross is directed —
Update of 10 am on August 16, 2016 —
See also Atiyah on the theology of
(Boolean) algebra vs. (Galois) geometry:
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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.
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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 —
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Saturday, October 31, 2015
Raiders of the Lost Crucible
Stanford Encyclopedia of Philosophy
on the date Friday, April 5, 2013 —
“First published Tue Sep 24, 1996;
substantive revision Fri Apr 5, 2013”
This journal on the date Friday, April 5, 2013 —
The object most closely resembling a “philosophers’ stone”
that I know of is the eightfold cube .
For some related philosophical remarks that may appeal
to a general Internet audience, see (for instance) a website
by I Ching enthusiast Andreas Schöter that displays a labeled
eightfold cube in the form of a lattice diagram —
Related material by Schöter —
A 20-page PDF, “Boolean Algebra and the Yi Jing.”
(First published in The Oracle: The Journal of Yijing Studies ,
Vol 2, No 7, Summer 1998, pp. 19–34.)
I differ with Schöter’s emphasis on Boolean algebra.
The appropriate mathematics for I Ching studies is,
I maintain, not Boolean algebra but rather Galois geometry.
See last Saturday’s post Two Views of Finite Space.
Unfortunately, that post is, unlike Schöter’s work, not
suitable for a general Internet audience.
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Wednesday, October 28, 2015
Symmetry Framed
The cover of the K. O. Friedrichs book From Pythagoras to Einstein
shown in the previous post suggests a review (click the Log24
images for webpages where they can be manipulated) ….
The "more sophisticated" link in the first image above
leads to a webpage by Alexander Bogomolny,
"Pythagoras' Theorem by Tessellation," that says
"This is a subtle and beautiful proof."
Bogomolny refers us to the Friedrichs book, from which one of
the illustrations of the proof by tessellation is as follows —
For a quite different use of superposition, see
The Lindbergh Manifesto (May 19, 2015).
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Tuesday, March 24, 2015
Hirzebruch
(Continued from July 16, 2014.)
Some background from Wikipedia:
"Friedrich Ernst Peter Hirzebruch ForMemRS[2]
(17 October 1927 – 27 May 2012)
was a German mathematician, working in the fields of topology,
complex manifolds and algebraic geometry, and a leading figure
in his generation. He has been described as 'the most important
mathematician in Germany of the postwar period.'
[3][4][5][6][7][8][9][10][11]"
A search for citations of the A. E. Brouwer paper in
the previous post yields a quotation from the preface
to the third ("2013") edition of Wolfgang Ebeling's
Lattices and Codes: A Course Partially Based
on Lectures by Friedrich Hirzebruch , a book
reportedly published on September 19, 2012 —
|
"Sadly, on May 27 this year, Friedrich Hirzebruch, Hannover, July 2012 Wolfgang Ebeling "
(Prof. Dr. Wolfgang Ebeling, Institute of Algebraic Geometry, |
Also sadly …
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Thursday, January 8, 2015
Gitterkrieg
From the abstract of a talk, "Extremal Lattices," at TU Graz
on Friday, Jan. 11, 2013, by Prof. Dr. Gabriele Nebe
(RWTH Aachen) —
"I will give a construction of the extremal even
unimodular lattice Γ of dimension 72 I discovered
in summer 2010. The existence of such a lattice
was a longstanding open problem. The
construction that allows to obtain the
minimum by computer is similar to the one of the
Leech lattice from E8 and of the Golay code from
the Hamming code (Turyn 1967)."
On an earlier talk by Nebe at Oberwolfach in 2011 —
"Exciting new developments were presented by
Gabriele Nebe (Extremal lattices and codes ) who
sketched the construction of her recently found
extremal lattice in 72 dimensions…."
Nebe's Oberwolfach slides include one on
"The history of Turyn's construction" —
Nebe's list omits the year 1976. This was the year of
publication for "A New Combinatorial Approach to M24"
by R. T. Curtis, the paper that defined Curtis's
"Miracle Octad Generator."
Turyn's 1967 construction, uncredited by Curtis, may have
been the basis for Curtis's octad-generator construction.
See Turyn in this journal.
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Monday, January 5, 2015
Gitterkrieg*
Wednesday, March 13, 2013
|
"I pondered deeply, then, over the
adventures of the jungle. And after
some work with a colored pencil
I succeeded in making my first drawing.
My Drawing Number One.
It looked something like this:
I showed my masterpiece to the
grown-ups, and asked them whether
the drawing frightened them.
But they answered: 'Why should
anyone be frightened by a hat?'"
* For the title, see Plato Thanks the Academy (Jan. 3).
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Saturday, January 3, 2015
Plato Thanks the Academy
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Thursday, September 11, 2014
A Class by Itself
The American Mathematical Society yesterday:
Harvey Cohn (1923-2014)
Wednesday September 10th 2014
Cohn, an AMS Fellow and a Putnam Fellow (1942), died May 16 at the age of 90. He served in the Navy in World War II and following the war received his PhD from Harvard University in 1948 under the direction of Lars Ahlfors. He was a member of the faculty at Wayne State University, Stanford University, Washington University in St. Louis, the University of Arizona, and at City College of New York, where he was a distinguished professor. After retiring from teaching, he also worked for the NSA. Cohn was an AMS member since 1942.
Paid death notice from The New York Times , July 27, 2014:
COHN–Harvey. Fellow of the American Mathematical Society and member of the Society since 1942, died on May 16 at the age of 90. He was a brilliant Mathematician, an adoring husband, father and grandfather, and faithful friend and mentor to his colleagues and students. Born in New York City in 1923, Cohn received his B.S. degree (Mathematics and Physics) from CCNY in 1942. He received his M.S. degree from NYU (1943), and his Ph.D. from Harvard (1948) after service in the Navy (Electronic Technicians Mate, 1944-46). He was a member of Phi Beta Kappa (Sigma Chi), won the William Lowell Putnam Prize in 1942, and was awarded the Townsend Harris Medal in 1972. A pioneer in the intensive use of computers in an innovative way in a large number of classical mathematical problems, Harvey Cohn held faculty positions at Wayne State University, Stanford, Washington University Saint Louis (first Director of the Computing Center 1956-58), University of Arizona (Chairman 1958-1967), University of Copenhagen, and CCNY (Distinguished Professor of Mathematics). After his retirement from teaching, he worked in a variety of capacities for the National Security Agency and its research arm, IDA Center for Computing Sciences. He is survived by his wife of 63 years, Bernice, of Laguna Woods, California and Ft. Lauderdale, FL, his son Anthony, daughter Susan Cohn Boros, three grandchildren and one great-granddaughter.
— Published in The New York Times on July 27, 2014
See also an autobiographical essay found on the web.
None of the above sources mention the following book, which is apparently by this same Harvey Cohn. (It is dedicated to "Tony and Susan.")
Advanced Number Theory, by Harvey Cohn
Courier Dover Publications, 1980 – 276 pages
(First published by Wiley in 1962 as A Second Course in Number Theory )
Publisher's description:
" 'A very stimulating book … in a class by itself.'— American Mathematical Monthly
Advanced students, mathematicians and number theorists will welcome this stimulating treatment of advanced number theory, which approaches the complex topic of algebraic number theory from a historical standpoint, taking pains to show the reader how concepts, definitions and theories have evolved during the last two centuries. Moreover, the book abounds with numerical examples and more concrete, specific theorems than are found in most contemporary treatments of the subject.
The book is divided into three parts. Part I is concerned with background material — a synopsis of elementary number theory (including quadratic congruences and the Jacobi symbol), characters of residue class groups via the structure theorem for finite abelian groups, first notions of integral domains, modules and lattices, and such basis theorems as Kronecker's Basis Theorem for Abelian Groups.
Part II discusses ideal theory in quadratic fields, with chapters on unique factorization and units, unique factorization into ideals, norms and ideal classes (in particular, Minkowski's theorem), and class structure in quadratic fields. Applications of this material are made in Part III to class number formulas and primes in arithmetic progression, quadratic reciprocity in the rational domain and the relationship between quadratic forms and ideals, including the theory of composition, orders and genera. In a final concluding survey of more recent developments, Dr. Cohn takes up Cyclotomic Fields and Gaussian Sums, Class Fields and Global and Local Viewpoints.
In addition to numerous helpful diagrams and tables throughout the text, appendices, and an annotated bibliography, Advanced Number Theory also includes over 200 problems specially designed to stimulate the spirit of experimentation which has traditionally ruled number theory."
User Review –
"In a nutshell, the book serves as an introduction to Gauss' theory of quadratic forms and their composition laws (the cornerstone of his Disquisitiones Arithmeticae) from the modern point of view (ideals in quadratic number fields). I strongly recommend it as a gentle introduction to algebraic number theory (with exclusive emphasis on quadratic number fields and binary quadratic forms). As a bonus, the book includes material on Dirichlet L-functions as well as proofs of Dirichlet's class number formula and Dirichlet's theorem in primes in arithmetic progressions (of course this material requires the reader to have the background of a one-semester course in real analysis; on the other hand, this material is largely independent of the subsequent algebraic developments).
Better titles for this book would be 'A Second Course in Number Theory' or 'Introduction to quadratic forms and quadratic fields'. It is not a very advanced book in the sense that required background is only a one-semester course in number theory. It does not assume prior familiarity with abstract algebra. While exercises are included, they are not particularly interesting or challenging (if probably adequate to keep the reader engaged).
While the exposition is *slightly* dated, it feels fresh enough and is particularly suitable for self-study (I'd be less likely to recommend the book as a formal textbook). Students with a background in abstract algebra might find the pace a bit slow, with a bit too much time spent on algebraic preliminaries (the entire Part I—about 90 pages); however, these preliminaries are essential to paving the road towards Parts II (ideal theory in quadratic fields) and III (applications of ideal theory).
It is almost inevitable to compare this book to Borevich-Shafarevich 'Number Theory'. The latter is a fantastic book which covers a large superset of the material in Cohn's book. Borevich-Shafarevich is, however, a much more demanding read and it is out of print. For gentle self-study (and perhaps as a preparation to later read Borevich-Shafarevich), Cohn's book is a fine read."
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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 —
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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 —
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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).
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Monday, December 9, 2013
Heaven Descending
An I Ching study quoted in Waiting for Ogdoad (St. Andrew's Day, 2013)—
(Click for clearer image.)
The author of the above I Ching study calls his lattice "Arising Heaven."
The following lattice might, therefore, be called "Heaven Descending."
Click for the source, mentioned in Anatomy of a Cube (Sept. 18, 2011).
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Monday, July 29, 2013
St. Walter’s Day
Today is the dies natalis of group theorist Walter Feit.
"The Steiner systems S (5,6,12) and S (5,8,24) are remarkable combinatorial
configurations unlike any others. Their automorphism groups are the Mathieu
groups M12 and M24. These are the only 5-transitive permutation groups other
than symmetric and alternating groups: (a fact long conjectured but only
proved as a consequence of the classification). The Leech lattice is a blown up
version of S (5,8,24). It is the unique even unimodular lattice in 24 dimensions
with no vectors of weight 2. This uniqueness is an essential reason why it is a
geometric object of fundamental importance. The automorphism group Co.O
of the Leech lattice involves about half of the sporadic groups and generally it
is felt that these are well understood."
— Walter Feit, book review, Bulletin of the American Mathematical Society ,
Vol. 8 (1983), 120-124, page 123
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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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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.
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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.
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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.”
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Monday, December 24, 2012
Post-Mortem for Quincy
Raiders of the Lost Trunk, or:
Stars in the Attic
See also A Glass for Klugman :
Context: Poetry and Truth, Eternal Recreation,
Solid Symmetry, and Stevens's Rock.
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Saturday, November 3, 2012
Rigor
A New Yorker weblog post from yesterday, All Souls' Day—
"As the mathematician Terence Tao has written,
math study has three stages:
the 'pre-rigorous,' in which basic rules are learned,
the theoretical 'rigorous' stage, and, last and most intriguing,
'the post-rigorous,' in which intuition suddenly starts to play a part."
Related material—
Rigor in a Log24 post of Sunday evening, May 25, 2008: "Hall of Mirrors."
Note in that post the tesseract viewed as the lattice of
the 16 subsets of a 4-element set.
Some further material related to tesseracts and time, in three stages
(roughly corresponding to Tao's, but not in chronological order):
- Bakhtin,
- Spaces as Hypercubes, and
- Pindar.
See also a recent Log24 post on remarks from Four Quartets .
(The vertices of a tesseract form, in various natural ways, four quartets.)
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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.
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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.)
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Sunday, March 18, 2012
Square-Triangle Diamond
The diamond shape of yesterday's noon post
is not wholly without mathematical interest …
"Every triangle is an n -replica" is true
if and only if n is a square.
The 16 subdiamonds of the above figure clearly
may be mapped by an affine transformation
to 16 subsquares of a square array.
(See the diamond lattice in Weyl's Symmetry .)
Similarly for any square n , not just 16.
There is a group of 322,560 natural transformations
that permute the centers of the 16 subsquares
in a 16-part square array. The same group may be
viewed as permuting the centers of the 16 subtriangles
in a 16-part triangular array.
(Updated March 29, 2012, to correct wording and add Weyl link.)
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Friday, September 16, 2011
Icons
Background: Jung's Aion in this journal discusses this
figure from finite geometry's diamond theorem—
Fig. A
This resembles a figure that served Jung
as a schema of the Self—
Fig. B
Fig. A, with color variations, serves as the core
of many automatically generated Identicons —
a different sort of self-symbol.
Examples from Sept. 6 at MathOverflow—
A user wanting to custom-tailor his self-symbol should consider
the following from the identicon service Gravatar—
1. User Submissions. "… you hereby do and shall grant to Automattic a worldwide, perpetual, irrevocable, royalty-free and fully-paid, transferable (including rights to sublicense) right to perform the Services (e.g., to use, modify, reproduce, distribute, prepare derivative works of, display, perform, and otherwise fully exercise and exploit all intellectual property, publicity, and moral rights with respect to any User Submissions, and to allow others to do so)."
Sounds rather Faustian.
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Friday, June 10, 2011
Hierophant
Some background for yesterday’s posts:
Midrash for Gnostics and related notes,
as well as yesterday’s New York Lottery.
…. “We seek
The poem of pure reality, untouched
By trope or deviation, straight to the word,
Straight to the transfixing object, to the object
At the exactest point at which it is itself,
Transfixing by being purely what it is….”
— Wallace Stevens (1879-1955),
“An Ordinary Evening in New Haven” IX
“Reality is the beginning not the end,
Naked Alpha, not the hierophant Omega,
of dense investiture, with luminous vassals.”
— Wallace Stevens,
“An Ordinary Evening in New Haven” VI
“A hierophant is a person who brings religious congregants into the presence of that which is deemed holy . The word comes from Ancient Greece, where it was constructed from the combination of ta hiera , ‘the holy,’ and phainein , ‘to show.’ In Attica it was the title of the chief priest at the Eleusinian Mysteries. A hierophant is an interpreter of sacred mysteries and arcane principles.”
Weyl as Alpha, Chern as Omega—
Postscript for Ellen Page, star of “Smart People”
and of “X-Men: The Last Stand“— a different page 679.
Your assignment, should you choose to accept it—
Interpret today’s NY lottery numbers— Midday 815, Evening 888.
My own bias is toward 815 as 8/15 and 888 as a trinity,
but there may be less obvious and more interesting approaches.
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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.
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Monday, May 2, 2011
The Vine*
See "Nine is a Vine" and "Hereafter" in this journal.
As quoted here last October 23—
Margaret Atwood on Lewis Hyde's Trickster Makes This World: Mischief, Myth, and Art—
"Trickster is among other things the gatekeeper who opens the door into the next world; those who mistake him for a psychopath never even know such a door exists." (159)
What is "the next world"? It might be the Underworld….
The pleasures of fabulation, the charming and playful lie– this line of thought leads Hyde to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the con-artist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation and art all come from the same ancient root, a word meaning "to join," "to fit," and "to make." (254) If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist. Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart.
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Wednesday, October 27, 2010
Review
A mathematical review—
— Gian-Carlo Rota
A science fiction—
— Alfred Bester
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Saturday, October 23, 2010
Paranormal Jackass
The follow-up to last year's runaway horror hit, "Paranormal Activity 2," kicked off its first weekend in theaters with a major haul. The creepy tale… pulled in $20.1 million on Friday.
Trailing behind "Paranormal" is last week's box-office busting debut "Jackass 3D. " The prank-fest, which landed about $50 million its first weekend in theaters, slipped to the second-place slot….
…
The Clint Eastwood-helmed ensemble drama "Hereafter" landed in fourth place. Exploring the lives of three people who are dealing with death and the afterlife in several ways, including the story of a psychic played by Matt Damon, the screen legend's latest turn in the director's chair made approximately $4.1 million on Friday.
Related material—
Margaret Atwood on Lewis Hyde's Trickster Makes This World: Mischief, Myth, and Art—
"Trickster is among other things the gatekeeper who opens the door into the next world; those who mistake him for a psychopath never even know such a door exists." (159)
What is "the next world"? It might be the Underworld….
The pleasures of fabulation, the charming and playful lie– this line of thought leads Hyde to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the con-artist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation and art all come from the same ancient root, a word meaning "to join," "to fit," and "to make." (254) If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist. Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart.
George P. Hansen on Martin Gardner and the paranormal.
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Saturday, October 16, 2010
The Mandelbrot Numbers
Benoît Mandelbrot died on Oct. 14.
— New York Lottery on Thursday, Oct. 14, 2010
Related material on 109: See 1/09, 2009.
Related material on 060: See Hexagram 60 of the I Ching and…
Margaret Atwood on Lewis Hyde's Trickster Makes This World: Mischief, Myth, and Art—
"Trickster is among other things the gatekeeper who opens the door into the next world; those who mistake him for a psychopath never even know such a door exists." (159)
What is "the next world"? It might be the Underworld….
The pleasures of fabulation, the charming and playful lie– this line of thought leads Hyde to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the con-artist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation and art all come from the same ancient root, a word meaning "to join," "to fit," and "to make." (254) If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist. Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart.
Comments Off on The Mandelbrot Numbers
Monday, September 27, 2010
The Social Network…
… In the Age of Citation
1. INTRODUCTION TO THE PROBLEM
Social network analysis is focused on the patterning of the social
relationships that link social actors. Typically, network data take the
form of a square-actor by actor-binary adjacency matrix, where
each row and each column in the matrix represents a social actor. A
cell entry is 1 if and only if a pair of actors is linked by some social
relationship of interest (Freeman 1989).
— "Using Galois Lattices to Represent Network Data,"
by Linton C. Freeman and Douglas R. White,
Sociological Methodology, Vol. 23, pp. 127–146 (1993)
From this paper's CiteSeer page—
Citations
| 766 | Social Network Analysis: Methods and Applications – WASSERMAN, FAUST – 1994 |
| 100 | The act of creation – Koestler – 1964 |
| 75 | Visual Thinking – Arnheim – 1969 |
Visual Image of the Problem—
From a Google search today:
Related material—
"It is better to light one candle…"
"… the early favorite for best picture at the Oscars" — Roger Moore
Comments Off on The Social Network…
Thursday, June 3, 2010
Trickster
Margaret Atwood (pdf) on Lewis Hyde’s
Trickster Makes This World: Mischief, Myth, and Art –
“Trickster,” says Hyde, “feels no anxiety when he deceives…. He… can tell his lies with creative abandon, charm, playfulness, and by that affirm the pleasures of fabulation.” (71) As Hyde says, “… almost everything that can be said about psychopaths can also be said about tricksters,” (158), although the reverse is not the case. “Trickster is among other things the gatekeeper who opens the door into the next world; those who mistake him for a psychopath never even know such a door exists.” (159)
What is “the next world”? It might be the Underworld….
The pleasures of fabulation, the charming and playful lie– this line of thought leads Hyde to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the con-artist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation and art all come from the same ancient root, a word meaning to join, to fit, and to make. (254) If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist. Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart.
See also George P. Hansen on Martin Gardner, Trickster.
Comments Off on Trickster
Tuesday, July 14, 2009
Tuesday July 14, 2009
For Galois on Bastille Day
Elements
of Finite Geometry
Some fans of the alchemy in
Katherine Neville’s novel
The Eight and in Dan Brown’s
novel Angels & Demons may
enjoy the following analogy–
Elements
of Finite Geometry
Some fans of the alchemy in
Katherine Neville’s novel
The Eight and in Dan Brown’s
novel Angels & Demons may
enjoy the following analogy–
Note that the alchemical structure
at left, suited more to narrative
than to mathematics, nevertheless
is mirrored within the pure
mathematics at right.
Related material
on Galois and geometry:
| Geometries of the group PSL(2, 11) by Francis Buekenhout, Philippe Cara, and Koen Vanmeerbeek. Geom. Dedicata, 83 (1-3): 169–206, 2000–
|
Comments Off on Tuesday July 14, 2009
Sunday, November 16, 2008
Sunday November 16, 2008
Art and Lies
Observations suggested by an article on author Lewis Hyde– "What is Art For?"– in today's New York Times Magazine:
Margaret Atwood (pdf) on Lewis Hyde's
Trickster Makes This World: Mischief, Myth, and Art —
"Trickster," says Hyde, "feels no anxiety when he deceives…. He… can tell his lies with creative abandon, charm, playfulness, and by that affirm the pleasures of fabulation." (71) As Hyde says, "… almost everything that can be said about psychopaths can also be said about tricksters," (158), although the reverse is not the case. "Trickster is among other things the gatekeeper who opens the door into the next world; those who mistake him for a psychopath never even know such a door exists." (159)
What is "the next world"? It might be the Underworld….
The pleasures of fabulation, the charming and playful lie– this line of thought leads Hyde to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the con-artist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation and art all come from the same ancient root, a word meaning to join, to fit, and to make. (254) If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist. Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart.
Related material:
The Trickster
and the Paranormal
and
Martin Gardner on
a disappearing cube —
|
"What happened to that… cube?"
Apollinax laughed until his eyes teared. "I'll give you a hint, my dear. Perhaps it slid off into a higher dimension." "Are you pulling my leg?" "I wish I were," he sighed. "The fourth dimension, as you know, is an extension along a fourth coordinate perpendicular to the three coordinates of three-dimensional space. Now consider a cube. It has four main diagonals, each running from one corner through the cube's center to the opposite corner. Because of the cube's symmetry, each diagonal is clearly at right angles to the other three. So why shouldn't a cube, if it feels like it, slide along a fourth coordinate?" — "Mr. Apollinax Visits New York," by Martin Gardner, Scientific American, May 1961, reprinted in The Night is Large |
For such a cube, see
ashevillecreative.com
this illustration in
The Religion of Cubism
(and the four entries
preceding it —
Log24, May 9, 2003).
Beware of Gardner's
"clearly" and other lies.
Comments Off on Sunday November 16, 2008
Wednesday, March 21, 2007
Wednesday March 21, 2007
Finite Relativity
continued
continued
This afternoon I added a paragraph to The Geometry of Logic that makes it, in a way, a sequel to the webpage Finite Relativity:
"As noted previously, in Figure 2 viewed as a lattice the 16 digital labels 0000, 0001, etc., may be interpreted as naming the 16 subsets of a 4-set; in this case the partial ordering in the lattice is the structure preserved by the lattice's group of 24 automorphisms– the same automorphism group as that of the 16 Boolean connectives. If, however, these 16 digital labels are interpreted as naming the 16 functions from a 4-set to a 2-set (of two truth values, of two colors, of two finite-field elements, and so forth), it is not obvious that the notion of partial order is relevant. For such a set of 16 functions, the relevant group of automorphisms may be the affine group of A mentioned above. One might argue that each Venn diagram in Fig. 3 constitutes such a function– specifically, a mapping of four nonoverlapping regions within a rectangle to a set of two colors– and that the diagrams, considered simply as a set of two-color mappings, have an automorphism group of order larger than 24… in fact, of order 322,560. Whether such a group can be regarded as forming part of a 'geometry of logic' is open to debate."
The epigraph to "Finite Relativity" is:
"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 added paragraph seems to fit this description.
Comments Off on Wednesday March 21, 2007
Saturday, January 6, 2007
Saturday January 6, 2007
An Epiphany
for the Birthday
of E. L. Doctorow,
Author of
City of God
(Doctorow wrote about
New York. A city more
closely associated with
God is Jerusalem.)
On the morning of January 2 this year, inspired by Sambin’s “basic picture,” I considered an entry dealing with Galois lattices (pdf). This train of thought was halted by news of the death earlier that morning of Teddy Kollek, 95, a founder of the Israeli intelligence service and six-term mayor of Jerusalem. (This led later to the entry “Damnation Morning“– a reference to the Fritz Leiber short story.)
This morning’s entry reboards the Galois train of thought.
Here are some relevant links:
Galois Connections (a French weblog entry providing an brief overview of Galois theory and an introduction to the use of Galois lattices in “formal concept analysis“)
Ontology (an introduction to formal concept analysis linked to on 3/31/06)
One motive for resuming consideration of Galois lattices today is to honor the late A. Richard Newton, a pioneer in engineering design who died at 55– also on Tuesday, Jan. 2, the date of Kollek’s death. Today’s New York Times obituary for Newton says that “most recently, Professor Newton championed the study of synthetic biology.”
A check of syntheticbiology.org leads to a web page on– again– ontology.
For the relationship between ontology (in the semantic-web sense) and Galois lattices, see (for instance)
“Knowledge Organisation and Information Retrieval Using Galois Lattices” (ps) and its references.
An epiphany within all this that Doctorow might appreciate is the following from Wikipedia, found by following a link to “upper ontology” in the syntheticbiology.org ontology page:
- There is no self-evident way of dividing the world up into concepts.
- There is no neutral ground that can serve as a means of translating between specialized (lower) ontologies.
- Human language itself is already an arbitrary approximation of just one among many possible conceptual maps. To draw any necessary correlation between English words and any number of intellectual concepts we might like to represent in our ontologies is just asking for trouble.
Related material:
The intellectual concepts
mentioned by Richard Powers
at the end of tomorrow’s
New York Times Book Review.
(See the links on these concepts
in yesterday’s “Goldberg Variation.”)
See also Old School Tie.
Comments Off on Saturday January 6, 2007
Thursday, October 19, 2006
Thursday October 19, 2006
For Sir Thomas Browne
(Born Oct. 19, 1605,
died Oct. 19, 1682)
Browne is noted for
Hydriotaphia (Urne-Buriall)
and The Garden of Cyrus.
Related material:
Comments Off on Thursday October 19, 2006
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.
Comments Off on Tuesday October 3, 2006
Thursday, April 27, 2006
Thursday April 27, 2006
The Blue Buildings
in the Summer Air
by Wallace Stevens
(Collected Poems, pp. 216-217)
I.
Cotton Mather died when I was a boy. The books
He read, all day, all night and all the nights,
Had got him nowhere. There was always the doubt,
That made him preach the louder, long for a church
In which his voice would roll its cadences,
After the sermon, to quiet that mouse in the wall.
II.
Over wooden Boston, the sparkling Byzantine
Was everything that Cotton Mather was
And more. Yet the eminent thunder from the mouse,
The grinding in the arches of the church,
The plaster dropping, even dripping, down,
The mouse, the moss, the woman on the shore…
III.
If the mouse should swallow the steeple, in its
time…
It was a theologian’s needle, much
Too sharp for that. The shore, the sea, the sun,
Their brilliance through the lattices, crippled
The chandeliers, their morning glazes spread
In opal blobs along the walls and floor.
IV.
Look down now, Cotton Mather, from the blank.
Was heaven where you thought? It must be there.
It must be where you think it is, in the light
On bed-clothes, in an apple on a plate.
It is the honey-comb of the seeing man.
It is the leaf the bird brings back to the boat.
V.
Go, mouse, go nibble at Lenin in his tomb.
Are you not le plus pur, you ancient one?
Cut summer down to find the honey-comb.
You are one… Go hunt for honey in his hair.
You are one of the not-numberable mice
Searching all day, all night, for the honey-comb.
Comments Off on Thursday April 27, 2006
Tuesday, March 14, 2006
Tuesday March 14, 2006
Fearful Symmetry
and Minkowski Space-Time
and Minkowski Space-Time
(For the tigers of Princeton,
a selection suggested by
the work of Richard Parker
on Lorentzian lattices)
Comments Off on Tuesday March 14, 2006
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.
Comments Off on Wednesday November 30, 2005
Sunday, November 21, 2004
Sunday November 21, 2004
Pictures at 11
From today's Maureen Dowd column:
"Trapped in their blue bell jar,
drowning in unfulfilled dreams,
Democrats are the
'Desperate Housewives' of politics."
 Sam Dean for The New York Times
Law and Religion The Rev. Jerry Falwell's Liberty University is part of a movement around the nation that brings a religious perspective to the law. |
Atticus Finch in To Kill a Mockingbird: "She's… broken a rigid and time-honored code of our society, a code so severe that whoever breaks it is hounded from our midst as unfit to live with." |
2004 Country Music Awards
"Every performance was a gem,
including 'Mockingbird'…
sung by Toby Keith with his
17-year-old daughter Krystal."
Monday, August 30, 2004
Monday August 30, 2004
Q.E.D.
A Log24 entry of Aug. 17, 2004, on the
three Semitic (or “Abrahamic”) religions:
|
From Scotsman.com News Ex-Priest Sentenced By Pat Hurst, PA News, in Athens An ex-priest who lives in Britain was given a 12-month suspended sentence today for disrupting the men’s Olympic marathon in Athens.
Cornelius Horan, 57, a former Catholic priest living in London, appeared before a Greek judge this morning, local police said. He was sentenced and released from custody but his whereabouts are unknown. Irishman Horan, originally from Kerry, dashed from the sidelines to attack the marathon front-runner during yesterday’s event. He told officers he staged the disruption to “prepare for the second coming”. A police spokesman said: “He has got mental problems. He is not very well. “His only explanation for his behaviour was that it was for the second coming.” Horan also disrupted last year’s Silverstone Formula One Grand Prix by dashing across the track. Leslie Broad, of Deunant Books, which publishes Mr Horan’s books on its website, said: “We publish two of his books on biblical prophecies and he seems to be fairly convinced that the second coming is due fairly shortly. “After the incident at Silverstone, he did say he would never do anything like that again. “He comes across as a shy, very intelligent and compassionate man but as is often the way with people who are very intelligent, it sometimes manifests itself in very strange ways. “I think he found prison a fairly uplifting experience. He came out feeling that he had met a lot of people he wouldn’t normally have met, people who had committed serious crimes.” Horan’s victim yesterday, Vanderlei De Lima, from Brazil, was at the head of the race just three miles from the finish. Horan grabbed him and bundled him into spectators at the side of the road. After a scuffle, the runner managed to get away, but he was clearly ruffled and finished third. The Brazilian Olympic Committee put in an official complaint to the Greeks and at one point the final medal ceremony to be staged during the closing ceremony was in doubt. Horan was arrested and taken to the General Police Division of Attica, where he stayed overnight. |
Author biography
from Deunant Books:
Father Cornelius (“Neil”) Horan
Horan
“Neil Horan was born in 1947, in Scartaglen, County Kerry, in the Republic of Ireland. After schooling in Ireland he was ordained a Catholic Priest in Saint Mary’s Cathedral, Killarney, in 1973.
He has served all his priestly life in the Southwark Diocese, covering London south of the River Thames and Kent, his first Parish being Bexley in Kent. His interest in Bible prophecy began when he attended a lecture in 1974, given by the Apostolic Fellowship of Christ, a group which had originated with the Christadelphians. Meaning ‘Brothers in Christ’, the Christadelphians were a small Church formed in 1861 by Dr John Thomas. Father Horan says he owes a debt of gratitude to the Christadelphian tradition for the understanding of the Bible which they gave him. He regards the Bible as the greatest Book in the world and has devoted his life to making it better known, especially the Prophecies.
He is not a prophet, considering himself to be merely an interpreter, has never received a Divine message or vision, and God has never spoken to him. He feels that he is right only in so far as he interprets the Book of Books correctly.
He is still a Catholic Priest, listed in the Catholic Directory under his full name of Cornelius Horan. Cornelius, a Centurian [sic] in the Roman army, was the first Christian convert; Father Horan is proud to bear that name and hopes to meet his famous namesake soon, when Jesus comes.”
A Glorious New World
by Father Neil Horan
“Are there passages in the Bible that foretell events that were, at the time it was written, far in the future? Father Neil Horan argues eloquently, knowledgeably and persuasively in this book, first published in 1985, that this is so. It is easy to scoff at predictions of events that were, according to the book, to have taken place a few years ago but which have not happened, but to do that would be wrong. With only the most subtle changes of emphasis in interpretation, it could just as easily be argued that events in the Middle East particularly have to a large degree fulfilled the prophecies for the years since 1985.
Then there are the events yet to come. They are, according to the author and his sources, to be the most significant in the history of mankind, and are going to happen soon. With a little thought, certain current-day world figures are a disconcertingly comfortable match for some of the characters who will act out the earth-shattering dramas to come. Even the most hardened cynic will get that prickly feeling down the back of his neck as he reads this book.
Taken together with Father Horan’s later work ‘Christ Will Soon Take Power From All Governments’ (also available from Deunant Books) the two books represent one of the most remarkable and significant bodies of work seen in this field for many, many years.”
|
Ludwig Wittgenstein,
Grammar and Geometry:
For more on Wittgenstein, theology, and grammar, see the Log24 Related material: |
Comments Off on Monday August 30, 2004
Tuesday, August 3, 2004
Tuesday August 3, 2004
Science and Fiction:
Attica to GATTACA
“There is no gene for fate.”
— Vincent, a character in
the 1997 film GATTACA
The film GATTACA was discussed in a Log24 entry for Saturday, July 31, 2004– the date of death of Frank Smith, also known as Big Black, a prominent figure in the events at Attica in 1971. He died in Kinston, North Carolina, a town of about 24,000 about halfway between Raleigh and the Atlantic Ocean.
See today’s 6:01 AM entry for some details of Mr. Smith’s life. In his memory, here are three links.
The first is to
Screening DNA:
Exploring the
Cinema-Genetics Interface,
by Stephen Nottingham
This online book, from which the above GATTACA quote was taken, discusses genetics in film more generally… Specifically, from Part 7 of Screening DNA:
In Star Wars Episode I: The Phantom Menace–
“Midi-chlorians are essentially genes for the force, which determine whether one will become either a Jedi or else a dark shadow of one. In particular, they evoke mitochondrian genes, as mitochondria once lived symbiotically in human cells. Mitochondria are a cell’s energy-producing ‘power plant,’ in which a positive mutation could lead to an individual having greater strength and stamina. Mitochondrial genes are also now known to control many critical stages in human development.“
The second link in memory of Mr. Smith, one he would probably prefer, is to another book, less academic in nature, that also deals with mitochondria:
|
A Wind in the Door, From Chapter 3, The stranger was dark, dark as night and tall as a tree, and there was something in the repose of his body, the quiet of his voice, which drove away fear. Charles Wallace stepped towards him. “Who are you?” “A Teacher.” Charles Wallace’s sigh was longing. “I wish you were my teacher.” “I am.” The cello-like voice was calm, slightly amused. |
The third link is to the aforementioned
Wind.
Comments Off on Tuesday August 3, 2004
Tuesday August 3, 2004
Death of Big Black
(Sequel to yesterday’s entry and to
the entries of Saturday, July 31,
feast day of St. Ignatius Loyola)
|
Current online information from The Free Press of Kinston, North Carolina: Frank Smith Frank Smith, 70, of 2609 Brookhaven Drive, died Saturday, July 31, 2004, at Lenoir Memorial Hospital. Arrangements are incomplete at Swinson Funeral Home. New York Times today: |
In the Heat |
|
Frank Smith, who as an inmate leader at Attica prison was tortured by officers in the aftermath of the prisoner uprising of 1971 and then spent a quarter century successfully fighting for legal damages, died Saturday in Kinston, N.C. He was 71. Mr. Smith, a huge man with a booming voice who was known as Big Black, figured large in the uprising at the Attica Correctional Facility, 30 miles east of Buffalo, during the second week of September 1971. He was chosen by other inmates to be chief of security with a principal responsibility to protect outsiders brought in to negotiate an end to the crisis. None were hurt. |
|
Saturday, October 18, 2003
Saturday October 18, 2003
For St. Gwen Verdon:
Enter Dancing
From Daily Quotational Lattice:
The story of the day is “Dance in America,” about a dancer who has dinner with some friends. Take note if you’re a dancer: Ariel, a bona fide dancer, deems the quotes about dancing to be “very powerful.”
“I tell them dance begins when a moment of hurt combines with a moment of boredom. I tell them it’s the body’s reaching, bringing air to itself. I tell them that it’s the heart’s triumph, the victory speech of the feet, the refinement of animal lunge and flight, the purest metaphor of tribe and self. It’s life flipping death the bird. I make this stuff up.”
“I am thinking of the dancing body’s magnificent and ostentatious scorn. This is how we offer ourselves, enter heaven, enter speaking: we say with motion, in space, This is what life’s done so far down here; this is all and what and everything it’s managed–this body, these bodies, that body– so what do you think, Heaven? What do you fucking think?”
Sunday, June 15, 2003
Sunday June 15, 2003
Readings for Trinity Sunday
-
Triune knot:
Problems in Combinatorial Group Theory, 7 and 8, in light of the remark in Section 8.3 of Lattice Polygons and the Number 12 -
Cardinal Newman:
Sermon 24 -
Simon Nickerson:
24=8×3.
For more on the structure
discussed by Nickerson, see
For theology in general, see
Comments Off on Sunday June 15, 2003
Monday, May 26, 2003
Monday May 26, 2003
Mental Health Month, Day 26:
Many Dimensions,
Part II — The Blue Matrix
But seriously…
John Baez in July 1999:
"…it's really the fact that the Leech lattice is 24-dimensional that lets us compactify 26-dimensional spacetime in such a way as to get a bosonic string theory with the Monster group as symmetries."
Well, maybe. I certainly hope so. If the Leech lattice and the Monster group turn out to have some significance in theoretical physics, then my own work, which deals with symmetries of substructures of the Leech lattice and the Monster, might be viewed in a different light. Meanwhile, I take (cold) comfort from some writers who pursue the "story" theory of truth, as opposed to the "diamond" theory. See the following from my journal:
Evariste Galois and the Rock that Changed Things, and
A Time to Gather Stones Together: Readings for Yom Kippur.
See, too, this web page on Marion Zimmer Bradley's fictional
Matrices, or Blue Star-Stones, and
the purely mathematical site Diamond Theory, which deals with properties of the above "blue matrix" and its larger relatives.
Comments Off on Monday May 26, 2003
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 |
Essay question:
If Jesus was God’s gift to man, and (as many men would agree) so was the young Sissy Spacek (also born on Christmas Day), was young Sissy’s existence in her 23rd year contingent or necessary? If the latter, should she be recognized as a Person of the Trinity? Quaternity? N-ity?
Talk amongst yourselves.
Comments Off on Friday May 23, 2003
Wednesday, April 2, 2003
Wednesday April 2, 2003
Symmetries…. May 15, 1998
The following journal note, from the day after Sinatra died, was written before I heard of his death. Note particularly the quote from Rilke. Other material was suggested, in part, by Alasdair Gray's Glasgow novel 1982 Janine. The "Sein Feld" heading is a reference to the Seinfeld final episode, which aired May 14, 1998. The first column contains a reference to angels — apparently Hell's Angels — and the second column provides a somewhat more serious look at this theological topic.
|
Sein Feld
1984 Janine
"But Angels love their own
"Logos means above all relation."
"Gesang ist Dasein…. |
Geometry and Theology
PA lottery May 14, 1998: |
|
|
Tape purchased 12/23/97:
|
"In the middle of 1982 Janine there are pages in which Jock McLeish is fighting with drugs and alcohol, attempting to either die or come through and get free of his fantasies. In his delirium, he hears the voice of God, which enters in small print, pushing against the larger type of his ravings. Something God says is repeated on the first and last pages of Unlikely Stories, Mostly, complete with illustration and the words 'Scotland 1984' beside it. God's statement is 'Work as if you were in the early days of a better nation.' It is the inherent optimism in that statement that perhaps best captures the strength of Aladair Gray's fiction, its straightforwardness and exuberance." |
The group of all projectivities and correlations of 
For another look at angels, see "Winging It," by Christopher R. Miller, The New York Times Book Review Bookend page for Sunday, May 24, 1998. May 24 is the feast day of Sara (also known by the Hindu name Kali), patron saint of Gypsies.
For another, later (July 16, 1998) reply to Dyson, from a source better known than myself, see Why Religion Matters, by Huston Smith, Harper Collins, 2001, page 66.
Comments Off on Wednesday April 2, 2003
Saturday, July 20, 2002
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Initial Xanga entry. Updated Nov. 18, 2006.