The structures of the title are the even subsets of a six-set and of
an eight-set, viewed modulo set complementation.
The "Brick Space" model of PG(5,2) —
.gif)
For the M24 relationship between these spaces, of 15 and of 63 points,
see G. M. Conwell's 1910 paper "The 3-Space PG (3,2) and Its Group,"
as well as Conwell heptads in this journal.
"As McCarthy peers through the screen, or veil, of technological modernity
to reveal the underlying symbolic structures of human experience,
The Making of Incarnation weaves a set of stories one inside the other,
rings within rings, a perpetual motion machine."
— Amazon.com description of a novel published on All Souls' Day
(Dia de los Muertos), 2021.
See also the underlying symbolic structures of Boolean functions . . .
as discussed, for instance, on Sept. 23 at medium.com —
Description of a book to be published in November —
Note the phrase "underlying combinatorial structure." AI scholium —
Rothko — "… the elimination of all obstacles between the painter and
the idea, and between the idea and the observer."
Walker Percy has similarly discussed elimination of obstacles between
the speaker and the word, and between the word and the hearer.
Click images to enlarge.
Related mathematics —
The source: http://finitegeometry.org/sc/gen/typednotes.html.
A document from the above image —
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AN INVARIANCE OF SYMMETRY BY STEVEN H. CULLINANE
We present a simple, surprising, and beautiful combinatorial
DEFINITION. A delta transform of a square array over a 4-set is
THEOREM. Every delta transform of the Klein group table has
PROOF (Sketch). The Klein group is the additive group of GF (4);
All delta transforms of the 45 matrices in the algebra generated by
THEOREM. If 1 m ≤ n2+2, there is an algebra of 4m
An induction proof constructs sets of basis matrices that yield REFERENCE S. H. Cullinane, Diamond theory (preprint). |
Update of 1:12 AM ET on Friday, Oct. 25, 2024 —
The above "invariance of symmetry" document was written in 1978
for submission to the "Research Announcements" section of the
Bulletin of the American Mathematical Society . This pro forma
submission was, of course, rejected. Though written before
I learned of similar underlying structures in the 1974 work of
R. T. Curtis on his "Miracle Octad Generator," it is not without
relevance to his work.
Today's description of Dartmouth College as a "gin-soaked gutter"
by Margaret Soltan (i.e., University Diaries) suggests a review:
Monday, November 14, 2022
|
See also "KenKen" and today's previous post.
For the significance of the date "February 20," see . . .
|
Google One: Gemini Advanced AI on 13 May 2024. Prompt: How can Weyl's 1946 remarks on coordinatization in "The Classical Groups" be applied to models of finite geometries? Response (after several drafts): In "The Classical Groups," Hermann Weyl discusses the concept of coordinatization, which is the process of assigning numerical coordinates to points in a geometric space. This process is fundamental in connecting abstract geometric concepts with concrete numerical representations, allowing for algebraic manipulation and analysis. Weyl's remarks on coordinatization have significant implications for models of finite geometries. Finite geometries are geometric systems with a finite number of points and lines, and they often exhibit different properties compared to traditional Euclidean geometry. Here's how Weyl's ideas can be applied to finite geometries: 1. **Relativity of Coordinatization:** Weyl emphasizes that the choice of coordinate system is arbitrary and that different coordinate systems can be used to describe the same geometric object. This relativity of coordinatization is particularly relevant in finite geometries, where multiple coordinate systems may exist, each with its own advantages and disadvantages. 2. **Equivalent Coordinatizations:** Weyl introduces the concept of "equivalent coordinatizations," which are different coordinate systems that can be transformed into one another through a group of transformations. In finite geometries, identifying and understanding these equivalent coordinatizations can provide deeper insights into the underlying structure and symmetries of the geometry. 3. **Objective Class of Coordinatizations:** Weyl discusses the problem of fixing an "objective class" of coordinatizations that capture essential features of the geometry while being independent of arbitrary choices. In finite geometries, this could involve identifying coordinatizations that best reflect the specific properties and relationships within the geometry. 4. **Group of Transformations:** Weyl emphasizes the role of the "group of transformations" that mediates between equivalent coordinatizations. In finite geometries, understanding these transformation groups can help classify different types of finite geometries and reveal their underlying symmetries. By applying Weyl's insights, researchers can gain a deeper understanding of finite geometries, including their classification, properties, and relationships. Additionally, exploring different coordinatization schemes can lead to new algebraic and geometric tools for studying finite geometries. For further exploration, you can delve into the works of other mathematicians like Steven H. Cullinane, who has written extensively on finite geometry and its connections to relativity and other areas of mathematics. I hope this explanation is helpful! Please let me know if you have any further questions. |
Update at about 5 PM ET the same day . . .
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Prompt: Correction: Cullinane has NOT written on finite geometry's "connections to relativity." He has, instead, quoted Weyl's remarks on what Weyl himself called "the relativity problem," but what was in fact his own term for problems involving coordinatization. Such problems are of interest both in physics and in finite geometry, but the latter is a field of pure mathematics, NOT of physics. Response: You are absolutely correct. I apologize for the error in my previous response. Steven H. Cullinane has indeed quoted Weyl's remarks on "the relativity problem" in the context of finite geometry. However, as you rightly pointed out, this term refers to Weyl's own terminology for problems involving coordinatization and not the theory of relativity in physics. Finite geometry is a field of pure mathematics and does not directly connect to the physical theories of relativity. While both areas involve concepts of space and transformations, they operate in fundamentally different domains. Cullinane's work focuses on applying Weyl's insights on coordinatization to the specific challenges and structures within finite geometries. This involves exploring different coordinatization schemes, understanding their equivalences, and identifying transformation groups that reveal the underlying symmetries of finite geometries. Thank you for pointing out this important distinction. It highlights the importance of precise language and accurate attribution when discussing complex mathematical concepts. |
Transcription —
(With one correction … The Meta.ai phrase "the group of 322,560"
was changed to "a group of 322,560" … and one addition — the
acronym MOG was defined. Superscript numerals for missing footnotes
were removed.)
|
You What is the Cullinane diamond theorem? Meta AI The Cullinane diamond theorem states that every image of D (a 4×4 array of two-color diagonally divided square tiles) under the group G (a group of 322,560 permutations of these 16 tiles) has some ordinary or color-interchange symmetry. Here are some key points about the theorem :
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According to Chu Hsi [Zhu Xi],
“Li” is
— Smith, Bol, Adler, and Wyatt, |
But very possibly the earliest use of li is the one instance that
it appears in the Classic of Poetry (Ode 210) where it refers to
the borders or boundary lines marking off areas in a field.
Here it appears in conjunction with chiang and is explained
as 'to divide into lots (or parcels of land)' (fen-ti )."
— P. 33 of "Li Revisited and Other Explorations"
by Allen Wittenborn, Bulletin of Sung and Yüan Studies
No. 17 (1981), pp. 32-48 (17 pages),
https://www.jstor.org/stable/23497457.
The Chinese concept of li in yesterday's post "Logos" is related,
if only by metaphor, to the underlying form (sets of "line diagrams")
of patterns in the Cullinane diamond theorem:
"But very possibly the earliest use of li is the one instance that
it appears in the Classic of Poetry (Ode 210) where it refers to
the borders or boundary lines marking off areas in a field.
Here it appears in conjunction with chiang and is explained
as 'to divide into lots (or parcels of land)' (fen-ti )."
— P. 33 of "Li Revisited and Other Explorations"
by Allen Wittenborn, Bulletin of Sung and Yüan Studies
No. 17 (1981), pp. 32-48 (17 pages),
https://www.jstor.org/stable/23497457.
A logo from the previous post —
This suggests a flashback to an image from Log24 on Nov. 6, 2003 —
Moral of the
Entertainment:
According to Chu Hsi [Zhu Xi],
“Li” is
“the principle or coherence
or order or pattern
underlying the cosmos.”
— Smith, Bol, Adler, and Wyatt,
Sung Dynasty Uses of the I Ching ,
Princeton University Press, 1990
|
“Isn’t your work—our work—all about accessing and deploying underlying sequences and patterns? Mapping particulars on to great universals? Isn’t that the art to which, in one way or another, we’ve both devoted our best years?”
— McCarthy, Tom. The Making of Incarnation: A Novel
The hardcover first edition was published by Knopf |
|
"It is said that the students of medieval Paris came to blows in the streets over the question of universals. The stakes are high, for at issue is our whole conception of our ability to describe the world truly or falsely, and the objectivity of any opinions we frame to ourselves. It is arguable that this is always the deepest, most profound problem of philosophy. It structures Plato's (realist) reaction to the sophists (nominalists). What is often called 'postmodernism' is really just nominalism, colourfully presented as the doctrine that there is nothing except texts. It is the variety of nominalism represented in many modern humanities, paralysing appeals to reason and truth."
— Simon Blackburn, Think, |
|
From Microsoft Copilot (with "Precise" setting)
on Dec. 18, 2023 —
How is the 1910 work of G. M. Conwell on PG(3,2) related to the Miracle Octad Generator of R. T. Curtis?
Searching for:
Searching for: Generating answers for you… The work of George M. Conwell on the finite projective 3-space PG(3,2) and the Miracle Octad Generator (MOG) of R. T. Curtis are both significant contributions to the field of combinatorial mathematics12. Conwell’s work on PG(3,2) provided a detailed study of the structure of this finite projective space1. On the other hand, the MOG is a tool for exploring the binary Golay code and its automorphism group M242. The connection between these two works lies in the underlying combinatorial structures they explore. The Rosenhain and Göpel tetrads, which are key structures in the finite geometry PG(3,2), play a significant role in the Curtis Miracle Octad Generator3. These structures provide a link between the finite geometric structures studied by Conwell and the combinatorial structures generated by the MOG3. For a more detailed understanding of this connection, I recommend checking out the resources mentioned in the references123. If you have any specific questions about the concepts in these works, feel free to ask! I’ll do my best to help. Learn more
5
_________________________________________________________
__________________________________________________________
Here is a summary of the document:
Reference:
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Correction:
"Subsets of four points in a projective 3-space" above is a Copilot error, and not
from the document being summarized. It should be "subsets of four points in an
affine 4-space."
"Unsocratic Dialogue" with Google DeepMind's Gemini AI continues . . .
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Do you remember a previous conversation about the question "What is the Cullinane diamond theorem?" If so, how would you answer the question now?
|
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What is the Cullinane diamond theorem?
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The conclusion of a Hungarian political figure's obituary in
tonight's online New York Times, written by Clay Risen —
"A quietly religious man, he spent his last years translating
works dealing with Roman Catholic canon law."
This journal on the Hungarian's date of death, October 8,
a Sunday, dealt in part with the submission to Wikipedia of
the following brief article . . . and its prompt rejection.
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The Cullinane diamond theorem is a theorem
The theorem also explains symmetry properties of the Reference
1. Cullinane diamond theorem at |
Some quotations I prefer to Catholic canon law —
|
Ludwig Wittgenstein,
97. Thought is surrounded by a halo. * See the post Wittgenstein's Diamond. Related language in Łukasiewicz (1937)—
|
See as well Diamond Theory in 1937.
"As McCarthy peers through the screen, or veil,
of technological modernity to reveal the underlying
symbolic structures of human experience,
The Making of Incarnation weaves a set of stories
one inside the other, rings within rings, a perpetual
motion machine." — Amazon.com description
of a novel published on All Souls' Day (Dia de los
Muertos), 2021.
The McCarthy novel is mentioned in The New York Times today —
For a simpler perpetual motion machine, see T. S. Eliot's "Chinese jar."
The previous post discussed the phrase "plot structure."
A different approach —
Textbook art from 1974 —
See as well a more interesting book I enjoyed reading in 1974.
The above 1975 book by Robert Greer Cohn, Modes of Art, is
Volume I of a planned three-volume work.
The passage below is from a review of Cohn's Vol. II, Ways of Art —
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Franklin, Ursula (1987) "Book Review: A Critical Work II. . . . . Those not familiar with the author's epistemology should begin with Appendix A of Ways of Art , a schematic demonstration of his tetrapolar-polypolar-dialectic, especially as it concerns the development of the French novel within the European tradition. But this dialectic, which has antecedents in Kierkegaard, Mallarme and Joyce, underlies all art, because: "this dimensional pulsation, or tetrapolar (and polypolar) higher vibrancy is, in short, the stuff of life: life is vibrant in this more complex way as well as in the more bipolar sense" (7). Cohn shows that "far out enough" the male or linear and the female or circular, the male vertical and the female horizontal dimensions "tend to merge as in relativity theory" (19). Ways of Art shows us the way through a historical becoming of art in its complex dialectic in which the metonymic (horizontal) axis constantly interrelates with the metaphoric (vertical). "Life is the mother, art the father" (vii); hence Cohn's quarrel with most contemporary Feminism, which is pronounced throughout the volume. Firmly grounded in its author's tetra-polypolar epistemology, this beautiful book becomes, however, at no point dryly abstract; it is the mature work of a true humanist who stands in clear and open opposition to the dehumanizing trend of "the quasi-scientific reductionism and abstract gimmickry of a great deal of current academic literary study, bellwethered by the structuralists, post-structuralists, and deconstructionists" (vi). Abundant footnotes constitute a substantial part of Ways of Art , on occasion developing insights almost into essays demonstrating crucial points along the general flow of the tradition from "Obscure Beginnings;' the opening chapter, to our "Contemporaries;' the last. Cohn reminds us that "In the Beginning was the Word;' for the Judaeo-Christian tradition at least, which his study fervently embraces; thus, for example, in Appendix 0 on "The Dance of the Sexes;' he censures "those who live by slogans, camps, and peer-opinion, the countless little bastard cults which characterize an era which has massively veered away from our free and beautiful Greco-Judaeo-Christian tradition" (332). Cohn traces man's way and that of his myths and rituals culminating in his art from that beginning along the lines of Freud, Neumann and Cassirer, and many others, always demonstrating the underlying polypolar dialectical rhythm. Thus in "From Barbarism to Young Culture;' we follow the Celts to Druidic ritual, Hebrew beginnings to the Psalms, Dionysian ritual to Greek tragedy, and thence to the beginnings of French dramatic literature originating in the Quem quaeritis sequence of the medieval Mass. Along the way arises artistic symbolism, for Cohn synonymous with "effective poetry;' to finally "ripen in France as never before" (99). Table I (134) graphs this development from the twelfth to the late nineteenth and early twentieth centuries. The author traces the rise of the artistic vocation from its antecedents in the double function of bard and priest, with the figure of Ronsard at the crossroads of that dying institution and the nascent concept of personal glory. "The Enlightenment Vocation" is exemplified in Montaigne, who humanizes the French cultural elite and points the way to French classicism and, farther down the road, after the moral collapse with the outgoing reign of Louis XIV, toward the Age of Reason. Clearly the most significant figure of the French Enlightenment for all of Western civilization is Rousseau, and Cohn beautifully shows us why this is so. Subsequently, "the nineteenth-century stage of the writer's journey will lead, starting from the crossroads of Rousseau, primarily in these two directions: the imperialistic and visionary prose of Balzac, the equally ambitious poetry of Mallarme", brothers under the skin" (199). And these two paths will then be reconciled in Proust's monumental A la recherche du temps perdu . . . . . |
The previous post's image illustrating the
ancient Lo Shu square as an affine transformation
suggests a similar view of Dürer's square.
That view illustrates the structural principle
underlying the diamond theorem —
See "Flusser and the I Ching," by Peter Zhang.
Zhang has written extensively on the concept of "interality,"
a term coined by his colleague Geling Shang.
For interality as the mathematics underlying the natural
automorphism group of the I Ching, see my own work.
The new URL supercube.space forwards to http://box759.wordpress.com/.
The term supercube is from a 1982 article by Solomon W. Golomb.
The related new URL supercube.group forwards to a page that
describes how the 2x2x2 (or eightfold, or "super") cube's natural
underlying automorphism group is Klein's simple group of order 168.
For further context, see the new URL supercube.art.
For some background, see the phrase Cube Space in this journal.
"… thinking starts with a problem, a difficulty, a contradiction."
— The late Arthur Mattuck of MIT in last night's post Gestalt .
"The 'technical support' is an underlying ground for aesthetic practice
that supports the work of art as canvas supported oil paint."
— MIT Press on the book by Rosalind Krauss titled Under Blue Cup .
From Grid View and List View (Log24, 15 April 2021) —
And then there is Bardo College . . .
"Katz approaches her subject from every angle,
its relationship to feminism, multiculturalism and
the counterculture, as well as its (now questionable)
cultural appropriation and even its underlying debt
to Minimalism (the use of repetition and the grid)."
This is from a Roberta Smith piece yesterday morning
in The New York Times print version:
"A version of this article appears in print on Aug. 6, 2021,
Section C, Page 6 of the New York edition with the headline:
Celebrating a Riotous Decor That Keeps Eyes Moving."
Well, perhaps not every angle.
“In a sense, we would see that change arises from
the structure of the object.”
— Nima Arkani-Hamed, quoted in
“A Jewel at the Heart of Quantum Physics,”
by Natalie Wolchover, Quanta Magazine , Sept. 17, 2013.
For those who prefer less barbaric entertainment —
"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."
— Thomas W. Tucker, review of Lyndon's Groups and Geometry
in The American Mathematical Monthly , Vol. 94, No. 4
(April 1987), pp. 392-394.
"The concept of group actions is very useful in the study of
isomorphisms of combinatorial structures."
— Olli Pottonen, "Classification of Steiner Quadruple Systems"
(Master's thesis, Helsinki, 2005, p. 48).
“In a sense, we would see that change arises from
the structure of the object.”
— Nima Arkani-Hamed, quoted in "A Jewel at the Heart of
Quantum Physics," by Natalie Wolchover, Quanta Magazine ,
Sept. 17, 2013.
See as well "Change Arises" in this journal.
From old posts tagged Change Arises —
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From Christmas 2005:
For the eightfold cube
For an rather more
Click on image for details. |
The phrase "change arises" is from Arkani-Hamed in 2013, describing
calculations in physics related to properties of the positive Grassmannian —
A related recent illustration from Quanta Magazine —
The above illustration of seven cells is not unrelated to
the eightfold-cube model of the seven projective points in
the Fano plane.
Robert M. Pirsig, Zen and the Art of Motorcycle Maintenance ,
Ch. 6 (italics are mine):
“A classical understanding sees the world primarily as underlying form itself.
A romantic understanding sees it primarily in terms of immediate appearance .”
Underlying Structure of the Design —
Schoolgirl Problem —
Various posts here on the geometry underlying the Mathieu group M24
are now tagged with the phrase “Geometry of Even Subsets.”
For example, a post with this diagram . . .
John Horgan in Scientific American magazine on October 8, 2019 —
"In the early 1990s, I came to suspect that the quest
for a unified theory is religious rather than scientific.
Physicists want to show that all things came from
one thing: a force, or essence, or membrane
wriggling in eleven dimensions, or something that
manifests perfect mathematical symmetry. In their
search for this primordial symmetry, however,
physicists have gone off the deep end . . . ."
Other approaches —
See "Story Theory of Truth" in this journal and, from the November 2019
Notices of the American Mathematical Society . . .
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More fundamental than the label of mathematician is that of human. And as humans, we’re hardwired to use stories to make sense of our world (story-receivers) and to share that understanding with others (storytellers) [2]. Thus, the framing of any communication answers the key question, what is the story we wish to share? Mathematics papers are not just collections of truths but narratives woven together, each participating in and adding to the great story of mathematics itself. The first endeavor for constructing a good talk is recognizing and choosing just one storyline, tailoring it to the audience at hand. Should the focus be on a result about the underlying structures of group actions? . . . .
[2] Gottschall, J. , The Storytelling Animal , — "Giving Good Talks," by Satyan L. Devadoss |
"Before time began, there was the Cube." — Optimus Prime
"The stars and galaxies seem static, eternal, or moving slowly
in deterministic patterns, becoming the background stage
on which we move. But if we could speed up the sequence,
we would see how dramatic and unpredictable this background
really is — an actor, director, script and stage all at once.
Moreover, it is a unified universe, a single unfolding event
of which we are an embedded part, a narrative of highly
dangerous and fine-tuned events, something more like
a detective thriller with many crimes and last-minute escapes
than the impersonal account of astronomy textbooks.
We are only just beginning to decipher the plot and figure out
the Cosmic Code, as Heinz Pagels puts it."
— Charles Jencks, The Architecture of the Jumping Universe :
A Polemic (How Complexity Science is Changing Architecture
and Culture), Academy Editions, 1995, rev. ed. 1997
"A Grand Unified Theory (GUT) is a model in particle physics…."
— Wikipedia
"Under the GUT symmetry operation these field components
transform into one another. The reason quantum particles
appear to have different properties in nature is that the unifying
symmetry is broken. The various gluons, quarks and leptons
are analogous to the facets of a cut diamond, which appear
differently according to the way the diamond is held but in
fact are all manifestations of the same underlying object."
— Heinz Pagels, Perfect Symmetry , Bantam paperback, 1986, p. 284
See also the recent post Multifaceted Narrative.
Finite Galois geometry with the underlying field the simplest one possible —
namely, the two-element field GF(2) — is a geometry of interstices :
For some less precise remarks, see the tags Interstice and Interality.
The rationalist motto "sincerity, order, logic and clarity" was quoted
by Charles Jencks in the previous post.
This post was suggested by some remarks from Queensland that
seem to exemplify these qualities —
From the series of posts tagged Kummerhenge —
A Wikipedia article relating the above 4×4 square to the work of Kummer —
A somewhat more interesting aspect of the geometry of the 4×4 square
is its relationship to the 4×6 grid underlying the Miracle Octad Generator
(MOG) of R. T. Curtis. Hudson's 1905 classic Kummer's Quartic Surface
deals with the Kummer properties above and also foreshadows, without
explicitly describing, the finite-geometry properties of the 4×4 square as
a finite affine 4-space — properties that are of use in studying the Mathieu
group M24 with the aid of the MOG.
A search this morning for articles mentioning the Miracle Octad Generator
of R. T. Curtis within the last year yielded an abstract for two talks given
at Hiroshima on March 8 and 9, 2018 —
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http://www.math.sci.hiroshima-u.ac.jp/ branched/files/2018/abstract/Aitchison.txt
Iain AITCHISON Title: Construction of highly symmetric Riemann surfaces, related manifolds, and some exceptional objects, I, II Abstract: Since antiquity, some mathematical objects have played a special role, underpinning new mathematics as understanding deepened. Perhaps archetypal are the Platonic polyhedra, subsequently related to Platonic idealism, and the contentious notion of existence of mathematical reality independent of human consciousness. Exceptional or unique objects are often associated with symmetry – manifest or hidden. In topology and geometry, we have natural base points for the moduli spaces of closed genus 2 and 3 surfaces (arising from the 2-fold branched cover of the sphere over the 6 vertices of the octahedron, and Klein's quartic curve, respectively), and Bring's genus 4 curve arises in Klein's description of the solution of polynomial equations of degree greater than 4, as well as in the construction of the Horrocks-Mumford bundle. Poincare's homology 3-sphere, and Kummer's surface in real dimension 4 also play special roles. In other areas: we have the exceptional Lie algebras such as E8; the sporadic finite simple groups; the division algebras: Golay's binary and ternary codes; the Steiner triple systems S(5,6,12) and S(5,8,24); the Leech lattice; the outer automorphisms of the symmetric group S6; the triality map in dimension 8; and so on. We also note such as: the 27 lines on a cubic, the 28 bitangents of a quartic curve, the 120 tritangents of a sextic curve, and so on, related to Galois' exceptional finite groups PSL2(p) (for p= 5,7,11), and various other so-called `Arnol'd Trinities'. Motivated originally by the `Eightfold Way' sculpture at MSRI in Berkeley, we discuss inter-relationships between a selection of these objects, illustrating connections arising via highly symmetric Riemann surface patterns. These are constructed starting with a labeled polygon and an involution on its label set. Necessarily, in two lectures, we will neither delve deeply into, nor describe in full, contexts within which exceptional objects arise. We will, however, give sufficient definition and detail to illustrate essential inter-connectedness of those exceptional objects considered. Our starting point will be simplistic, arising from ancient Greek ideas underlying atomism, and Plato's concepts of space. There will be some overlap with a previous talk on this material, but we will illustrate with some different examples, and from a different philosophical perspective. Some new results arising from this work will also be given, such as an alternative graphic-illustrated MOG (Miracle Octad Generator) for the Steiner system S(5,8,24), and an alternative to Singerman – Jones' genus 70 Riemann surface previously proposed as a completion of an Arnol'd Trinity. Our alternative candidate also completes a Trinity whose two other elements are Thurston's highly symmetric 6- and 8-component links, the latter related by Thurston to Klein's quartic curve. |
See also yesterday morning's post, "Character."
Update: For a followup, see the next Log24 post.
Underlying the I Ching structure is the finite affine space
of six dimensions over the Galois field with two elements.
In this field, "1 + 1 = 0," as noted here Wednesday.
See also other posts now tagged Interstice.
From "The Principle of Sufficient Reason," by George David Birkhoff,
in "Three Public Lectures on Scientific Subjects,"
delivered at the Rice Institute, March 6, 7, and 8, 1940 —
From the same lecture —
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Up to the present point my aim has been to consider a variety of applications of the Principle of Sufficient Reason, without attempting any precise formulation of the Principle itself. With these applications in mind I will venture to formulate the Principle and a related Heuristic Conjecture in quasi-mathematical form as follows: PRINCIPLE OF SUFFICIENT REASON. If there appears in any theory T a set of ambiguously determined ( i e . symmetrically entering) variables, then these variables can themselves be determined only to the extent allowed by the corresponding group G. Consequently any problem concerning these variables which has a uniquely determined solution, must itself be formulated so as to be unchanged by the operations of the group G ( i e . must involve the variables symmetrically). HEURISTIC CONJECTURE. The final form of any scientific theory T is: (1) based on a few simple postulates; and (2) contains an extensive ambiguity, associated symmetry, and underlying group G, in such wise that, if the language and laws of the theory of groups be taken for granted, the whole theory T appears as nearly self-evident in virtue of the above Principle. The Principle of Sufficient Reason and the Heuristic Conjecture, as just formulated, have the advantage of not involving excessively subjective ideas, while at the same time retaining the essential kernel of the matter. In my opinion it is essentially this principle and this conjecture which are destined always to operate as the basic criteria for the scientist in extending our knowledge and understanding of the world. It is also my belief that, in so far as there is anything definite in the realm of Metaphysics, it will consist in further applications of the same general type. This general conclusion may be given the following suggestive symbolic form:
While the skillful metaphysical use of the Principle must always be regarded as of dubious logical status, nevertheless I believe it will remain the most important weapon of the philosopher. |
Related remarks by a founding member of the Metaphysical Club:
See also the previous post, "Seven Types of Interality."
The above title was suggested by a film trailer quoted here Saturday —
" Jeremy Irons' dry Alfred Pennyworth:
'One misses the days when one's biggest concerns
were exploding wind-up penguins.' "
"Penguin Classics Deluxe Edition" describes, among other books,
an edition of the I Ching published on December 1, 2015.
Excerpt from this journal on that date —
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Tuesday, December 1, 2015
Verhexung
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Related material —
"I think you should be more explicit here in step two."
— Caption to a cartoon by Sidney Harris,
American Scientist , November-December 1977
"If any perfection is indicated in the work
it is recognized by the artist as truly miraculous
so he feels that he can take no credit for its
sudden appearance."
— Agnes Martin, 1973, "On the Perfection Underlying Life"
Two views of nested sequences of projective structures —
From this journal in April 2013:
From the arXiv in September 2014 —
Saniga's reference [6] is to a paper submitted to the arXiv in May 2014.
My own note of April 30, 2013, concludes with an historical reference
that indicates the mathematics underlying both my own and Saniga's
remarks —
The exercises at the end of Ch. II in Veblen and Young's
Projective Geometry, Vol. I (Ginn, 1910). For instance:
The previous post quoted Tom Wolfe on Chomsky's use of
the word "array."
An example of particular interest is the 4×4 array
(whether of dots or of unit squares) —
.
Some context for the 4×4 array —
The following definition indicates that the 4×4 array, when
suitably coordinatized, underlies the Kummer lattice .
Further background on the Kummer lattice:
Alice Garbagnati and Alessandra Sarti,
"Kummer Surfaces and K3 surfaces
with $(Z/2Z)^4$ symplectic action."
To appear in Rocky Mountain J. Math. —
The above article is written from the viewpoint of traditional
algebraic geometry. For a less traditional view of the underlying
affine 4-space from finite geometry, see the website
Finite Geometry of the Square and Cube.
Some further context …
"To our knowledge, the relation of the Golay code
to the Kummer lattice … is a new observation."
— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of
Kummer surfaces in the Mathieu group M24 "
As noted earlier, Taormina and Wendland seem not to be aware of
R. W. H. T. Hudson's use of the (uncoordinatized*) 4×4 array in his
1905 book Kummer's Quartic Surface. The array was coordinatized,
i.e. given a "vector space structure," by Cullinane eight years prior to
the cited remarks of Curtis.
* Update of Sept. 14: "Uncoordinatized," but parametrized by 0 and
the 15 two-subsets of a six-set. See the post of Sept. 13.
From this journal —
See (for instance) Sacred Order, July 18, 2006 —
From a novel published July 26, 2016, and reviewed
in yesterday's (print) New York Times Book Review —
|
The doors open slowly. I step into a hangar. From the rafters high above, lights blaze down, illuminating a twelve-foot cube the color of gunmetal. My pulse rate kicks up. I can’t believe what I’m looking at. Leighton must sense my awe, because he says, “Beautiful, isn’t it?” It is exquisitely beautiful. At first, I think the hum inside the hangar is coming from the lights, but it can’t be. It’s so deep I can feel it at the base of my spine, like the ultralow-frequency vibration of a massive engine. I drift toward the box, mesmerized.
— Crouch, Blake. Dark Matter: A Novel |
See also Log24 on the publication date of Dark Matter .
A note related to the diamond theorem and to the site
Finite Geometry of the Square and Cube —
The last link in the previous post leads to a post of last October whose
final link leads, in turn, to a 2009 post titled Summa Mythologica .
Some may view the above web page as illustrating the
Glasperlenspiel passage quoted here in Summa Mythologica —
“"I suddenly realized that in the language, or at any rate
in the spirit of the Glass Bead Game, everything actually
was all-meaningful, that every symbol and combination of
symbols led not hither and yon, not to single examples,
experiments, and proofs, but into the center, the mystery
and innermost heart of the world, into primal knowledge.
Every transition from major to minor in a sonata, every
transformation of a myth or a religious cult, every classical
or artistic formulation was, I realized in that flashing moment,
if seen with a truly meditative mind, nothing but a direct route
into the interior of the cosmic mystery, where in the alternation
between inhaling and exhaling, between heaven and earth,
between Yin and Yang, holiness is forever being created.”
A less poetic meditation on the above web page* —
"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."
Update of Sept. 5, 2016 — See also a related remark
by Lévi-Strauss in 1955: "…three different readings
become possible: left to right, top to bottom, front
to back."
* For the underlying mathematics, see a June 21, 1983, research note.
An old version of the Wikipedia article "Group theory"
(pictured in the previous post) —
"More poetically …"
From Hermann Weyl's 1952 classic Symmetry —
"Galois' ideas, which for several decades remained
a book with seven seals but later exerted a more
and more profound influence upon the whole
development of mathematics, are contained in
a farewell letter written to a friend on the eve of
his death, which he met in a silly duel at the age of
twenty-one. This letter, if judged by the novelty and
profundity of ideas it contains, is perhaps the most
substantial piece of writing in the whole literature
of mankind."
The seven seals from the previous post, with some context —
These models of projective points are drawn from the underlying
structure described (in the 4×4 case) as part of the proof of the
Cullinane diamond theorem .
(Continued from November 13)
The work of Ron Shaw in this area, ca. 1994-1995, does not
display explicitly the correspondence between anticommutativity
in the set of Dirac matrices and skewness in a line complex of
PG(3,2), the projective 3-space over the 2-element Galois field.
Here is an explicit picture —
References:
Arfken, George B., Mathematical Methods for Physicists , Third Edition,
Academic Press, 1985, pages 213-214
Cullinane, Steven H., Notes on Groups and Geometry, 1978-1986
Shaw, Ron, "Finite Geometry, Dirac Groups, and the Table of
Real Clifford Algebras," undated article at ResearchGate.net
Update of November 23:
See my post of Nov. 23 on publications by E. M. Bruins
in 1949 and 1959 on Dirac matrices and line geometry,
and on another author who gives some historical background
going back to Eddington.
Some more-recent related material from the Slovak school of
finite geometry and quantum theory —
The matrices underlying the Saniga paper are those of Pauli, not
those of Dirac, but these two sorts of matrices are closely related.
A quote that appeared here on April 14, 2013 —
"I know what 'nothing' means." — Joan Didion
Dirac on the 4×4 matrices of an underlying nothingness —
"Corresponding to the four rows and columns,
the wave function ψ must contain a variable
that takes on four values, in order that the matrices
shall be capable of being multiplied into it."
— P. A. M. Dirac, Principles of Quantum Mechanics,
Fourth Edition, Oxford University Press, 1958,
page 257
Two transitions from last Monday, Oct. 26, 2015,
according to the online New York Times today —
|
Leo P. Kadanoff, a physicist who provided critical insights into the transformations of matter from one state to another, died last Monday in Chicago. He was 78. The cause was respiratory failure, said the University of Chicago, where he was a professor from 1978 until his retirement in 2003. A member of the National Academy of Sciences and a fellow of the American Academy of Arts and Sciences, he received the National Medal of Science in 1999. “He won basically every prize except the Nobel Prize, and many people thought he should have won the Nobel,” said Emil Martinec, a physics professor at the University of Chicago who directs the university’s Kadanoff Center for Theoretical Physics. Dr. Kadanoff’s biggest scientific contribution came in the 1960s as scientists were trying to understand phase transitions, when matter changes from one form to another. A Cornell chemist, Benjamin Widom, had come up with mathematical relationships that described behavior associated with second-order phase transitions, which include the boiling of water to steam at a particular temperature and pressure. But Dr. Widom did not have an underlying physical explanation for why these relationships existed. |
Willis Carto, a reclusive behind-the-scenes wizard of the far-right fringe of American politics who used lobbying and publishing to denigrate Jews and other minorities and galvanize the movement to deny the Holocaust, died last Monday at his home in Virginia. He was 89. His death was announced by The American Free Press, a newspaper he helped found. Mr. Carto raised funds to finance a right-wing military dictatorship in the United States, campaigned to persuade blacks to voluntarily return to Africa and, most influentially, started newsletters, a journal and conferences of academics and others to deny the scale, and even the existence, of the Holocaust. The Anti-Defamation League called him “one of the most influential American anti-Semitic propagandists” and “the mastermind of the hate network.” His associates included neo-Nazis, Christian vigilantes, John Birch Society members and Ku Klux Klansmen, and his extreme views alienated mainstream conservatives. After William F. Buckley sued Mr. Carto for libel and won in 1985, Mr. Buckley said Mr. Carto epitomized “the fever swamps of the crazed right.’’ |
Related remarks:
Posts tagged "Steam," the post "On Ice-Breaking" from Oct. 27,
the post "Expanding the Seagram Spielraum" from Oct. 26, and
a 2008 article on the subject of the obituary at right above.
For the title, see posts from August 2007 tagged Gyges.
Related theological remarks:
Boolean spaces (old) vs. Galois spaces (new) in
“The Quality Without a Name”
(a post from August 26, 2015) and the…
Related literature: A search for Borogoves in this journal will yield
remarks on the 1943 tale underlying the above film.
(A review)
Continued from June 17, 2013
(John Baez as a savior for atheists):
As an atheists-savior, I prefer Galois…
The geometry underlying a figure that John Baez
posted four days ago, "A Hypercube of Bits," is
Galois geometry —
See The Galois Tesseract and an earlier
figure from Log24 on May 21, 2007:
For the genesis of the figure,
see The Geometry of Logic.
Two news items on art as a tool:
Two Log24 posts related to the 3×3 grid, the underlying structure for China’s
ancient Lo Shu “magic” square:
Finally, leftist art theorist Rosalind Krauss in this journal
on AntiChristmas, 2010:
Which is the tool here, the grid or Krauss?
The fictional zero theorem of Terry Gilliam's current film
by that name should not be confused with the zero system
underlying the diamond theorem.
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:
“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
The titles of the previous three posts refer to
Hermann Weyl’s 1918 book Raum, Zeit, Materie
(Space, Time, Matter).
This suggests a look at a poetically parallel 1950 title —
The Lion, the Witch and the Wardrobe —
and at its underlying philosophy:
I am among “those who do not know that this great myth became Fact.”
I do, however, note that some other odd things have become fact.
Those who wish more on this topic may consult:
Click image for a larger, clearer version.
* For related remarks, see posts of May 26-28, 2012.
The webpage Galois.us, on Galois matrices , has been created as
a starting point for remarks on the algebra (as opposed to the geometry)
underlying the rings of matrices mentioned in AMS abstract 79T-A37,
“Symmetry invariance in a diamond ring.”
See also related historical remarks by Weyl and by Atiyah.
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.
|
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.
From the LA Times online obituaries today:
|
Michael Feran Baigent was born in Nelson, New Zealand,
From 1998 he lectured on and led tours of the temples and Elliott Reid Longtime film, TV actor with a comic touch
Elliott "Ted" Reid, 93, a longtime character actor in films |
From a post last Saturday, June 22, and the earlier
post last Friday, June 21, that preceded it:
|
The Eliade passage was quoted in a 1971 Ph.D. thesis Some context— Stevens's Rock in this journal. Friday, June 21, 2013
Lexicon
|
|
"… a fundamental language
"… the questions raised by R. Lowell |
See also, in this journal, Big Rock.
Passage quoted in A Philosopher's Stone (April 4, 2013)—
This passage from Heidegger suggested the lexicon excerpt on
to hypokeimenon (the underlying) in yesterday's post Lexicon.
A related passage:
The Eliade passage was quoted in a 1971 Ph.D. thesis
on Wallace Stevens.
Some context— Stevens's Rock in this journal.
From the final pages of the new novel
Lexicon , by Max Barry:
|
"… a fundamental language
"… the questions raised by R. Lowell |
"… the clocks were striking thirteen." — 1984
Anne Taormina on Mathieu Moonshine —
This is, of course, the same group (of order 322,560) underlying the Diamond 16 Puzzle.
Some historical background for today's note on the geometry
underlying the Curtis Miracle Octad Generator (MOG):
The above incidence diagram recalls those in today's previous post
on the MOG, which is used to construct the large Mathieu group M24.
For some related material that is more up-to-date, search the Web
for Mathieu + Kummer .
A commenter on Saturday's "Seize the Dia" has
suggested a look at the work of one Mark Collins.
Here is such a look (click to enlarge):
I find attempts to associate pure mathematics with the words
"magic" or "mystic" rather nauseating. (H. F. Baker's work
on Pascal's mystic hexagram is no exception; Baker was
stuck with Pascal's obnoxious adjective, but had no truck
with any mystic aspects of the hexagram.)
The remarks above by Clifford Pickover on Collins, Dürer, and
binary representations may interest some non-mathematicians,
who should not be encouraged to waste their time on this topic.
For the mathematics underlying the binary representation of
Dürer's square, see, for instance, my 1984 article "Binary
Coordinate Systems."
Those without the background to understand that article
may enjoy, instead of Pickover's abortive attempts above at
mathematical vulgarization, his impressively awful 2009 novel
Jews in Hyperspace .
Pickover's 2002 book on magic squares was, unfortunately,
published by the formerly reputable Princeton University Press.
Related material from today's Daily Princetonian :
See also Nash + Princeton in this journal.
Click image for some background.
Shown above is a rearranged version of the
Miracle Octad Generator (MOG) of R. T. Curtis
("A new combinatorial approach to M24,"
Math. Proc. Camb. Phil. Soc., 79 (1976), 25-42.)
The 8-subcell rectangles in the left part of the figure may be
viewed as illustrating (if the top left subcell is disregarded)
the thirty-five 3-subsets of a 7-set.
Such a view relates, as the remarks below show, the
MOG's underlying Galois geometry, that of PG(3,2), to
the hexagrammum mysticum of Pascal.
On Danzer's 354 Configuration:
"Combinatorially, Danzer’s configuration can be interpreted
as defined by all 3-sets and all 4-sets that can be formed
by the elements of a 7-element set; each 'point' is represented
by one of the 3-sets, and it is incident with those lines
(represented by 4-sets) that contain the 3-set."
— Branko Grünbaum, "Musings on an Example of Danzer's,"
European Journal of Combinatorics , 29 (2008),
pp. 1910–1918 (online March 11, 2008)
"Danzer's configuration is deeply rooted in
Pascal's Hexagrammum Mysticum ."
— Marko Boben, Gábor Gévay, and Tomaž Pisanski,
"Danzer's Configuration Revisited," arXiv.org, Jan. 6, 2013
For an approach to such configurations that differs from
those of Grünbaum, Boben, Gévay, and Pisanski, see
Classical Geometry in Light of Galois Geometry.
Grünbaum has written little about Galois geometry.
Pisanski has recently touched on the subject;
see Configurations in this journal (Feb. 19, 2013).
"Core" (in the original, Kern ) is perhaps
not the best translation of hypokeimenon :
See also Heidegger's original German:
Related material: In this journal, "underlie" and "underlying."
"…a fundamental cognitive ability known as 'fluid' intelligence: the capacity to solve novel problems, to learn, to reason, to see connections and to get to the bottom of
…matrices are considered the gold standard of fluid-intelligence tests. Anyone who has taken an intelligence test has seen matrices like those used in the Raven’s: three rows, with three graphic items in each row, made up of squares, circles, dots or the like. Do the squares get larger as they move from left to right? Do the circles inside the squares fill in, changing from white to gray to black, as they go downward? One of the nine items is missing from the matrix, and the challenge is to find the underlying patterns— up, down and across— from six possible choices. Initially the solutions are readily apparent to most people, but they get progressively harder to discern. By the end of the test, most test takers are baffled."
— Dan Hurley, "Can You Make Yourself Smarter?," NY Times , April 18, 2012
See also "Raven Steals the Light" in this journal.
Related material:
Plan 9 from MIT and, perhaps exemplifying crystallized rather than fluid intelligence, Black Diamond.
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.
"… we have taken the first steps
in decoding the uniquely human
fascination with visual patterns…."
— W. Tecumseh Fitch et al. , July 2012
Fitch cites the following as a reference:
Washburn and Crowe discuss symmetries in general, but
not the Galois geometry underlying patterns like some of
those shown on their book's cover.
The concept of "deep structure," once a popular meme,
has long been abandoned by Chomskians.
It still applies, however, to the 1976 mathematics, diamond theory ,
underlying the formal patterns discussed in a Royal Society paper
this year.
A review of deep structure, from the Wikipedia article Cartesian linguistics—
|
[Numbers in parentheses refer to pages in the original 1966 Harper edition of Chomsky's book Cartesian Linguistics .] Deep structure vs. surface structure "Pursuing the fundamental distinction between body and mind, Cartesian linguistics characteristically assumes that language has two aspects" (32). These are namely the sound/character of a linguistic sign and its significance (32). Semantic interpretation or phonetic interpretation may not be identical in Cartesian linguistics (32). Deep structures are often only represented in the mind (a mirror of thought), as opposed to surface structures, which are not. Deep structures vary less between languages than surface structures. For instance, the transformational operations to derive surface forms of Latin and French may obscure common features of their deep structures (39). Chomsky proposes, "In many respects, it seems to me quite accurate, then, to regard the theory of transformational generative grammar, as it is developing in current work, as essentially a modern and more explicit version of the Port-Royal theory" (39). Summary of Port Royal Grammar The Port Royal Grammar is an often cited reference in Cartesian Linguistics and is considered by Chomsky to be a more than suitable example of Cartesian linguistic philosophy. "A sentence has an inner mental aspect (a deep structure that conveys its meaning) and an outer, physical aspect as a sound sequence"***** This theory of deep and surface structures, developed in Port Royal linguistics, meets the formal requirements of language theory. Chomsky describes it in modern terms as "a base system that generates deep structures and a transformational system that maps these into surface structures", essentially a form of transformational grammar akin to modern studies (42). |
The corresponding concepts from diamond theory are…
"Deep structure"— The line diagrams indicating the underlying
structure of varying patterns
"A base system that generates deep structures"—
Group actions on square arrays… for instance, on the 4×4 square
"A transformational system"— The decomposition theorem
that maps deep structure into surface structure (and vice-versa)
… Chomsky vs. Santa
From a New Yorker weblog yesterday—
"Happy Birthday, Noam Chomsky." by Gary Marcus—
"… two titans facing off, with Chomsky, as ever,
defining the contest"
"Chomsky sees himself, correctly, as continuing
a conversation that goes back to Plato, especially
the Meno dialogue, in which a slave boy is
revealed by Socrates to know truths about
geometry that he hadn’t realized he knew."
See Meno Diamond in this journal. For instance, from
the Feast of Saint Nicholas (Dec. 6th) this year—
The Meno Embedding
For related truths about geometry, see the diamond theorem.
For a related contest of language theory vs. geometry,
see pattern theory (Sept. 11, 16, and 17, 2012).
See esp. the Sept. 11 post, on a Royal Society paper from July 2012
claiming that
"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.’ "
The sorts of patterns discussed in the 2012 paper —
"First steps"? The mathematics underlying such patterns
was presented 35 years earlier, in Diamond Theory.
* See Gombrich-Douat in this journal.
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."
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) —
A web search for the author Cameron McEwen mentioned
in today's noon post was unsuccessful, but it did yield an
essay, quite possibly by a different Cameron McEwen, on
"The fundamental difference between analog
and digital systems may be understood as
underlying philosophical discourse since the Greeks."
The University of Bergen identifies the Wittgenstein
McEwen as associated with InteLex of Charlottesville.
The title of this post may serve to point out an analogy*
between the InteLex McEwen's analog-digital contrast
and the Euclidean-Galois contrast discussed previously
in this journal.
The latter contrast is exemplified in Pilate Goes to Kindergarten.
* An analogy, as it were, between analogies.
Title of a treatise by Dominique Douat—
"Méthode pour faire une infinité de desseins différens avec des carreaux mi-partis de deux couleurs par une ligne diagonale : ou observations du Père Dominique Doüat Religieux Carmes de la Province de Toulouse sur un mémoire inséré dans l'Histoire de l'Académie Royale des Sciences de Paris l'année 1704, présenté par le Révérend Père Sébastien Truchet religieux du même ordre, Académicien honoraire " (Paris, 1722)
"The earliest (and perhaps the rarest) treatise on the theory of design"
— E. H. Gombrich, 1979, in The Sense of Order
A facsimile version (excerpts, 108 pp., Feb. 5, 2010) of this treatise is available from
http://jacques-andre.fr/ed/ in a 23.1 MB pdf.
Sample page—
For a treatise on the finite geometry underlying such designs (based on a monograph I wrote in 1976, before I had heard of Douat or his predecessor Truchet), see Diamond Theory.
Stanley Fish in The New York Times yesterday evening—
From the MLA program Fish discussed—
Above: An MLA session, “Defining Form,” led
by Colleen Rosenfeld of Pomona College
An example from Pomona College in 1968—
The same underlying geometries (i.e., “form”) may be modeled with
a square figure and a cubical figure rather than with the triangular
figures of 1968 shown above.
See Finite Geometry of the Square and Cube.
Those who prefer a literary approach to form may enjoy the recent post As Is.
(For some context, see Game of Shadows.)
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.
A search on the word "innermost" in a PDF copy of a book
by Suzanne Gieser on Jung and Pauli yields no definite meaning
for the book's title, The Innermost Kernel (Springer, 2005).
The author does, however, devote a section (pp. 36-41) to the
influence of Schopenhauer on Jung and Pauli, and that section at least
suggests that the historical origin of her title is in Schopenhauer's
reformulation of Kant's "Ding an sich."
The Innermost Kernel , p. 37—
"… an expression of an underlying invisible world,
the one that forms the innermost essence of reality,
the thing-in-itself. This is the will, a blind existence
that forms an omnipresent entity beyond time, space
and individuality." *
* Arthur Schopenhauer, "Über die Vierfache Wurzel
des Satzes vom zureichenden Grunde" (1813),
Kleinere Schriften, SämtlicheWerke III
(Stuttgart, 1962), 805–806.
* See also Mann on Schopenhauer and an "innermost kernel."
A chess set previously mentioned in this journal—
These chessmen appeared in the weblog Minimalissimo
on Sept. 20, 2010. In Log24 on that date, the issue was
not so much the chessmen as the underlying board.
See "The Unfolding." See also the following from
the Occupy Space gallery in Limerick today—
| C A V E S – Anthony Murphy Solo Exhibition Opening 7 pm Thursday 1st Dec Exhibition 2nd – 22nd Dec 2011 Plato's allegory of the cave describes prisoners, inhabiting the cave since childhood, immobile, facing an interior wall. A large fire burns behind the prisoners, and as people pass this fire their shadows are cast upon the cave's wall, and these shadows of the activity being played out behind the prisoner become the only version of reality that the prisoner knows. C A V E S is an exhibition of three large scale works, each designed to immerse the viewer, and then to confront the audience with a question regarding how far they, as privileged viewers of the shadows and reflections being played out upon the walls, are willing to allow themselves to believe what they know to be a false reality. The works are based on explorations of simple 2D shapes; regular polygons are exploded to create fractured pattern, or layered upon one another until intricate forms emerge, upon which the projections can begin to draw out a third dimension. |
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"
(Continued from Abel Prize, August 26)
The situation is rather different when the
underlying Galois field has two rather than
three elements… See Galois Geometry.
The coffee scene from "Bleu"
Related material from this journal:
The Dream of
the Expanded Field
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.
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—
Perfect Symmetry (Oct. 2008) and Perfect Symmetry single (Dec. 2008)—
Related science…
Heinz Pagels in Perfect Symmetry (paperback, 1985), p. xvii—
The penultimate chapter of this third part of the book—
as far as speculation is concerned— describes some
recent mathematical models for the very origin of the
universe—how the fabric of space, time and matter can
be created out of absolutely nothing. What could have more
perfect symmetry than absolute nothingness? For the first
time in history, scientists have constructed mathematical
models that account for the very creation of the universe
out of nothing.
On Grand Unified Theories (GUT's) of physics (ibid., 284)—
In spite of the fact that GUTs leave deep puzzles unsolved,
they have gone a long way toward unifying the various
quantum particles. For example, many people are disturbed
by the large numbers of gluons, quarks and leptons. Part of
the appeal of the GUT idea is that this proliferation of
quantum particles is really superficial and that all the gluons
as well at the quarks and leptons may be simply viewed as
components of a few fundamental unifying fields. Under the
GUT symmetry operation these field components transform
into one another. The reason quantum particles appear to
have different properties in nature is that the unifying
symmetry is broken. The various gluons, quarks and leptons
are analogous to the facets of a cut diamond, which appear
differently according to the way the diamond is held but in
fact are all manifestations of the same underlying object.
Related art— Puzzle and Particles…
The Diamond 16 Puzzle (compare with Keane art above)—
—and The Standard Model of particle theory—
The fact that both the puzzle and the particles appear
within a 4×4 array is of course completely coincidental.
See also a more literary approach— "The Still Point and the Wheel"—
"Anomalies must be expected along the conceptual frontier between the temporal and the eternal."
— The Death of Adam , by Marilynne Robinson, Houghton Mifflin, 1998, essay on Marguerite de Navarre
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
Two characters named “Black” and “White” debate religion and the afterlife in the Cormac McCarthy play “The Sunset Limited.”
The play opened in Chicago in a Steppenwolf Theatre production on May 18, 2006.
A New York Times theater review from All Hallows’ Eve, 2006—
“…there is an abstract power in the mysteriousness of Mr. McCarthy’s
vision’s allowing for a multitude of interpretations.” –Jason Zinoman
The current New Yorker (Feb.14) has a note
by Lillian Ross on the same play— “Two-Man Show: O Death”
Some purely visual black-and-white variations that are less dramatic, but have their own “abstract power”—
A book cover pictured here last November to contrast with
“the sound and fury of the rarified Manhattan art world”—
and a web page with multiple interpretations of the book cover’s pattern—
A synchronicity— The first version of “Symmetry Framed” was done
on May 18, 2006— the day “The Sunset Limited” opened.
Another synchronicity relates the mathematics underlying
such patterns to the Halloween date of the above review.
See “To Announce a Faith,” from October 31, 2006.
Continued from yesterday's Church Diamond and from Dec. 17's Fare Thee Well —
|
The San Francisco Examiner last year
on New Year's Eve — Entertainment
Discover the modern art of Amish quilts By: Leslie Katz 12/31/09 1:00 AM Arts editor
Quilts made by Amish women in Pennsylvania, Household handicrafts and heirlooms made by American women seen as precursors to modern art is one underlying thesis of “Amish Abstractions: Quilts from the Collection of Faith and Stephen Brown,” a provocative exhibit on view at the de Young Museum through June. Curated by Jill D’Alessandro of the Fine Arts Museums of San Francisco, the show features about 50 full-size and crib quilts made between 1880 and 1940 in Pennsylvania and the Midwest during what experts consider the apex of Amish quilt-making production. Faith and Stephen Brown, Bay Area residents who began collecting quilts in the 1970s after seeing one in a shop window in Chicago and being bowled over by its bold design, say their continued passion for the quilts as art is in part because they’re so reminiscent of paintings by modern masters like Mark Rothko, Josef Albers, Sol LeWitt and Ellsworth Kelly — but the fabric masterpieces came first. “A happy visual coincidence” is how the Browns and D’Alessandro define the connection, pointing to the brilliance in color theory, sophisticated palettes and complex geometry that characterize both the quilts and paintings. “There’s an insane symmetry to these quilts,” says D’Alessandro…. Read more at the San Francisco Examiner . |
The festive nature of the date of the above item, New Year's Eve, suggests Stephen King's
All work and no play makes Jack a dull boy.
and also a (mis)quotation from a photographer's weblog—
"Art, being bartender, is never drunk."
— Quotation from Peter Viereck misattributed to Randall Jarrell in
Art as Bartender and the Golden Gate.
By a different photographer —
See also…
We may imagine the bartender above played by Louis Sullivan.
This morning's post mentioned the new film "Black Swan," starring Natalie Portman, that opens December 3.
Portman also starred in the 2006 film "V for Vendetta," based very loosely on today's date— November 5, Guy Fawkes Day.
Some background on Alan Moore, the creator of the graphic novel underlying that film—
1. The New York Times , March 12, 2006
2. Panelling Parallax: The Fearful Symmetry of William Blake and Alan Moore
3. This journal on March 24, 2009
Also from March 24, 2009— An image for what Thomas Pynchon, in this morning's post, called "the watchful scavengers of Epiphany."
"A world of made
is not a world of born— pity poor flesh
and trees, poor stars and stones, but never this
fine specimen of hypermagical
ultraomnipotence."
— e. e. cummings, 1944
For one such specimen, see The Matrix of Abraham—
a 5×5 square that is hypermagical… indeed, diabolical.
Related material on the algebra and geometry underlying some smaller structures
that have also, unfortunately, become associated with the word "magic"—
" … listen: there's a hell
of a good universe next door; let's go"
— e. e. cummings
Happy birthday, e. e.
The Dick Medal
Review of the film "Knowing" from 2009—
Nicolas Cage's character, an astrophysicist, looks at a chart (written 50 years earlier by a child) with a colleague and points out a chronologically correct prediction of the date and number of dead in world wide tragedies over the last fifty years, and his colleague's response is "Systems that find meaning in numbers are a dime a dozen. Why? Because people see what they want to see." Well that would be a pretty neat trick. You could build a career on that in a Vegas showroom.
Film Title: Next
Based on the 1954 short story
"The Golden Man" by Philip K. Dick
Release Date:
April 27, 2007
About the Film:
Nicolas Cage stars as Cris Johnson, a Las Vegas magician with a secret gift that is both a blessing and a curse: He has the uncanny ability to tell you what happens next.
Related material from this journal on the release date of "Next"— April 27, 2007—
|
Production Credits: Thanks to the
– and to
|
"It’s almost enough to make you think that time present and time past might both be present in time future. As someone may have said."
— David Orr, "The Age of Citation"
Or— Childhood's Rear End
This post was suggested by…
Related material:
The Zeppelin album cover, featuring rear views of nude children, was shot at the Giant's Causeway.
From a page at led-zeppelin.org—
See also Richard Rorty on Heidegger—
Safranski, the author of ''Schopenhauer and the Wild Years of Philosophy,'' never steps back and pronounces judgment on Heidegger, but something can be inferred from the German title of his book: ''Ein Meister aus Deutschland'' (''A Master From Germany''). Heidegger was, undeniably, a master, and was very German indeed. But Safranski's spine-chilling allusion is to Paul Celan's best-known poem, ''Death Fugue.'' In Michael Hamburger's translation, its last lines are:
death is a master from Germany his eyes are blue
he strikes you with leaden bullets his aim is true
a man lives in the house your golden hair Margarete
he sets his pack on us he grants us a grave in the air
he plays with the serpents and daydreams death is a master from Germany
your golden hair Margarete
your ashen hair Shulamith.
No one familiar with Heidegger's work can read Celan's poem without recalling Heidegger's famous dictum: ''Language is the house of Being. In its home man dwells.'' Nobody who makes this association can reread the poem without having the images of Hitler and Heidegger — two men who played with serpents and daydreamed — blend into each other. Heidegger's books will be read for centuries to come, but the smell of smoke from the crematories — the ''grave in the air'' — will linger on their pages.
Heidegger is the antithesis of the sort of philosopher (John Stuart Mill, William James, Isaiah Berlin) who assumes that nothing ultimately matters except human happiness. For him, human suffering is irrelevant: philosophy is far above such banalities. He saw the history of the West not in terms of increasing freedom or of decreasing misery, but as a poem. ''Being's poem,'' he once wrote, ''just begun, is man.''
For Heidegger, history is a sequence of ''words of Being'' — the words of the great philosophers who gave successive historical epochs their self-image, and thereby built successive ''houses of Being.'' The history of the West, which Heidegger also called the history of Being, is a narrative of the changes in human beings' image of themselves, their sense of what ultimately matters. The philosopher's task, he said, is to ''preserve the force of the most elementary words'' — to prevent the words of the great, houses-of-Being-building thinkers of the past from being banalized.
Related musical meditations—
Shine On (Saturday, April 21, 2007), Shine On, Part II, and Built (Sunday, April 22, 2007).
Related pictorial meditations—
The Giant's Causeway at Peter J. Cameron's weblog
and the cover illustration for Diamond Theory (1976)—
The connection between these two images is the following from Cameron's weblog today—
… as we saw, there are two different Latin squares of order 4;
one, but not the other, can be extended to a complete set
of 3 MOLS [mutually orthogonal Latin squares].
The underlying structures of the square pictures in the Diamond Theory cover are those of the two different Latin squares of order 4 mentioned by Cameron.
Connection with childhood—
The children's book A Wind in the Door, by Madeleine L'Engle. See math16.com. L'Engle's fantasies about children differ from those of Arthur C. Clarke and Led Zeppelin.
"The present study is closely connected with a lecture* given by Prof. Ernst Cassirer at the Warburg Library whose subject was 'The Idea of the Beautiful in Plato's Dialogues'…. My investigation traces the historical destiny of the same concept…."
* See Cassirer's Eidos und Eidolon : Das Problem des Schönen und der Kunst in Platons Dialogen, in Vorträge der Bibliothek Warburg II, 1922/23 (pp. 1–27). Berlin and Leipzig, B.G. Teubner, 1924.
— Erwin Panofsky, Idea: A Concept in Art Theory, foreword to the first German edition, Hamburg, March 1924
On a figure from Plato's Meno—
The above figures illustrate Husserl's phrase "eidetic variation"—
a phrase based on Plato's use of eidos, a word
closely related to the word "idea" in Panofsky's title.
For remarks by Cassirer on the theory of groups, a part of
mathematics underlying the above diamond variations, see
his "The Concept of Group and the Theory of Perception."
Sketch of some further remarks—
The Waterfield question in the sketch above
is from his edition of Plato's Theaetetus
(Penguin Classics, 1987).
The "design theory" referred to in the sketch
is that of graphic design, which includes the design
of commercial logos. The Greek word logos
has more to do with mathematics and theology.
"If there is one thread of warning that runs
through this dialogue, from beginning to end,
it is that verbal formulations as such are
shot through with ambiguity."
— Rosemary Desjardins, The Rational Enterprise:
Logos in Plato's Theaetetus, SUNY Press, 1990
Related material—
The Principle of Sufficient Reason
from "Three Public Lectures on Scientific Subjects,"
delivered at the Rice Institute, March 6, 7, and 8, 1940
EXCERPT 1—
My primary purpose will be to show how a properly formulated
Principle of Sufficient Reason plays a fundamental
role in scientific thought and, furthermore, is to be regarded
as of the greatest suggestiveness from the philosophic point
of view.2
In the preceding lecture I pointed out that three branches
of philosophy, namely Logic, Aesthetics, and Ethics, fall
more and more under the sway of mathematical methods.
Today I would make a similar claim that the other great
branch of philosophy, Metaphysics, in so far as it possesses
a substantial core, is likely to undergo a similar fate. My
basis for this claim will be that metaphysical reasoning always
relies on the Principle of Sufficient Reason, and that
the true meaning of this Principle is to be found in the
“Theory of Ambiguity” and in the associated mathematical
“Theory of Groups.”
If I were a Leibnizian mystic, believing in his “preestablished
harmony,” and the “best possible world” so
satirized by Voltaire in “Candide,” I would say that the
metaphysical importance of the Principle of Sufficient Reason
and the cognate Theory of Groups arises from the fact that
God thinks multi-dimensionally3 whereas men can only
think in linear syllogistic series, and the Theory of Groups is
2 As far as I am aware, only Scholastic Philosophy has fully recognized and ex-
ploited this principle as one of basic importance for philosophic thought
3 That is, uses multi-dimensional symbols beyond our grasp.
______________________________________________________________________
the appropriate instrument of thought to remedy our deficiency
in this respect.
The founder of the Theory of Groups was the mathematician
Evariste Galois. At the end of a long letter written in
1832 on the eve of a fatal duel, to his friend Auguste
Chevalier, the youthful Galois said in summarizing his
mathematical work,4 “You know, my dear Auguste, that
these subjects are not the only ones which I have explored.
My chief meditations for a considerable time have been
directed towards the application to transcendental Analysis
of the theory of ambiguity. . . . But I have not the time, and
my ideas are not yet well developed in this field, which is
immense.” This passage shows how in Galois’s mind the
Theory of Groups and the Theory of Ambiguity were
interrelated.5
Unfortunately later students of the Theory of Groups
have all too frequently forgotten that, philosophically
speaking, the subject remains neither more nor less than the
Theory of Ambiguity. In the limits of this lecture it is only
possible to elucidate by an elementary example the idea of a
group and of the associated ambiguity.
Consider a uniform square tile which is placed over a
marked equal square on a table. Evidently it is then impossible
to determine without further inspection which one
of four positions the tile occupies. In fact, if we designate
its vertices in order by A, B, C, D, and mark the corresponding
positions on the table, the four possibilities are for the
corners A, B, C, D of the tile to appear respectively in the
positions A, B, C, D; B, C, D, A; C, D, A, B; and D, A, B, C.
These are obtained respectively from the first position by a
4 My translation.
5 It is of interest to recall that Leibniz was interested in ambiguity to the extent
of using a special notation v (Latin, vel ) for “or.” Thus the ambiguously defined
roots 1, 5 of x2-6x+5=0 would be written x = l v 5 by him.
______________________________________________________________________
null rotation ( I ), by a rotation through 90° (R), by a rotation
through 180° (S), and by a rotation through 270° (T).
Furthermore the combination of any two of these rotations
in succession gives another such rotation. Thus a rotation R
through 90° followed by a rotation S through 180° is equivalent
to a single rotation T through 270°, Le., RS = T. Consequently,
the "group" of four operations I, R, S, T has
the "multiplication table" shown here:
This table fully characterizes the group, and shows the exact
nature of the underlying ambiguity of position.
More generally, any collection of operations such that
the resultant of any two performed in succession is one of
them, while there is always some operation which undoes
what any operation does, forms a "group."
__________________________________________________
EXCERPT 2—
Up to the present point my aim has been to consider a
variety of applications of the Principle of Sufficient Reason,
without attempting any precise formulation of the Principle
itself. With these applications in mind I will venture to
formulate the Principle and a related Heuristic Conjecture
in quasi-mathematical form as follows:
PRINCIPLE OF SUFFICIENT REASON. If there appears
in any theory T a set of ambiguously determined ( i e .
symmetrically entering) variables, then these variables can themselves
be determined only to the extent allowed by the corresponding
group G. Consequently any problem concerning these variables
which has a uniquely determined solution, must itself be
formulated so as to be unchanged by the operations of the group
G ( i e . must involve the variables symmetrically).
HEURISTIC CONJECTURE. The final form of any
scientific theory T is: (1) based on a few simple postulates; and
(2) contains an extensive ambiguity, associated symmetry, and
underlying group G, in such wise that, if the language and laws
of the theory of groups be taken for granted, the whole theory T
appears as nearly self-evident in virtue of the above Principle.
The Principle of Sufficient Reason and the Heuristic Conjecture,
as just formulated, have the advantage of not involving
excessively subjective ideas, while at the same time
retaining the essential kernel of the matter.
In my opinion it is essentially this principle and this
conjecture which are destined always to operate as the basic
criteria for the scientist in extending our knowledge and
understanding of the world.
It is also my belief that, in so far as there is anything
definite in the realm of Metaphysics, it will consist in further
applications of the same general type. This general conclu-
sion may be given the following suggestive symbolic form:
While the skillful metaphysical use of the Principle must
always be regarded as of dubious logical status, nevertheless
I believe it will remain the most important weapon of the
philosopher.
___________________________________________________________________________
A more recent lecture on the same subject —
by Jean-Pierre Ramis (Johann Bernoulli Lecture at U. of Groningen, March 2005)
Where Three Worlds Meet
From an obituary for David Brown, who died at 93 on Monday–
"David Brown was a force in the entertainment, literary and journalism worlds," Frank A. Bennack, Jr., vice chairman and chief executive officer of Hearst Corporation, said in a statement Tuesday. —Polly Anderson of the Associated Press
Mark Kramer, "Breakable Rules for Literary Journalists," Section 8–
"Readers are likely to care about how a situation came about and what happens next when they are experiencing it with the characters. Successful literary journalists never forget to be entertaining. The graver the writer's intentions, and the more earnest and crucial the message or analysis behind the story, the more readers ought to be kept engaged. Style and structure knit story and idea alluringly.
If the author does all this storytelling and digressing and industrious structure-building adroitly, readers come to feel they are heading somewhere with purpose, that the job of reading has a worthy destination. The sorts of somewheres that literary journalists reach tend to marry eternal meanings and everyday scenes. Richard Preston's 'The Mountains of Pi,' for instance, links the awkward daily lives of two shy Russian emigre mathematicians to their obscure intergalactic search for hints of underlying order in a chaotic universe."
Hints:
“Logic is all about the entertaining of possibilities.”
— Colin McGinn, Mindsight: Image, Dream, Meaning, Harvard U. Press, 2004
"According to the Buddha, scholars speak in sixteen ways of the state of the soul after death…. While I hesitate to disagree with the Compassionate One, I think there are more than sixteen possibilities described here…."
"That's entertainment!"
| “Zhu Xi maintained that all things are brought into being by the union of two universal aspects of reality: qi, sometimes translated as vital (or physical, material) force; and li, sometimes translated as rational principle (or law).” —Wikipedia |
| “Drop off the key, Lee” — Paul Simon
Reference frame (Click for details.) According to Chu Hsi [Zhu Xi],
“Li” is “the principle or coherence or order or pattern underlying the cosmos.” – Smith, Bol, Adler, and Wyatt, Sung Dynasty Uses of the I Ching, Related material: Dynasty and |
Text
Online Etymology Dictionary
|
“Discuss the geometry
underlying the above picture.”
— Log24, June 11, 2009
Geometry for Jews
(continued from Michelangelo’s birthday, 2003)
“Discuss the geometry underlying the above picture.”
Abstraction and the Holocaust (Mark Godfrey, Yale University Press, 2007) describes one approach to such a discussion: Bochner “took a photograph of a new arrangement of blocks, cut it up, reprinted it as a negative, and arranged the four corners in every possible configuration using the serial principles of rotation and reversal to make Sixteen Isomorphs (Negative) of 1967, which he later illustrated alongside works by Donald Judd, Sol LeWitt and Eva Hesse in his Artforum article ‘The Serial Attitude.’ [December 1967, pp. 28-33]” Bochner’s picture of “every possible configuration”–
Compare with the 24 figures in Frame Tales
(Log24, Nov. 10, 2008) and in Theme and Variations.
New collection release:
|
Those who actually try to view
the Wade collection will
encounter the following warning:
| To access the images in the ARTstor Digital Library you need to be affiliated with a participating institution (university, college, museum, public library or K-12 school). |
"… in some autistic enchantment,
pure as one of Bach's inverted canons or Euler's formula for polyhedra."
— George Steiner, "A Death of Kings," in The New Yorker, issue dated Sept. 7, 1968
A correspondence underlying
The Steiner here is
Jakob, not George.
See "Pope to Pray on
Autism Sunday 2009."
See also Log24 on that
Sunday– February 8:
Other knight figures:
Click on the SpringerLink
knight for a free copy
(pdf, 1.2 mb) of
the following paper
dealing with the geometry
underlying the R.T. Curtis
knight figures above:
Context:
Literature and Chess and
Sporadic Group References
Details:
| Adapted (for HTML) from the opening paragraphs of the above paper, W. Jonsson's 1970 "On the Mathieu Groups M22, M23, M24…"–
"[A]… uniqueness proof is offered here based upon a detailed knowledge of the geometric aspects of the elementary abelian group of order 16 together with a knowledge of the geometries associated with certain subgroups of its automorphism group. This construction was motivated by a question posed by D.R. Hughes and by the discussion Edge [5] (see also Conwell [4]) gives of certain isomorphisms between classical groups, namely
where A8 is the alternating group on eight symbols, S6 the symmetric group on six symbols, Sp(4,2) and PSp(4,2) the symplectic and projective symplectic groups in four variables over the field GF(2) of two elements, [and] PGL, PSL and SL are the projective linear, projective special linear and special linear groups (see for example [7], Kapitel II). The symplectic group PSp(4,2) is the group of collineations of the three dimensional projective space PG(3,2) over GF(2) which commute with a fixed null polarity tau…." References 4. Conwell, George M.: The three space PG(3,2) and its group. Ann. of Math. (2) 11, 60-76 (1910). 5. Edge, W.L.: The geometry of the linear fractional group LF(4,2). Proc. London Math. Soc. (3) 4, 317-342 (1954). 7. Huppert, B.: Endliche Gruppen I. Berlin-Heidelberg-New York: Springer 1967. |
A Wealth of
Algebraic Structure
A 1987 article by R. T. Curtis on the geometry of his Miracle Octad Generator (MOG) as it relates to the geometry of the 4×4 square is now available online ($20):
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, doi:10.1017/S0013091500004600.
(Published online by Cambridge University Press 19 Dec 2008.)
In the above article, Curtis explains how two-thirds of his 4×6 MOG array may be viewed as the 4×4 model of the four-dimensional affine space over GF(2). (His earlier 1974 paper (below) defining the MOG discussed the 4×4 structure in a purely combinatorial, not geometric, way.)
For further details, see The Miracle Octad Generator as well as Geometry of the 4×4 Square and Curtis’s original 1974 article, which is now also available online ($20):
A new combinatorial approach to M24, by R. T. Curtis. Abstract:
“In this paper, we define M24 from scratch as the subgroup of S24 preserving a Steiner system S(5, 8, 24). The Steiner system is produced and proved to be unique and the group emerges naturally with many of its properties apparent.”
(Received June 15 1974)
— Mathematical Proceedings of the Cambridge Philosophical Society (1976), 79: 25-42, doi:10.1017/S0305004100052075.
(Published online by Cambridge University Press 24 Oct 2008.)
* For instance:
Click for details.
Bertram Kostant, Professor Emeritus of Mathematics at MIT, on an object discussed in this week's New Yorker:
Hermann Weyl on the hard core of objectivity:
Steven H. Cullinane on the symmetries of a 4×4 array of points:
|
A Structure-Endowed Entity
"A guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity S, try to determine its 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 S in this way." — Hermann Weyl in Symmetry Let us apply Weyl's lesson to the following "structure-endowed entity."
What is the order of the resulting group of automorphisms? |
The above group of
automorphisms plays
a role in what Weyl,
following Eddington,
called a "colorful tale"–
This puzzle shows
that the 4×4 array can
also be viewed in
thousands of ways.
"You can make 322,560
pairs of patterns. Each
pair pictures a different
symmetry of the underlying
16-point space."
— Steven H. Cullinane,
July 17, 2008
For other parts of the tale,
see Ashay Dharwadker,
the Four-Color Theorem,
and Usenet Postings.
Rudolf Arnheim, a student of Gestalt psychology (which, an obituary notes, emphasizes "the perception of forms as organized wholes") was the first Professor of the Psychology of Art at Harvard. He died at 102 on Saturday, June 9, 2007.
The conclusion of yesterday's New York Times obituary of Arnheim:
"… in The New York Times Book Review in 1986, Celia McGee called Professor Arnheim 'the best kind of romantic,' adding, 'His wisdom, his patient explanations and lyrical enthusiasm are those of a teacher.'"
A related quotation:
"And you are teaching them a thing or two about yourself. They are learning that you are the living embodiment of two timeless characterizations of a teacher: 'I say what I mean, and I mean what I say' and 'We are going to keep doing this until we get it right.'"
Here, yet again, is an illustration that has often appeared in Log24– notably, on the date of Arnheim's death:
Related quotations:
"We have had a gutful of fast art and fast food. What we need more of is slow art: art that holds time as a vase holds water: art that grows out of modes of perception and whose skill and doggedness make you think and feel; art that isn't merely sensational, that doesn't get its message across in 10 seconds, that isn't falsely iconic, that hooks onto something deep-running in our natures. In a word, art that is the very opposite of mass media. For no spiritually authentic art can beat mass media at their own game."
— Robert Hughes, speech of June 2, 2004
"Whether the 3×3 square grid is fast art or slow art, truly or falsely iconic, perhaps depends upon the eye of the beholder."
If the beholder is Rudolf Arnheim, whom we may now suppose to be viewing the above figure in the afterlife, the 3×3 square is apparently slow art. Consider the following review of his 1982 book The Power of the Center:
"Arnheim deals with the significance of two kinds of visual organization, the concentric arrangement (as exemplified in a bull's-eye target) and the grid (as exemplified in a Cartesian coordinate system)….
It is proposed that the two structures of grid and target are the symbolic vehicles par excellence for two metaphysical/psychological stances. The concentric configuration is the visual/structural equivalent of an egocentric view of the world. The self is the center, and all distances exist in relation to the focal spectator. The concentric arrangement is a hermetic, impregnable pattern suited to conveying the idea of unity and other-worldly completeness. By contrast, the grid structure has no clear center, and suggests an infinite, featureless extension…. Taking these two ideal types of structural scaffold and their symbolic potential (cosmic, egocentric vs. terrestrial, uncentered) as given, Arnheim reveals how their underlying presence organizes works of art."
— Review of Rudolf Arnheim's The Power of the Center: A Study of Composition in the Visual Arts (Univ. of Calif. Press, 1982). Review by David A. Pariser, Studies in Art Education, Vol. 24, No. 3 (1983), pp. 210-213
Arnheim himself says in this book (pp. viii-ix) that "With all its virtues, the framework of verticals and horizontals has one grave defect. It has no center, and therefore it has no way of defining any particular location. Taken by itself, it is an endless expanse in which no one place can be distinguished from the next. This renders it incomplete for any mathematical, scientific, and artistic purpose. For his geometrical analysis, Descartes had to impose a center, the point where a pair of coordinates [sic] crossed. In doing so he borrowed from the other spatial system, the centric and cosmic one."
Students of art theory should, having read the above passages, discuss in what way the 3×3 square embodies both "ideal types of structural scaffold and their symbolic potential."
We may imagine such a discussion in an afterlife art class– in, perhaps, Purgatory rather than Heaven– that now includes Arnheim as well as Ernst Gombrich and Kirk Varnedoe.
Such a class would be one prerequisite for a more advanced course– Finite geometry of the square and cube.
Thanks to the
Pennsylvania Lottery for
today's suggestion of links
to the dates 9/15 and 6/06–
— and to
Hermann Weyl
for the illustration
from 6/06 (D-Day)
underlying the
following "gold medal"
from 9/15, 2006:
.
Conceit
at Harvard
“… there is some virtue in tracking cultural trends in terms of their relation to the classic Trinitarian framework of Christian thought.”
— Description of lectures to be given Monday, Tuesday, and Wednesday of this week (on Father, Son, and Holy Spirit, respectively, and their relationship to “cultural trends”) at Harvard’s Memorial Church
I prefer more-classic trinitarian frameworks– for example,
and the structural trinity
underlying
classic quilt patterns:
Click on pictures for further details.
These mathematical trinities are
conceits in the sense of concepts
or notions; examples of the third
kind of conceit are easily
found, especially at Harvard.
For a possible corrective to
examples of the third kind,
see
To Measure the Changes.
An application of the finite geometry underlying the diamond theorem:
“Qubits in phase space: Wigner function approach to quantum error correction and the mean king problem,” by Juan Pablo Paz, Augusto Jose Roncaglia, and Marcos Saraceno (arXiv:quant-ph/0410117 v2 4 Nov 2004) (pdf)
Thanks to Peter Woit’s weblog
for a link to the above illustration.
This picture of
“Coxeter Exhuming Geometry”
suggests the following comparison:
For the second tombstone,
see this morning’s entry,
Birth, Death, and Symmetry.
Further details on the geometry
underlying the second tombstone:
The above is from
Variable Resolution 4–k Meshes:
Concepts and Applications (pdf),
by Luiz Velho and Jonas Gomes.
See also Symmetry Framed
and The Garden of Cyrus.
| “That corpse you planted last year in your garden, Has it begun to sprout? Will it bloom this year? Or has the sudden frost disturbed its bed?” — T. S. Eliot, “The Waste Land“ |
Story
There is one story
and one story only
That will prove
worth your telling….
— Robert Graves,
“To Juan at the Winter Solstice”
“To many, mathematicians have come to resemble an esoteric sect, whose members alone have access to secret otherworldly mysteries.
All of us who came to Mykonos believed that this is an unfortunate situation. Mathematics is an inseparable part of human culture, and should be viewed and treated as such. Our underlying assumption was that mathematical reasoning had something important in common with that quintessential human activity – story-telling. But what this means, and what kind of connections can be drawn between the two, remained to be sorted out.”
Flashback to
Harrison Ford’s birthday
a year earlier:
“He’s a Mad Scientist and
I’m his Beautiful Daughter.”
— Deety in Heinlein’s
The Number of the Beast.
“If you have ever loved a book
so much that you began to
believe that it continued on
in its own world
even after you put it down,
this book could be for you.”
— Jodi Russell, review of
Number of the Beast
These last two quotations
are from
Story Theory and
the Number of the Beast,
by Steven H. Cullinane on
December 21, 2001.
Related material:
See Lucky(?) Numbers,
yesterday’s Pennsylvania lottery,
and the previous entry.
“There is an
underlying timelessness
in the basic conversation
that is mathematics.”
— Barry Mazur (pdf)
“The authors of the etiquette book The Art of Civilized Conversation say that conversation’s versatility makes it ‘the Swiss Army knife of social skills.'”
Today is the birthday of Helmut Wielandt (Dec. 19,
"In his speech accepting membership of the Heidelberg Academy in 1960 he said:-
It is to one of Schur's seminars that I owe the stimulus to work with permutation groups, my first research area. At that time the theory had nearly died out. It had developed last century, but at about the turn of the century had been so completely superseded by the more generally applicable theory of abstract groups that by 1930 even important results were practically forgotten – to my mind unjustly."
Permutation groups are still not without interest. See today's updates (Notes [01] and [02]) to Pattern Groups.
A Miniature
Rosetta Stone
John Baez discussed (Sept. 6, 2003) the analogies of Weil, and he himself furnished another such Rosetta stone on a much smaller scale:
“… a 24-element group called the ‘binary tetrahedral group,’ a 24-element group called ‘SL(2,Z/3),’ and the vertices of a regular polytope in 4 dimensions called the ’24-cell.’ The most important fact is that these are all the same thing!”
For further details, see Wikipedia on the 24-cell, on special linear groups, and on Hurwitz quaternions,
The group SL(2,Z/3), also known as “SL(2,3),” is of course derived from the general linear group GL(2,3). For the relationship of this group to the quaternions, see the Log24 entry for August 4 (the birthdate of the discoverer of quaternions, Sir William Rowan Hamilton).
The 3×3 square shown above may, as my August 4 entry indicates, be used to picture the quaternions and, more generally, the 48-element group GL(2,3). It may therefore be regarded as the structure underlying the miniature Rosetta stone described by Baez.
“The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled.”
— J. L. Alperin, book review,
Bulletin (New Series) of the American
Mathematical Society 10 (1984), 121
Two Poles
From today’s New York Times:
From erraticimpact.com on Paul Ricoeur:
| From today’s NYT obituary of Sol Stetin:
“Mr. Stetin, who emigrated from Poland at the age of 10 and dropped out of high school in the ninth grade, was fond of saying he got his education in the labor movement.” |
 |
“… it is not in isolation that the rhetorical power of such oppositions resides, but in their articulation in relation to other oppositions. In Aristotle’s Physics the four elements of earth, air, fire and water were said to be opposed in pairs. For more than two thousand years oppositional patterns based on these four elements were widely accepted as the fundamental structure underlying surface reality….
The structuralist semiotician Algirdas Greimas introduced the semiotic square (which he adapted from the ‘logical square’ of scholastic philosophy) as a means of analysing paired concepts more fully….”
— Daniel Chandler, Semiotics for Beginners
Poetry’s Bones and
Theme and Variations.
Other readings on polarity:
Log24, May 24, 2003, and
from July 26, 2003:
|
Bright Star and Dark Lady “Mexico is a solar country — but it is also a black country, a dark country. This duality of Mexico has preoccupied me since I was a child.” — Octavio Paz, |
||
|
Bright Star
|
Amen. |
Dark Lady
|
"Professor Krauss even uses many of the same decorations with which she festooned earlier volumes. Bataille’s photograph of a big toe, for example, which I like to think of as her mascot, reappears. As does her favorite doodle, a little graph known as a 'Klein Group' or 'L Schema' whose sides and diagonals sport arrows pointing to corners labeled with various opposing pairs: e.g., 'ground' and 'not ground,' 'figure' and 'not figure.' Professor Krauss seems to believe that this device, lifted from the pages of structuralist theory, illuminates any number of deep mysteries: the nature of modernism, to begin with, but also the essence of gender relations, self-consciousness, perception, vision, castration anxiety, and other pressing conundrums that, as it happens, she has trouble distinguishing from the nature of modernism. Altogether, the doodle is a handy thing to have around. One is not surprised that Professor Krauss reproduces it many times in her new book."
 A Jungian on this six-line figure: "They are the same six lines |
"People have believed in the fundamental character of binary oppositions since at least classical times. For instance, in his Metaphysics Aristotle advanced as primary oppositions: form/matter, natural/unnatural, active/passive, whole/part, unity/variety, before/after and being/not-being.* But it is not in isolation that the rhetorical power of such oppositions resides, but in their articulation in relation to other oppositions. In Aristotle's Physics the four elements of earth, air, fire and water were said to be opposed in pairs. For more than two thousand years oppositional patterns based on these four elements were widely accepted as the fundamental structure underlying surface reality….
The structuralist semiotician Algirdas Greimas introduced the semiotic square (which he adapted from the 'logical square' of scholastic philosophy) as a means of analysing paired concepts more fully…."
— Daniel Chandler, Semiotics for Beginners.
* Compare Chandler's list of Aristotle's primary oppositions with Aristotle's list (also in the Metaphysics) of Pythagorean oppositions (see Midrash Jazz Quartet).
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