Tom Hanks as Indiana Langdon in Raiders of the Lost Articulation :
An unarticulated (but colored) cube:
A 2x2x2 articulated cube:
A 4x4x4 articulated cube built from subcubes like
the one viewed by Tom Hanks above:
Wednesday, September 17, 2014
Raiders of the Lost Articulation
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Monday, September 15, 2014
A Seventh Seal
This post was suggested by the two previous posts, Sermon and Structure.
Vide below the final paragraph— in Chapter 7— of Cameron’s Parallelisms ,
as well as Baudelaire in the post Correspondences :
Comme de longs échos qui de loin se confondent
Dans une ténébreuse et profonde unité….
— Baudelaire, “Correspondances “
A related image search (click to enlarge):
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Sunday, September 14, 2014
Sensibility
Structured gray matter:
Graphic symmetries of Galois space:
The reason for these graphic symmetries in affine Galois space —
symmetries of the underlying projective Galois space:
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Saturday, September 13, 2014
Sense
“A simple grid structure makes both evolutionary and developmental sense.”
— Van Wedeen, MD, of the Martinos Center for Biomedical Imaging at
Massachusetts General Hospital, Science Daily , March 29, 2012
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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, August 22, 2014
Review
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Friday, August 15, 2014
The Omega Matrix
The webpage Rosenhain and Göpel Tetrads in PG(3,2)
has been updated to include more material on symplectic structure.
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Wednesday, August 13, 2014
Stranger than Dreams*
Illustration from a discussion of a symplectic structure
in a 4×4 array quoted here on January 17, 2014 —
See symplectic structure in this journal.
* The final words of Point Omega , a 2010 novel by Don DeLillo.
See also Omega Matrix in this journal.
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Monday, August 11, 2014
Syntactic/Symplectic
(Continued from August 9, 2014.)
Syntactic:
Symplectic:
"Visual forms— lines, colors, proportions, etc.— are just as capable of
articulation , i.e. of complex combination, as words. But the laws that govern
this sort of articulation are altogether different from the laws of syntax that
govern language. The most radical difference is that visual forms are not
discursive . They do not present their constituents successively, but
simultaneously, so the relations determining a visual structure are grasped
in one act of vision."
– Susanne K. Langer, Philosophy in a New Key
For examples, see The Diamond-Theorem Correlation
in Rosenhain and Göpel Tetrads in PG(3,2).
This is a symplectic correlation,* constructed using the following
visual structure:
.
* Defined in (for instance) Paul B. Yale, Geometry and Symmetry ,
Holden-Day, 1968, sections 6.9 and 6.10.
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Sunday, August 10, 2014
Sermon
From The Mathematics of Language:
10th and 11th Biennial Conference….
Berlin, Springer, 2010 —
“Creation Myths of Generative Grammar
and the Mathematics of Syntactic Structures”
by Geoffrey K. Pullum, University of Edinburgh
Abstract
“Syntactic Structures (Chomsky [6]) is widely believed to have laid
the foundations of a cognitive revolution in linguistic science, and
to have presented (i) the first use in linguistics of powerful new ideas
regarding grammars as generative systems, (ii) a proof that English
was not a regular language, (iii) decisive syntactic arguments against
context-free phrase structure grammar description, and (iv) a
demonstration of how transformational rules could provide a formal
solution to those problems. None of these things are true. This paper
offers a retrospective analysis and evaluation.”
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Saturday, August 9, 2014
Syntactic/Symplectic
Syntactic Structure —
See the Lightfoot of today’s previous post:
Symplectic Structure —
See the plaited, or woven, structure of August 6:
.
See also Deep Structure (Dec. 9, 2012).
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Thursday, August 7, 2014
Abuse of Language
From Wikipedia — Abuse of language —
“… in mathematics, a use of terminology in a way that is not formally correct
but that simplifies exposition or suggests the correct intuition.”
The phrase “symplectic structure” in the previous post
was a deliberate abuse of language. The real definition:
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Friday, August 1, 2014
The Diamond-Theorem Correlation
Click image for a larger, clearer version.
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Thursday, July 31, 2014
Zero System
The title phrase (not to be confused with the film 'The Zero Theorem')
means, according to the Encyclopedia of Mathematics,
a null system , and
"A null system is also called null polarity,
a symplectic polarity or a symplectic correlation….
it is a polarity such that every point lies in its own
polar hyperplane."
See Reinhold Baer, "Null Systems in Projective Space,"
Bulletin of the American Mathematical Society, Vol. 51
(1945), pp. 903-906.
An example in PG(3,2), the projective 3-space over the
two-element Galois field GF(2):
See also the 10 AM ET post of Sunday, June 8, 2014, on this topic.
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Saturday, July 26, 2014
OOPs
Or: Two Rivets Short of a Paradigm
Detail from an author photo:
|
From rivet-rivet.net:
The philosopher Graham Harman is invested in re-thinking the autonomy of objects and is part of a movement called Object-Oriented-Philosophy (OOP). Harman wants to question the authority of the human being at the center of philosophy to allow the insertion of the inanimate into the equation. With the aim of proposing a philosophy of objects themselves, Harman puts the philosophies of Bruno Latour and Martin Heidegger in dialogue. Along these lines, Harman proposes an unconventional reading of the tool-being analysis made by Heidegger. For Harman, the term tool does not refer only to human-invented tools such as hammers or screwdrivers, but to any kind of being or thing such as a stone, dog or even a human. Further, he uses the terms objects, beings, tools and things, interchangeably, placing all on the same ontological footing. In short, there is no “outside world.” Harman distinguishes two characteristics of the tool-being: invisibility and totality. Invisibility means that an object is not simply used but is: “[an object] form(s) a cosmic infrastructure of artificial and natural and perhaps supernatural forces, power by which our last action is besieged.” For instance, nails, wooden boards and plumbing tubes do their work to keep a house “running” silently (invisibly) without being viewed or noticed. Totality means that objects do not operate alone but always in relation to other objects–the smallest nail can, for example, not be disconnected from wooden boards, the plumbing tubes or from the cement. Depending on the point of view of each entity (nail, tube, etc.) a different reality will emerge within the house. For Harman, “to refer to an object as a tool-being is not to say that it is brutally exploited as a means to an end, but only that it is torn apart by the universal duel between the silent execution of an object’s reality and the glistening aura of its tangible surface.” — From "The Action of Things," an M.A. thesis at the Center for Curatorial Studies, Bard College, by Manuela Moscoso, May 2011, edited by Sarah Demeuse |
From Wikipedia, a programming paradigm:
See also posts tagged Turing's Cathedral, and Alley Oop (Feb. 11, 2003).
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Sunday, July 20, 2014
Sunday School
Paradigms of Geometry:
Continuous and Discrete
The discovery of the incommensurability of a square’s
side with its diagonal contrasted a well-known discrete
length (the side) with a new continuous length (the diagonal).
The figures below illustrate a shift in the other direction.
The essential structure of the continuous configuration at
left is embodied in the discrete unit cells of the square at right.
See Desargues via Galois (August 6, 2013).
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Saturday, July 12, 2014
Sequel
A sequel to the 1974 film
Thunderbolt and Lightfoot :
Contingent and Fluky
Some variations on a thunderbolt theme:
These variations also exemplify the larger
Verbum theme:
A search today for Verbum in this journal yielded
a Georgetown University Chomskyite, Professor
David W. Lightfoot.
"Dr. Lightfoot writes mainly on syntactic theory,
language acquisition and historical change, which
he views as intimately related. He argues that
internal language change is contingent and fluky,
takes place in a sequence of bursts, and is best
viewed as the cumulative effect of changes in
individual grammars, where a grammar is a
'language organ' represented in a person's
mind/brain and embodying his/her language
faculty."
Some syntactic work by another contingent and fluky author
is related to the visual patterns illustrated above.
See Tecumseh Fitch in this journal.
For other material related to the large Verbum cube,
see posts for the 18th birthday of Harry Potter.
That birthday was also the upload date for the following:
See esp. the comments section.
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Friday, July 4, 2014
The Hallowed Crucible
“A physicist who played a central role in developing
the theory of supersymmetry – often known as SUSY –
has died.”
— Times Higher Education , July 3, 2014
In honor of the above physicist, Bruno Zumino,
here are two sets of Log24 posts:
Structure, May 2-4, 2013 (the dates of a physicists’ celebration
for Zumino’s 90th birthday)
Hallmark, June 21, 2014 (the date of Zumino’s death)
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Thursday, July 3, 2014
Core Mathematics: Arrays
Mathematics vulgarizer Keith Devlin on July 1
posted an essay on Common Core math education.
His essay was based on a New York Times story from June 29,
“Math Under Common Core Has Even Parents Stumbling.”
An image from that story:
The Times gave no source, other than the photographer’s name,
for the above image. Devlin said,
“… the image of a Common Core math worksheet
the Times chose to illustrate its story showed
a very sensible, and deep use of dot diagrams,
to understand structure in arithmetic.”
Devlin seems ignorant of the fact that there is
no such thing as a “Common Core math worksheet.”
The Core is a set of standards without worksheets
(one of its failings).
Neither the Times nor whoever filled out the worksheet
nor Devlin seemed to grasp that the image the Times used
shows some multiplication word problems that are more
advanced than the topic that Devlin called the
“deep use of dot diagrams to understand structure in arithmetic.”
This Core topic is as follows:
For some worksheets that are (purportedly) relevant, see,
for instance…
http://search.theeducationcenter.com/search/
_Common_Core_Label-2.OA.C.4–keywords-math_worksheets,
in particular the worksheet
http://www.theeducationcenter.com/editorial_content/multipli-city:
Some other exercises said to be related to standard 2.OA.C.4:
| http://www.ixl.com/standards/ common-core/math/grade-2
|
The Common Core of course fails to provide materials for parents
that are easily findable on the Web and that give relevant background
for the above second-grade topic. It leaves this crucial task up to
individual states and school districts, as well as to private enterprise.
This, and not the parents’ ignorance described in Devlin’s snide remarks,
accounts for the frustration that the Times story describes.
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Sunday, June 8, 2014
Vide
"The relevance of a geometric theorem is determined by what the theorem
tells us about space, and not by the eventual difficulty of the proof."
— Gian-Carlo Rota discussing the theorem of Desargues
What space tells us about the theorem :
In the simplest case of a projective space (as opposed to a plane ),
there are 15 points and 35 lines: 15 Göpel lines and 20 Rosenhain lines.*
The theorem of Desargues in this simplest case is essentially a symmetry
within the set of 20 Rosenhain lines. The symmetry, a reflection
about the main diagonal in the square model of this space, interchanges
10 horizontally oriented (row-based) lines with 10 corresponding
vertically oriented (column-based) lines.
Vide Classical Geometry in Light of Galois Geometry.
* Update of June 9: For a more traditional nomenclature, see (for instance)
R. Shaw, 1995. The "simplest case" link above was added to point out that
the two types of lines named are derived from a natural symplectic polarity
in the space. The square model of the space, apparently first described in
notes written in October and December, 1978, makes this polarity clearly visible:
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Wednesday, May 21, 2014
The Tetrahedral Model of PG(3,2)
The page of Whitehead linked to this morning
suggests a review of Polster's tetrahedral model
of the finite projective 3-space PG(3,2) over the
two-element Galois field GF(2).
|
The above passage from Whitehead's 1906 book suggests
that the tetrahedral model may be older than Polster thinks.
Shown at right below is a correspondence between Whitehead's
version of the tetrahedral model and my own square model,
based on the 4×4 array I call the Galois tesseract (at left below).
(Click to enlarge.)
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Saturday, May 10, 2014
Happy Birthday
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Wednesday, April 2, 2014
Change Arises
For a different view of change arising, click on the tag above.
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Friday, March 21, 2014
Three Constructions of the Miracle Octad Generator
See also a Log24 post on this subject from Dec. 14, 2013,
especially (scroll down) the update of March 9, 2014.
Related material on the Turyn-Curtis construction
from the University of Cambridge —
— Slide by "Dr. Parker" — Apparently Richard A. Parker —
Lecture 4, "Discovering M24," in slides for lectures 1-8 from lectures
at Cambridge in 2010-2011 on "Sporadic and Related Groups."
See also the Parker lectures of 2012-2013 on the same topic.
A third construction of Curtis's 35 4×6 1976 MOG arrays would use
Cullinane's analysis of the 4×4 subarrays' affine and projective structure,
and point out the fact that Conwell's 1910 correspondence of the 35
4+4-partitions of an 8-set with the 35 lines of the projective 3-space
over the 2-element field, PG(3, 2), is essentially the same correspondence
as that constituting Curtis's 1976 MOG.
See The Diamond Theorem, Finite Relativity, Galois Space,
Generating the Octad Generator, and The Klein Correspondence.
Update of March 22-March 23 —
Adding together as (0,1)-matrices over GF(2) the black parts (black
squares as 1's, all other squares as 0's) of the 35 4×6 arrays of the 1976
Curtis MOG would then reveal* the symmetric role played in octads
by what Curtis called the heavy brick , and so reveal also the action of
S3 on the three Curtis bricks that leaves invariant the set of all 759
octads of the S(5, 8, 24) constructed from the 35 MOG arrays. For more
details of this "by-hand" construction, see Geometry of the 4×4 Square.
For the mathematical properties of the S(5, 8, 24), it is convenient to
have a separate construction (such as Turyn's), not by hand, of the
extended binary Golay code. See the Brouwer preprint quoted above.
* "Then a miracle occurs," as in the classic 1977 Sidney Harris cartoon.
Illustration of array addition from March 23 —
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Friday, February 28, 2014
Score
"Once a verbal structure is read, and reread
often enough to be possessed, it 'freezes.'
It turns into a unity in which all parts exist at
once, without regard to the specific movement
of the narrative. We may compare it to the study
of a music score, where we can turn to any
part without regard to sequential performance."
— Northrop Frye in The Great Code
Gardner reportedly died at 65 on February 19.
A post linked to here on that date suggests some
musical remarks.
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Code
|
From Northrop Frye's The Great Code: The Bible and Literature , Ch. 3: Metaphor I — "In the preceding chapter we considered words in sequence, where they form narratives and provide the basis for a literary theory of myth. Reading words in sequence, however, is the first of two critical operations. Once a verbal structure is read, and reread often enough to be possessed, it 'freezes.' It turns into a unity in which all parts exist at once, without regard to the specific movement of the narrative. We may compare it to the study of a music score, where we can turn to any part without regard to sequential performance. The term 'structure,' which we have used so often, is a metaphor from architecture, and may be misleading when we are speaking of narrative, which is not a simultaneous structure but a movement in time. The term 'structure' comes into its proper context in the second stage, which is where all discussion of 'spatial form' and kindred critical topics take their origin." |
Related material:
|
"The Great Code does not end with a triumphant conclusion or the apocalypse that readers may feel is owed them or even with a clear summary of Frye’s position, but instead trails off with a series of verbal winks and nudges. This is not so great a fault as it would be in another book, because long before this it has been obvious that the forward motion of Frye’s exposition was illusory, and that in fact the book was devoted to a constant re-examination of the same basic data from various closely related perspectives: in short, the method of the kaleidoscope. Each shake of the machine produces a new symmetry, each symmetry as beautiful as the last, and none of them in any sense exclusive of the others. And there is always room for one more shake."
— Charles Wheeler, "Professor Frye and the Bible," South Atlantic Quarterly 82 (Spring 1983), pp. 154-164, reprinted in a collection of reviews of the book. |
For code in a different sense, but related to the first passage above,
see Diamond Theory Roullete, a webpage by Radamés Ajna.
For "the method of the kaleidoscope" mentioned in the second
passage above, see both the Ajna page and a webpage of my own,
Kaleidoscope Puzzle.
Sunday, February 23, 2014
Sunday School
Lang to Langlands
|
Lang — “Elliptic functions parametrize elliptic curves, and the intermingling of the analytic and algebraic-arithmetic theory has been at the center of mathematics since the early part of the nineteenth century.” — Serge Lang, preface to Elliptic Functions (second edition, 1987) Langlands — “The theory of modular functions and modular forms, defined on the upper half-plane H and subject to appropriate tranformation laws with respect to the group Gamma = SL(2, Z) of fractional linear transformations, is closely related to the theory of elliptic curves, because the family of all isomorphism classes of elliptic curves over C can be parametrized by the quotient Gamma\H. This is an important, although formal, relation that assures that this and related quotients have a natural structure as algebraic curves X over Q. The relation between these curves and elliptic curves predicted by the Taniyama-Weil conjecture is, on the other hand, far from formal.” — Robert P. Langlands, review of Elliptic Curves , by Anthony W. Knapp. (The review appeared in Bulletin of the American Mathematical Society , January 1994.) |
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Sunday, January 26, 2014
A Midnight Exorcism
The summoning of the spirit of Bertrand Russell
yesterday by Peter J. Cameron at his weblog
suggests a review of this weblog’s posts of
Christmas Eve, December 24-25, 2013.
(Recall that Robert D. Carmichael, who, in a book
linked to at midnight last Christmas Eve discusses
some “magic” mathematical structures,
reportedly was trained as a Presbyterian minister.
See also The Presbyterian Exorcist.)
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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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Tuesday, December 31, 2013
Christmas Ornaments
Continued from December 25—
A link from Sunday afternoon to Nov. 26, 2012,
suggests a review of one of the above structures.
The Dreaming Jewels cover at left is taken from a review
by Jo Walton at Tor.com—
"This is a book that it’s clearly been difficult
for publishers to market. The covers have been
generally pretty awful, and also very different.
I own a 1975 Corgi SF Collectors Library
paperback that I bought new for 40p in the later
seventies. It’s purple, and it has a slightly grainy
cover, and it matches my editions of The Menace
From Earth and A Canticle for Leibowitz .
(Dear old Corgi SF Collectors Editions with their
very seventies fonts! How I imprinted on them at
an early age!) I mention this, however, because
the (uncredited) illustration actually represents and
illustrates the book much better than any of the other
cover pictures I’ve seen. It shows a hexagon with an
attempt at facets, a man, a woman, hands, a snake,
and stars, all in shades of green. It isn’t attractive,
but it wouldn’t put off people who’d enjoy what’s inside
either."
The "hexagon with an attempt at facets" is actually
an icosahedron, as the above diagram shows.
(The geometric part of the diagram is from a Euclid webpage.)
For Plato's dream about these jewels, see his Timaeus.
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Thursday, December 26, 2013
How It Works
“Design is how it works.” — Steve Jobs
“By far the most important structure in design theory
is the Steiner system S(5, 8, 24).”
— “Block Designs,” by Andries E. Brouwer (Ch. 14 (pp. 693-746),
Section 16 (p. 716) of Handbook of Combinatorics, Vol. I ,
MIT Press, 1995, edited by Ronald L. Graham, Martin Grötschel,
and László Lovász)
For some background on that Steiner system, see the footnote to
yesterday’s Christmas post.
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Tuesday, December 24, 2013
Through a Mirror, Darkly
Review of a book first published in 1989—
Reality's Mirror: Exploring the Mathematics of Symmetry —
"Here is a book that explains in laymen language
what symmetry is all about, from the lowliest snowflake
and flounder to the lofty group structures whose
astonishing applications to the Old One are winning
Nobel prizes. Bunch's book is a marvel of clear, witty
science writing, as delightful to read as it is informative
and up-to-date. The author is to be congratulated on
a job well done." — Martin Gardner
A completely different person whose name
mirrors that of the Mathematics of Symmetry author —
See also this journal on the date mentioned in the Princetonian .
"Always with a little humor." — Yen Lo
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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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Saturday, November 30, 2013
Waiting for Ogdoad
Continued from October 30 (Devil’s Night), 2013.
“In a sense, we would see that change
arises from the structure of the object.”
— Theoretical physicist quoted in a
Simons Foundation article of Sept. 17, 2013
This suggests a review of mathematics and the
“Classic of Change ,” the I Ching .
The physicist quoted above was discussing a rather
complicated object. His words apply to a much simpler
object, an embodiment of the eight trigrams underlying
the I Ching as the corners of a cube.
See also…
(Click for clearer image.)
The Cullinane image above illustrates the seven points of
the Fano plane as seven of the eight I Ching trigrams and as
seven natural ways of slicing the cube.
For a different approach to the mathematics of cube slices,
related to Gauss’s composition law for binary quadratic forms,
see the Bhargava cube in a post of April 9, 2012.
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Friday, November 29, 2013
Odd Facts
"These are odd facts…." — G. H. Hardy,
quoted in the previous post, "Centered"
Other odd facts:
If n is odd, then the object at the center
of the n×n square is a square.
Similarly for the n×n×n cube.
Related meditation:
“In a sense, we would see that change
arises from the structure of the object,” he said.
“But it’s not from the object changing.
The object is basically timeless.”
— Theoretical physicist quoted in a
Simons Foundation article of Sept. 17, 2013,
"A Jewel at the Heart of Quantum Physics"
See also "My God, it's full of… everything."
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Saturday, November 16, 2013
Raiders of the Lost Theorem
Yes. See …
The 48 actions of GL(2,3) on a 3×3 coordinate-array A,
when matrices of that group right-multiply the elements of A,
with A =
| (1,1) (1,0) (1,2) (0,1) (0,0) (0,2) (2,1) (2,0) (2,2) |
Actions of GL(2,p) on a pxp coordinate-array have the
same sorts of symmetries, where p is any odd prime.
Note that A, regarded in the Sallows manner as a magic square,
has the constant sum (0,0) in rows, columns, both diagonals, and
all four broken diagonals (with arithmetic modulo 3).
For a more sophisticated approach to the structure of the
ninefold square, see Coxeter + Aleph.
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Thursday, October 10, 2013
A Space for Scarlett
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Saturday, September 21, 2013
Mathematics and Narrative (continued)
Mathematics:
A review of posts from earlier this month —
Wednesday, September 4, 2013
|
Narrative:
Aooo.
Happy birthday to Stephen King.
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Sunday, September 8, 2013
The Importance of Being Ernst
For John Cusack and Menno Meyjes —
"I still haven't found what I'm looking for." — Bono
"In fact Surrealism found what it had been looking for
from the first in the 1920 collages [by Max Ernst],
which introduced an entirely original scheme of visual structure…."
— Rosalind Krauss quoting André Breton*
in "The Master's Bedroom"
See also tonight's 10 PM post.
* "Artistic Genesis and Perspective of Surrealism" (1941),
in Surrealism and Painting (New York, Harper & Row, 1972, p. 64).
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Wednesday, September 4, 2013
Tuesday, September 3, 2013
“The Stone” Today Suggests…
|
The Philosopher's Gaze , by David Michael Levin, The post-metaphysical question—question for a post-metaphysical phenomenology—is therefore: Can the perceptual field, the ground of perception, be released from our historical compulsion to represent it in a way that accommodates our will to power and its need to totalize and reify the presencing of being? In other words: Can the ground be experienced as ground? Can its hermeneutical way of presencing, i.e., as a dynamic interplay of concealment and unconcealment, be given appropriate respect in the receptivity of a perception that lets itself be appropriated by the ground and accordingly lets the phenomenon of the ground be what and how it is? Can the coming-to-pass of the ontological difference that is constitutive of all the local figure-ground differences taking place in our perceptual field be made visible hermeneutically, and thus without violence to its withdrawal into concealment? But the question concerning the constellation of figure and ground cannot be separated from the question concerning the structure of subject and object. Hence the possibility of a movement beyond metaphysics must also think the historical possibility of breaking out of this structure into the spacing of the ontological difference: différance , the primordial, sensuous, ekstatic écart . As Heidegger states it in his Parmenides lectures, it is a question of "the way historical man belongs within the bestowal of being (Zufügung des Seins ), i.e., the way this order entitles him to acknowledge being and to be the only being among all beings to see the open" (PE* 150, PG** 223. Italics added). We might also say that it is a question of our response-ability, our capacity as beings gifted with vision, to measure up to the responsibility for perceptual responsiveness laid down for us in the "primordial de-cision" (Entscheid ) of the ontological difference (ibid.). To recognize the operation of the ontological difference taking place in the figure-ground difference of the perceptual Gestalt is to recognize the ontological difference as the primordial Riß , the primordial Ur-teil underlying all our perceptual syntheses and judgments—and recognize, moreover, that this rift, this division, decision, and scission, an ekstatic écart underlying and gathering all our so-called acts of perception, is also the only "norm" (ἀρχή ) by which our condition, our essential deciding and becoming as the ones who are gifted with sight, can ultimately be judged. * PE: Parmenides of Heidegger in English— Bloomington: Indiana University Press, 1992 ** PG: Parmenides of Heidegger in German— Gesamtausgabe , vol. 54— Frankfurt am Main: Vittorio Klostermann, 1992 |
Examples of "the primordial Riß " as ἀρχή —
For an explanation in terms of mathematics rather than philosophy,
see the diamond theorem. For more on the Riß as ἀρχή , see
Function Decomposition Over a Finite Field.
Comments Off on “The Stone” Today Suggests…
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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Wednesday, June 19, 2013
Ein Eck
"Da hats ein Eck" —
"you've/she's (etc.) got problems there"
St. Galluskirche:
St. Gallus's Day, 2012:
Click image for a St. Gallus's Day post.
A related problem:
Discuss the structure of the 4x4x4 "magic" cube
sent by Pierre de Fermat to Father Marin Mersenne
on April 1, 1640, in light of the above post.
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Sunday, June 16, 2013
Mathematical Review
From a weblog post on June 11, 2013, by one Pete Trbovich:
|
Here again, I don't think Steven Cullinane is really unhinged per se. At the very least, his geometric study is fun to play with, particularly when you find this toy. And I'm not really sure that anything he says is wrong per se. But you might find yourself asking "So what?" or more to the point, "Why is this supposed to be the central theory to explaining life, the universe, and everything?" |
It isn't supposed to be such a theory.
I do not know why Trbovich thinks it is .
— Steven H. Cullinane
Update of 11 PM June 16:
For one such central theory of everything, see
the I Ching . Diamond theory is, unlike that
Chinese classic, pure mathematics, but the larger
of the binary-coordinate structures it is based on
are clearly isomorphic, simply as structures , to
the I Ching 's 64 hexagrams.
Make of this what you will.
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Saturday, June 15, 2013
At the Still Point
(Continued from yesterday's posts, "Object of Beauty"
and "Amy's Shadow")
A winner of a Nobel Prize for X-ray crystallography stands
at the head of the New York Times obituary list today.
In memoriam —
|
X-Ray Vision "Crystal Engineering in Kindergarten," by Bart Kahr:
"If the reader is beginning to suspect that Froebel’s
Click images for some backstories. |
Some further background:
The Times follows yesterday's egregious religious error
with an egregious scientific error:
"The technique developed by Dr. Karle and Herbert A. Hauptman,
called X-ray crystallography, is now routinely used by scientists…."
Karle was reportedly born in 1918, Hauptman in 1917.
Wikipedia on the history of X-ray crystallography:
"The idea that crystals could be used as a
diffraction grating for X-rays arose in 1912…."
The Nobel Foundation:
"The Nobel Prize in Physics 1914 was awarded to
Max von Laue 'for his discovery of the diffraction of
X-rays by crystals.'"
"The Nobel Prize in Physics 1915 was awarded jointly to
Sir William Henry Bragg and William Lawrence Bragg
'for their services in the analysis of crystal structure
by means of X-rays.'"
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Friday, June 14, 2013
Object of Beauty
This journal on July 5, 2007 —
“It is not clear why MySpace China will be successful."
— The Chinese magazine Caijing in 2007, quoted in
Asia Sentinel on July 12, 2011
This journal on that same date, July 12, 2011 —
See also the eightfold cube and kindergarten blocks
at finitegeometry.org/sc.
Friedrich Froebel, Froebel's Chief Writings on Education ,
Part II, "The Kindergarten," Ch. III, "The Third Play":
"The little ones, who always long for novelty and change,
love this simple plaything in its unvarying form and in its
constant number, even as they love their fairy tales with
the ever-recurring dwarfs…."
This journal, Group Actions, Nov. 14, 2012:
"Those who insist on vulgarizing their mathematics
may regard linear and affine group actions on the eight
cubes as the dance of Snow White (representing (0,0,0))
and the Seven Dwarfs—
."
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Thursday, June 13, 2013
Gate
"Eight is a Gate." — Mnemonic rhyme
Today's previous post, Window, showed a version
of the Chinese character for "field"—
This suggests a related image—
The related image in turn suggests…
Unlike linear perspective, axonometry has no vanishing point,
and hence it does not assume a fixed position by the viewer.
This makes axonometry 'scrollable'. Art historians often speak of
the 'moving' or 'shifting' perspective in Chinese paintings.
Axonometry was introduced to Europe in the 17th century by
Jesuits returning from China.
As was the I Ching. A related structure:
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Friday, June 7, 2013
Rubric’s Cuber
From Night of Lunacy (Sunday, May 5, 2013):
Related posts: Rubric, Cuber, and Pound Sign.
Click image for some background.
See also Story Theory and Princeton Apocalypse.
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Tuesday, May 28, 2013
Codes
The hypercube model of the 4-space over the 2-element Galois field GF(2):
The phrase Galois tesseract may be used to denote a different model
of the above 4-space: the 4×4 square.
MacWilliams and Sloane discussed the Miracle Octad Generator
(MOG) of R. T. Curtis further on in their book (see below), but did not
seem to realize in 1977 that the 4×4 structures within the MOG are
based on the Galois-tesseract model of the 4-space over GF(2).
The thirty-five 4×4 structures within the MOG:
Curtis himself first described these 35 square MOG patterns
combinatorially, (as his title indicated) rather than
algebraically or geometrically:
A later book co-authored by Sloane, first published in 1988,
did recognize the 4×4 MOG patterns as based on the 4×4
Galois-tesseract model.
Between the 1977 and 1988 Sloane books came the diamond theorem.
Update of May 29, 2013:
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977
(the year the above MacWilliams-Sloane book was first published):
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Sunday, May 19, 2013
Priority Claim
From an arXiv preprint submitted July 18, 2011,
and last revised on March 11, 2013 (version 4):
"By our construction, this vector space is the dual
of our hypercube F24 built on I \ O9. The vector space
structure of the latter, to our knowledge, is first
mentioned by Curtis in [Cur89]. Hence altogether
our proposition 2.3.4 gives a novel geometric
meaning in terms of Kummer geometry to the known
vector space structure on I \ O9."
[Cur89] reference:
R. T. Curtis, "Further elementary techniques using
the miracle octad generator," Proc. Edinburgh
Math. Soc. 32 (1989), 345-353 (received on
July 20, 1987).
— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group M 24 ,"
arXiv.org > hep-th > arXiv:1107.3834
"First mentioned by Curtis…."
No. I claim that to the best of my knowledge, the
vector space structure was first mentioned by me,
Steven H. Cullinane, in an AMS abstract submitted
in October 1978, some nine years before the
Curtis article.
|
Update of the above paragraph on July 6, 2013—
No. The vector space structure was described by
The vector space structure as it occurs in a 4×4 array |
See Notes on Finite Geometry for some background.
See in particular The Galois Tesseract.
For the relationship of the 1978 abstract to Kummer
geometry, see Rosenhain and Göpel Tetrads in PG(3,2).
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Sunday, May 5, 2013
Night of Lunacy*
Structure vs. Character continued…
|
Structure |
|
Related vocabulary:
Nick Tosches on the German word “Quell “
* The title is from Heidegger.
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Saturday, May 4, 2013
Master Class
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Tuesday, February 19, 2013
Configurations
Yesterday's post Permanence dealt with the cube
as a symmetric model of the finite projective plane
PG(2,3), which has 13 points and 13 lines. The points
and lines of the finite geometry occur in the cube as
the 13 axes of symmetry and the 13 planes through
the center perpendicular to those axes. If the three
axes lying in a plane that cuts the cube in a hexagon
are supplemented by the axis perpendicular to that
plane, each plane is associated with four axes and,
dually, each axis is associated with four planes.
My web page on this topic, Cubist Geometries, was
written on February 27, 2010, and first saved to the
Internet Archive on Oct. 4, 2010.
For a more recent treatment of this topic that makes
exactly the same points as the 2010 page, see p. 218
of Configurations from a Graphical Viewpoint , by
Tomaž Pisanski and Brigitte Servatius, published by
Springer on Sept. 23, 2012 (date from both Google
Books and Amazon.com):
For a similar 1998 treatment of the topic, see Burkard Polster's
A Geometrical Picture Book (Springer, 1998), pp. 103-104.
The Pisanski-Servatius book reinforces my argument of Jan. 13, 2013,
that the 13 planes through the cube's center that are perpendicular
to the 13 axes of symmetry of the cube should be called the cube's
symmetry planes , contradicting the usual use of of that term.
That argument concerns the interplay between Euclidean and
Galois geometry. Pisanski and Servatius (and, in 1998, Polster)
emphasize the Euclidean square and cube as guides* to
describing the structure of a Galois space. My Jan. 13 argument
uses Galois structures as a guide to re-describing those of Euclid .
(For a similar strategy at a much more sophisticated level,
see a recent Harvard Math Table.)
Related material: Remarks on configurations in this journal
during the month that saw publication of the Pisanski-Servatius book.
* Earlier guides: the diamond theorem (1978), similar theorems for
2x2x2 (1984) and 4x4x4 cubes (1983), and Visualizing GL(2,p)
(1985). See also Spaces as Hypercubes (2012).
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Wednesday, February 13, 2013
Form:
Story, Structure, and the Galois Tesseract
Recent Log24 posts have referred to the
"Penrose diamond" and Minkowski space.
The Penrose diamond has nothing whatever
to do with my 1976 monograph "Diamond Theory,"
except for the diamond shape and the connection
of the Penrose diamond to the Klein quadric—
The Klein quadric occurs in the five-dimensional projective space
over a field. If the field is the two-element Galois field GF(2), the
quadric helps explain certain remarkable symmetry properties
of the R. T. Curtis Miracle Octad Generator (MOG), hence of
the large Mathieu group M24. These properties are also
relevant to the 1976 "Diamond Theory" monograph.
For some background on the quadric, see (for instance)…
See also The Klein Correspondence,
Penrose Space-Time, and a Finite Model.
Related material:
|
"… one might crudely distinguish between philosophical – J. M. E. Hyland. "Proof Theory in the Abstract." (pdf) |
Those who prefer story to structure may consult
- today's previous post on the Penrose diamond
- the remarks of Scott Aaronson on August 17, 2012
- the remarks in this journal on that same date
- the geometry of the 4×4 array in the context of M24.
Comments Off on Form:
Tuesday, February 5, 2013
Arsenal
The previous post discussed some fundamentals of logic.
The name “Boole” in that post naturally suggests the
concept of Boolean algebra . This is not the algebra
needed for Galois geometry . See below.
Some, like Dan Brown, prefer to interpret symbols using
religion, not logic. They may consult Diamond Mandorla,
as well as Blade and Chalice, in this journal.
See also yesterday’s Universe of Discourse.
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Wednesday, January 30, 2013
Abstract Possibility
Today's NY Times "Stone Links" to philosophy include
a link to a review of a collection of Hilary Putnam's papers.
Related material, from Putnam's "What is Mathematical
Truth?" (Historia Mathematica 2 (1975): 529-543)—
"In this paper I argue that mathematics should be interpreted realistically – that is, that mathematics makes assertions that are objectively true or false, independently of the human mind, and that something answers to such mathematical notions as ‘set’ and ‘function’. This is not to say that reality is somehow bifurcated – that there is one reality of material things, and then, over and above it, a second reality of ‘mathematical things’. A set of objects, for example, depends for its existence on those objects: if they are destroyed, then there is no longer such a set. (Of course, we may say that the set exists ‘tenselessly’, but we may also say the objects exist ‘tenselessly’: this is just to say that in pure mathematics we can sometimes ignore the important difference between ‘exists now’ and ‘did exist, exists now, or will exist’.) Not only are the ‘objects’ of pure mathematics conditional upon material objects; they are, in a sense, merely abstract possibilities. Studying how mathematical objects behave might better be described as studying what structures are abstractly possible and what structures are not abstractly possible."
See also Wittgenstein's Diamond and Plato's Diamond.
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Wednesday, January 23, 2013
DNA and a Galois Field
From Ewan Birney's weblog today:
| WEDNESDAY, 23 JANUARY 2013
Using DNA as a digital archive media Today sees the publication in Nature of “Toward practical high-capacity low-maintenance storage of digital information in synthesised DNA,” a paper spearheaded by my colleague Nick Goldman and in which I played a major part, in particular in the germination of the idea. |
Birney appeared in Log24 on Dec. 30, 2012, quoted as follows:
"It is not often anyone will hear the phrase 'Galois field' and 'DNA' together…."
— Birney's weblog on July 3, 2012, "Galois and Sequencing."
Birney's widespread appearance in news articles today about the above Nature publication suggests a review of the "Galois-field"-"DNA" connection.
See, for instance, the following papers:
- Gail Rosen and Jeff Moore. "Investigation of Coding Structure in DNA," IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Hong Kong, April 2003. [pdf]
- Gail Rosen. "Finding Near-Periodic DNA Regions using a Finite-Field Framework," 2nd IEEE Genomic Signal Processing Workshop (GENSIPS), Baltimore, MD, May 2004. [pdf]
- Gail Rosen. "Examining Coding Structure and Redundancy in DNA," IEEE Engineering in Medicine and Biology Magazine, Volume 25, Issue 1, January/February 2006. [pdf]
A Log24 post of Sept. 17, 2012, also mentions the phrases "Galois field" and "DNA" together.
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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 10, 2012
Review of Leonardo Article
Review of an often-cited Leonardo article that is
now available for purchase online…
|
The Tiling Patterns of Sebastien Truchet Authors: Cyril Stanley Smith and Pauline Boucher
Source: Leonardo , Vol. 20, No. 4, Published by: The MIT Press Stable URL: http://www.jstor.org/stable/1578535 . |
Smith and Boucher give a well-illustrated account of
the early history of Truchet tiles, but their further remarks
on the mathematics underlying patterns made with
these tiles (see the diamond theorem* of 1976) are
worthless.
For instance…
Excerpt from pages 383-384—
"A detailed analysis of Truchet's
patterns touches upon the most fundamental
questions of the relation between
mathematical formalism and the structure
of the material world. Separations
between regions differing in density
require that nothing be as important as
something and that large and small cells of
both must coexist. The aggregation of
unitary choice of directional distinction
at interfaces lies at the root of all being
and becoming."
* This result is about Truchet-tile patterns, but the
underlying mathematics was first discovered by
investigating superimposed patterns of half-circles .
See Half-Circle Patterns at finitegeometry.org.
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Wednesday, December 5, 2012
Arte Programmata*
The 1976 monograph "Diamond Theory" was an example
of "programmed art" in the sense established by, for
instance, Karl Gerstner. The images were produced
according to strict rules, and were in this sense
"programmed," but were drawn by hand.
Now an actual computer program has been written,
based on the Diamond Theory excerpts published
in the Feb. 1977 issue of Computer Graphics and Art
(Vol. 2, No. 1, pp. 5-7), that produces copies of some of
these images (and a few malformed images not in
Diamond Theory).
See Isaac Gierard's program at GitHub—
https://github.com/matthewepler/ReCode_Project/
blob/dda7b23c5ad505340b468d9bd707fd284e6c48bf/
isaac_gierard/StevenHCullinane_DiamondTheory/
StevenHCullinane_DiamondTheory.pde
As the suffix indicates, this program is in the
Processing Development Environment language.
It produces the following sketch:
The rationale for selecting and arranging these particular images is not clear,
and some of the images suffer from defects (exercise: which ones?), but the
overall effect of the sketch is pleasing.
For some background for the program, see The ReCode Project.
It is good to learn that the Processing language is well-adapted to making the
images in such sketches. The overall structure of the sketch gives, however,
no clue to the underlying theory in "Diamond Theory."
For some related remarks, see Theory (Sept. 30, 2012).
* For the title, see Darko Fritz, "Notions of the Program in 1960s Art."
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Saturday, November 24, 2012
Will and Representation*
Robert A. Wilson, in an inaugural lecture in April 2008—
Representation theory
A group always arises in nature as the symmetry group of some object, and group
theory in large part consists of studying in detail the symmetry group of some
object, in order to throw light on the structure of the object itself (which in some
sense is the “real” object of study).
But if you look carefully at how groups are used in other areas such as physics
and chemistry, you will see that the real power of the method comes from turning
the whole procedure round: instead of starting from an object and abstracting
its group of symmetries, we start from a group and ask for all possible objects
that it can be the symmetry group of .
This is essentially what we call Representation theory . We think of it as taking a
group, and representing it concretely in terms of a symmetrical object.
Now imagine what you can do if you combine the two processes: we start with a
symmetrical object, and find its group of symmetries. We now look this group up
in a work of reference, such as our big red book (The ATLAS of Finite Groups),
and find out about all (well, perhaps not all) other objects that have the same
group as their group of symmetries.
We now have lots of objects all looking completely different, but all with the same
symmetry group. By translating from the first object to the group, and then to
the second object, we can use everything we know about the first object to tell
us things about the second, and vice versa.
As Poincaré said,
Mathematicians do not study objects, but relations between objects.
Thus they are free to replace some objects by others, so long as the
relations remain unchanged.
Fano plane transformed to eightfold cube,
and partitions of the latter as points of the former:
* For the "Will" part, see the PyrE link at Talk Amongst Yourselves.
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Thursday, November 22, 2012
Finite Relativity
(Continued from 1986)
|
S. H. Cullinane This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.
— H. Weyl, The Classical Groups , In finite geometry "points" are often defined as ordered n-tuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such n-tuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S. This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame. The frame: A 4×4 array. The invariant structure: The following set of 15 partitions of the frame into two 8-sets.
A representative coordinatization:
0000 0001 0010 0011
The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2). |
S. H. Cullinane This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.
— H. Weyl, The Classical Groups , In finite geometry "points" are often defined as ordered n-tuples of a finite (i.e., Galois) field GF(q). What geometric structures ("frames of reference," in Weyl's terms) are coordinatized by such n-tuples? Weyl's use of "objectively" seems to mean that such structures should have certain objective— i.e., purely geometric— properties invariant under each S. This note suggests such a frame of reference for the affine 4-space over GF(2), and a class of 322,560 equivalent coordinatizations of the frame. The frame: An array of 16 congruent equilateral subtriangles that make up a larger equilateral triangle. The invariant structure: The following set of 15 partitions of the frame into two 8-sets.
The group: The group AGL(4,2) of 322,560 regular affine transformations of the ordered 4-tuples over GF(2). |
For some background on the triangular version,
see the Square-Triangle Theorem,
noting particularly the linked-to coordinatization picture.
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The Red Line vs. the Red Eye
"Any sufficiently advanced technology is indistinguishable from magic."
"The HP/Autonomy Debacle," by John C. Dvorak at pcmag.com on Tuesday, Nov. 20, 2012—
"The whole Autonomy thing was weird since the company seemed to be performing magic. On co-founder Michael Richard Lynch's Wikipedia page, the company is described as 'a leader in the area of computer understanding of unstructured information, an area which is becoming known as meaning-based computing .'
I do not know how gullible HP's board of directors is, but when I see the sudden emergence of something called 'meaning-based computing,' the alarms sound and the bullcrap meter begins to tag the red line."
A story by Terence K. Huwe in Online magazine, Sept.-Oct. 2011, defines meaning-based computing (MBC), discusses Autonomy , and llnks to…
John Markoff in The New York Times , March 4, 2011—
"Engineers and linguists at Cataphora, an information-sifting company based in Silicon Valley, have their software mine documents for the activities and interactions of people— who did what when, and who talks to whom. The software seeks to visualize chains of events. It identifies discussions that might have taken place across e-mail, instant messages and telephone calls.
Then the computer pounces, so to speak, capturing 'digital anomalies' that white-collar criminals often create in trying to hide their activities.
For example, it finds 'call me' moments— those incidents when an employee decides to hide a particular action by having a private conversation. This usually involves switching media, perhaps from an e-mail conversation to instant messaging, telephone or even a face-to-face encounter."
For example…
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Wednesday, November 14, 2012
Group Actions
The December 2012 Notices of the American
Mathematical Society has an ad on page 1564
(in a review of two books on vulgarized mathematics)
for three workshops next year on “Low-dimensional
Topology, Geometry, and Dynamics”—
(Only the top part of the ad is shown; for further details
see an ICERM page.)
(ICERM stands for Institute for Computational
and Experimental Research in Mathematics.)
The ICERM logo displays seven subcubes of
a 2x2x2 eight-cube array with one cube missing—
The logo, apparently a stylized image of the architecture
of the Providence building housing ICERM, is not unlike
a picture of Froebel’s Third Gift—
© 2005 The Institute for Figuring
Photo by Norman Brosterman from the Inventing Kindergarten
exhibit at The Institute for Figuring (co-founded by Margaret Wertheim)
The eighth cube, missing in the ICERM logo and detached in the
Froebel Cubes photo, may be regarded as representing the origin
(0,0,0) in a coordinatized version of the 2x2x2 array—
in other words the cube invariant under linear , as opposed to
more general affine , permutations of the cubes in the array.
These cubes are not without relevance to the workshops’ topics—
low-dimensional exotic geometric structures, group theory, and dynamics.
See The Eightfold Cube, A Simple Reflection Group of Order 168, and
The Quaternion Group Acting on an Eightfold Cube.
Those who insist on vulgarizing their mathematics may regard linear
and affine group actions on the eight cubes as the dance of
Snow White (representing (0,0,0)) and the Seven Dwarfs—
.
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Wednesday, October 10, 2012
Melancholia, Depression, Ambiguity
Occurrences of the phrase "magic square" in Lowe-Porter's translation of the Thomas Mann novel Doctor Faustus—
"On the wall above the piano was an arithmetical diagram fastened with drawing-pins, something he had found in a second-hand shop: a so-called magic square, such as appears also in Dürer's Melancolia , along with the hour-glass, the circle, the scale, the polyhedron, and other symbols. Here as there, the figure was divided into sixteen Arabic-numbered fields, in such a way that number one was in the right-hand lower corner, sixteen in the upper left; and the magic, or the oddity, simply consisted in the fact that the sum of these numerals, however you added them, straight down, crosswise, or diagonally, always came to thirty-four. What the principle was upon which this magic uniformity rested I never made out, but by virtue of the prominent place Adrian had given it over the piano, it always attracted the eye, and I believe I never visited his room without giving a quick glance, slanting up or straight down and testing once more the invariable, incredible result."
….
"Adrian kept without changing during the whole four and a half years he spent in Leipzig his two-room quarters in Peterstrasse near the Collegium Beatae Virginis, where he had again pinned the magic square above his cottage piano."
….
" 'The decisive factor is that every note, without exception, has significance and function according to its place in the basic series or its derivatives. That would guarantee what I call the indifference to harmony and melody.'
'A magic square,' I said. 'But do you hope to have people hear all that?' "
….
" 'Extraordinarily Dürerish. You love it. First "how will I shiver after the sun"; and then the houre-glasse of the Melancolia . Is the magic square coming too?' "
….
"Here I will remind the reader of a conversation I had with Adrian on a long-ago day, the day of his sister's wedding at Buchel, as we walked round the Cow Trough. He developed for me— under pressure of a headache— his idea of the 'strict style,' derived from the way in which, as in the lied 'O lieb Madel, wie schlecht bist du ' melody and harmony are determined by the permutation of a fundamental five-note motif, the symbolic letters h, e, a, e, e-flat. He showed me the 'magic square' of a style of technique which yet developed the extreme of variety out of identical material and in which there is no longer anything unthematic, anything that could not prove itself to be a variation of an ever constant element. This style, this technique, he said, admitted no note, not one, which did not fulfil its thematic function in the whole structure— there was no longer any free note."
Review of related material—
Last night's midnight post (disambiguation), the followup 1 AM post (ambiguation), today's noon post (ambiguity), and Dürer in this journal.
The tesseracts of the noon post are related to the Dürer magic square by a well-known adjacency property.
"… the once stable 'father's depression' has been transmuted into a shifting reality that shimmered in a multiplicity of facets."
— Haim Omer, Tel-Aviv University, on Milanese ambiguation therapy,
p. 321 in "Three Styles of Constructive Therapy,"
Constructive Therapies, Vol. 2 , pp. 319-333,
ed. by Michael F. Hoyt (Guilford Press paperback, 1998)
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Saturday, September 22, 2012
Occupy Space
"The word 'space' has, as you suggest, a large number of different meanings."
— Nanavira Thera in [Early Letters. 136] 10.xii.1958
From that same letter (links added to relevant Wikipedia articles)—
|
Space (ākāsa) is undoubtedly used in the Suttas
Your second letter seems to suggest that the space |
A simpler metaphysical system along the same lines—
|
The theory, he had explained, was that the persona
— The Gameplayers of Zan , |
"I am glad you have discovered that the situation is comical:
ever since studying Kummer I have been, with some difficulty,
refraining from making that remark."
— Nanavira Thera, [Early Letters, 131] 17.vii.1958
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Monday, September 17, 2012
Pattern Conception
( Continued from yesterday's post FLT )
Context Part I —
"In 1957, George Miller initiated a research programme at Harvard University to investigate rule-learning, in situations where participants are exposed to stimuli generated by rules, but are not told about those rules. The research program was designed to understand how, given exposure to some finite subset of stimuli, a participant could 'induce' a set of rules that would allow them to recognize novel members of the broader set. The stimuli in question could be meaningless strings of letters, spoken syllables or other sounds, or structured images. Conceived broadly, the project was a seminal first attempt to understand how observers, exposed to a set of stimuli, could come up with a set of principles, patterns, rules or hypotheses that generalized over their observations. Such abstract principles, patterns, rules or hypotheses then allow the observer to recognize not just the previously seen stimuli, but a wide range of other stimuli consistent with them. Miller termed this approach 'pattern conception ' (as opposed to 'pattern perception'), because the abstract patterns in question were too abstract to be 'truly perceptual.'….
…. the 'grammatical rules' in such a system are drawn from the discipline of formal language theory (FLT)…."
— W. Tecumseh Fitch, Angela D. Friederici, and Peter Hagoort, "Pattern Perception and Computational Complexity: Introduction to the Special Issue," Phil. Trans. R. Soc. B (2012) 367, 1925-1932
Context Part II —
Context Part III —
A four-color theorem describes the mathematics of
general structures, not just symbol-strings, formed from
four kinds of things— for instance, from the four elements
of the finite Galois field GF(4), or the four bases of DNA.
Context Part IV —
A quotation from William P. Thurston, a mathematician
who died on Aug. 21, 2012—
"It may sound almost circular to say that
what mathematicians are accomplishing
is to advance human understanding of mathematics.
I will not try to resolve this
by discussing what mathematics is,
because it would take us far afield.
Mathematicians generally feel that they know
what mathematics is, but find it difficult
to give a good direct definition.
It is interesting to try. For me,
'the theory of formal patterns'
has come the closest, but to discuss this
would be a whole essay in itself."
Related material from a literate source—
"So we moved, and they, in a formal pattern"
Formal Patterns—
Not formal language theory but rather
finite projective geometry provides a graphic grammar
of abstract design—
See also, elsewhere in this journal,
Crimson Easter Egg and Formal Pattern.
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Sunday, September 16, 2012
Master Class
Wikipedia (links added)—
"Hubbard coined Dianetics from the Greek stems dia ,
meaning through, and nous , meaning mind."
"The snow kept falling on the world,
big white flakes like white gloves."
— Frederick Seidel, "House Master,"
poem in The New Yorker of Sept. 3, 2012
Detail of Aug. 30 illustration, with added arrow—
The part of the illustration at upper right is from a post of
Friday, July 13th, 2012, on the death of producer Richard Zanuck.
"Pay no attention to the shadow behind the curtain."
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Tuesday, September 11, 2012
Symmetry and Hierarchy
A followup to Intelligence Test (April 2, 2012).
Philosophical Transactions of the Royal Society
B (2012) 367, 2007–2022
(theme issue of July 19, 2012) —
Gesche Westphal-Fitch [1], Ludwig Huber [2],
Juan Carlos Gómez [3], and W. Tecumseh Fitch [1]
Juan Carlos Gómez [3], and W. Tecumseh Fitch [1]
[1] Department of Cognitive Biology, University of Vienna,
Althanstrasse 14, 1090 Vienna, Austria
Althanstrasse 14, 1090 Vienna, Austria
[2] Messerli Research Institute, University of Veterinary Medicine Vienna,
Medical University of Vienna and University of Vienna,
Veterinärplatz 1, 1210 Vienna, Austria
Medical University of Vienna and University of Vienna,
Veterinärplatz 1, 1210 Vienna, Austria
[3] School of Psychology, St Mary’s College, University of St Andrews,
South Street, St Andrews, KY16 9JP, UK
South Street, St Andrews, KY16 9JP, UK
Excerpt (added in an update on Dec. 8, 2012) —
Conclusion —
"… We believe that applying the theoretical
framework of formal language theory to two-dimensional
patterns offers a rich new perspective on the
human capacity for producing regular, hierarchically
organized structures. Such visual patterns may actually
prove more flexible than music or language for probing
the full extent of human pattern processing abilities.
With the results presented here, we have taken the
first steps in decoding the uniquely human fascination
with visual patterns, what Gombrich termed our
‘sense of order’.
Although the patterns we studied are most similar
to tilings or mosaics, they are examples of a much
broader type of abstract plane pattern, a type found
in virtually all of the world’s cultures [4]. Given that
such abstract visual patterns seem to represent
human universals, they have received astonishingly
little attention from psychologists. This neglect is particularly
unfortunate given their democratic nature,
their popular appeal and the ease with which they
can be generated and analysed in the laboratory.
With the current research, we hope to spark renewed
scientific interest in these ‘unregarded arts’, which
we believe have much to teach us about the nature of
the human mind."
the human mind."
[4] Washburn, D. K. & Crowe, D. W.,1988
Symmetries of Culture:
Theory and Practice of Plane Pattern Analysis.
Seattle, WA: University of Washington Press.
Symmetries of Culture:
Theory and Practice of Plane Pattern Analysis.
Seattle, WA: University of Washington Press.
Commentary —
For symmetry , see Finite Geometry of the Square and Cube.
For hierarchy , see my assessment of Gombrich.
For culture , see T. S. Eliot and Russell Kirk on Eliot.
For culture , see T. S. Eliot and Russell Kirk on Eliot.
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Sunday, September 9, 2012
Decomposition Sermon
(Continued from Walpurgisnacht 2012)
Wikipedia article on functional decomposition—
"Outside of purely mathematical considerations,
perhaps the greatest value of functional decomposition
is the insight it provides into the structure of the world."
Certainly this is true for the sort of decomposition
known as harmonic analysis .
It is not, however, true of my own decomposition theorem,
which deals only with structures made up of at most four
different sorts of elementary parts.
But my own approach has at least some poetic value.
See the four elements of the Greeks in (for instance)
Eliot's Four Quartets and in Auden's For the Time Being .
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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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Defining Form
Background: Square-Triangle Theorem.
For a more literary approach, see "Defining Form" in this journal
and a bibliography from the University of Zaragoza.
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Wednesday, July 18, 2012
Incommensurables
(Continued from Midsummer Eve)
"At times, bullshit can only be countered with superior bullshit."
— Norman Mailer, March 3, 1992, PBS transcript
"Just because it is a transition between incommensurables, the transition between competing paradigms cannot be made a step at a time, forced by logic and neutral experience. Like the gestalt switch, it must occur all at once (though not necessarily in an instant) or not at all."
— Thomas Kuhn, The Structure of Scientific Revolutions , 1962, as quoted in The Enneagram of Paradigm Shifting
"In the spiritual traditions from which Jung borrowed the term, it is not the SYMMETRY of mandalas that is all-important, as Jung later led us to believe. It is their capacity to reveal the asymmetry that resides at the very heart of symmetry."
I have little respect for Enneagram enthusiasts, but they do at times illustrate Mailer's maxim.
My own interests are in the purely mathematical properties of the number nine, as well as those of the next square, sixteen.
Those who prefer bullshit may investigate non-mathematical properties of sixteen by doing a Google image search on MBTI.
For bullshit involving nine, see (for instance) Einsatz in this journal.
For non-bullshit involving nine, sixteen, and "asymmetry that resides at the very heart of symmetry," see Monday's Mapping Problem continued. (The nine occurs there as the symmetric figures in the lower right nine-sixteenths of the triangular analogs diagram.)
For non-bullshit involving psychological and philosophical terminology, see James Hillman's Re-Visioning Psychology .
In particular, see Hillman's "An Excursion on Differences Between Soul and Spirit."
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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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Thursday, July 12, 2012
Galois Space
An example of lines in a Galois space * —
The 35 lines in the 3-dimensional Galois projective space PG(3,2)—
_lines_as_arrays.jpg)
There are 15 different individual linear diagrams in the figure above.
These are the points of the Galois space PG(3,2). Each 3-set of linear diagrams
represents the structure of one of the 35 4×4 arrays and also represents a line
of the projective space.
The symmetry of the linear diagrams accounts for the symmetry of the
840 possible images in the kaleidoscope puzzle.
* For further details on the phrase "Galois space," see
Beniamino Segre's "On Galois Geometries," Proceedings of the
International Congress of Mathematicians, 1958 [Edinburgh].
(Cambridge U. Press, 1960, 488-499.)
(Update of Jan. 5, 2013— This post has been added to finitegeometry.org.)
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Tuesday, June 26, 2012
Looking Deeply
Last night's post on The Trinity of Max Black and the use of
the term "eightfold" by the Mathematical Sciences Research Institute
at Berkeley suggest a review of an image from Sept. 22, 2011—
The triskele detail above echoes a Buddhist symbol found,
for instance, on the Internet in an ad for meditation supplies—
Related remarks—
http://www.spencerart.ku.edu/about/dialogue/fdpt.shtml—
Mary Dusenbury (Radcliffe '64)—
"… I think a textile, like any work of art, holds a tremendous amount of information— technical, material, historical, social, philosophical— but beyond that, many works of art are very beautiful and they speak to us on many layers— our intellect, our heart, our emotions. I've been going to museums since I was a very small child, thinking about what I saw, and going back to discover new things, to see pieces that spoke very deeply to me, to look at them again, and to find more and more meaning relevant to me in different ways and at different times of my life. …
… I think I would suggest to people that first of all they just look. Linger by pieces they find intriguing and beautiful, and look deeply. Then, if something interests them, we have tried to put a little information around the galleries to give a bit of history, a bit of context, for each piece. But the most important is just to look very deeply."
http://en.wikipedia.org/wiki/Nikaya_Buddhism—
According to Robert Thurman, the term "Nikāya Buddhism" was coined by Professor Masatoshi Nagatomi of Harvard University, as a way to avoid the usage of the term Hinayana.[12] "Nikaya Buddhism" is thus an attempt to find a more neutral way of referring to Buddhists who follow one of the early Buddhist schools, and their practice.
12. The Emptiness That is Compassion:
An Essay on Buddhist Ethics, Robert A. F. Thurman, 1980
[Religious Traditions , Vol. 4 No. 2, Oct.-Nov. 1981, pp. 11-34]
http://dsal.uchicago.edu/cgi-bin/philologic/getobject.pl?c.2:1:6.pali—
Nikāya [Sk. nikāya, ni+kāya]
collection ("body") assemblage, class, group
http://en.wiktionary.org/wiki/नि—
Sanskrit etymology for नि (ni)
1 From Proto-Indo-European *ni …
Prefix
नि (ni)
- down
- back
- in, into
http://www.rigpawiki.org/index.php?title=Kaya—
Kaya (Skt. kāya ; སྐུ་, Tib. ku ; Wyl. sku ) —
the Sanskrit word kaya literally means ‘body’
but can also signify dimension, field or basis.
• structure, existentiality, founding stratum ▷HVG KBEU
Note that The Trinity of Max Black is a picture of a set—
i.e., of an "assemblage, class, group."
Note also the reference above to the word "gestalt."
"Was ist Raum, wie können wir ihn
erfassen und gestalten?"
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Friday, June 22, 2012
Bowling in Diagon Alley
Josefine Lyche bowling (Facebook, June 12, 2012)
A professor of philosophy in 1984 on Socrates's geometric proof in Plato's Meno dialogue—
"These recondite issues matter because theories about mathematics have had a big place in Western philosophy. All kinds of outlandish doctrines have tried to explain the nature of mathematical knowledge. Socrates set the ball rolling…."
— Ian Hacking in The New York Review of Books , Feb. 16, 1984
The same professor introducing a new edition of Kuhn's Structure of Scientific Revolutions—
"Paradigms Regained" (Los Angeles Review of Books , April 18, 2012)—
"That is the structure of scientific revolutions: normal science with a paradigm and a dedication to solving puzzles; followed by serious anomalies, which lead to a crisis; and finally resolution of the crisis by a new paradigm. Another famous word does not occur in the section titles: incommensurability. This is the idea that, in the course of a revolution and paradigm shift, the new ideas and assertions cannot be strictly compared to the old ones."
The Meno proof involves inscribing diagonals in squares. It is therefore related, albeit indirectly, to the classic Greek discovery that the diagonals of a square are incommensurable with its sides. Hence the following discussion of incommensurability seems relevant.
See also von Fritz and incommensurability in The New York Times (March 8, 2011).
For mathematical remarks related to the 10-dot triangular array of von Fritz, diagonals, and bowling, see this journal on Nov. 8, 2011— "Stoned."
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Wednesday, June 13, 2012
State of the Art
The new June/July issue of the AMS Notices
on a recent Paris exhibit of art and mathematics—
Mathématiques, un dépaysement soudain
Exhibit at the Fondation Cartier, Paris
October 21, 2011–March 18, 2012
… maybe walking
into the room was supposed to evoke the kind of
dépaysement for which the exhibition is named
(the word dépaysement refers to the sometimes
disturbing feeling one gets when stepping outside
of one’s usual reference points). I was with
my six-year-old daughter, who quickly gravitated
toward the colorful magnetic tiles on the wall that
visitors could try to fit together. She spent a good
half hour there, eventually joining forces with a
couple of young university students. I would come
and check on her every once in a while and heard
some interesting discussions about whether or not
it was worth looking for patterns to help guide the
placing of the tiles. The fifteen-year age difference
didn’t seem to bother anyone.
The tiles display was one of the two installations
here that offered the visitor a genuine chance to
engage in mathematical activity, to think about
pattern and structure while satisfying an aesthetic
urge to make things fit and grow….
The Notices included no pictures with this review.
A search to find out what sort of tiles were meant
led, quite indirectly, to the following—
The search indicated it is unlikely that these Truchet tiles
were the ones on exhibit.
Nevertheless, the date of the above French weblog post,
1 May 2011, is not without interest in the context of
today's previous post. (That post was written well before
I had seen the new AMS Notices issue online.)
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Thursday, May 10, 2012
For Thor’s Day
Today's previous post was "Midnight in Oslo (continued)."
The link "a 4-element set" in "Midnight"
was to a more elaborate structure in a post titled "Tesseract."
In memory of an Oslo "hero of midnight"
(a phrase quoted here last September 1)—
A search for material that is more entertaining—
Odin 's Tesseract.
See also a related Hollywood story in The Washington Post .
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Friday, April 27, 2012
Paradigms Lost continues…
This post was suggested by Paradigms Lost
(a post cited here a year ago today),
by David Weinberger's recent essay "Shift Happens,"
and by today's opening of "The Raven."
David Weinberger in The Chronicle of Higher Education , April 22—
"… Kuhn was trying to understand how Aristotle could be such a brilliant natural scientist except when it came to understanding motion. Aristotle's idea that stones fall and fire rises because they're trying to get to their natural places seems like a simpleton's animism.
Then it became clear to Kuhn all at once. Ever since Newton, we in the West have thought movement changes an object's position in neutral space but does not change the object itself. For Aristotle, a change in position was a change in a quality of the object, and qualitative change tended toward an asymmetric actualization of potential: an acorn becomes an oak, but an oak never becomes an acorn. Motion likewise expressed a tendency for things to actualize their essence by moving to their proper place. With that, 'another initially strange part of Aristotelian doctrine begins to fall into place,' Kuhn wrote in The Road Since Structure ."
Dr. John Raven (of Raven's Progressive Matrices)—
"… these tools cannot be immediately applied within our current workplaces, educational systems, and public management systems because the operation of these systems is determined, not by personal developmental or societal needs, but by a range of latent, rarely discussed, and hard to influence sociological forces.
But this is not a cry of despair: It points to another topic which has been widely neglected by psychologists: It tells us that human behaviour is not mainly determined by internal properties— such as talents, attitudes, and values— but by external social forces. Such a transformation in psychological thinking and theorising is as great as the transformation Newton introduced into physics by noting that the movement of inanimate objects is not determined by internal, 'animistic,' properties of the objects but by invisible external forces which act upon them— invisible forces that can nevertheless be mapped, measured, and harnessed to do useful work for humankind.
So this brings us to our fourth conceptualisation and measurement topic: How are these social forces to be conceptualised, mapped, measured, and harnessed in a manner analogous to the way in which Newton made it possible to harness the destructive forces of the wind and the waves to enable sailing boats to get to their destinations?"
Before Newton, boats never arrived?
Comments Off on Paradigms Lost continues…
Thursday, March 1, 2012
Block That Metaphor:
The Cube Model and Peano Arithmetic
The eightfold cube model of the Fano plane may or may not have influenced a new paper (with the date Feb. 10, 2011, in its URL) on an attempted consistency proof of Peano arithmetic—
The Consistency of Arithmetic, by Storrs McCall
"Is Peano arithmetic (PA) consistent? This paper contains a proof that it is. …
Axiomatic proofs we may categorize as 'syntactic', meaning that they concern only symbols and the derivation of one string of symbols from another, according to set rules. 'Semantic' proofs, on the other hand, differ from syntactic proofs in being based not only on symbols but on a non-symbolic, non-linguistic component, a domain of objects. If the sole paradigm of 'proof ' in mathematics is 'axiomatic proof ', in which to prove a formula means to deduce it from axioms using specified rules of inference, then Gödel indeed appears to have had the last word on the question of PA-consistency. But in addition to axiomatic proofs there is another kind of proof. In this paper I give a proof of PA's consistency based on a formal semantics for PA. To my knowledge, no semantic consistency proof of Peano arithmetic has yet been constructed.
The difference between 'semantic' and 'syntactic' theories is described by van Fraassen in his book The Scientific Image :
"The syntactic picture of a theory identifies it with a body of theorems, stated in one particular language chosen for the expression of that theory. This should be contrasted with the alternative of presenting a theory in the first instance by identifying a class of structures as its models. In this second, semantic, approach the language used to express the theory is neither basic nor unique; the same class of structures could well be described in radically different ways, each with its own limitations. The models occupy centre stage." (1980, p. 44)
Van Fraassen gives the example on p. 42 of a consistency proof in formal geometry that is based on a non-linguistic model. Suppose we wish to prove the consistency of the following geometric axioms:
A1. For any two lines, there is at most one point that lies on both.
A2. For any two points, there is exactly one line that lies on both.
A3. On every line there lie at least two points.
The following diagram shows the axioms to be consistent:
Figure 1
The consistency proof is not a 'syntactic' one, in which the consistency of A1-A3 is derived as a theorem of a deductive system, but is based on a non-linguistic structure. It is a semantic as opposed to a syntactic proof. The proof constructed in this paper, like van Fraassen's, is based on a non-linguistic component, not a diagram in this case but a physical domain of three-dimensional cube-shaped blocks. ….
… The semantics presented in this paper I call 'block semantics', for reasons that will become clear…. Block semantics is based on domains consisting of cube-shaped objects of the same size, e.g. children's wooden building blocks. These can be arranged either in a linear array or in a rectangular array, i.e. either in a row with no space between the blocks, or in a rectangle composed of rows and columns. A linear array can consist of a single block, and the order of individual blocks in a linear or rectangular array is irrelevant. Given three blocks A, B and C, the linear arrays ABC and BCA are indistinguishable. Two linear arrays can be joined together or concatenated into a single linear array, and a rectangle can be re-arranged or transformed into a linear array by successive concatenation of its rows. The result is called the 'linear transformation' of the rectangle. An essential characteristic of block semantics is that every domain of every block model is finite. In this respect it differs from Tarski’s semantics for first-order logic, which permits infinite domains. But although every block model is finite, there is no upper limit to the number of such models, nor to the size of their domains.
It should be emphasized that block models are physical models, the elements of which can be physically manipulated. Their manipulation differs in obvious and fundamental ways from the manipulation of symbols in formal axiomatic systems and in mathematics. For example the transformations described above, in which two linear arrays are joined together to form one array, or a rectangle of blocks is re-assembled into a linear array, are physical transformations not symbolic transformations. …"
— Storrs McCall, Department of Philosophy, McGill University
See also…
- Finite Geometry and Physical Space,
- McCall in the book search of this morning's Salvation for Gopnik, and
- the post Declarations of Jan. 29, 2012.
Comments Off on Block That Metaphor:
Friday, February 17, 2012
Physics vs. Geometry
Physics
The February 2012 issue of Scientific American
has a cover article titled "Is Space Digital?".
The article discusses whether physical space
"is made of chunks. Blocks. Bits."
Maybe it is, maybe it isn't.
Geometry
The word "space" in pure mathematics
(as opposed to physics) applies to
a great variety of structures.
Some are continuous, some are not.
For some purely mathematical structures
that are not continuous, (i.e., are made of
"chunks, blocks, bits") see finitegeometry.org/sc —
in particular, the pages on Finite Geometry and Physical Space
and on Noncontinuous Groups.
The geometry of these structures may or may not eventually
be relevant to the "21st-century physics" discussed
in the February Scientific American.
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Thursday, February 2, 2012
Die Lichtung
See January 4th, 2012.
(This link resulted from an application of Heidegger's
philosophy of "the opening" and "the shining" (Die Lichtung ).)
See also The Shining of May 29.
Update of 12:19 AM Feb. 3, 2012—
The undated (but cached by Google on January 4th, 2012)
unsigned post from a deleted weblog linked to above as
"an application" is also available in a version that is signed
(but still undated).
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Tuesday, January 10, 2012
Defining Form
(Continued from Epiphany and from yesterday.)
Detail from the current American Mathematical Society homepage—
Further detail, with a comparison to Dürer’s magic square—
 |
 |
The three interpenetrating planes in the foreground of Donmoyer‘s picture
provide a clue to the structure of the the magic square array behind them.
Group the 16 elements of Donmoyer’s array into four 4-sets corresponding to the
four rows of Dürer’s square, and apply the 4-color decomposition theorem.
Note the symmetry of the set of 3 line diagrams that result.
Now consider the 4-sets 1-4, 5-8, 9-12, and 13-16, and note that these
occupy the same positions in the Donmoyer square that 4-sets of
like elements occupy in the diamond-puzzle figure below—
Thus the Donmoyer array also enjoys the structural symmetry,
invariant under 322,560 transformations, of the diamond-puzzle figure.
Just as the decomposition theorem’s interpenetrating lines explain the structure
of a 4×4 square , the foreground’s interpenetrating planes explain the structure
of a 2x2x2 cube .
For an application to theology, recall that interpenetration is a technical term
in that field, and see the following post from last year—
| Saturday, June 25, 2011
— m759 @ 12:00 PM “… the formula ‘Three Hypostases in one Ousia ‘
Ousia
|
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Sunday, January 8, 2012
Big Apple
“…the nonlinear characterization of Billy Pilgrim
emphasizes that he is not simply an established
identity who undergoes a series of changes but
all the different things he is at different times.”
This suggests that the above structure
be viewed as illustrating not eight parts
but rather 8! = 40,320 parts.
"The Cardinal seemed a little preoccupied today."
The New Yorker , May 13, 2002
See also a note of May 14 , 2002.
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Sunday, January 1, 2012
Sunday Shul
"… myths are stories, and like all narratives
they unravel through time, whereas grids
are not only spatial to start with,
they are visual structures that explicitly reject
a narrative or sequential reading of any kind."
— Rosalind Krauss in "Grids,"
October (Summer 1979), 9: 50-64.
Counterexample—
The Ninefold Square
See Coxeter and the Aleph and Ayn Sof—
| Mathematics and Narrative, Illustrated |
|
 Mathematics |
 Narrative |
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Saturday, December 24, 2011
Stevens for Christmas Eve
A search for Wallace Stevens ebooks
today at Alibris yielded 24 results.
I selected one to order—
Wallace Stevens: A World of Transforming Shapes .
From that book—
Stevens's phrase "diamond globe" in this context suggests an image search
on permutahedron + stone + log24 .
For the results of that search (2 MB), click here.
Some background for the phrase used in the search—
See a photo by Mike Zabrocki from June 4, 2011.
See also a Log24 image and a generalization of the underlying structure.
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Monday, November 21, 2011
Random Reference
Joseph T. Clark, S. J., Conventional Logic and Modern Logic:
A Prelude to Transition (Philosophical Studies of the American
Catholic Philosophical Association, III) Woodstock, Maryland:
Woodstock College Press, 1952—
| Alonzo Church, "Logic: formal, symbolic, traditional," Dictionary of Philosophy (New York: Philosophical Library, 1942), pp. 170-182. | The contents of this ambitious Dictionary are most uneven. Random reference to its pages is dangerous. But this contribution is among its best. It is condensed. But not dense. A patient and attentive study will pay big dividends in comprehension. Church knows the field and knows how to depict it. A most valuable reference. |
Another book to which random reference is dangerous—
For greater depth, see "Cassirer and Eddington on Structures,
Symmetry and Subjectivity" in Steven French's draft of
"Symmetry, Structure and the Constitution of Objects"
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Sunday, November 13, 2011
Sermon–
The Space Case
"A generation lost in space"
— Don McLean, "American Pie"
Last night's post discussed Jim Dodge's fictional vision of a "spherical diamond" related to physics.
For some background, see Poetry and Physics (April 25, 2011).
That post quotes a July 2008 New Yorker article —
By Benjamin Wallace-Wells, contributing editor at Rolling Stone
and sometime writer on space—
“There’s a dream that underlying the physical universe is some beautiful mathematical structure, and that the job of physics is to discover that,” Smolin told me later. “The dream is in bad shape,” he added. “And it’s a dream that most of us are like recovering alcoholics from.” Lisi’s talk, he said, “was like being offered a drink.”
Or a toke.
"Now John at the bar is a friend of mine
He gets me my drinks for free
And he's quick with a joke or to light up your smoke
But there's someplace that he'd rather be"
— Billy Joel, "Piano Man"
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Wednesday, November 2, 2011
The Poetry of Universals
A search today, All Souls Day, for relevant learning
at All Souls College, Oxford, yields the person of
Sir Michael Dummett and the following scholarly page—
My own background is in mathematics rather than philosophy.
From a mathematical point of view, the cells discussed above
seem related to some "universals" in an example of Quine.
In Quine's example,* universals are certain equivalence classes
(those with the "same shape") of a family of figures
(33 convex regions) selected from the 28 = 256 subsets
of an eight-element set of plane regions.
A smaller structure, closer to Wright's concerns above,
is a universe of 24 = 16 subsets of a 4-element set.
The number of elements in this universe of Concepts coincides,
as it happens, with the number obtained by multiplying out
the title of T. S. Eliot's Four Quartets .
For a discussion of functions that map "cells" of the sort Wright
discusses— in the quartets example, four equivalence classes,
each with four elements, that partition the 16-element universe—
onto a four-element set, see Poetry's Bones.
For some philosophical background to the Wright passage
above, see "The Concept Horse," by Harold W. Noonan—
Chapter 9, pages 155-176, in Universals, Concepts, and Qualities ,
edited by P. F. Strawson and Arindam Chakrabarti,
Ashgate Publishing, 2006.
For a different approach to that concept, see Devil's Night, 2011.
* Admittedly artificial. See From a Logical Point of View , IV, 3
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Monday, October 31, 2011
Beauty, Truth, Halloween
On Halloween…
"Remember that for Ockham there is nothing in the universe that is
in any way universal except a concept or word: there are no real
natures shared by many things. However, things do resemble one
another, some things more closely than others. So the various
degrees of resemblance give a foundation in reality for our conceptual
structures, such as Porphyry's tree.
Now resemblance (or similitude or likeness) is a relation.
If such relations are realities, then we can say that there are realities
out there that correspond to our conceptual structures."
— R.J. Kilcullen at Macquarie University, course labeled Phil360
"The kernel of a homomorphism is always a congruence.
Indeed, every congruence arises as a kernel."
— Congruence Relation, section on Universal Algebra, in Wikipedia
"Beauty then is a relation."
"An Attempt to Understand the Problem of Universals"
is the title of a talk by Fabian Geier, University of Bamberg—
"The talk was held at Gdańsk University on May 26th 2008."
Related material— Stevie Nicks turns 60.
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Wednesday, October 26, 2011
Erlanger and Galois
Peter J. Cameron yesterday on Galois—
"He was killed in a duel at the age of 20…. His work languished for another 14 years until Liouville published it in his Journal; soon it was recognised as the foundation stone of modern algebra, a position it has never lost."
Here Cameron is discussing Galois theory, a part of algebra. Galois is known also as the founder* of group theory, a more general subject.
Group theory is an essential part of modern geometry as well as of modern algebra—
"In der Galois'schen Theorie, wie hier, concentrirt sich das Interesse auf Gruppen von Änderungen. Die Objecte, auf welche sich die Änderungen beziehen, sind allerdings verschieden; man hat es dort mit einer endlichen Zahl discreter Elemente, hier mit der unendlichen Zahl von Elementen einer stetigen Mannigfaltigkeit zu thun."
— Felix Christian Klein, Erlanger Programm , 1872
("In the Galois theory, as in ours, the interest centres on groups of transformations. The objects to which the transformations are applied are indeed different; there we have to do with a finite number of discrete elements, here with the infinite number of elements in a continuous manifoldness." (Translated by M.W. Haskell, published in Bull. New York Math. Soc. 2, (1892-1893), 215-249))
Related material from Hermann Weyl, Symmetry , Princeton University Press, 1952 (paperback reprint of 1982, pp. 143-144)—
"A field is perhaps the simplest algebraic structure we can invent. Its elements are numbers…. Space is another example of an entity endowed with a structure. Here the elements are points…. What we learn from our whole discussion and what has indeed become a guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity Σ try to determine is group of automorphisms , the group of those element-wise transformations which leave all structural relations undisturbed. You can expect to gain a deep insight into the constitution of Σ in this way."
For a simple example of a group acting on a field (of 8 elements) that is also a space (of 8 points), see Generating the Octad Generator and Knight Moves.
* Joseph J. Rotman, An Introduction to the Theory of Groups , 4th ed., Springer, 1994, page 2
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Possibly Related
 |
Peter Woit, phrase from a weblog post on October 25th, 2011—
"In possibly related news…."
For 779, see post 779 in this weblog.
For 8974, see Hollywood Endings.
For 082, see page 82 of Culture and Value , ed. G.H. von Wright, tr. Peter Winch (Oxford 1980) (as quoted by M. Jamie Ferreira in "The Point Outside the World: Kierkegaard and Wittgenstein on Nonsense, Paradox and Religion," Religious Studies , Vol. 30, March 1994, pp. 29-44, reprinted in Wittgenstein Studies (1997))—
Wittgenstein: “God’s essence is supposed to guarantee his existence— but what this really means is that what is here at issue is not the existence of something.”
For 0372, see page 372 in Essays of Three Decades , by Thomas Mann, translated by H. T. Lowe-Porter, Alfred A. Knopf, 1947 ("Schopenhauer," 1938, pp. 372-410)—
THE PLEASURE we take in a metaphysical system, the gratification purveyed by the intellectual organization of the world into a closely reasoned, complete, and balanced structure of thought, is always of a pre-eminently aesthetic kind. It flows from the same source as the joy, the high and ever happy satisfaction we get from art, with its power to shape and order its material, to sort out life's manifold confusions so as to give us a clear and general view.
Truth and beauty must always be referred the one to the other. Each by itself, without the support given by the other, remains a very fluctuating value. Beauty that has not truth on its side and cannot have reference to it, does not live in it and through it, would be an empty chimera— and "What is truth?"
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Wednesday, October 12, 2011
High White Noon
Grid from a post linked to in yesterday's 24 Hour DeLillo—
For an example of this grid as slow art , consider the following—
"One can show that the binary tetrahedral group
is isomorphic to the special linear group SL(2,3)—
the group of all 2×2 matrices over the finite field F3
with unit determinant." —Wikipedia
As John Baez has noted, these two groups have the same structure as the geometric 24-cell.
For the connection of the grid to the groups and the 24-cell, see Visualizing GL(2,p).
Related material—
The 3×3 grid has been called a symbol of Apollo (Greek god of reason and of the sun).
"This is where we sat through his hushed hour,
a torchlit sky, the closeness of hills barely visible
at high white noon." — Don DeLillo, Point Omega
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Wednesday, September 21, 2011
Symmetric Generation
Suggested by yesterday's Relativity Problem Revisited and by Cassirer on Objectivity—
From Symmetric Generation of Groups , by R.T. Curtis (Cambridge U. Press, 2007)—
"… we are saying much more than that
some set of seven involutions, which would be a very weak
requirement. We are asserting that M 24 is generated by a set
of seven involutions which possesses all the symmetries of
acting on the points of the 7-point projective plane…."
— Symmetric Generation , p. 41
"It turns out that this approach is particularly revealing and that
many simple groups, both sporadic and classical, have surprisingly
simple definitions of this type."
— Symmetric Generation , p. 42
See also (click to enlarge)—
Cassirer's remarks connect the concept of objectivity with that of object .
The above quotations perhaps indicate how the Mathieu group
"This is the moment which I call epiphany. First we recognise that the object is one integral thing, then we recognise that it is an organised composite structure, a thing in fact: finally, when the relation of the parts is exquisite, when the parts are adjusted to the special point, we recognise that it is that thing which it is. Its soul, its whatness, leaps to us from the vestment of its appearance. The soul of the commonest object, the structure of which is so adjusted, seems to us radiant. The object achieves its epiphany."
— James Joyce, Stephen Hero
For a simpler object "which possesses all the symmetries of
For symmetric generation of
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Sunday, September 18, 2011
What Rough Beast
Lurching Toward Decision
"Suskind… nails, I think, Obama's intellectual blind spot. Indeed, Obama himself nails it, telling Suskind that he was too inclined to search for 'the perfect technical answer' to the myriad of complex issues coming at him."
— Frank Rich on Ron Suskind's new book about the White House, Confidence Men
Very distantly related material—
From "Confidence Game," an Oct. 12, 2008, post in this journal, a quasi-European perspective—
|
Kaleidoscope turning… – Roger Zelazny, Eye of Cat |
See also …
Gravity’s Rainbow , Penguin Classics, 1995, page 742:
"… knowing his Tarot, we would expect to look among the Humility, among the gray and preterite souls, to look for him adrift in the hostile light of the sky, the darkness of the sea….
Now there’s only a long cat’s-eye of bleak sunset left over the plain tonight, bright gray against a purple ceiling of clouds, with an iris of
742"
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Anatomy of a Cube
R.D. Carmichael's seminal 1931 paper on tactical configurations suggests
a search for later material relating such configurations to block designs.
Such a search yields the following—
"… it seems that the relationship between
BIB [balanced incomplete block ] designs
and tactical configurations, and in particular,
the Steiner system, has been overlooked."
— D. A. Sprott, U. of Toronto, 1955
The figure by Cullinane included above shows a way to visualize Sprott's remarks.
For the group actions described by Cullinane, see "The Eightfold Cube" and
"A Simple Reflection Group of Order 168."
Update of 7:42 PM Sept. 18, 2011—
From a Summer 2011 course on discrete structures at a Berlin website—
A different illustration of the eightfold cube as the Steiner system S(3, 4, 8)—
Note that only the static structure is described by Felsner, not the
168 group actions discussed (as above) by Cullinane. For remarks on
such group actions in the literature, see "Cube Space, 1984-2003."
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Saturday, September 3, 2011
The Galois Tesseract (continued)
A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
two-thirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79T-A37, Notices , Feb. 1979).
Here is some supporting material—
The passage from Carmichael above emphasizes the importance of
the 4×4 square within the MOG.
The passage from Conway and Sloane, in a book whose first edition
was published in 1988, makes explicit the structure of the MOG's
4×4 square as the affine 4-space over the 2-element Galois field.
The passage from Curtis (1974, published in 1976) describes 35 sets
of four "special tetrads" within the 4×4 square of the MOG. These
correspond to the 35 sets of four parallel 4-point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 "special tetrads" rather by the parity
of their intersections with the square's rows and columns.
The affine structure appears in the 1979 abstract mentioned above—
The "35 structures" of the abstract were listed, with an application to
Latin-square orthogonality, in a note from December 1978—
See also a 1987 article by R. T. Curtis—
Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:
“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”
(Received July 20 1987)
– Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353
* For instance:
Update of Sept. 4— This post is now a page at finitegeometry.org.
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Friday, September 2, 2011
Rigged?
Sarah Tomlin in a Nature article on the July 12-15 2005 Mykonos meeting on Mathematics and Narrative—
"Today, Mazur says he has woken up to the power of narrative, and in Mykonos gave an example of a 20-year unsolved puzzle in number theory which he described as
Michel Chaouli in "How Interactive Can Fiction Be?" (Critical Inquiry 31, Spring 2005), pages 613-614—
"…a simple thought experiment….*
… If the cliffhanger is done well, it will not simply introduce a wholly unprepared turn into the narrative (a random death, a new character, an entirely unanticipated obstacle) but rather tighten the configuration of known elements to such a degree that the next step appears both inevitable and impossible. We feel the pressure rising to a breaking point, but we simply cannot foresee where the complex narrative structure will give way. This interplay of necessity and contingency produces our anxious— and highly pleasurable— speculation about the future path of the story. But if we could determine that path even slightly, we would narrow the range of possible outcomes and thus the uncertainty in the play of necessity and contingency. The world of the fiction would feel, not open, but rigged."
* The idea of the thought experiment emerged in a conversation with Barry Mazur.
Barry Mazur in the preface to his 2003 book Imagining Numbers—
"But the telltale adjective real suggests two things: that these numbers are somehow real to us and that, in contrast, there are unreal numbers in the offing. These are the imaginary numbers .
The imaginary numbers are well named, for there is some imaginative work to do to make them as much a part of us as the real numbers we use all the time to measure for bookshelves.
This book began as a letter to my friend Michel Chaouli. The two of us had been musing about whether or not one could 'feel' the workings of the imagination in its various labors. Michel had also mentioned that he wanted to 'imagine imaginary numbers.' That very (rainy) evening, I tried to work up an explanation of the idea of these numbers, still in the mood of our conversation."
See also The Galois Quaternion and 2/19.
New York Lottery last evening
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Thursday, September 1, 2011
How It Works
“Design is how it works.” — Steven Jobs (See Symmetry and Design.)
“By far the most important structure in design theory is the Steiner system S(5, 8, 24).”
— “Block Designs,” by Andries E. Brouwer
The name Carmichael is not to be found in Booher’s thesis. A book he does cite for the history of S(5,8,24) gives the date of Carmichael’s construction of this design as 1937. It should be dated 1931, as the following quotation shows—
From Log24 on Feb. 20, 2010—
“The linear fractional group modulo 23 of order 24•23•11 is often represented as a doubly transitive group of degree 24 on the symbols ∞, 0, 1, 2,…, 22. This transitive group contains a subgroup of order 8 each element of which transforms into itself the set ∞, 0, 1, 3, 12, 15, 21, 22 of eight elements, while the whole group transforms this set into 3•23•11 sets of eight each. This configuration of octuples has the remarkable property that any given set of five of the 24 symbols occurs in one and just one of these octuples. The largest permutation group Γ on the 24 symbols, each element of which leaves this configuration invariant, is a five-fold transitive group of degree 24 and order 24•23•22•21•20•48. This is the Mathieu group of degree 24.”
– R. D. Carmichael, “Tactical Configurations of Rank Two,” in American Journal of Mathematics, Vol. 53, No. 1 (Jan., 1931), pp. 217-240
Epigraph from Ch. 4 of Design Theory , Vol. I:
“Es is eine alte Geschichte,
doch bleibt sie immer neu ”
—Heine (Lyrisches Intermezzo XXXIX)
See also “Do you like apples?“
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Sunday, August 28, 2011
The Cosmic Part
Yesterday's midday post, borrowing a phrase from the theology of Marvel Comics,
offered Rubik's mechanical contrivance as a rather absurd "Cosmic Cube."
A simpler candidate for the "Cube" part of that phrase:
The Eightfold Cube
As noted elsewhere, a simple reflection group* of order 168 acts naturally on this structure.
"Because of their truly fundamental role in mathematics,
even the simplest diagrams concerning finite reflection groups
(or finite mirror systems, or root systems—
the languages are equivalent) have interpretations
of cosmological proportions."
— Alexandre V. Borovik in "Coxeter Theory: The Cognitive Aspects"
Borovik has a such a diagram—
The planes in Borovik's figure are those separating the parts of the eightfold cube above.
In Coxeter theory, these are Euclidean hyperplanes. In the eightfold cube, they represent three of seven projective points that are permuted by the above group of order 168.
In light of Borovik's remarks, the eightfold cube might serve to illustrate the "Cosmic" part of the Marvel Comics phrase.
For some related theological remarks, see Cube Trinity in this journal.
Happy St. Augustine's Day.
* I.e., one generated by reflections : group actions that fix a hyperplane pointwise. In the eightfold cube, viewed as a vector space of 3 dimensions over the 2-element Galois field, these hyperplanes are certain sets of four subcubes.
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Wednesday, August 24, 2011
Symmetry
An article from cnet.com tonight —
For Jobs, design is about more than aesthetics
By: Jay Greene
… The look of the iPhone, defined by its seamless pane of glass, its chrome border, its perfect symmetry, sparked an avalanche of copycat devices that tried to mimic its aesthetic.
Virtually all of them failed. And the reason is that Jobs understood that design wasn't merely about what a product looks like. In a 2003 interview with the New York Times' Rob Walker detailing the genesis of the iPod, Jobs laid out his vision for product design.
''Most people make the mistake of thinking design is what it looks like,'' Jobs told Walker. "People think it's this veneer— that the designers are handed this box and told, 'Make it look good!' That's not what we think design is. It's not just what it looks like and feels like. Design is how it works.''
Related material: Open, Sesame Street (Aug. 19) continues… Brought to you by the number 24—
"By far the most important structure in design theory is the Steiner system
— "Block Designs," by Andries E. Brouwer (Ch. 14 (pp. 693-746) of Handbook of Combinatorics , Vol. I, MIT Press, 1995, edited by Ronald L. Graham, Martin Grötschel, and László Lovász, Section 16 (p. 716))
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Wednesday, August 10, 2011
Objectivity
From math16.com—
Quotations on Realism
|
The story of the diamond mine continues
(see Coordinated Steps and Organizing the Mine Workers)—
From The Search for Invariants (June 20, 2011):
The conclusion of Maja Lovrenov's
"The Role of Invariance in Cassirer’s Interpretation of the Theory of Relativity"—
"… physical theories prove to be theories of invariants
with regard to certain groups of transformations and
it is exactly the invariance that secures the objectivity
of a physical theory."
— SYNTHESIS PHILOSOPHICA 42 (2/2006), pp. 233–241
Related material from Sunday's New York Times travel section—
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Saturday, August 6, 2011
Correspondences
Comme de longs échos qui de loin se confondent
Dans une ténébreuse et profonde unité….
— Baudelaire, “Correspondances ”
From “A Four-Color Theorem”—
Figure 1
Note that this illustrates a natural correspondence
between
(A) the seven highly symmetrical four-colorings
of the 4×2 array at the left of Fig. 1, and
(B) the seven points of the smallest
projective plane at the right of Fig. 1.
To see the correspondence, add, in binary
fashion, the pairs of projective points from the
“points” section that correspond to like-colored
squares in a four-coloring from the left of Fig. 1.
(The correspondence can, of course, be described
in terms of cosets rather than of colorings.)
A different correspondence between these 7 four-coloring
structures and these 7 projective-line structures appears in
a structural analysis of the Miracle Octad Generator
(MOG) of R.T. Curtis—
Figure 2
| Here the correspondence between the 7 four-coloring structures (left section) and the 7 projective-line structures (center section) is less obvious, but more fruitful. It yields, as shown, all of the 35 partitions of an 8-element set (an 8-set ) into two 4-sets. The 7 four-colorings in Fig. 2 also appear in the 35 4×4 parts of the MOG that correspond, in a way indicated by Fig. 2, to the 35 8-set paritions. This larger correspondence— of 35 4×2 arrays with 35 4×4 arrays— is the MOG, at least as it was originally defined. See The MOG, Generating the Octad Generator, and Eightfold Geometry.
For some applications of the Curtis MOG, see |
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Tuesday, July 26, 2011
The Case of the Missing Smile
In today's online New York Times , Roger Cohen quotes a manifesto—
A more complete excerpt—
Note that Cohen omits the concluding punctuation—
three exclamation points and a smile emoticon—
!!!:-)
(Compare and contrast with the smile of Hannibal Lecter.)
Related material from this journal on the following day, Flag Day, June 14—
Note that the structure of the central flag above |
See also the remark of author Siri Hustvedt (of Norwegian-American
background) that was quoted here Sunday.
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Friday, July 22, 2011
The Aristophanic View
"Some mathematicians are birds, others are frogs.
Birds fly high in the air and survey broad vistas of
mathematics out to the far horizon. They delight in
concepts that unify our thinking and bring together
diverse problems from different parts of the
landscape. Frogs live in the mud below and see
only the flowers that grow nearby. They delight in
the details of particular objects, and they solve
problems one at a time. I happen to be a frog, but
many of my best friends are birds. The main theme
of my talk tonight is this. Mathematics needs both
birds and frogs. Mathematics is rich and beautiful
because birds give it broad visions and frogs give it
intricate details. Mathematics is both great art and
important science, because it combines generality
of concepts with depth of structures. It is stupid to
claim that birds are better than frogs because they
see farther, or that frogs are better than birds
because they see deeper. The world of mathematics
is both broad and deep, and we need birds and
frogs working together to explore it.
This talk is called the Einstein lecture…."
— Freeman Dyson, Notices of the American
Mathematical Society , February 2009
The Didion reading was suggested by the "6212" in yesterday evening's New York Lottery.
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Monday, July 11, 2011
And/Or Problem
"It was the simultaneous emergence
and mutual determination
of probability and logic
that von Neumann found intriguing
and not at all well understood."
Context:
Update of 7 AM ET July 12, 2011—
Freeman Dyson on John von Neumann's
Sept. 2, 1954, address to the International
Congress of Mathematicians on
"Unsolved Problems in Mathematics"—
…."The hall was packed with
mathematicians, all expecting to hear a brilliant
lecture worthy of such a historic occasion. The
lecture was a huge disappointment. Von Neumann
had probably agreed several years earlier to give
a lecture about unsolved problems and had then
forgotten about it. Being busy with many other
things, he had neglected to prepare the lecture.
Then, at the last moment, when he remembered
that he had to travel to Amsterdam and say something
about mathematics, he pulled an old lecture
from the 1930s out of a drawer and dusted it off.
The lecture was about rings of operators, a subject
that was new and fashionable in the 1930s. Nothing
about unsolved problems. Nothing about the
future."
— Notices of the American Mathematical Society ,
February 2009, page 220
For a different account, see Giovanni Valente's
2009 PhD thesis from the University of Maryland,
Chapter 2, "John von Neumann's Mathematical
'Utopia' in Quantum Theory"—
"During his lecture von Neumann discussed operator theory and its con-
nections with quantum mechanics and noncommutative probability theory,
pinpointing a number of unsolved problems. In his view geometry was so tied
to logic that he ultimately outlined a logical interpretation of quantum prob-
abilities. The core idea of his program is that probability is invariant under
the symmetries of the logical structure of the theory. This is tantamount to
a formal calculus in which logic and probability arise simultaneously. The
problem that exercised von Neumann then was to construct a geometrical
characterization of the whole theory of logic, probability and quantum me-
chanics, which could be derived from a suitable set of axioms…. As he
himself finally admitted, he never managed to set down the sought-after
axiomatic formulation in a way that he felt satisfactory."
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Accentuate the Positive
An image that may be viewed as
a cube with a “+“ on each face—
The eightfold cube
Underlying structure
For the Pope and others on St. Benedict’s Day
who prefer narrative to mathematics—
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Wednesday, July 6, 2011
Nordstrom-Robinson Automorphisms
A 2008 statement on the order of the automorphism group of the Nordstrom-Robinson code—
"The Nordstrom-Robinson code has an unusually large group of automorphisms (of order 8! = 40,320) and is optimal in many respects. It can be found inside the binary Golay code."
— Jürgen Bierbrauer and Jessica Fridrich, preprint of "Constructing Good Covering Codes for Applications in Steganography," Transactions on Data Hiding and Multimedia Security III, Springer Lecture Notes in Computer Science, 2008, Volume 4920/2008, 1-22
A statement by Bierbrauer from 2004 has an error that doubles the above figure—
The automorphism group of the binary Golay code G is the simple Mathieu group M24 of order
— Jürgen Bierbrauer, "Nordstrom-Robinson Code and A7-Geometry," preprint dated April 14, 2004, published in Finite Fields and Their Applications , Volume 13, Issue 1, January 2007, Pages 158-170
The error is corrected (though not detected) later in the same 2004 paper—
In fact the symmetry group of the octacode is a semidirect product of an elementary abelian group of order 16 and the simple group GL(3, 2) of order 168. This constitutes a large automorphism group (of order 2688), but the automorphism group of NR is larger yet as we saw earlier (order 40,320).
For some background, see a well-known construction of the code from the Miracle Octad Generator of R.T. Curtis—
For some context, see the group of order 322,560 in Geometry of the 4×4 Square.
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Monday, June 27, 2011
Galois Cube Revisited
The 3×3×3 Galois Cube
This cube, unlike Rubik's, is a
purely mathematical structure.
Its properties may be compared
with those of the order-2 Galois
cube (of eight subcubes, or
elements ) and the order-4 Galois
cube (of 64 elements). The
order-3 cube (of 27 elements)
lacks, because it is based on
an odd prime, the remarkable
symmetry properties of its smaller
and larger cube neighbors.
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Tuesday, June 14, 2011
Dawn’s Early Light
Note that the structure of the central flag above is not unlike that of the skull and crossbones flag.
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Sunday, June 12, 2011
For Commencement Day at Stanford
See also June 2, 2007, and June 19, 2010,
as well as Kernel of Eternity in this journal.
Some background— Square of Opposition
in the Stanford Encyclopedia of Philosophy
and Deep Structures in this journal.
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Sunday, June 5, 2011
Edifice Complex
"Total grandeur of a total edifice,
Chosen by an inquisitor of structures
For himself. He stops upon this threshold,
As if the design of all his words takes form
And frame from thinking and is realized."
— Wallace Stevens, "To an Old Philosopher in Rome"
The following edifice may be lacking in grandeur,
and its properties as a configuration were known long
before I stumbled across a description of it… still…
"What we do may be small, but it has
a certain character of permanence…."
— G.H. Hardy, A Mathematician's Apology
The Kummer 166 Configuration
as seen by Kantor in 1969— (pdf, 2.5 MB)
For some background, see Configurations and Squares.
For some quite different geometry of the 4×4 square that is
original with me, see a page with that title. (The geometry's
importance depends in part on its connection with the
Miracle Octad Generator (MOG) of R.T. Curtis. I of course
had nothing to do with the MOG's discovery, but I do claim credit
for discovering some geometric properties of the 4×4 square
that constitutes two-thirds of the MOG as originally defined .)
Related material— The Schwartz Notes of June 1.
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Thursday, May 12, 2011
But Seriously…
Yesterday's "Succor" cited the New York Lottery of Tuesday— Midday 489, Evening 886.
One interpretation of these numbers—
- 489 as the number of a page in the Collected Poems of Wallace Stevens
with verses that suggested to one author the following questions:
"How can one express one's sense of the ground of things?
What is the structure of Being itself…?" — Thomas Jensen Hines - 886 as a number applied recently to a poem by Gerard Manley Hopkins
with the notable phrase "the unchanging register of change"
Some background from Tuesday—
- 489 background: See "Groups Acting" (10:10 AM EDT)
- 886 background: See a different Hopkins poem (linked to in "Rhyme Scheme," 1:23 AM EDT)
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Tuesday, May 10, 2011
Groups Acting
The LA Times on last weekend's film "Thor"—
"… the film… attempts to bridge director Kenneth Branagh's high-minded Shakespearean intentions with Marvel Entertainment's bottom-line-oriented need to crank out entertainment product."
Those averse to Nordic religion may contemplate a different approach to entertainment (such as Taymor's recent approach to Spider-Man).
A high-minded— if not Shakespearean— non-Nordic approach to groups acting—
"What was wrong? I had taken almost four semesters of algebra in college. I had read every page of Herstein, tried every exercise. Somehow, a message had been lost on me. Groups act . The elements of a group do not have to just sit there, abstract and implacable; they can do things, they can 'produce changes.' In particular, groups arise naturally as the symmetries of a set with structure. And if a group is given abstractly, such as the fundamental group of a simplical complex or a presentation in terms of generators and relators, then it might be a good idea to find something for the group to act on, such as the universal covering space or a graph."
— Thomas W. Tucker, review of Lyndon's Groups and Geometry in The American Mathematical Monthly , Vol. 94, No. 4 (April 1987), pp. 392-394
"Groups act "… For some examples, see
- The 2×2×2 Cube,
- The Diamond 16 Puzzle,
- The Diamond Theorem, and
- Finite Geometry of the Square and Cube.
Related entertainment—
High-minded— Many Dimensions—
Not so high-minded— The Cosmic Cube—
One way of blending high and low—
The high-minded Charles Williams tells a story
in his novel Many Dimensions about a cosmically
significant cube inscribed with the Tetragrammaton—
the name, in Hebrew, of God.
The following figure can be interpreted as
the Hebrew letter Aleph inscribed in a 3×3 square—
The above illustration is from undated software by Ed Pegg Jr.
For mathematical background, see a 1985 note, "Visualizing GL(2,p)."
For entertainment purposes, that note can be generalized from square to cube
(as Pegg does with his "GL(3,3)" software button).
For the Nordic-averse, some background on the Hebrew connection—
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Sunday, May 8, 2011
Matrix
From Thomas Mann, "Schopenhauer," 1938, in Essays of Three Decades , translated by H. T. Lowe-Porter, Alfred A. Knopf, 1947, pp. 372-410—
Page 372: THE PLEASURE we take in a metaphysical system, the gratification purveyed by the intellectual organization of the world into a closely reasoned, complete, and balanced structure of thought, is always of a pre-eminently aesthetic kind. It flows from the same source as the joy, the high and ever happy satisfaction we get from art, with its power to shape and order its material, to sort out life's manifold confusions so as to give us a clear and general view.
Truth and beauty must always be referred the one to the other. Each by itself, without the support given by the other, remains a very fluctuating value. Beauty that has not truth on its side and cannot have reference to it, does not live in it and through it, would be an empty chimera— and "What is truth?"
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Saturday, May 7, 2011
Annals of Mathematics
University Diaries praised today the late Robert Nozick's pedagogical showmanship.
His scholarship was less praiseworthy. His 2001 book Invariances: The Structure of the Objective World failed, quite incredibly, to mention Hermann Weyl's classic summary of the connection between invariance and objectivity. See a discussion of Nozick in The New York Review of Books of December 19, 2002—
"… one should mention, first and foremost, the mathematician Hermann Weyl who was almost obsessed by this connection. In his beautiful little book Symmetry he tersely says, 'Objectivity means invariance with respect to the group of automorphisms….'"
See also this journal on Dec. 3, 2002, and Feb. 20, 2007.
For some context, see a search on the word stem "objectiv-" in this journal.
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Wednesday, May 4, 2011
Unity and Multiplicity
Continued from Crimson Walpurgisnacht.
Epigraphs— Two quotations from
Shakespeare's Birthday last year—
|
"I was reading Durant's section on Plato, struggling to understand his theory of the ideal Forms that lay in inviolable perfection out beyond the phantasmagoria. (That was the first, and I think the last, time that I encountered that word.)" |
|
"We tell ourselves stories in order to live…. We interpret what we see, select the most workable of multiple choices. We live entirely, especially if we are writers, by the imposition of a narrative line upon disparate images, by the 'ideas' with which we have learned to freeze the shifting phantasmagoria which is our actual experience." |
From Thomas Mann, "Schopenhauer," 1938, in Essays of Three Decades , translated by H. T. Lowe-Porter, Alfred A. Knopf, 1947, pp. 372-410—
Page 372: THE PLEASURE we take in a metaphysical system, the gratification purveyed by the intellectual organization of the world into a closely reasoned, complete, and balanced structure of thought, is always of a pre-eminently aesthetic kind. It flows from the same source as the joy, the high and ever happy satisfaction we get from art, with its power to shape and order its material, to sort out life's manifold confusions so as to give us a clear and general view.
Truth and beauty must always be referred the one to the other. Each by itself, without the support given by the other, remains a very fluctuating value. Beauty that has not truth on its side and cannot have reference to it, does not live in it and through it, would be an empty chimera— and "What is truth?"
….
Page 376: … the life of Plato was a very great event in the history of the human spirit; and first of all it was a scientific and a moral event. Everyone feels that something profoundly moral attaches to this elevation of the ideal as the only actual, above the ephemeralness and multiplicity of the phenomenal, this devaluation of the senses to the advantage of the spirit, of the temporal to the advantage of the eternal— quite in the spirit of the Christianity that came after it. For in a way the transitory phenomenon, and the sensual attaching to it, are put thereby into a state of sin: he alone finds truth and salvation who turns his face to the eternal. From this point of view Plato's philosophy exhibits the connection between science and ascetic morality.
But it exhibits another relationship: that with the world of art. According to such a philosophy time itself is merely the partial and piecemeal view which an individual holds of ideas— the latter, being outside time, are thus eternal. "Time"— so runs a beautiful phrase of Plato— "is the moving image of eternity." And so this pre-Christian, already Christian doctrine, with all its ascetic wisdom, possesses on the other hand extraordinary charm of a sensuous and creative kind; for a conception of the world as a colourful and moving phantasmagoria of pictures, which are transparencies for the ideal and the spiritual, eminently savours of the world of art, and through it the artist, as it were, first comes into his own.
From last night's online NY Times obituaries index—
"How much story do you want?" — George Balanchine
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Monday, April 25, 2011
Poetry and Physics
One approach to the storied philosophers' stone, that of Jim Dodge in Stone Junction , was sketched in yesterday's Easter post. Dodge described a mystical "spherical diamond." The symmetries of the sphere form what is called in mathematics a Lie group . The "spherical" of Dodge therefore suggests a review of the Lie group E8 in Garrett Lisi's poetic theory of everything.
A check of the Wikipedia article on Lisi's theory yields…
Diamond and E8 at Wikipedia
Related material — E8 as "a diamond with thousands of facets"—
Also from the New Yorker article—
“There’s a dream that underlying the physical universe is some beautiful mathematical structure, and that the job of physics is to discover that,” Smolin told me later. “The dream is in bad shape,” he added. “And it’s a dream that most of us are like recovering alcoholics from.” Lisi’s talk, he said, “was like being offered a drink.”
A simpler theory of everything was offered by Plato. See, in the Timaeus , the Platonic solids—
Figure from this journal on August 19th, 2008.
See also July 19th, 2008.
“It’s all in Plato, all in Plato:
bless me, what do they
teach them at these schools!”
— C. S. Lewis
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Wednesday, April 20, 2011
Ready When You Are, C.B.
This journal at 5:48 PM EST on Thursday, March 10, 2011—
Paradigms Lost
(Continued from February 19)
The cover of the April 1, 1970 second edition of
The Structure of Scientific Revolutions , by Thomas S. Kuhn—
Note the quote on the cover—
"A landmark in intellectual history."— Science
This afternoon's online New York Times—
Google today, asked to "define:landmark," yields—
- A boundary line indicated by a stone, stake, etc.
(Deu 19:14; Deu 27:17; Pro 22:28; Pro 23:10; Job 24:2).
Landmarks could not be removed without incurring the severe displeasure of God.
sacred-texts.com/bib/ebd/ebd223.htm
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