Thursday, February 7, 2019
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 finitegeometry properties of the 4×4 square as
a finite affine 4space — properties that are of use in studying the Mathieu
group M_{24 }with the aid of the MOG.
Comments Off on Geometry of the 4×4 Square: The Kummer Configuration
Wednesday, December 12, 2018
Those pleased by what Ross Douthat today called
"The Return of Paganism" are free to devise rituals
involving what might be called "the sacred geometry
of the Kummer 16_{6 }configuration."
As noted previously in this journal,
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
See also earlier posts also tagged "Kummerhenge" and
another property of the remarkable Kummer 16_{6} —
For some related literary remarks, see "Transposed" in this journal.
Some background from 2001 —
Comments Off on Kummerhenge Continues.
Thursday, November 22, 2018
Comments Off on Rosenhain and Göpel Meet Kummer in Projective 3Space
Thursday, July 12, 2018
“… the utterly real thing in writing is the only thing that counts…."
— Maxwell Perkins to Ernest Hemingway, Aug. 30, 1935
"Omega is as real as we need it to be."
— Burt Lancaster in "The Osterman Weekend"
Comments Off on Kummerhenge Illustrated
Thursday, June 21, 2018
See also the Omega Matrix in this journal.
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Saturday, June 16, 2018
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
See too "The Ruler of Reality" in this journal.
Related material —
A more esoteric artifact: The Kummer 16_{6} Configuration . . .
An array of Göpel tetrads appears in the background below.
"As you can see, we've had our eye on you
for some time now, Mr. Anderson."
Comments Off on Kummer’s (16, 6) (on 6/16)
Monday, September 12, 2016
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 4space 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 M_{24 }"
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 twosubsets of a sixset. See the post of Sept. 13.
Comments Off on The Kummer Lattice
Tuesday, July 12, 2016
Further details from Edmund Hess in 1889* related to
last night's remarks on the Klein 60_{15} configuration
and the Kummer 16_{6} configuration —
* Edmund Hess, "Beiträge zur Theorie der räumlichen Configurationen.
Ueber die Klein'sche Configuration Cf. (60₁₅, 30₆) und einige
bemerkenswerthe aus dieser ableitbare räumliche Configurationen."
Verhandlungen der Kaiserlichen LeopoldinischCarolinischen
Deutschen Akademie der Naturforscher, Vol.55, No. 2, pp. 98167
Comments Off on Klein and Kummer Configurations in 1889
Wednesday, May 25, 2016
From "Projective Geometry and PTSymmetric Dirac Hamiltonian,"
Y. Jack Ng and H. van Dam,
Physics Letters B , Volume 673, Issue 3,
23 March 2009, Pages 237–239
(http://arxiv.org/abs/0901.2579v2, last revised Feb. 20, 2009)
" Studies of spin½ theories in the framework of projective geometry
have been undertaken before. See, e.g., Ref. [4]. ^{1 }"
" ^{1} These papers are rather mathematical and technical.
The authors of the first two papers discuss the Dirac equation
in terms of the PluckerKlein correspondence between lines of
a threedimensional projective space and points of a quadric
in a fivedimensional projective space. The last paper shows
that the Dirac equation bears a certain relation to Kummer’s
surface, viz., the structure of the Dirac ring of matrices is
related to that of Kummer’s 16_{6} configuration . . . ."
[4]
O. Veblen
Proc. Natl. Acad. Sci. USA , 19 (1933), p. 503
Full Text via CrossRef
E.M. Bruins
Proc. Nederl. Akad. Wetensch. , 52 (1949), p. 1135
F.C. Taylor Jr., Master thesis, University of North Carolina
at Chapel Hill (1968), unpublished
A remark of my own on the structure of Kummer’s 16_{6} configuration . . . .
See that structure in this journal, for instance —
See as well yesterday morning's post.
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Friday, March 7, 2014
The Dream of the Expanded Field continues…
From Klein's 1893 Lectures on Mathematics —
"The varieties introduced by Wirtinger may be called Kummer varieties…."
— E. Spanier, 1956
From this journal on March 10, 2013 —
From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
"The Universal Kummer Threefold," by
Qingchun Ren, Steven V Sam, Gus Schrader, and Bernd Sturmfels —
Two such considerations —
Update of 10 PM ET March 7, 2014 —
The following slides by one of the "Kummer Threefold" authors give
some background related to the above 64point vector space and
to the Weyl group of type E_{7}, W (E_{7}):
The Cayley reference is to "Algorithm for the characteristics of the
triple ϑfunctions," Journal für die Reine und Angewandte
Mathematik 87 (1879): 165169. <http://eudml.org/doc/148412>.
To read this in the context of Cayley's other work, see pp. 441445
of Volume 10 of his Collected Mathematical Papers .
Comments Off on Kummer Varieties
Thursday, September 27, 2012
Denote the ddimensional hypercube by γ_{d} .
"… after coloring the sixtyfour vertices of γ_{6}
alternately red and blue, we can say that
the sixteen pairs of opposite red vertices represent
the sixteen nodes of Kummer's surface, while
the sixteen pairs of opposite blue vertices
represent the sixteen tropes."
— From "Kummer's 16_{6 }," section 12 of Coxeter's 1950
"Selfdual Configurations and Regular Graphs"
Just as the 4×4 square represents the 4dimensional
hypercube γ_{4 }over the twoelement Galois field GF(2),
so the 4x4x4 cube represents the 6dimensional
hypercube γ_{6} over GF(2).
For religious interpretations, see
Nanavira Thera (Indian) and
I Ching geometry (Chinese).
See also two professors in The New York Times
discussing images of the sacred in an oped piece
dated Sept. 26 (Yom Kippur).
Comments Off on Kummer and the Cube
Saturday, May 4, 2019
The title refers to CalabiYau spaces.
T. S. Eliot —
Four Quartets
. . . Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.
A less "cosmic" but still noteworthy code — The Golay code.
This resides in a 12dimensional space over GF(2).
Related material from Plato and R. T. Curtis —
A related CalabiYau "Chinese jar" first described in detail in 1905 —
A figure that may or may not be related to the 4x4x4 cube that
holds the classical Chinese "cosmic code" — the I Ching —
ftp://ftp.cs.indiana.edu/pub/hanson/forSha/AK3/old/K3pix.pdf
Comments Off on The Chinese Jars of ShingTung Yau
Friday, March 29, 2019
"This outer automorphism can be regarded as
the seed from which grow about half of the
sporadic simple groups…." — Noam Elkies
Closely related material —
The top two cells of the Curtis "heavy brick" are also
the key to the diamondtheorem correlation.
Comments Off on FrontRow Seed
Thursday, March 28, 2019
The previous post, "Dream of Plenitude," suggests . . .
"So here's to you, NordstromRobinson . . . ."
Comments Off on Culture
Tuesday, February 26, 2019
Some related material in this journal — See a search for k6.gif.
Some related material from Harvard —
Elkies' "15 simple transpositions" clearly correspond to the 15 edges of
the complete graph K_{6} and to the 15 2subsets of a 6set.
For the connection to PG(3,2), see Finite Geometry of the Square and Cube.
The following "manifestation" of the 2subsets of a 6set might serve as
the desired Wikipedia citation —
See also the above 1986 construction of PG(3,2) from a 6set
in the work of other authors in 1994 and 2002 . . .

GonzalezDorrego, Maria R. (Maria del Rosario),
(16,6) Configurations and Geometry of Kummer Surfaces in P^{3}.
American Mathematical Society, Providence, RI, 1994.

Dolgachev, Igor, and Keum, JongHae,
"Birational Automorphisms of Quartic Hessian Surfaces."
Trans. Amer. Math. Soc. 354 (2002), 30313057.
Comments Off on Citation
Monday, February 18, 2019
From yesterday's post on sacerdotal jargon —
A related note from May 1986 —
Comments Off on Sacerdotal K_{6}, Continued
Saturday, February 16, 2019
The title refers to a 1514 engraving.
See also Angel Particle in this journal.
Comments Off on Melancholy for Dürer
Tuesday, February 12, 2019
This journal on the above date, October 17, 2008 —
“Every musician wants to do something of lasting quality,
something which will hold up for a long time, and
I guess we did it with ‘Stairway.'”
— Jimmy Page on “Stairway to Heaven“
Scholium —
"Kummer " in German means "sorrow."
Related material —
Other posts now tagged Dolmen.
Comments Off on A Long Time
Saturday, January 26, 2019
The above cryptic search result indicates that there may
soon be a new Norwegian art installation based on this page
of Eddington (via Log24) —
See also other posts tagged Kummerhenge.
Comments Off on Installasjon
Friday, December 14, 2018
References in recent posts to physical space and
to mathematical space suggest a comparison.
Physical space is well known, at least in the world
of mass entertainment.
Mathematical space, such as the 12dimensional
finite space of the Golay code, is less well known.
A figure from each space —
The source of the ConwaySloane brick —
Quote from a mathematics writer —
“Looking carefully at Golay’s code is like staring into the sun.”
— Richard Evan Schwartz
The former practice yields reflections like those of Conway and Sloane.
The latter practice is not recommended.
Comments Off on Small Space Odyssey
Thursday, December 13, 2018
For Oslo artist Josefine Lyche, excerpts
from a Google image search today —
Material related to Lyche's experience as an adolescent with a ZX Spectrum computer —
Click "Hello World" for a larger image.
Comments Off on Space Art
Wednesday, December 12, 2018
Some images, and a definition, suggested by my remarks here last night
on Apollo and Ross Douthat's remarks today on "The Return of Paganism" —
In finite geometry and combinatorics,
an inscape is a 4×4 array of square figures,
each figure picturing a subset of the overall 4×4 array:
Related material — the phrase
"Quantum Tesseract Theorem" and …
A. An image from the recent
film "A Wrinkle in Time" —
B. A quote from the 1962 book —
"There's something phoney
in the whole setup, Meg thought.
There is definitely something rotten
in the state of Camazotz."
Comments Off on An Inscape for Douthat
Friday, December 7, 2018
(Continued from this morning)
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
See also other Log24 posts tagged Kummerhenge.
Comments Off on The Angel Particle
Sunday, December 2, 2018
A search this morning for articles mentioning the Miracle Octad Generator
of R. T. Curtis within the last year yielded an abstract for two talks given
at Hiroshima on March 8 and 9, 2018 —
http://www.math.sci.hiroshimau.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 2fold 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 HorrocksMumford bundle. Poincare's homology 3sphere, 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 socalled `Arnol'd Trinities'.
Motivated originally by the `Eightfold Way' sculpture at MSRI in Berkeley, we discuss interrelationships 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 interconnectedness 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 graphicillustrated 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 8component 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.
Comments Off on Symmetry at Hiroshima
Friday, November 23, 2018
Comments Off on Artfield Studio
Thursday, November 22, 2018
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
Note also the four 4×4 arrays surrounding the central diamond
in the chi of the chirho page of the Book of Kells —
From a Log24 post
of March 17, 2012
"Interlocking, interlacing, interweaving"
— Condensed version of page 141 in Eddington's
1939 Philosophy of Physical Science
Comments Off on Geometric Incarnation
Sunday, November 18, 2018
Update of Nov. 19 —
"Design is how it works." — Steve Jobs
See also www.cullinane.design.
Comments Off on Space Music
Saturday, September 22, 2018
Comments Off on Minimalist Configuration
Monday, July 16, 2018
"The novel has a parallel narrative that eventually
converges with the main story."
— Wikipedia on a book by Foer's novelist brother
Public Squares
An image from the online New York Times
on the date, July 6,
of the above Atlantic article —
An image from "Blackboard Jungle," 1955 —
"Through the unknown, remembered gate . . . ."
— T. S. Eliot, Four Quartets
Comments Off on Greatly Exaggerated Report
Friday, July 6, 2018
"… Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness."
— T. S. Eliot, "Burnt Norton," 1936
"Read something that means something."
— Advertising slogan for The New Yorker
The previous post quoted some mystic meditations of Octavio Paz
from 1974. I prefer some less mystic remarks of Eddington from
1938 (the Tanner Lectures) published by Cambridge U. Press in 1939 —
"… we have sixteen elements with which to form a groupstructure" —
See as well posts tagged Dirac and Geometry.
Comments Off on Something
Friday, June 29, 2018
The phrase "Blue Dream" in the previous post
suggests a Web search for Traumnovelle .
That search yields an interesting weblog post
from 2014 commemorating the 1999 dies natalis
(birth into heaven) of St. Stanley Kubrick.
Related material from March 7, 2014,
in this journal —
That 2014 post was titled "Kummer Varieties." It is now tagged
"Kummerhenge." For some backstory, see other posts so tagged.
Comments Off on For St. Stanley
Wednesday, June 27, 2018
"Summerfield, Kummerhenge. Kummerhenge, Summerfield."
Comments Off on The Letterman Intro
A passage that may or may not have influenced Madeleine L'Engle's
writings about the tesseract :
From Mere Christianity , by C. S. Lewis (1952) —
"Book IV – Beyond Personality:
or First Steps in the Doctrine of the Trinity"
. . . .
I warned you that Theology is practical. The whole purpose for which we exist is to be thus taken into the life of God. Wrong ideas about what that life is, will make it harder. And now, for a few minutes, I must ask you to follow rather carefully.
You know that in space you can move in three ways—to left or right, backwards or forwards, up or down. Every direction is either one of these three or a compromise between them. They are called the three Dimensions. Now notice this. If you are using only one dimension, you could draw only a straight line. If you are using two, you could draw a figure: say, a square. And a square is made up of four straight lines. Now a step further. If you have three dimensions, you can then build what we call a solid body, say, a cube—a thing like a dice or a lump of sugar. And a cube is made up of six squares.
Do you see the point? A world of one dimension would be a straight line. In a twodimensional world, you still get straight lines, but many lines make one figure. In a threedimensional world, you still get figures but many figures make one solid body. In other words, as you advance to more real and more complicated levels, you do not leave behind you the things you found on the simpler levels: you still have them, but combined in new ways—in ways you could not imagine if you knew only the simpler levels.
Now the Christian account of God involves just the same principle. The human level is a simple and rather empty level. On the human level one person is one being, and any two persons are two separate beings—just as, in two dimensions (say on a flat sheet of paper) one square is one figure, and any two squares are two separate figures. On the Divine level you still find personalities; but up there you find them combined in new ways which we, who do not live on that level, cannot imagine.
In God's dimension, so to speak, you find a being who is three Persons while remaining one Being, just as a cube is six squares while remaining one cube. Of course we cannot fully conceive a Being like that: just as, if we were so made that we perceived only two dimensions in space we could never properly imagine a cube. But we can get a sort of faint notion of it. And when we do, we are then, for the first time in our lives, getting some positive idea, however faint, of something superpersonal—something more than a person. It is something we could never have guessed, and yet, once we have been told, one almost feels one ought to have been able to guess it because it fits in so well with all the things we know already.
You may ask, "If we cannot imagine a threepersonal Being, what is the good of talking about Him?" Well, there isn't any good talking about Him. The thing that matters is being actually drawn into that threepersonal life, and that may begin any time —tonight, if you like.
. . . .

But beware of being drawn into the personal life of the Happy Family .
https://www.jstor.org/stable/24966339 —
"The colorful story of this undertaking begins with a bang."
And ends with …
Martin Gardner on Galois—
"Galois was a thoroughly obnoxious nerd,
suffering from what today would be called
a 'personality disorder.' His anger was
paranoid and unremitting."
Comments Off on Taken In
Sunday, June 24, 2018
A clue to the relationship between the Kummer (16, 6)
configuration and the large Mathieu group M_{24} —
Related material —
See too the diamondtheorem correlation.
Comments Off on For 6/24
Saturday, June 23, 2018
Backstory for fiction fans, from Log24 on June 11 —
Related non fiction —
See as well the structure discussed in today's previous post.
Comments Off on Meanwhile …
From Nanavira Thera, "Early Letters," in Seeking the Path —
"nine possibilities arising quite naturally" —
Compare and contrast with Hudson's parametrization of the
4×4 square by means of 0 and the 15 2subsets of a 6set —
Comments Off on Plan 9 from Inner Space
Friday, June 22, 2018
"… lo lidchok et haketz …."
— Acceptance speech, Guardian of Zion award, 2002
Also on February 20, 2012 —
Comments Off on For the Late Charles Krauthammer
Thursday, June 21, 2018
A Buddhist view —
"Just fancy a scale model of Being
made out of string and cardboard."
— Nanavira Thera, 1 October 1957,
on a model of Kummer's Quartic Surface
mentioned by Eddington
A Christian view —
A formal view —
From a Log24 search for High Concept:
See also Galois Tesseract.
Comments Off on Models of Being
"Just fancy a scale model of Being
made out of string and cardboard."
— Nanavira Thera, 1 October 1957,
on a model of Kummer's Quartic Surface
mentioned by Eddington
"… a treatise on Kummer's quartic surface."
The "supermathematician" Eddington did not see fit to mention
the title or the author of the treatise he discussed.
See Hudson + Kummer in this journal.
See also posts tagged Dirac and Geometry.
Comments Off on Dirac and Geometry (continued)
From an obituary for Stanley Cavell, Harvard philosopher
who reportedly died at 91 on Tuesday, June 19:
The London Review of Books weblog yesterday —
"Michael Wood reviewed [Cavell’s]
Philosophy the Day after Tomorrow in 2005:
'The ordinary slips away from us. If we ignore it, we lose it.
If we look at it closely, it becomes extraordinary, the way
words or names become strange if we keep staring at them.
The very notion turns into a baffling riddle.' "
See also, in this journal, Tuesday morning's Ici vient M. Jordan and
this morning's previous post.
Update of 3:24 AM from my RSS feed —
Comments Off on Cavell’s Matrix
Tuesday, June 19, 2018
Comments Off on Ici vient M. Jordan
Saturday, June 16, 2018
"But perhaps the desire for story
is what gets us into trouble to begin with."
— Sarah Marshall on June 5, 2018
"Beckett wrote that Joyce believed fervently in
the significance of chance events and of
random connections. ‘To Joyce reality was a paradigm,
an illustration of a possibly unstateable rule…
According to this rule, reality, no matter how much
we try to manipulate it, can only shift about
in continual movement, yet movement
limited in its possibilities…’ giving rise to
‘the notion of the world where unexpected simultaneities
are the rule.’ In other words, a coincidence … is actually
just part of a continually moving pattern, like a kaleidoscope.
Or Joyce likes to put it, a ‘collideorscape’."
— Gabrielle Carey, "Breaking Up with James Joyce,"
Sydney Review of Books , 15 June 2018
Carey's carelessness with quotations suggests a look at another
author's quoting of Ellmann on Joyce —
Comments Off on For June 16
Monday, June 11, 2018
The title was suggested by the name "ARTI" of an artificial
intelligence in the new film 2036: Origin Unknown.
The Eye of ARTI —
See also a post of May 19, "UhOh" —
— and a post of June 6, "Geometry for Goyim" —
Mystery box merchandise from the 2011 J. J. Abrams film Super 8
An arty fact I prefer, suggested by the triangular computereye forms above —
This is from the July 29, 2012, post The Galois Tesseract.
See as well . . .
Comments Off on Arty Fact
Friday, February 16, 2018
The Institute for Advanced Study (IAS) at Princeton in its Fall 2015 Letter
revived "Beautiful Mathematics" as a title:
This ugly phrase was earlier used by Truman State University
professor Martin Erickson as a book title. See below.
In the same IAS Fall 2015 Letter appear the following remarks
by Freeman Dyson —
". . . a special case of a much deeper connection that Ian Macdonald
discovered between two kinds of symmetry which we call modular and affine.
The two kinds of symmetry were originally found in separate parts of science,
modular in pure mathematics and affine in physics. Modular symmetry is
displayed for everyone to see in the drawings of flying angels and devils
by the artist Maurits Escher. Escher understood the mathematics and got the
details right. Affine symmetry is displayed in the peculiar groupings of particles
created by physicists with highenergy accelerators. The mathematician
Robert Langlands was the first to conjecture a connection between these and
other kinds of symmetry. . . ." (Wikipedia link added.)
The adjective "modular" might aptly be applied to . . .
The adjective "affine" might aptly be applied to . . .
The geometry of the 4×4 square combines modular symmetry
(i.e., related to theta functions) with the affine symmetry above.
Hudson's 1905 discussion of modular symmetry (that of Rosenhain
tetrads and Göpel tetrads) in the 4×4 square used a parametrization
of that square by the digit 0 and the fifteen 2subsets of a 6set, but
did not discuss the 4×4 square as an affine space.
For the connection of the 15 Kummer modular 2subsets with the 16
element affine space over the twoelement Galois field GF(2), see my note
of May 26, 1986, "The 2subsets of a 6set are the points of a PG(3,2)" —
— and the affine structure in the 1979 AMS abstract
"Symmetry invariance in a diamond ring" —
For some historical background on the symmetry investigations by
Dyson and Macdonald, see Dyson's 1972 article "MIssed Opportunities."
For Macdonald's own use of the words "modular" and "affine," see
Macdonald, I. G., "Affine Lie algebras and modular forms,"
Séminaire N. Bourbaki , Vol. 23 (19801981), Talk no. 577, pp. 258276.
Comments Off on Two Kinds of Symmetry
Friday, December 22, 2017
From a Log24 post of October 10, 2017 —
Related material from May 25, 2016 —
Comments Off on IT
Thursday, November 30, 2017
Scholia on the title — See Quantum + Mystic in this journal.
The Matrix of LéviStrauss —
"In Vol. I of Structural Anthropology , p. 209, I have shown that
this analysis alone can account for the double aspect of time
representation in all mythical systems: the narrative is both
'in time' (it consists of a succession of events) and 'beyond'
(its value is permanent)." — Claude LéviStrauss, 1976
I prefer the earlier, betterknown, remarks on time by T. S. Eliot
in Four Quartets , and the following four quartets (from
The Matrix Meets the Grid) —
.
From a Log24 post of June 2627, 2017:
A work of Eddington cited in 1974 by von Franz —
See also Dirac and Geometry and Kummer in this journal.
Ron Shaw on Eddington's triads "associated in conjugate pairs" —
For more about hyperbolic and isotropic lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.
For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.
Comments Off on The Matrix for Quantum Mystics
Wednesday, November 29, 2017
See also Inscape in this journal and, for a related Chapel Hill thesis,
the post Kummer and Dirac.
Comments Off on Definitions
Tuesday, October 31, 2017
https://www.shmoop.com/nocountryforoldmen/coinsymbol.html —
"You know the date on this coin?"
Related material —
This journal on March 7, 2014 —
From Klein’s 1893 Lectures on Mathematics —
“The varieties introduced by Wirtinger may be called
Kummer varieties….” — E. Spanier, 1956
From the "varieties introduced by Wirtinger" link above —
.
Comments Off on One Fell Shmoop
Friday, September 29, 2017
(Some Remarks for Science Addicts)
Principles —
Personalities —
* See "Tradition Twelve."
Comments Off on Principles Before Personalities*
Saturday, September 9, 2017
"For use of the Kummer surface in Buddhist metaphysics . . ."
is a phrase from a search in this journal for Nanavira.
See as well Buddhism in the previous post.
Comments Off on Mathematics and Metaphysics
Saturday, September 2, 2017
This post was suggested by the names* (if not the very abstruse
concepts ) in the Aug. 20, 2013, preprint "A Panoramic Overview
of Interuniversal Teichmuller Theory," by S. Mochizuki.
* Specifically, Jacobi and Kummer (along with theta functions).
I do not know of any direct connection between these names'
relevance to the writings of Mochizuki and their relevance
(via Hudson, 1905) to my own much more elementary studies of
the geometry of the 4×4 square.
Comments Off on A Touchstone
Wednesday, July 26, 2017
For the title, see Icon Parking in a search for 54th in this journal.
For related iconic remarks, click on either image below.
.
This post was suggested by the Dec. 30, 2016, date of the
death in Nuremberg of mathematician Wolf Barth. The first
image above is from a mathematicsrelated work by
John von Neumann discussed here on that date.
See also Wolf Barth in this journal for posts that largely
concern not the above Barth, but an artist of the same name.
For posts on the mathematician only, see Barth + Kummer.
Comments Off on Icon Parking
Monday, June 26, 2017
This post was suggested by the previous post — Four Dots —
and by the phrase "smallest perfect" in this journal.
Related material (click to enlarge) —
Detail —
From the work of Eddington cited in 1974 by von Franz —
See also Dirac and Geometry and Kummer in this journal.
Updates from the morning of June 27 —
Ron Shaw on Eddington's triads "associated in conjugate pairs" —
For more about hyperbolic and isotropic lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.
For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.
Comments Off on Upgrading to Six
Tuesday, May 23, 2017
Comments Off on Pursued by a Biplane
Friday, April 14, 2017
The above fourelement sets of black subsquares of a 4×4 square array
are 15 of the 60 Göpel tetrads , and 20 of the 80 Rosenhain tetrads , defined
by R. W. H. T. Hudson in his 1905 classic Kummer's Quartic Surface .
Hudson did not view these 35 tetrads as planes through the origin in a finite
affine 4space (or, equivalently, as lines in the corresponding finite projective
3space).
In order to view them in this way, one can view the tetrads as derived,
via the 15 twoelement subsets of a sixelement set, from the 16 elements
of the binary Galois affine space pictured above at top left.
This space is formed by taking symmetricdifference (Galois binary)
sums of the 15 twoelement subsets, and identifying any resulting four
element (or, summing three disjoint twoelement subsets, sixelement)
subsets with their complements. This process was described in my note
"The 2subsets of a 6set are the points of a PG(3,2)" of May 26, 1986.
The space was later described in the following —
Comments Off on Hudson and Finite Geometry
Wednesday, March 29, 2017
"And as the characters in the meme twitch into the abyss
that is the sky, this meme will disappear into whatever
internet abyss swallowed MySpace."
—Staff writer Kamila Czachorowski, Harvard Crimson today
From Log24 posts tagged Art Space —
From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
“The Universal Kummer Threefold,” by
Qingchun Ren, Steven V Sam, Gus Schrader, and
Bernd Sturmfels —
Two such considerations —
Comments Off on Art Space, Continued
Sunday, December 11, 2016
*The Hudson of the title is the author of Kummer's Quartic Surface (1905).
The Rosenhain of the title is the author for whom Hudson's 4×4 diagrams
of "Rosenhain tetrads" are named. For the "complexity to simplicity" of
the title, see Roger Fry in the previous post.
Comments Off on Complexity to Simplicity via Hudson and Rosenhain*
Thursday, November 3, 2016
(Continued … See the title in this journal, as well as Cube Bricks.)
Cube Bricks 1984 —
Related material —
Dirac and Geometry in this journal,
Kummer's Quartic Surface in this journal,
Nanavira Thera in this journal, and
The Razor's Edge and Nanavira Thera.
See as well Bill Murray's 1984 film "The Razor's Edge" …
Movie poster from 1984 —
"A thin line separates
love from hate,
success from failure,
life from death."
Three other dualities, from Nanavira Thera in 1959 —
"I find that there are, in every situation,
three independent dualities…."
(Click to enlarge.)
Comments Off on Triple Cross
Tuesday, September 13, 2016
The previous post discussed the parametrization of
the 4×4 array as a vector 4space over the 2element
Galois field GF(2).
The 4×4 array may also be parametrized by the symbol
0 along with the fifteen 2subsets of a 6set, as in Hudson's
1905 classic Kummer's Quartic Surface —
Hudson in 1905:
These two ways of parametrizing the 4×4 array — as a finite space
and as an array of 2element sets — were related to one another
by Cullinane in 1986 in describing, in connection with the Curtis
"Miracle Octad Generator," what turned out to be 15 of Hudson's
1905 "Göpel tetrads":
A recap by Cullinane in 2013:
Click images for further details.
Comments Off on Parametrizing the 4×4 Array
Tuesday, July 12, 2016
The following passage by Igor Dolgachev (Good Friday, 2003)
seems somewhat relevant (via its connection to Kummer's 16_{6} )
to previous remarks here on Dirac matrices and geometry —
Note related remarks from E. M. Bruins in 1959 —
Comments Off on Group Elements and Skew Lines
Friday, June 3, 2016
A review of some recent posts on Dirac and geometry,
each of which mentions the late physicist Hendrik van Dam:
The first of these posts mentions the work of E. M. Bruins.
Some earlier posts that cite Bruins:
Comments Off on Bruins and van Dam
Tuesday, May 24, 2016
The authors Taormina and Wendland in the previous post
discussed some mathematics they apparently did not know was
related to a classic 1905 book by R. W. H. T. Hudson, Kummer's
Quartic Surface .
"This famous book is a prototype for the possibility
of explaining and exploring a manyfaceted topic of
research, without focussing on general definitions,
formal techniques, or even fancy machinery. In this
regard, the book still stands as a highly recommendable,
unparalleled introduction to Kummer surfaces, as a
permanent source of inspiration and, last but not least,
as an everlasting symbol of mathematical culture."
— Werner Kleinert, Mathematical Reviews ,
as quoted at Amazon.com
Some 4×4 diagrams from that book are highly relevant to the
discussion by Taormina and Wendland of the 4×4 squares within
the 1974 Miracle Octad Generator of R. T. Curtis that were later,
in 1987, described by Curtis as pictures of the vector 4space over
the twoelement Galois field GF(2).
Hudson did not think of his 4×4 diagrams as illustrating a vector space,
but he did use them to picture certain subsets of the 16 cells in each
diagram that he called Rosenhain and Göpel tetrads .
Some related work of my own (click images for related posts)—
Rosenhain tetrads as 20 of the 35 projective lines in PG(3,2)
Göpel tetrads as 15 of the 35 projective lines in PG(3,2)
Related terminology describing the Göpel tetrads above
Comments Off on Rosenhain and Göpel Revisited
Tuesday, February 9, 2016
The hexagons above appear also in Gary W. Gibbons,
"The Kummer Configuration and the Geometry of Majorana Spinors,"
1993, in a cube model of the Kummer 16_{6} configuration —
Related material — The Religion of Cubism (May 9, 2003).
Comments Off on Cubism
Thursday, December 3, 2015
(Continued)
From p. 34 of the preprint "Snapshots of Conformal Field Theory,"
by Katrin Wendland, arXiv, 11 April 2014 —
50. Gannon, T.: Much ado about Mathieu (arXiv:1211.5531 [math.RT])
85. Taormina, A., Wendland, K.: The overarching finite symmetry group
of Kummer surfaces in the Mathieu group M24. JHEP 08, 125 (2013)
86. Taormina, A., Wendland, K.: Symmetrysurfing the moduli space
of Kummer K3s (arXiv:1303.2931 [hepth])
87. Taormina, A., Wendland, K.: A twist in the M24 moonshine story
(arXiv:1303.3221 [hepth])
The Wendland paper was published on Jan. 7, 2015, in
Mathematical Aspects of Quantum Field Theories ,
edited by Damien Calaque and Thomas Strobl
(Springer Mathematical Physics Studies), pages 89129.
Comments Off on Overarching Symmetry
Monday, November 23, 2015
Some background for my post of Nov. 20,
"Anticommuting Dirac Matrices as Skew Lines" —
His earlier paper that Bruins refers to, "Line Geometry
and Quantum Mechanics," is available in a free PDF.
For a biography of Bruins translated by Google, click here.
For some additional historical background going back to
Eddington, see Gary W. Gibbons, "The Kummer
Configuration and the Geometry of Majorana Spinors,"
pages 3952 in Oziewicz et al., eds., Spinors, Twistors,
Clifford Algebras, and Quantum Deformations:
Proceedings of the Second Max Born Symposium held
near Wrocław, Poland, September 1992 . (Springer, 2012,
originally published by Kluwer in 1993.)
For morerecent remarks on quantum geometry, see a
paper by Saniga cited in today's update to my Nov. 20 post.
Comments Off on Dirac and Line Geometry
Friday, September 25, 2015
A note on the somewhat distant relation —
See also Kummer in this journal.
Comments Off on A Great Moonshine
Saturday, July 4, 2015
Some context for yesterday's post on a symplectic polarity —
This 1986 note may or may not have inspired some remarks
of Wolf Barth in his foreword to the 1990 reissue of Hudson's
1905 Kummer's Quartic Surface .
See also the diamondtheorem correlation.
Comments Off on Context
Wednesday, June 17, 2015
The title of the previous post, "Slow Art," is a phrase
of the late art critic Robert Hughes.
Example from mathematics:

Göpel tetrads as subsets of a 4×4 square in the classic
1905 book Kummer's Quartic Surface by R. W. H. T. Hudson.
These subsets were constructed as helpful schematic diagrams,
without any reference to the concept of finite geometry they
were later to embody.

Göpel tetrads (not named as such), again as subsets of
a 4×4 square, that form the 15 isotropic projective lines of the
finite projective 3space PG(3,2) in a note on finite geometry
from 1986 —

Göpel tetrads as these figures of finite geometry in a 1990
foreword to the reissued 1905 book of Hudson:
Click the Barth passage to see it with its surrounding text.
Related material:
Comments Off on Slow Art, Continued
Wednesday, June 10, 2015
"Those that can be obtained…." —
Related music video: Waterloo.
* "In defense of the epistemic view of quantum states:
a toy theory," by Robert W. Spekkens, Perimeter Institute
for Theoretical Physics, Waterloo, Canada
Comments Off on Epistemic* Tetrads
Tuesday, February 3, 2015
A short poem by several authors:
"The role of
the 16 singular points
on the Kummer surface
is now played by
the 64 singular points
on the Kummer threefold."
— From Remark 2.4 on page 9 of
"The Universal Kummer Threefold,"
by Qingchun Ren, Steven V Sam,
Gus Schrader, and Bernd Sturmfels,
http://arxiv.org/abs/1208.1229v3,
August 6, 2012 — June 12, 2013.
See also "Expanded Field" in this journal.
Illustration from "Sunday School," July 20, 2014.
Other Log24 background: Kummer, Spielraum, Art Space.
Comments Off on Expanding the Spielraum
Thursday, August 14, 2014
Robin Williams and the Stages of Grief
A weblog post from Jan. 25, 2014 (click image to enlarge)—
Clues for a Mystery (Click links for more details)—
Clue 1: A June 11,* 2014, math death.
Clue 2: The answer is a surname.
Midrash for Will Hunting:
See Nanavira Thera at Wikipedia and space notes from September 2012.
* According to a (perhaps inaccurate) math department.
June 10, according to other sources cited by the department.
Comments Off on MIT Mystery
Tuesday, March 25, 2014
(Continued)
“Richard Hughes’s celebrated short novel is
a masterpiece of concentrated narrative.”
— New York Review of Books on
A High Wind in Jamaica
As perhaps were, in their way, parts of the life
of the late Patrice Wymore Flynn, who reportedly
died at 87 on Saturday.
Deep backstory: See Colony of Santiago (Jamaica).
For the “mathematics” part of this post’s title, see
Saturday’s Log24 post on Kummersurface terms
and a post of September 23, 2012.
Comments Off on Narrative and Mathematics
Saturday, March 22, 2014
Mathematical
Literary (also from May 18, 2010)
Comments Off on Two Types of Symmetry
Saturday, March 8, 2014
“Charting the Real FourQubit Pauli Group
via Ovoids of a Hyperbolic Quadric of PG(7,2),”
by Metod Saniga, Péter Lévay and Petr Pracna,
arXiv:1202.2973v2 [mathph] 26 Jun 2012 —
P. 4— “It was found that Q ^{+}(5,2) (the Klein quadric)
has, up to isomorphism, a unique one — also known,
after its discoverer, as a Conwell heptad [18].
The set of 28 points lying off Q ^{+}(5,2) comprises
eight such heptads, any two having exactly one
point in common.”
P. 11— “This split reminds us of a similar split of
63 points of PG(5,2) into 35/28 points lying on/off
a Klein quadric Q ^{+}(5,2).”
[18] G. M. Conwell, Ann. Math. 11 (1910) 60–76
A similar split occurs in yesterday’s Kummer Varieties post.
See the 63 = 28 + 35 vectors of R^{8} discussed there.
For more about Conwell heptads, see The Klein Correspondence,
Penrose SpaceTime, and a Finite Model.
For my own remarks on the date of the above arXiv paper
by Saniga et. al., click on the image below —
Walter Gropius
Comments Off on Conwell Heptads in Eastern Europe
Wednesday, March 5, 2014
See also Rosenhain and Göpel in the Wikipedia
article Kummer surface, and in this journal.
Related material: user @hyperelliptic on Twitter.
Comments Off on Rosenhain and Göpel Again
Monday, February 10, 2014
(Continued from Mystery Box, Feb. 4, and Mystery Box II, Feb. 5.)
The Box
Inside the Box
Outside the Box
For the connection of the inside notation to the outside geometry,
see Desargues via Galois.
(For a related connection to curves and surfaces in the outside
geometry, see Hudson's classic Kummer's Quartic Surface and
Rosenhain and Göpel Tetrads in PG(3,2).)
Comments Off on Mystery Box III: Inside, Outside
Friday, September 27, 2013
"Let X equal the month of full bookstores."
— Proof by David Auburn
"Harvard Book Store is thrilled to welcome Stephen King
for a special event to celebrate the release of Doctor Sleep ,
the longanticipated sequel to 1977’s The Shining .
Mr. King will read from and discuss the new book,
followed by a Q&A with the audience at Memorial Church
in Harvard Yard on September 27."

See as well Corpse + Eliot and some remarks
on the mathematics of Kummer
from this September and from last September.
Comments Off on Proof (continued)
Sunday, September 22, 2013
From yesterday —
"… a list of group theoretic invariants
and their geometric incarnation…"
— David Lehavi on the Kummer 16_{6} configuration in 2007
Related material —
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
"This is not theology; this is mathematics."
— Steven H. Cullinane on four quartets
To wit:
Click to enlarge.
Comments Off on Incarnation, Part 2
Saturday, September 21, 2013
The Kummer 16_{6} configuration is the configuration of sixteen
6sets within a 4×4 square array of points in which each 6set
is determined by one of the 16 points of the array and
consists of the 3 other points in that point's row and the
3 other points in that point's column.
See Configurations and Squares.
The Wikipedia article Kummer surface uses a rather poetic
phrase* to describe the relationship of the 16_{6} to a number
of other mathematical concepts — "geometric incarnation."
Related material from finitegeometry.org —
* Apparently from David Lehavi on March 18, 2007, at Citizendium .
Comments Off on Geometric Incarnation
Mathematics:
A review of posts from earlier this month —
Wednesday, September 4, 2013
Unexpected connections between areas of mathematics
previously thought to be unrelated are sometimes referred
to as "moonshine." An example— the apparent connections
between parts of complex analysis and groups related to the
large Mathieu group M_{24}. Some recent work on such apparent
connections, by Anne Taormina and Katrin Wendland, among
others (for instance, Miranda C.N. Cheng and John F.R. Duncan),
involves structures related to Kummer surfaces .
In a classic book, Kummer's Quartic Surface (1905),
R.W.H.T. Hudson pictured a set of 140 structures, the 80
Rosenhain tetrads and the 60 Göpel tetrads, as 4element
subsets of a 16element 4×4 array. It turns out that these
140 structures are the planes of the finite affine geometry
AG(4,2) of four dimensions over the twoelement Galois field.
(See Diamond Theory in 1937.)
Thursday, September 5, 2013
(Continued from yesterday)
The foreword by Wolf Barth in the 1990 Cambridge U. Press
reissue of Hudson's 1905 classic Kummer's Quartic Surface
covers some of the material in yesterday's post Moonshine.
The distinction that Barth described in 1990 was also described, and illustrated,
in my 1986 note "Picturing the smallest projective 3space." The affine 4space
over the the finite Galois field GF(2) that Barth describes was earlier described—
within a 4×4 array like that pictured by Hudson in 1905— in a 1979 American
Mathematical Society abstract, "Symmetry invariance in a diamond ring."
"The distinction between Rosenhain and Goepel tetrads
is nothing but the distinction between isotropic and
nonisotropic planes in this affine space over the finite field."
The 1990 paragraph of Barth quoted above may be viewed as a summary
of these facts, and also of my March 17, 2013, note "Rosenhain and Göpel
Tetrads in PG(3,2)."

Narrative:
Aooo.
Happy birthday to Stephen King.
Comments Off on Mathematics and Narrative (continued)
Thursday, September 5, 2013
(Continued from yesterday)
The foreword by Wolf Barth in the 1990 Cambridge U. Press
reissue of Hudson's 1905 classic Kummer's Quartic Surface
covers some of the material in yesterday's post Moonshine.
The distinction that Barth described in 1990 was also described, and illustrated,
in my 1986 note "Picturing the smallest projective 3space." The affine 4space
over the the finite Galois field GF(2) that Barth describes was earlier described—
within a 4×4 array like that pictured by Hudson in 1905— in a 1979 American
Mathematical Society abstract, "Symmetry invariance in a diamond ring."
"The distinction between Rosenhain and Goepel tetrads
is nothing but the distinction between isotropic and
nonisotropic planes in this affine space over the finite field."
The 1990 paragraph of Barth quoted above may be viewed as a summary
of these facts, and also of my March 17, 2013, note "Rosenhain and Göpel
Tetrads in PG(3,2)."
Comments Off on Moonshine II
Wednesday, September 4, 2013
Unexpected connections between areas of mathematics
previously thought to be unrelated are sometimes referred
to as "moonshine." An example— the apparent connections
between parts of complex analysis and groups related to the
large Mathieu group M_{24}. Some recent work on such apparent
connections, by Anne Taormina and Katrin Wendland, among
others (for instance, Miranda C.N. Cheng and John F.R. Duncan),
involves structures related to Kummer surfaces .
In a classic book, Kummer's Quartic Surface (1905),
R.W.H.T. Hudson pictured a set of 140 structures, the 80
Rosenhain tetrads and the 60 Göpel tetrads, as 4element
subsets of a 16element 4×4 array. It turns out that these
140 structures are the planes of the finite affine geometry
AG(4,2) of four dimensions over the twoelement Galois field.
(See Diamond Theory in 1937.)
A Google search documents the moonshine
relating Rosenhain's and Göpel's 19thcentury work
in complex analysis to M_{24 } via the book of Hudson and
the geometry of the 4×4 square.
Comments Off on Moonshine
Saturday, August 17, 2013
The following excerpt from a January 20, 2013, preprint shows that
a Galoisgeometry version of the large Desargues 15_{4}20_{3} configuration,
although based on the nineteenthcentury work of Galois* and of Fano,**
may at times have twentyfirstcentury applications.
Some context —
Atkinson's paper does not use the square model of PG(3,2), which later
in 2013 provided a natural view of the large Desargues 15_{4}20_{3} configuration.
See my own Classical Geometry in Light of Galois Geometry. Atkinson's
"subset of 20 lines" corresponds to 20 of the 80 Rosenhain tetrads
mentioned in that later article and pictured within 4×4 squares in Hudson's
1905 classic Kummer's Quartic Surface.
* E. Galois, definition of finite fields in "Sur la Théorie des Nombres,"
Bulletin des Sciences Mathématiques de M. Férussac,
Vol. 13, 1830, pp. 428435.
** G. Fano, definition of PG(3,2) in "Sui Postulati Fondamentali…,"
Giornale di Matematiche, Vol. 30, 1892, pp. 106132.
Comments Off on UptoDate Geometry
Monday, June 10, 2013
Today's previous post on coordinate systems
suggests a look at the phrase "Galois coordinates."
A search shows that the phrase, though natural,
has apparently not been used before 2011* for solutions
to what Hermann Weyl called "the relativity problem."
A thorough historical essay on Galois coordinatization
in this sense would require more academic resources
than I have available. It would likely describe a number
of applications of Galoisfield coordinates to square
(and perhaps to cubical) arrays that were studied before
1976, the date of my Diamond Theory monograph.
But such a survey might not find any such pre1976
coordinatization of a 4×4 array by the 16 elements
of the vector 4space over the Galois field with two
elements, GF(2).
Such coordinatizations are important because of their
close relationship to the Mathieu group M _{24 }.
See a preprint by Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group M _{24} ," with its remark
denying knowledge of any such coordinatization
prior to a 1989 paper by R. T. Curtis.
Related material:
Some images related to Galois coordinates, excerpted
from a Google search today (click to enlarge)—
* A rather abstract 2011 paper that uses the phrase
"Galois coordinates" may have some implications
for the naive form of the relativity problem
related to square and cubical arrays.
Comments Off on Galois Coordinates
Sunday, May 19, 2013
From an arXiv preprint submitted July 18, 2011,
and last revised on March 11, 2013 (version 4):
"By our construction, this vector space is the dual
of our hypercube F_{2}^{4} built on I \ O_{9}. The vector space
structure of the latter, to our knowledge, is first
mentioned by Curtis in [Cur89]. Hence altogether
our proposition 2.3.4 gives a novel geometric
meaning in terms of Kummer geometry to the known
vector space structure on I \ O_{9}."
[Cur89] reference:
R. T. Curtis, "Further elementary techniques using
the miracle octad generator," Proc. Edinburgh
Math. Soc. 32 (1989), 345353 (received on
July 20, 1987).
— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group M _{24 },"
arXiv.org > hepth > arXiv:1107.3834
"First mentioned by Curtis…."
No. I claim that to the best of my knowledge, the
vector space structure was first mentioned by me,
Steven H. Cullinane, in an AMS abstract submitted
in October 1978, some nine years before the
Curtis article.
Update of the above paragraph on July 6, 2013—
No. The vector space structure was described by
(for instance) Peter J. Cameron in a 1976
Cambridge University Press book —
Parallelisms of Complete Designs .
See the proof of Theorem 3A.13 on pages 59 and 60.
The vector space structure as it occurs in a 4×4 array
of the sort that appears in the Curtis Miracle Octad
Generator may first have been pointed out by me,
Steven H. Cullinane, in an AMS abstract submitted in
October 1978, some nine years before the Curtis article.

See Notes on Finite Geometry for some background.
See in particular The Galois Tesseract.
For the relationship of the 1978 abstract to Kummer
geometry, see Rosenhain and Göpel Tetrads in PG(3,2).
Comments Off on Priority Claim
Thursday, April 25, 2013
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 M_{24}.
For some related material that is more uptodate, search the Web
for Mathieu + Kummer .
Comments Off on Rosenhain and Göpel Revisited
Friday, April 19, 2013
Desargues' theorem according to a standard textbook:
"If two triangles are perspective from a point
they are perspective from a line."
The converse, from the same book:
"If two triangles are perspective from a line
they are perspective from a point."
Desargues' theorem according to Wikipedia
combines the above statements:
"Two triangles are in perspective axially [i.e., from a line]
if and only if they are in perspective centrally [i.e., from a point]."
A figure often used to illustrate the theorem,
the Desargues configuration , has 10 points and 10 lines,
with 3 points on each line and 3 lines on each point.
A discussion of the "if and only if" version of the theorem
in light of Galois geometry requires a larger configuration—
15 points and 20 lines, with 3 points on each line
and 4 lines on each point.
This large Desargues configuration involves a third triangle,
needed for the proof (though not the statement ) of the
"if and only if" version of the theorem. Labeled simply
"Desargues' Theorem," the large configuration is the
frontispiece to Volume I (Foundations) of Baker's 6volume
Principles of Geometry .
Pointline incidence in this larger configuration is,
as noted in a post of April 1, 2013, described concisely
by 20 Rosenhain tetrads (defined in 1905 by
R. W. H. T. Hudson in Kummer's Quartic Surface ).
The third triangle, within the larger configuration,
is pictured below.
Comments Off on The Large Desargues Configuration
Monday, April 1, 2013
Background: Rosenhain and Göpel Tetrads in PG(3,2)
Introduction:
The Large Desargues Configuration
Added by Steven H. Cullinane on Friday, April 19, 2013
Desargues' theorem according to a standard textbook:
"If two triangles are perspective from a point
they are perspective from a line."
The converse, from the same book:
"If two triangles are perspective from a line
they are perspective from a point."
Desargues' theorem according to Wikipedia
combines the above statements:
"Two triangles are in perspective axially [i.e., from a line]
if and only if they are in perspective centrally [i.e., from a point]."
A figure often used to illustrate the theorem,
the Desargues configuration , has 10 points and 10 lines,
with 3 points on each line and 3 lines on each point.
A discussion of the "if and only if" version of the theorem
in light of Galois geometry requires a larger configuration—
15 points and 20 lines, with 3 points on each line
and 4 lines on each point.
This large Desargues configuration involves a third triangle,
needed for the proof (though not the statement ) of the
"if and only if" version of the theorem. Labeled simply
"Desargues' Theorem," the large configuration is the
frontispiece to Volume I (Foundations) of Baker's 6volume
Principles of Geometry .
Pointline incidence in this larger configuration is,
as noted in the post of April 1 that follows
this introduction, described concisely
by 20 Rosenhain tetrads (defined in 1905 by
R. W. H. T. Hudson in Kummer's Quartic Surface ).
The third triangle, within the larger configuration,
is pictured below.

A connection discovered today (April 1, 2013)—
(Click to enlarge the image below.)
Update of April 18, 2013
Note that Baker's Desarguestheorem figure has three triangles,
ABC, A'B'C', A"B"C", instead of the two triangles that occur in
the statement of the theorem. The third triangle appears in the
course of proving, not just stating, the theorem (or, more precisely,
its converse). See, for instance, a note on a standard textbook for
further details.
(End of April 18, 2013 update.)
Update of April 14, 2013
See Baker's Proof (Edited for the Web) for a detailed explanation
of the above picture of Baker's Desarguestheorem frontispiece.
(End of April 14, 2013 update.)
Update of April 12, 2013
A different figure, from a site at National Tsing Hua University,
shows the three triangles of Baker's figure more clearly:
(End of update of April 12, 2013)
Update of April 13, 2013
Another in a series of figures illustrating
Desargues's theorem in light of Galois geometry:
See also the original VeblenYoung figure in context.
(End of update of April 13, 2013)
Rota's remarks, while perhaps not completely accurate, provide some context
for the above DesarguesRosenhain connection. For some other context,
see the interplay in this journal between classical and finite geometry, i.e.
between Euclid and Galois.
For the recent context of the above finitegeometry version of Baker's Vol. I
frontispiece, see Sunday evening's finitegeometry version of Baker's Vol. IV
frontispiece, featuring the Göpel, rather than the Rosenhain, tetrads.
For a 1986 illustration of Göpel and Rosenhain tetrads (though not under
those names), see Picturing the Smallest Projective 3Space.
In summary… the following classicalgeometry figures
are closely related to the Galois geometry PG(3,2):
Volume I of Baker's Principles
has a cover closely related to
the Rosenhain tetrads in PG(3,2)

Volume IV of Baker's Principles
has a cover closely related to
the Göpel tetrads in PG(3,2)

Foundations
(click to enlarge)

Higher Geometry
(click to enlarge)

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Sunday, March 10, 2013
(Continued)
The 16point affine Galois space:
Further properties of this space:
In Configurations and Squares, see the
discusssion of the Kummer 16_{6} configuration.
Some closely related material:
Comments Off on Galois Space
Tuesday, February 19, 2013
Yesterday's post Permanence dealt with the cube
as a symmetric model of the finite projective plane
PG(2,3), which has 13 points and 13 lines. The points
and lines of the finite geometry occur in the cube as
the 13 axes of symmetry and the 13 planes through
the center perpendicular to those axes. If the three
axes lying in a plane that cuts the cube in a hexagon
are supplemented by the axis perpendicular to that
plane, each plane is associated with four axes and,
dually, each axis is associated with four planes.
My web page on this topic, Cubist Geometries, was
written on February 27, 2010, and first saved to the
Internet Archive on Oct. 4, 2010.
For a more recent treatment of this topic that makes
exactly the same points as the 2010 page, see p. 218
of Configurations from a Graphical Viewpoint , by
Tomaž Pisanski and Brigitte Servatius, published by
Springer on Sept. 23, 2012 (date from both Google
Books and Amazon.com):
For a similar 1998 treatment of the topic, see Burkard Polster's
A Geometrical Picture Book (Springer, 1998), pp. 103104.
The PisanskiServatius book reinforces my argument of Jan. 13, 2013,
that the 13 planes through the cube's center that are perpendicular
to the 13 axes of symmetry of the cube should be called the cube's
symmetry planes , contradicting the usual use of of that term.
That argument concerns the interplay between Euclidean and
Galois geometry. Pisanski and Servatius (and, in 1998, Polster)
emphasize the Euclidean square and cube as guides* to
describing the structure of a Galois space. My Jan. 13 argument
uses Galois structures as a guide to redescribing those of Euclid .
(For a similar strategy at a much more sophisticated level,
see a recent Harvard Math Table.)
Related material: Remarks on configurations in this journal
during the month that saw publication of the PisanskiServatius book.
* Earlier guides: the diamond theorem (1978), similar theorems for
2x2x2 (1984) and 4x4x4 cubes (1983), and Visualizing GL(2,p)
(1985). See also Spaces as Hypercubes (2012).
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Tuesday, October 16, 2012
Last Wednesday's 11 PM post mentioned the
adjacencyisomorphism relating the 4dimensional
hypercube over the 2element Galois field GF(2) to
the 4×4 array made up of 16 square cells, with
opposite edges of the 4×4 array identified.
A web page illustrates this property with diagrams that
enjoy the Karnaugh property— adjacent vertices, or cells,
differ in exactly one coordinate. A brief paper by two German
authors relates the Karnaugh property to the construction
of a magic square like that of Dürer (see last Wednesday).
In a similar way (search the Web for Karnaugh + cube ),
vertex adjacency in the 6dimensional hypercube over GF(2)
is isomorphic to cell adjacency in the 4x4x4 cube, with
opposite faces of the 4x4x4 cube identified.
The above cube may be used to illustrate some properties
of the 64point Galois 6space that are more advanced
than those studied by enthusiasts of "magic" squares
and cubes.
See
Those who prefer narrative to mathematics may
consult posts in this journal containing the word "Cuber."
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Sunday, September 23, 2012
"We live together, we act on, and react to, one another; but always and in all circumstances we are by ourselves. The martyrs go hand in hand into the arena; they are crucified alone. Embraced, the lovers desperately try to fuse their insulated ecstasies into a single selftranscendence; in vain. By its very nature every embodied spirit is doomed to suffer and enjoy in solitude. Sensations, feelings, insights, fancies – all these are private and, except through symbols and at second hand, incommunicable. We can pool information about experiences, but never the experiences themselves. From family to nation, every human group is a society of island universes. Most island universes are sufficiently like one another to permit of inferential understanding or even of mutual empathy or "feeling into." Thus, remembering our own bereavements and humiliations, we can condole with others in analogous circumstances, can put ourselves (always, of course, in a slightly Pickwickian sense) in their places. But in certain cases communication between universes is incomplete or even nonexistent. The mind is its own place, and the places inhabited by the insane and the exceptionally gifted are so different from the places where ordinary men and women live, that there is little or no common ground of memory to serve as a basis for understanding or fellow feeling. Words are uttered, but fail to enlighten. The things and events to which the symbols refer belong to mutually exclusive realms of experience."
— The Doors of Perception
"Greet guests with a touch of glass."
— The Perception of Doors
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In Like Flynn
From the Wall Street Journal site Friday evening—
ESSAY September 21, 2012, 9:10 p.m. ET
Are We Really Getting Smarter?
Americans' IQ scores have risen steadily over the past century.
James R. Flynn examines why.

No, thank you. I prefer the ninth configuration as is—
Why? See Josefine Lyche's art installation "Grids, you say?"
Her reference there to "High White Noon" is perhaps
related to the use of that phrase in this journal.
The phrase is from a 2010 novel by Don DeLillo.
See "Point Omega," as well as Lyche's "Omega Point,"
in this journal.
The Wall Street Journal author above, James R. Flynn (born in 1934),
"is famous for his discovery of the Flynn effect, the continued
yearafteryear increase of IQ scores in all parts of the world."
—Wikipedia
His son Eugene Victor Flynn is a mathematician, coauthor
of the following chapter on the Kummer surface—
For use of the Kummer surface in Buddhist metaphysics, see last night's
post "Occupy Space (continued)" and the letters of Nanavira Thera from the
late 1950s at nanavira.blogspot.com.
These letters, together with Lyche's use of the phrase "high white noon,"
suggest a further quotation—
You know that it would be untrue
You know that I would be a liar
If I was to say to you
Girl, we couldn't get much higher
See also the Kummer surface at the web page Configurations and Squares.
Comments Off on Plan 9 (continued)–
Saturday, September 22, 2012
(Continued)
"The word 'space' has, as you suggest, a large number of different meanings."
— Nanavira Thera in [Early Letters. 136] 10.xii.1958
From that same letter (links added to relevant Wikipedia articles)—
Space (ākāsa) is undoubtedly used in the Suttas
to mean 'what/where the four mahābhūtas are not',
or example, the cavities in the body are called ākāsa
M.62—Vol. I, p. 423). This, clearly, is the everyday
'space' we all experience—roughly, 'What I can move
bout in', the empty part of the world. 'What you can't
ouch.' It is the 'space' of what Miss Lounsberry has so
appily described as 'the visible world of our five
senses'. I think you agree with this. And, of course, if
this is the only meaning of the word that we are
going to use, my 'superposition of several spaces' is
disqualified. So let us say 'superposition of several
extendednesses'. But when all these
extendednesses have been superposed, we get
'space'—i.e. our normal spacecontaining visible
world 'of the five senses'. But now there is another
point. Ākāsa is the negative of the four mahābhūtas,
certainly, but of the four mahābhūtas understood
in the same everyday sense—namely, solids (the
solid parts of the body, hair, nails, teeth, etc.),
liquids (urine, blood, etc.), heat and processes
(digestion) and motion or wind (N.B. not 'air').
These four, together with space, are the normal
furniture of our visible world 'of the five senses',
and it is undoubtedly thus that they are intended
in many Suttas. But there is, for example, a Sutta
(I am not sure where) in which the Ven. Sariputta
Thera is said to be able to see a pile of logs
successively as paṭhavi, āpo, tejo, and vāyo; and
it is evident that we are not on the same level.
On the everyday level a log of wood is solid and
therefore pathavi (like a bone), and certainly not
āpo, tejo, or vāyo. I said in my last letter that I
think that, in this second sense—i.e. as present in,
or constitutive of, any object (i.e. = rupa)—they
are structural and strictly parallel to nama and can
be defined exactly in terms of the Kummer
triangle. But on this fundamental level ākāsa has
no place at all, at least in the sense of our normal
everyday space. If, however, we take it as equivalent
to extendedness then it would be a given arbitrary
content—defining one sense out of many—of which
the four mahābhūtas (in the fundamental sense) are
the structure. In this sense (but only in this sense—
and it is probably an illegitimate sense of ākāsa)
the four mahābhūtas are the structure of space
(or spatial things). Quite legitimately, however, we
can say that the four mahābhūtas are the structure
of extended things—or of coloured things, or of smells,
or of tastes, and so on. We can leave the scientists'
space (full of right angles and without reference to the
things in it) to the scientists. 'Space' (= ākāsa) is the
space or emptiness of the world we live in; and this,
when analyzed, is found to depend on a complex
superposition of different extendednesses (because
all these extendednesses define the visible world
'of the five senses'—which will include, notably,
tangible objects—and this world 'of the five
senses' is the four mahābhūtas [everyday space]
and ākāsa).
Your second letter seems to suggest that the space
of the world we live in—the set of patterns
(superimposed) in which “we” are—is scientific space.
This I quite disagree with—if you do suggest it—,
since scientific space is a pure abstraction, never
experienced by anybody, whereas the superimposed
set of patterns is exactly what I experience—the set
is different for each one of us—, but in all of these
sets 'space' is infinite and undifferentiable, since it is,
by definition, in each set, 'what the four mahābhūtas
are not'.

A simpler metaphysical system along the same lines—
The theory, he had explained, was that the persona
was a fourdimensional figure, a tessaract in space,
the elementals Fire, Earth, Air, and Water permutating
and pervolving upon themselves, making a cruciform
(in threespace projection) figure of equal lines and
ninety degree angles.
— The Gameplayers of Zan ,
a 1977 novel by M. A. Foster

"I am glad you have discovered that the situation is comical:
ever since studying Kummer I have been, with some difficulty,
refraining from making that remark."
— Nanavira Thera, [Early Letters, 131] 17.vii.1958
Comments Off on Occupy Space
Tuesday, February 14, 2012
The showmanship of Nicki Minaj at Sunday's
Grammy Awards suggested the above title,
that of a novel by the author of The Exorcist .
The Ninth Configuration —
The ninth* in a list of configurations—
"There is a (2^{d1})_{d} configuration
known as the Cox configuration."
— MathWorld article on "Configuration"
For further details on the Cox 32_{6} configuration's Levi graph,
a model of the 64 vertices of the sixdimensional hypercube γ_{6 },
see Coxeter, "SelfDual Configurations and Regular Graphs,"
Bull. Amer. Math. Soc. Vol. 56, pages 413455, 1950.
This contains a discussion of Kummer's 16_{6} as it
relates to γ_{6 }, another form of the 4×4×4 Galois cube.
See also Solomon's Cube.
* Or tenth, if the fleeting reference to 11_{3} configurations is counted as the seventh—
and then the ninth would be a 15_{3} and some related material would be Inscapes.
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Sunday, June 5, 2011
"Total grandeur of a total edifice,
Chosen by an inquisitor of structures
For himself. He stops upon this threshold,
As if the design of all his words takes form
And frame from thinking and is realized."
— Wallace Stevens, "To an Old Philosopher in Rome"
The following edifice may be lacking in grandeur,
and its properties as a configuration were known long
before I stumbled across a description of it… still…
"What we do may be small, but it has
a certain character of permanence…."
— G.H. Hardy, A Mathematician's Apology
The Kummer 16_{6} Configuration
as seen by Kantor in 1969— (pdf, 2.5 MB)
For some background, see Configurations and Squares.
For some quite different geometry of the 4×4 square that is
original with me, see a page with that title. (The geometry's
importance depends in part on its connection with the
Miracle Octad Generator (MOG) of R.T. Curtis. I of course
had nothing to do with the MOG's discovery, but I do claim credit
for discovering some geometric properties of the 4×4 square
that constitutes twothirds of the MOG as originally defined .)
Related material— The Schwartz Notes of June 1.
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Sunday, March 27, 2011
On this date 106 years ago…
Prefatory note from Hudson's classic Kummer's Quartic Surface ,
Cambridge University Press, 1905—
RONALD WILLIAM HENRY TURNBULL HUDSON would have
been twentynine years old in July of this year; educated at
St Paul's School, London, and at St John's College, Cambridge,
he obtained the highest honours in the public examinations of the
University, in 1898, 1899, 1900; was elected a Fellow of St John's
College in 1900; became a Lecturer in Mathematics at University
College, Liverpool, in 1902; was D.Sc. in the University of London
in 1903; and died, as the result of a fall while climbing in Wales,
in the early autumn of 1904….
A manysided theory such as that of this volume is
generally to be won only by the work of many lives;
one who held so firmly the faith that the time is well spent
could ill be spared.
— H. F. Baker, 27 March 1905
For some more recent remarks related to the theory, see
Defining Configurations and its updates, March 2027, 2011.
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Friday, March 18, 2011
The OnLine Encyclopedia of Integer Sequences has an article titled "Number of combinatorial configurations of type (n_3)," by N.J.A. Sloane and D. Glynn.
From that article:
 DEFINITION: A combinatorial configuration of type (n_3) consists of an (abstract) set of n points together with a set of n triples of points, called lines, such that each point belongs to 3 lines and each line contains 3 points.
 EXAMPLE: The unique (8_3) configuration consists of the triples 125, 148, 167, 236, 278, 347, 358, 456.
The following corrects the word "unique" in the example.
* This post corrects an earlier post, also numbered 14660 and dated 7 PM March 18, 2011, that was in error.
The correction was made at about 11:50 AM on March 20, 2011.
_____________________________________________________________
Update of March 21
The problem here is of course with the definition. Sloane and Glynn failed to include in their definition a condition that is common in other definitions of configurations, even abstract or purely "combinatorial" configurations. See, for instance, Configurations of Points and Lines , by Branko Grunbaum (American Mathematical Society, 2009), p. 17—
In the most general sense we shall consider combinatorial (or abstract) configurations; we shall use the term setconfigurations as well. In this setting "points" are interpreted as any symbols (usually letters or integers), and "lines" are families of such symbols; "incidence" means that a "point" is an element of a "line". It follows that combinatorial configurations are special kinds of general incidence structures. Occasionally, in order to simplify and clarify the language, for "points" we shall use the term marks, and for "lines" we shall use blocks. The main property of geometric configurations that is preserved in the generalization to setconfigurations (and that characterizes such configurations) is that two marks are incident with at most one block, and two blocks with at most one mark.
Whether or not omitting this "at most one" condition from the definition is aesthetically the best choice, it dramatically changes the number of configurations in the resulting theory, as the above (8_3) examples show.
Update of March 22 (itself updated on March 25)
For further background on configurations, see Dolgachev—
Note that the two examples Dolgachev mentions here, with 16 points and 9 points, are not unrelated to the geometry of 4×4 and 3×3 square arrays. For the Kummer and related 16point configurations, see section 10.3, "The Three Biplanes of Order 4," in Burkard Polster's A Geometrical Picture Book (Springer, 1998). See also the 4×4 array described by Gordon Royle in an undated web page and in 1980 by Assmus and Sardi. For the Hesse configuration, see (for instance) the passage from Coxeter quoted in Quaternions in an Affine Galois Plane.
Update of March 27
See the above link to the (16,6) 4×4 array and the (16,6) exercises using this array in R.D. Carmichael's classic Introduction to the Theory of Groups of Finite Order (1937), pp. 4243. For a connection of this sort of 4×4 geometry to the geometry of the diamond theorem, read "The 2subsets of a 6set are the points of a PG(3,2)" (a note from 1986) in light of R.W.H.T. Hudson's 1905 classic Kummer's Quartic Surface , pages 89, 1617, 4445, 7677, 7879, and 80.
Comments Off on Defining Configurations*
Wednesday, September 3, 2003
Reciprocity
From my entry of Sept. 1, 2003:
"…the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity….
… E. M. Forster famously advised his readers, 'Only connect.' 'Reciprocity' would be Michael Kruger's succinct philosophy, with all that the word implies."
— William Boyd, review of Himmelfarb, New York Times Book Review, October 30, 1994
Last year's entry on this date:
The picture above is of the complete graph K_{6 }… Six points with an edge connecting every pair of points… Fifteen edges in all.
Diamond theory describes how the 15 twoelement subsets of a sixelement set (represented by edges in the picture above) may be arranged as 15 of the 16 parts of a 4×4 array, and how such an array relates to grouptheoretic concepts, including Sylvester's synthematic totals as they relate to constructions of the Mathieu group M_{24}.
If diamond theory illustrates any general philosophical principle, it is probably the interplay of opposites…. "Reciprocity" in the sense of Lao Tzu. See
Reciprocity and Reversal in Lao Tzu.
For a sense of "reciprocity" more closely related to Michael Kruger's alleged philosophy, see the Confucian concept of Shu (Analects 15:23 or 24) described in
Shu: Reciprocity.
Kruger's novel is in part about a Jew: the quintessential Jewish symbol, the star of David, embedded in the K_{6} graph above, expresses the reciprocity of male and female, as my May 2003 archives illustrate. The star of David also appears as part of a graphic design for cubes that illustrate the concepts of diamond theory:
Click on the design for details.
Those who prefer a Jewish approach to physics can find the star of David, in the form of K_{6}, applied to the sixteen 4×4 Dirac matrices, in
A Graphical Representation
of the Dirac Algebra.
The star of David also appears, if only as a heuristic arrangement, in a note that shows generating partitions of the affine group on 64 points arranged in two opposing triplets.
Having thus, as the New York Times advises, paid tribute to a Jewish symbol, we may note, in closing, a much more sophisticated and subtle concept of reciprocity due to Euler, Legendre, and Gauss. See
The Jewel of Arithmetic and
The Golden Theorem.
Thursday, December 5, 2002
Sacerdotal Jargon
From the website
Abstracts and Preprints in Clifford Algebra [1996, Oct 8]:
Paper: clfalg/good9601
From: David M. Goodmanson
Address: 2725 68th Avenue S.E., Mercer Island, Washington 98040
Title: A graphical representation of the Dirac Algebra
Abstract: The elements of the Dirac algebra are represented by sixteen 4×4 gamma matrices, each pair of which either commute or anticommute. This paper demonstrates a correspondence between the gamma matrices and the complete graph on six points, a correspondence that provides a visual picture of the structure of the Dirac algebra. The graph shows all commutation and anticommutation relations, and can be used to illustrate the structure of subalgebras and equivalence classes and the effect of similarity transformations….
Published: Am. J. Phys. 64, 870880 (1996)
The following is a picture of K_{6}, the complete graph on six points. It may be used to illustrate various concepts in finite geometry as well as the properties of Dirac matrices described above.
From
"The Relations between Poetry and Painting,"
by Wallace Stevens:
"The theory of poetry, that is to say, the total of the theories of poetry, often seems to become in time a mystical theology or, more simply, a mystique. The reason for this must by now be clear. The reason is the same reason why the pictures in a museum of modern art often seem to become in time a mystical aesthetic, a prodigious search of appearance, as if to find a way of saying and of establishing that all things, whether below or above appearance, are one and that it is only through reality, in which they are reflected or, it may be, joined together, that we can reach them. Under such stress, reality changes from substance to subtlety, a subtlety in which it was natural for Cézanne to say: 'I see planes bestriding each other and sometimes straight lines seem to me to fall' or 'Planes in color. . . . The colored area where shimmer the souls of the planes, in the blaze of the kindled prism, the meeting of planes in the sunlight.' The conversion of our Lumpenwelt went far beyond this. It was from the point of view of another subtlety that Klee could write: 'But he is one chosen that today comes near to the secret places where original law fosters all evolution. And what artist would not establish himself there where the organic center of all movement in time and space—which he calls the mind or heart of creation— determines every function.' Conceding that this sounds a bit like sacerdotal jargon, that is not too much to allow to those that have helped to create a new reality, a modern reality, since what has been created is nothing less."
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