Saturday, May 21, 2011
Heisenberg on Heraclitus
From Physics and Philosophy , by Werner Heisenberg, 1958, reprinted by Penguin Classics, 2003—
Page 28—
… In the philosophy of Heraclitus of Ephesus the concept of Becoming occupies the foremost
place. He regarded that which moves, the fire, as the basic element. The difficulty, to reconcile
the idea of one fundamental principle with the infinite variety of phenomena, is solved for him by
recognizing that the strife of the opposites is really a kind of harmony. For Heraclitus the world is
at once one and many, it is just 'the opposite tension' of the opposites that constitutes the unity
of the One. He says: 'We must know that war is common to all and strife is justice, and that all
things come into being and pass away through strife.'
Looking back to the development of Greek philosophy up to this point one realizes that it has
been borne from the beginning to this
Page 29—
stage by the tension between the One and the Many. For our senses the world consists of an
infinite variety of things and events, colors and sounds. But in order to understand it we have to
introduce some kind of order, and order means to recognize what is equal, it means some sort
of unity. From this springs the belief that there is one fundamental principle, and at the same
time the difficulty to derive from it the infinite variety of things. That there should be a material
cause for all things was a natural starting point since the world consists of matter. But when one
carried the idea of fundamental unity to the extreme one came to that infinite and eternal
undifferentiated Being which, whether material or not, cannot in itself explain the infinite variety
of things. This leads to the antithesis of Being and Becoming and finally to the solution of
Heraclitus, that the change itself is the fundamental principle; the 'imperishable change, that
renovates the world,' as the poets have called it. But the change in itself is not a material cause
and therefore is represented in the philosophy of Heraclitus by the fire as the basic element,
which is both matter and a moving force.
We may remark at this point that modern physics is in some way extremely near to the
doctrines of Heraclitus. If we replace the word 'fire' by the word 'energy' we can almost repeat
his statements word for word from our modern point of view. Energy is in fact the substance
from which all elementary particles, all atoms and therefore all things are made, and energy is
that which moves. Energy is a substance, since its total amount does not change, and the
elementary particles can actually be made from this substance as is seen in many experiments on
the creation of elementary particles. Energy can be changed into motion, into heat, into light
and into tension. Energy may be called the fundamental cause for all change in the world. But this
comparison of Greek philosophy with the ideas of modern science will be discussed later.
* See earlier uses of the phrase in this journal. Further background— Hopkins and Heraclitus.
Comments Off on Unity and Multiplicity (continued*)
Wednesday, May 4, 2011
Continued from Crimson Walpurgisnacht.
Epigraphs— Two quotations from
Shakespeare's Birthday last year—
"I was reading Durant's section on Plato, struggling to understand his theory of the ideal Forms that lay in inviolable perfection out beyond the phantasmagoria. (That was the first, and I think the last, time that I encountered that word.)"
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Screenwriter Joan Didion —
"We tell ourselves stories in order to live….
We interpret what we see, select the most workable of multiple choices. We live entirely, especially if we are writers, by the imposition of a narrative line upon disparate images, by the 'ideas' with which we have learned to freeze the shifting phantasmagoria which is our actual experience."
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From Thomas Mann, "Schopenhauer," 1938, in Essays of Three Decades , translated by H. T. Lowe-Porter, Alfred A. Knopf, 1947, pp. 372-410—
Page 372: THE PLEASURE we take in a metaphysical system, the gratification purveyed by the intellectual organization of the world into a closely reasoned, complete, and balanced structure of thought, is always of a pre-eminently aesthetic kind. It flows from the same source as the joy, the high and ever happy satisfaction we get from art, with its power to shape and order its material, to sort out life's manifold confusions so as to give us a clear and general view.
Truth and beauty must always be referred the one to the other. Each by itself, without the support given by the other, remains a very fluctuating value. Beauty that has not truth on its side and cannot have reference to it, does not live in it and through it, would be an empty chimera— and "What is truth?"
….
Page 376: … the life of Plato was a very great event in the history of the human spirit; and first of all it was a scientific and a moral event. Everyone feels that something profoundly moral attaches to this elevation of the ideal as the only actual, above the ephemeralness and multiplicity of the phenomenal, this devaluation of the senses to the advantage of the spirit, of the temporal to the advantage of the eternal— quite in the spirit of the Christianity that came after it. For in a way the transitory phenomenon, and the sensual attaching to it, are put thereby into a state of sin: he alone finds truth and salvation who turns his face to the eternal. From this point of view Plato's philosophy exhibits the connection between science and ascetic morality.
But it exhibits another relationship: that with the world of art. According to such a philosophy time itself is merely the partial and piecemeal view which an individual holds of ideas— the latter, being outside time, are thus eternal. "Time"— so runs a beautiful phrase of Plato— "is the moving image of eternity." And so this pre-Christian, already Christian doctrine, with all its ascetic wisdom, possesses on the other hand extraordinary charm of a sensuous and creative kind; for a conception of the world as a colourful and moving phantasmagoria of pictures, which are transparencies for the ideal and the spiritual, eminently savours of the world of art, and through it the artist, as it were, first comes into his own.
From last night's online NY Times obituaries index—
"How much story do you want?" — George Balanchine
Comments Off on Unity and Multiplicity
Tuesday, April 26, 2011
Today's earlier post mentions one approach to the concepts of unity and multiplicity. Here is another.

Unity:
The 3×3×3 Galois Cube

Multiplicity:
One of a group, GL(3,3), of 11,232
natural transformations of the 3×3×3 Cube
See also the earlier 1985 3×3 version by Cullinane.
Comments Off on Unity and Multiplicity
Thursday, September 17, 2020
Continues in The New York Times :

“One day — ‘I don’t know exactly why,’ he writes — he tried to
put together eight cubes so that they could stick together but
also move around, exchanging places. He made the cubes out
of wood, then drilled a hole in the corners of the cubes to link
them together. The object quickly fell apart.
Many iterations later, Rubik figured out the unique design
that allowed him to build something paradoxical:
a solid, static object that is also fluid….” — Alexandra Alter
Another such object: the eightfold cube .
Comments Off on Structure and Mutability . . .
Thursday, August 6, 2020
Continues.
See a Log24 search for Beadgame Space.
This post might be regarded as a sort of “checked cell”
for the above concepts listed as tags . . .

Related material from a Log24 search for Structuralism —

Comments Off on Structure and Mutability . . .
Thursday, May 28, 2020
“Old men ought to be explorers.” — T. S. Eliot
“Everybody’s lost but me!” — Young Indiana Jones, quoted
in a book review (“Knox Peden on Martin Hägglund”) in
Sydney Review of Books on May 26 . . .
” Here I am reminded of the words of
the young Indiana Jones alone in the desert,
decades before the Last Crusade:
‘Everybody’s lost but me.’ “


Related remarks — Now You See It, Now You Don’t.
Comments Off on Unity Game
Monday, October 21, 2019
Related entertainment —
Detail:
George Steiner —
"Perhaps an insane conceit."
Perhaps.
See Quantum Tesseract Theorem .
Perhaps Not.
See Dirac and Geometry .
Comments Off on Algebra and Space… Illustrated.
Wednesday, September 18, 2019
“I” as black monolith:

Unity
Roman numeral I
as well as capital I.
Dimensions: 6×1.
Comments Off on Unity
Tuesday, August 20, 2019
From my reading Monday morning —
From the online New York Times this afternoon —
Related literature —
For the Church of Synchronology —
The Gigantomachia page above is dated September 20, 2003.
See as well my own webpage from that date: "The Form, the Pattern."
Comments Off on Ice Giants and Fire Gods: Mind the Gap
Monday, August 19, 2019

For “the Pergamum altar,” see Pergamon in this journal.
See also . . .

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Tuesday, June 25, 2019
Comments Off on Medium and Message
Wednesday, May 1, 2019
In memory of Quentin Fiore — from a Log24 search for McLuhan,
an item related to today's previous post . . .
Related material from Log24 on the above-reported date of death —
See also, from a search for Analogy in this journal . . .
.
Comments Off on The Medium and the Message
Friday, March 1, 2019
"Maybe an image is too strong
Or maybe is not strong enough."
— "Solomon and the Witch,"
by William Butler Yeats

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Sunday, August 26, 2018
Mathematics —
Narrative —
See also other recent posts tagged Queensland.
Comments Off on Mathematics and Narrative: Queensland
Thursday, June 21, 2018
"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 "super-mathematician" 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)
Wednesday, May 16, 2018
Comments Off on Church and Temple
Friday, June 23, 2017
The life of Mr. Breder is not unrelated to that of Carl Andre.
See also, in this journal, Bulk Apperception.
Comments Off on Annals of Art and Design
Wednesday, May 10, 2017
In memory of an art dealer who
reportedly died on Sunday, May 7—
Decorations for a Cartoon Graveyard

Comments Off on In the Park with Yin and Yang
Tuesday, May 9, 2017
Comments Off on Text and Context
Tuesday, July 12, 2016
The following passage by Igor Dolgachev (Good Friday, 2003)
seems somewhat relevant (via its connection to Kummer's 166 )
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
Wednesday, May 25, 2016
From "Projective Geometry and PT-Symmetric 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 spin-½ 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 Plucker-Klein correspondence between lines of
a three-dimensional projective space and points of a quadric
in a five-dimensional 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 166 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 166 configuration . . . .
See that structure in this journal, for instance —
See as well yesterday morning's post.
Comments Off on Kummer and Dirac
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 many-faceted 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 4-space over
the two-element 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

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Monday, December 14, 2015
(Continued)
See a post by Peter Woit from Sept. 24, 2005 — Dirac's Hidden Geometry.
The connection, if any, with recent Log24 posts on Dirac and Geometry
is not immediately apparent. Some related remarks from a novel —
From Broken Symmetries by Paul Preuss
(first published by Simon and Schuster in 1983) —
"He pondered the source of her fascination with the occult, which sooner or later seemed to entangle a lot of thoughtful people who were not already mired in establishmentarian science or religion. It was the religious impulse, at base. Even reason itself could function as a religion, he supposed— but only for those of severely limited imagination.
He’d toyed with 'psi' himself, written a couple of papers now much quoted by crackpots, to his chagrin. The reason he and so many other theoretical physicists were suckers for the stuff was easy to understand— for two-thirds of a century an enigma had rested at the heart of theoretical physics, a contradiction, a hard kernel of paradox. Quantum theory was inextricable from the uncertainty relations.
The classical fox knows many things, but the quantum-mechanical hedgehog knows only one big thing— at a time. 'Complementarity,' Bohr had called it, a rubbery notion the great professor had stretched to include numerous pairs of opposites. Peter Slater was willing to call it absurdity, and unlike some of his older colleagues who, following in Einstein’s footsteps, demanded causal explanations for everything (at least in principle), Peter had never thirsted after 'hidden variables' to explain what could not be pictured. Mathematical relationships were enough to satisfy him, mere formal relationships which existed at all times, everywhere, at once. It was a thin nectar, but he was convinced it was the nectar of the gods.
The psychic investigators, on the other hand, demanded to know how the mind and the psychical world were related. Through ectoplasm, perhaps? Some fifth force of nature? Extra dimensions of spacetime? All these naive explanations were on a par with the assumption that psi is propagated by a species of nonlocal hidden variables, the favored explanation of sophisticates; ignotum per ignotius .
'In this connection one should particularly remember that the human language permits the construction of sentences which do not involve any consequences and which therefore have no content at all…' The words were Heisenberg’s, lecturing in 1929 on the irreducible ambiguity of the uncertainty relations. They reminded Peter of Evan Harris Walker’s ingenious theory of the psi force, a theory that assigned psi both positive and negative values in such a way that the mere presence of a skeptic in the near vicinity of a sensitive psychic investigation could force null results. Neat, Dr. Walker, thought Peter Slater— neat, and totally without content.
One had to be willing to tolerate ambiguity; one had to be willing to be crazy. Heisenberg himself was only human— he’d persuasively woven ambiguity into the fabric of the universe itself, but in that same set of 1929 lectures he’d rejected Dirac’s then-new wave equations with the remark, 'Here spontaneous transitions may occur to the states of negative energy; as these have never been observed, the theory is certainly wrong.' It was a reasonable conclusion, and that was its fault, for Dirac’s equations suggested the existence of antimatter: the first antiparticles, whose existence might never have been suspected without Dirac’s crazy results, were found less than three years later.
Those so-called crazy psychics were too sane, that was their problem— they were too stubborn to admit that the universe was already more bizarre than anything they could imagine in their wildest dreams of wizardry."
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Particularly relevant …
"Mathematical relationships were enough to satisfy him,
mere formal relationships which existed at all times,
everywhere, at once."
Some related pure mathematics —

Comments Off on Dirac and Geometry
Friday, November 27, 2015
(A Prequel to Dirac and Geometry)
"So Einstein went back to the blackboard.
And on Nov. 25, 1915, he set down
the equation that rules the universe.
As compact and mysterious as a Viking rune,
it describes space-time as a kind of sagging mattress…."
— Dennis Overbye in The New York Times online,
November 24, 2015
Some pure mathematics I prefer to the sagging Viking mattress —
Readings closely related to the above passage —
Thomas Hawkins, "From General Relativity to Group Representations:
the Background to Weyl's Papers of 1925-26," in Matériaux pour
l'histoire des mathématiques au XXe siècle: Actes du colloque
à la mémoire de Jean Dieudonné, Nice, 1996 (Soc. Math.
de France, Paris, 1998), pp. 69-100.
The 19th-century algebraic theory of invariants is discussed
as what Weitzenböck called a guide "through the thicket
of formulas of general relativity."
Wallace Givens, "Tensor Coordinates of Linear Spaces," in
Annals of Mathematics Second Series, Vol. 38, No. 2, April 1937,
pp. 355-385.
Tensors (also used by Einstein in 1915) are related to
the theory of line complexes in three-dimensional
projective space and to the matrices used by Dirac
in his 1928 work on quantum mechanics.
For those who prefer metaphors to mathematics —
"We acknowledge a theorem's beauty
when we see how the theorem 'fits' in its place,
how it sheds light around itself, like a Lichtung ,
a clearing in the woods."
— Gian-Carlo Rota, Indiscrete Thoughts ,
Birkhäuser Boston, 1997, page 132
Rota fails to cite the source of his metaphor.
It is Heidegger's 1964 essay, "The End of Philosophy
and the Task of Thinking" —
"The forest clearing [ Lichtung ] is experienced
in contrast to dense forest, called Dickung
in our older language."
— Heidegger's Basic Writings ,
edited by David Farrell Krell,
Harper Collins paperback, 1993, page 441
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Comments Off on Einstein and Geometry
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 39-52 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 more-recent 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, November 13, 2015
Note that the six anticommuting sets of Dirac matrices listed by Arfken
correspond exactly to the six spreads in the above complex of 15 projective
lines of PG(3,2) fixed under a symplectic polarity (the diamond theorem
correlation ). As I noted in 1986, this correlation underlies the Miracle
Octad Generator of R. T. Curtis, hence also the large Mathieu group.
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
Related material:
The 6-set in my 1986 note above also appears in a 1996 paper on
the sixteen Dirac matrices by David M. Goodmanson —
Background reading:
Ron Shaw on finite geometry, Clifford algebras, and Dirac groups
(undated compilation of publications from roughly 1994-1995)—

Comments Off on A Connection between the 16 Dirac Matrices and the Large Mathieu Group
Sunday, October 25, 2015
(Continued.)
Featured on this morning's online front page of
The New York Times —
Some further details —
An example of New York Times culture is shown above —
"… Mondrian paintings at the Museum of Modern Art
blend symmetry with a tensile volatility."
(To be fair, this contemptible bullshit is from a picture caption,
not from the art review being summarized.)
Related cultural observations —
Math for Child Buyers and Fiction for Child Sellers.
Comments Off on Buyers and Sellers of Children
Friday, May 22, 2015
Donald in Mathmagic Land
Manly P. Hall
Comments Off on Mathmagic Land
Saturday, February 7, 2015
Comments Off on Words and Pictures, continued
From actor James Spader, whose birthday is today —
"… my father taught English. My mother taught art…."
— Spader in a 2014 interview
See as well the 2013 film "Words and Pictures"
and Log24 posts on a 2007 film, "The Last Mimzy."
Above: A scene from Spader's TV series "The Blacklist"
that was aired on Thursday, February 5, 2015.
Comments Off on Word and Object
Monday, November 3, 2014
A weblog reports Chris Rock's remarks
on Saturday Night Live this past weekend:
"It’s America, we commercialize everything.
Look at what we did to Christmas.
Christmas. Christmas is Jesus’ birthday.
It’s Jesus’ birthday. Now, I don’t know Jesus
but from what I’ve read, Jesus is the least
materialistic person to ever roam the earth.
No bling on Jesus.
Jesus kept a low profile and we turned his
birthday into the most materialistic day of the
year. Matter of fact, we have the Jesus birthday
season. It’s a whole season of materialism.
Then, at the end of the Jesus birthday season
we have the nerve to have an economist come
on TV and tell you how horrible the Jesus birthday
season was this year. Oh, we had a horrible Jesus’
birthday this year. Hopefully, business will pick up
by his Crucifixion.”
Related music and image:
"Show us the way to the next little girl …"
Natalie Wood in "Miracle on 34th Street" (1947)
Related non-materialistic meditations:
The Rhetoric of Abstract Concepts and Gods and Giants.
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Friday, October 10, 2014
(Continued from Grids and Space and posts tagged Riddle for Caltech)

Comments Off on Both Hands and an Ass Map
Tuesday, September 9, 2014
This post is continued from a March 12, 2013, post titled
"Smoke and Mirrors" on art in Tromsø, Norway, and from
a June 22, 2014, post on the nineteenth-century
mathematicians Rosenhain and Göpel.
The latter day was the day of death for
mathematician Loren D. Olson, Harvard '64.
For some background on that June 22 post, see the tag
Rosenhain and Göpel in this journal.
Some background on Olson, who taught at the
University of Tromsø, from the American Mathematical
Society yesterday:
Olson died not long after attending the 50th reunion of the
Harvard Class of 1964.
For another connection between that class (also my own)
and Tromsø, see posts tagged "Elegantly Packaged."
This phrase was taken from today's (print)
New York Times review of a new play titled "Smoke."
The phrase refers here to the following "package" for
some mathematical objects that were named after
Rosenhain and Göpel — a 4×4 array —
For the way these objects were packaged within the array
in 1905 by British mathematician R. W. H. T. Hudson, see
a page at finitegometry.org/sc. For the connection to the art
in Tromsø mentioned above, see the diamond theorem.
Comments Off on Smoke and Mirrors
Thursday, July 24, 2014
Comments Off on Death in Mathmagic Land
Sunday, June 22, 2014
(Continued)
Some bizarre remarks on “purity” in the previous post
suggest a review of some pure mathematics.

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Wednesday, May 14, 2014
"It was our old friend Pythagoras who discovered
that the pentagram was full of mathematics."
— Narrator, "Donald in Mathmagic Land," Disney, 1959
… and it was Peter J. Cameron who discovered that
mathematics was full of pentagrams.
From Log24 on May 3: Gray Space —
Robert J. Stewart (left) and a pentagram photo posted May 2
by Oslo artist Josefine Lyche. See also Lyche in this journal.
From Log24 on May 13: An Artist's Memorial —
The death mentioned in the above May 13 post occurred on
May 12, the date of a scheduled Black Mass at Harvard.
Related material:

Comments Off on Death in Mathmagic Land
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
Saturday, February 15, 2014
(Continued)
See The Oslo Version in this journal and the New Year’s Day 2014 post.
The pictures of the 56 spreads in that post (shown below) are based on
the 20 Rosenhain and 15 Göpel tetrads that make up the 35 lines of
PG(3,2), the finite projective 3-space over the 2-element Galois field.

Click for a larger image.
Comments Off on Rosenhain and Göpel
Saturday, September 21, 2013
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 M24. 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 4-element
subsets of a 16-element 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 two-element 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 3-space." The affine 4-space
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
non-isotropic 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)."
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Narrative:
Aooo.
Happy birthday to Stephen King.
Comments Off on Mathematics and Narrative (continued)
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 M24.
For some related material that is more up-to-date, search the Web
for Mathieu + Kummer .
Comments Off on Rosenhain and Göpel Revisited
Tuesday, March 19, 2013
Angels & Demons meet Hudson Hawk
Dan Brown's four-elements diamond in Angels & Demons :
The Leonardo Crystal from Hudson Hawk :
Hudson:
Mathematics may be used to relate (very loosely)
Dan Brown's fanciful diamond figure to the fanciful
Leonardo Crystal from Hudson Hawk …
"Giving himself a head rub, Hawk bears down on
the three oddly malleable objects. He TANGLES
and BENDS and with a loud SNAP, puts them together,
forming the Crystal from the opening scene."
— A screenplay of Hudson Hawk
Happy birthday to Bruce Willis.
Comments Off on Mathematics and Narrative (continued)
Tuesday, August 5, 2014
"We tell ourselves stories in order to live…. We interpret what we see, select the most workable of multiple choices. We live entirely, especially if we are writers, by the imposition of a narrative line upon disparate images, by the 'ideas' with which we have learned to freeze the shifting phantasmagoria which is our actual experience."
— Joan Didion
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See also a post from May 4, 2011 (the date, according to a Google
search, of untitled notes regarding a matrix called Omega).
Comments Off on The Omega Story
Monday, June 27, 2011

The 3×3×3 Galois Cube
See Unity and Multiplicity.
This cube, unlike Rubik's, is a
purely mathematical structure.
Its properties may be compared
with those of the order-2 Galois
cube (of eight subcubes, or
elements ) and the order-4 Galois
cube (of 64 elements). The
order-3 cube (of 27 elements)
lacks, because it is based on
an odd prime, the remarkable
symmetry properties of its smaller
and larger cube neighbors.
Comments Off on Galois Cube Revisited
Saturday, April 30, 2011
Part I — Unity and Multiplicity
(Continued from The Talented and Galois Cube)

Part II — "A feeling, an angel, the moon, and Italy"—
Click for details

Comments Off on Crimson Walpurgisnacht
Thursday, April 1, 2021


That’s Careful How , not Careful Glow , Magoo.
See also Magoo at the Tate .
Comments Off on For Mr. Magoo
Continues.


“Some day, when I’m awfully low . . .
I will feel a glow just thinking of you”
— Sinatra
Comments Off on Sex Textiles . . .
Friday, March 26, 2021
Columbus in Zombieland

Comments Off on Cards of Identity Continues:

Some images from Feb. 5, 2021, in a search for “ABC Art” —

A colored version using CSS —

See https://codepen.io/m759/pen/wvoGwzx .
“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.
Comments Off on ABC Art
Thursday, March 18, 2021


Related material —
The Teacup of Experience




Comments Off on Significant Form
Monday, March 1, 2021
“Jerome, Jeremiah . . . Jeremiah, Jerome.”

See Das Nichts Nichtet , a post on February 13th.
See also a death on that date from The New York Times today —
“He opened his booth in the diamond district on 47th Street
between Fifth and Sixth Avenues, in 1948 . . . .”
Comments Off on The Point
Comments Off on Choice of Viewpoint . . .
Wednesday, February 17, 2021
“The number one , then, has become Husserl’s touchstone
for discriminating between psychological processes and
logical laws. It is his reality detector. What is
psychological (or empirical) comes on in discrete
individual instances– ones– and you can examine their
edges. What is logical (or ideal) comes on as a
seamless oceanic unity without temporal edges….”
— Marianne Sawicki, “Edmund Husserl (1859—1938),”
Internet Encyclopedia of Philosophy
See also Roman Numeral in this journal.
Comments Off on Eins und Einheit
Wednesday, January 27, 2021
Continues.


Related vocabulary —

See as well the word facet in this journal.
Analogously, one might write . . .
A Hiroshima cube consists of 6 faces ,
each with 4 squares called facets ,
for a total of 24 facets. . . .”
(See Aitchison’s Octads , a post of Feb. 19, 2020.)


Click image to enlarge. Background: Posts tagged ‘Aitchison.'”
Comments Off on Adoration of the Cube . . .
Thursday, January 21, 2021
The New York Times this afternoon reports
the death of an author last year
“somewhere between May 2 and May 15.”
“Down and down I go, round and round I go”
— Kevin Spacey in the soundtrack album for
“Midnight in the Garden of Good and Evil”
Comments Off on Man Date
Comments Off on Cards of Identity
Tuesday, October 20, 2020
The Ballad of Goo Ballou —
the Sequel to . . .

“Let me count the ways” is an appropriate request
for students of the discrete , as opposed to the
continuous , which instead requires measurement .
Related academic material —
Raymond Cattell on crystallized vs. fluid intelligence.
For a more literary approach, see Crystal and Dragon
and For Trevanian.
This post was inspired in part by
the American Sequel Society and . . .

Comments Off on The Browning Methods
Wednesday, September 9, 2020



For a Jedi holocron of sorts, see this journal on the above YouTube date —

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Monday, September 7, 2020
From the subtitles to “A Discovery of Witches,”
Season 1, Episode 2 —
An actor playing a contemporary (2018) fictional Oxford professor —
378
00:35:54,235 –> 00:35:56,593
We’re among hundreds of laboratories
using genetics
379
00:35:56,595 –> 00:35:59,713
to study species origin,
but in our lab
380
00:35:59,715 –> 00:36:02,315
humans aren’t the only species
we’re studying.
An earlier non-fictional Oxford student writes —

Related material: Other posts tagged Structure and Mutability.
Comments Off on A Discovery of Species
Saturday, August 8, 2020
In memory of Wilford Brimley:
“The polymorphic Thing, capable of absorbing the human
as but one among other morphological possibilities in its
seemingly infinite repertoire, can be understood, that is,
as the embodiment of evolution.”
— Eric White, Science Fiction Studies #61 (Vol. 20, Part 3, Nov. 1993),
“The Erotics of Becoming: XENOGENESIS and The Thing“

Comments Off on Bullshit Studies

See also other posts now tagged
Natural Diagram .
Related remarks by J. H. Conway —

Comments Off on A Natural Diagram
Thursday, July 30, 2020
An article yesterday at Quanta Magazine suggests a review . . .
From Diamond Theorem images at Pinterest —

Some background —

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Wednesday, July 22, 2020
“The pattern of the thing precedes the thing.
I fill in the gaps of the crossword at any spot
I happen to choose. These bits I write on
index cards until the novel is done.”
— Vladimir Nabokov, interview,
Paris Review No. 41 (Summer-Fall 1967).

Another story —


Related material: Mathematics as a Black Art.
Comments Off on Card
Friday, July 17, 2020
Comments Off on Poetic as Well as Prosaic
Comments Off on Theoretical as Well as Practical Value
Saturday, July 11, 2020
Comments Off on Raiders of the Lost Arkenstone
The previous post contained a passage from Iris Murdoch’s
1961 essay “Against Dryness.” Some related philosophy —

For those who prefer pure mathematics to philosophical ruminations
there are some relevant remarks in my webpage of August 27, 2003.
Comments Off on Philosophy for Murdoch Fans
Thursday, July 9, 2020

In memoriam —
Click the quotation below for “Foster’s Space” posts.

Comments Off on Fashion Space

For those who prefer fiction —
“Twenty-four glyphs, each one representing not a letter, not a word,
but a concept, arranged into four groups, written in Boris’s own hand,
an artifact that seemed to have resurrected him from the dead. It was
as if he were sitting across from Bourne now, in the dim antiquity of
the museum library.
This was what Bourne was staring at now, written on the unfolded
bit of onionskin.”
— “Robert Ludlum’s” The Bourne Enigma , published on June 21, 2016
Passing, on June 21, 2016, into a higher dimension —

Comments Off on The Enigma Glyphs
Thursday, May 28, 2020
My website on finite geometry is now available
on GitHub at http://m759.github.io/ . The part
of greatest interest to coders is also at
https://repl.it/@m759/View-4x4x4#index.html .
Comments Off on Finite Geometry at GitHub
Thursday, May 7, 2020
“Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to explain
how art fits into our subject and what we mean by beauty.”
— Sir Michael Atiyah, quoted here on April 4, 2016

Illustrations, from the American Mathematical Society Spring
2020 book sale, of a book scheduled to be published May 28.
Comments Off on Now You See It, Now You Don’t
Sunday, January 19, 2020
Richard H. Masland and I were in a Harvard freshman seminar together
in the 1960-61 academic year. From a Crimson article on the program —
"Freshman Seminar Program Department Administrator Corinna S. Rohse
described the program’s courses, which allow students to study subjects
that vary from Sanskrit to the mathematical basis for chess, as
'jewel-like: small and incredibly well-cut.' "
This suggests a review of Log24 posts now tagged Four Gates.
Comments Off on For a Harvard Classmate Who Died on St. Lucia’s Day
Sunday, December 29, 2019
Related reading —
"I closed my eyes and saw the number 137—
so very close to the reciprocal of alpha—
on the chest of the runner in Van Cortlandt Park.
Should I start the story there? "
— Alpert, Mark. Saint Joan of New York
(Science and Fiction) (p. 103).
Springer International Publishing. Kindle edition.
Cover detail:
See as well St. Joan in this journal.
Comments Off on Springer Link
Friday, December 27, 2019
Booklist on Final Theory :
"Alpert, an editor for Scientific American , laces his high-IQ
doomsday thriller with clearly explicated and hauntingly beautiful
scientific theories…."
Booklist on The Omega Theory :
"Alpert’s follow-up to his acclaimed first novel, Final Theory (2008),
continues the adventures of science historian David Swift."
See as well this journal on June 1, 2008.
Comments Off on The Secret Life of Mark Alpert
. . . Pace Wallace Stevens.
"The history of the universe can thus be seen as
an endless chain of changes, but Aquinas argued
that there must be some transcendent entity that
initiated the chain, something that is itself
unchanging and that already possesses all of the
properties that worldly objects can come to possess.
He also claimed that this entity must be eternal;
because it is the root of all causes, nothing else
could’ve caused it. And unlike all worldly objects,
the transcendent entity is necessary—it must exist."
— Mark Alpert in Scientific American, 12/23/2019
Comments Off on God in the Object…
Wednesday, October 9, 2019
Note that in the pictures below of the 15 two-subsets of a six-set,
the symbols 1 through 6 in Hudson's square array of 1905 occupy the
same positions as the anticommuting Dirac matrices in Arfken's 1985
square array. Similarly occupying these positions are the skew lines
within a generalized quadrangle (a line complex) inside PG(3,2).
Related narrative — The "Quantum Tesseract Theorem."
Comments Off on The Joy of Six
Tuesday, August 27, 2019
See also Tiger in this journal, esp. —
and . . .
other "Death and the Spirit" posts.
Comments Off on Tiger Leaps
Monday, August 19, 2019
The title is from the post "Child's Play" of May 21, 2012 . . .
"It seems that only one course is open to the philosopher
who values knowledge and truth above all else. He must
refuse to accept from the champions of the forms the
doctrine that all reality is changeless [and exclusively
immaterial], and he must turn a deaf ear to the other party
who represent reality as everywhere changing [and as only
material]. Like a child begging for 'both', he must declare
that reality or the sum of things is both at once [το όν τε και
το παν συναμφότερα] (Sophist 246a-249d)."
Related material —
"Schoolgirl Space: 1984 Revisited" (July 9, 2019) and
posts tagged Tetrahedron vs. Square.
Comments Off on A Couple of Tots
Sunday, August 18, 2019
Some background for The Epstein Chronicles —

“What modern painters
are trying to do,
if they only knew it,
is paint invariants.”
— James J. Gibson, Leonardo,
Vol. 11, pp. 227-235.
Pergamon Press Ltd., 1978
|
See also Robert Maxwell,
Frank Oppenheimer,
and the history of Leonardo .

Click the above Pergamon Press image
for Pergamon-related material.
Comments Off on Structure at Pergamon
Wednesday, August 14, 2019
"If you want to win, you need to know just one thing and not to waste your time
on anything else: the pleasures of erudition are reserved for losers. The more
a person knows, the more things have gone wrong."
— Eco, Umberto. Numero Zero (p. 9). HMH Books. Kindle Edition.
"All this notwithstanding, I dreamed what all losers dream, about one day writing
a book that would bring me fame and fortune. To learn how to become a great
writer, I became what in the last century was called the nègre (or ghostwriter,
as they say today, to be politically correct) for an author of detective stories who
gave himself an American name to improve sales, like the actors in spaghetti
westerns. But I enjoyed working in the shadows, hidden behind a double veil
(the Other’s and the Other’s other name)."
— Eco, Umberto. Numero Zero (pp. 9-10). HMH Books. Kindle Edition.
How's this for a double veil, Umberto? —
Boeing/Being Meets the Flying Spaghetti Monster . . .

Comments Off on Double Veil
For the title, see Zero: Both Real and Imaginary (a Log24 search).
The title was suggested by the previous post, by Zorro Ranch,
by the classic 1967 film The Producers , and by . . .
Related material —
Vanity Fair on Sept. 8, 2017, celebrated the young actress
who played Beverly Marsh in the 2017 film version of
Stephen King's IT . See a post from her 12th birthday —
"Winter's Game" — that touches upon Maori themes.
More generally, see Bester + Deceivers in this journal.
And for the Church of Synchronology . . .
See posts related to the above Vanity Fair date.
Comments Off on The Mask of Zero
Tuesday, August 13, 2019
The Matrix of Lévi-Strauss —
(From his “Structure and Form: Reflections on a Work by Vladimir Propp.”
Translated from a 1960 work in French. It appeared in English as
Chapter VIII of Structural Anthropology, Volume 2 (U. of Chicago Press, 1976).
Chapter VIII was originally published in Cahiers de l’Institut de Science
Économique Appliquée , No. 9 (Series M, No. 7) (Paris: ISEA, March 1960).)

The structure of the matrix of Lévi-Strauss —

Illustration from Diamond Theory , by Steven H. Cullinane (1976).
The relevant field of mathematics is not Boolean algebra, but rather
Galois geometry.
Comments Off on Putting the Structure in Structuralism
Sunday, July 14, 2019
The Quantum Tesseract Theorem Revisited
From page 274 —
"The secret is that the super-mathematician expresses by the anticommutation
of his operators the property which the geometer conceives as perpendicularity
of displacements. That is why on p. 269 we singled out a pentad of anticommuting
operators, foreseeing that they would have an immediate application in describing
the property of perpendicular directions without using the traditional picture of space.
They express the property of perpendicularity without the picture of perpendicularity.
Thus far we have touched only the fringe of the structure of our set of sixteen E-operators.
Only by entering deeply into the theory of electrons could I show the whole structure
coming into evidence."
A related illustration, from posts tagged Dirac and Geometry —
Compare and contrast Eddington's use of the word "perpendicular"
with a later use of the word by Saniga and Planat.
Comments Off on Old Pathways in Science:
Wednesday, July 3, 2019
Comments Off on Best Meets Bester
Tuesday, June 25, 2019
Monday, June 24, 2019
Comments Off on Ex Fano Apollinis
Saturday, March 9, 2019
Commemorating a talk given by Iain Aitchison
at Hiroshima a year ago today.

Comments Off on The Hiroshima Model
Saturday, December 22, 2018
The following are some notes on the history of Clifford algebras
and finite geometry suggested by the "Clifford Modules" link in a
Log24 post of March 12, 2005 —
A more recent appearance of the configuration —

Comments Off on Cremona-Richmond
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
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
Tuesday, September 25, 2018
See some posts related to three names
associated with Trinity College, Cambridge —
Atiyah + Shaw + Eddington .
Comments Off on Trinity
Saturday, September 1, 2018
The date of Ron Shaw's 2016 death appears to be June 21:
All other Internet sources I have seen omit the June 21 date.
This journal on that date —

Comments Off on Ron Shaw — D. 21 June 2016
Wednesday, August 22, 2018
See the title in a TV review* from io9 this morning and in
this journal.
* Spoiler alert
Comments Off on Castle Rock
Tuesday, August 21, 2018
A search for "IMA" yields, at ima.org.au . . .
"Since 1975, the Institute of Modern Art (IMA) has been Queensland’s
leading independent forum for art and its discourses." … And …
"Clog, therefore, purple Jack and crimson Jill."
— Wallace Stevens
Comments Off on Purported Endgame
Click the image above for an interview dated Nov. 9, 2015.
Also on that date —

Comments Off on Available Light
Wednesday, August 1, 2018
“… I realized that to me, Gödel and Escher and Bach
were only shadows cast in different directions
by some central solid essence.
I tried to reconstruct the central object . . . ."
— Douglas Hofstadter (1979)
See also posts of July 23, 2007, and April 7, 2018.
* Term from a visual-culture lexicon —

Comments Off on Abschattungen*
O AS IS
See also other posts tagged "Is and As."
Comments Off on “Ready Player One” Mnemonic:
Tuesday, July 31, 2018
Comments Off on A is for Abschattungen
"A blank underlies the trials of device." — Wallace Stevens
.
Comments Off on Lexicon
Wednesday, May 16, 2018
The title of the previous post, "Church and Temple," together
with today's online New York Times obituaries for singer
Lara Saint Paul (d. May 8) and playwright Leah Rose Napolin
(d. May 13), suggests a review…
See as well a Log24 search for Isaac Singer.
Comments Off on Review
. . . as time goes by.

Comments Off on Same Old Story
Friday, April 13, 2018
See also Log24 posts now tagged "Is and As."
Comments Off on Philosophy 101
Monday, March 12, 2018
Remarks related to a recent film and a not-so-recent film.
For some historical background, see Dirac and Geometry in this journal.
Also (as Thas mentions) after Saniga and Planat —
The Saniga-Planat paper was submitted on December 21, 2006.
Excerpts from this journal on that date —
"Open the pod bay doors, HAL."
Comments Off on “Quantum Tesseract Theorem?”
Monday, February 26, 2018
From the website of Richard P. Gabriel —
" As part of my studies, I came up with a 'theory of poetry'
based loosely on Christopher Alexander’s 'Nature of Order.' "
[The Alexander link is mine, not Gabriel’s.]
A phrase from this journal a year ago today — "poetic order" —
links to the theory of Gabriel —
From Gabriel's "The Nature of Poetic Order" —
Positive Space
• Positive space is the characteristic of a center
that moves outward from itself, seemingly oozing life
rather than collapsing on itself
• An image that resonates is showing positive space
• A word that has many connotations that fit with the
other centers in the poem is showing positive space
• It is an expansion outward rather than a contraction
inward, and it shows that the poem is unfolding
in front of us and not dying
Related material —
From a post of April 26, 2017 —

Comments Off on The Unfolding
Sunday, February 25, 2018
A sequel to Friday's "The Dark, Seductive Art of Phillips Exeter."
From Twitter today —
Groton School @GrotonSchool Feb 1
Groton School Theater Department Presents: CABARET
Friday, February 23rd at 7:30 pm
Saturday, February 24th at 7:30 pm
Sunday, February 25th at 3 pm
All performances are free to the public. Please make ticket
reservations in advance online: http://bit.ly/CABARET2018.
Comments Off on The Dark, Seductive Art of Groton
Friday, February 23, 2018
From two pedagogues in Montana —
https://www.nctm.org/Publications/
Mathematics-Teaching-in-Middle-School/
2016/Vol22/Issue1/Quilt-Block-Symmetries/
Related material from author Dan Brown's father,
a pedagogue who taught at Phillips Exeter Academy :
Click the above image for some background.
Related material from Log24 —
Verwandlungslehre.
Compare and contrast with the above
Transformational Geometry cover:
Related material from Vienna — The previous post and
Wittgenstein on Bewitchment.
See as well . . .
Click to enlarge.
Comments Off on The Dark, Seductive Art of Phillips Exeter
Related material — (Click to enlarge) —
"Risin' up to the challenge of our rival"
— Eye of the Tiger
Detail —

Comments Off on Snow Games
Saturday, February 17, 2018
Michael Atiyah on the late Ron Shaw —

Phrases by Atiyah related to the importance in mathematics
of the two-element Galois field GF(2) —
- “The digital revolution based on the 2 symbols (0,1)”
- “The algebra of George Boole”
- “Binary codes”
- “Dirac’s spinors, with their up/down dichotomy”
These phrases are from the year-end review of Trinity College,
Cambridge, Trinity Annual Record 2017 .
I prefer other, purely geometric, reasons for the importance of GF(2) —
- The 2×2 square
- The 2x2x2 cube
- The 4×4 square
- The 4x4x4 cube
See Finite Geometry of the Square and Cube.
See also today’s earlier post God’s Dice and Atiyah on the theology of
(Boolean) algebra vs. (Galois) geometry:

Comments Off on The Binary Revolution
On a Trinity classmate of Ian Macdonald (see previous post)—
Atiyah's eulogy of Shaw in Trinity Annual Record 2017
is on pages 137 through 146. The conclusion —

Comments Off on God’s Dice
Sunday, December 10, 2017
See also Symplectic in this journal.
From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens 54, 59-79 (1992):
“… what is the origin of the unusual name ‘symplectic’? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure ‘line complex group’ the ‘symplectic group.’
… the adjective ‘symplectic’ means ‘plaited together’ or ‘woven.’
This is wonderfully apt….”
The above symplectic figure appears in remarks on
the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2). See also
related remarks on the notion of linear (or line ) complex
in the finite projective space PG(3,2) —

Comments Off on Geometry
Thursday, November 30, 2017
Scholia on the title — See Quantum + Mystic in this journal.
The Matrix of Lévi-Strauss —
"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 Lévi-Strauss, 1976
I prefer the earlier, better-known, 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 26-27, 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
Friday, November 24, 2017
The Matrix —
The Grid —
Picturing the Witt Construction —
"Read something that means something." — New Yorker ad
Comments Off on The Matrix Meets the Grid
Thursday, November 23, 2017
David Brooks in The New York Times today —
"We once had a unifying national story, celebrated each Thanksgiving.
It was an Exodus story. Americans are the people who escaped oppression,
crossed a wilderness and are building a promised land. The Puritans brought
this story with them. Each wave of immigrants saw themselves in this story.
The civil rights movement embraced this story.
But we have to admit that many today do not resonate with this story. . . .
Today, we have no common national narrative, no shared way
of interpreting the flow of events. Without a common story,
we don’t know what our national purpose is. We have no
common set of goals or ideals.
We need a new national narrative."
From a post of August 15, 2010 —
For some background, see Java Jive and Today's Theology.
Related readings —
From 1928:
From the previous post:
"Thus, instead of Propp's chronological scheme,
in which the order of succession of events
is a feature of the structure . . .
another scheme should be adopted, which would present
a structural model defined as the group of transformations
of a small number of elements. This scheme would appear
as a matrix . . . ."
— Claude Lévi-Strauss, 1960

Comments Off on The Matrix
Claude Lévi-Strauss
From his “Structure and Form:
Reflections on a Work by Vladimir Propp” *
To maintain. as I have done. that the permutability of contents is not arbitrary amounts to saying that, if the analysis is carried to a sufficiently deep level, behind diversity we will discover constancy. And, of course. the avowed constancy of form must not hide from us that functions are also permutable.
The structure of the folktale as it is illustrated by Propp presents a chronological succession of qualitatively distinct functions. each constituting an independent genre. One can wonder whether—as with dramatis personae and their attributes— Propp does not stop too soon, seeking the form too close to the level of empirical observation. Among the thirty-one functions that he distinguishes, several are reducible to the same function reappearing at different moments of the narrative but after undergoing one or a number of transformations . I have already suggested that this could be true of the false hero (a transformation of the villain), of assigning a difficult task (a transformation of the test), etc. (see p. 181 above), and that in this case the two parties constituting the fundamental tale would themselves be transformations of each other.
Nothing prevents pushing this reduction even further and analyzing each separate partie into a small number of recurrent functions, so that several of Propp’s functions would constitute groups of transformations of one and the same function. We could treat the “violation” as the reverse of the “prohibition” and the latter as a negative transformation of the “injunction.” The “departure” of the hero and his “return” would appear as the negative and positive expressions of the same disjunctive function. The “quest” of the hero (hero pursues someone or something) would become the opposite of “pursuit” (hero is pursued by something or someone), etc.

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). With regard to Propp’s theories my analysis offers another advantage: I can reconcile much better than Propp himself his principle of a permanent order of wondertale elements with the fact that certain functions or groups of functions are shifted from one tale to the next (pp. 97-98. p. 108) If my view is accepted, the chronological succession will come to be absorbed into an atemporal matrix structure whose form is indeed constant. The shifting of functions is then no more than a mode of permutation (by vertical columns or fractions of columns).
These critical remarks are certainly valid for the method used by Propp and for his conclusions. However. it cannot be stressed enough that Propp envisioned them and in several places formulated with perfect clarity the solutions I have just suggested. Let us take up again from this viewpoint the two essential themes of our discussion: constancy of the content (in spite of its permutability) and permutability of functions (in spite of their constancy).
* Translated from a 1960 work in French. It appeared in English as Chapter VIII
of Structural Anthropology, Volume 2 (U. of Chicago Press, 1976). Chapter VIII
was originally published in Cahiers de l’Institut de Science Économique Appliquée ,
No. 9 (Series M, No. 7) (Paris: ISEA, March 1960).
|
See also “Lévi-Strauss” + Formula in this journal.
Some background related to the previous post —

Comments Off on Lévi-Strauss vs. Propp
Wednesday, November 22, 2017
David E. Wellbery on Goethe
From an interview published on 2 November 2017 at
http://literaturwissenschaft-berlin.de/interview-with-david-wellbery/
as later republished in
https://thepointmag.com/2017/dialogue/
irreducible-significance-david-wellbery-literature-goethe-cavell —
The logo at left above is that of The Point .
The menu icon at right above is perhaps better
suited to illustrate Verwandlungslehre .

Comments Off on Goethe on All Souls’ Day
Sunday, October 29, 2017
Comments Off on Sunday in the Park
Related material —
Faust Vivifies Death with Wit and Humor
by April H. N. Yee, Harvard Crimson , Feb. 7, 2008.
See as well all posts now tagged Willow and Mandorla.
Comments Off on Damnation… Or Not?
Tuesday, October 17, 2017
The title is a phrase by Octavio Paz from today's post
"Status Symbols."
Other phrases from a link target in Sunday's post
The Strength at the Centre —
… a single world
In which he is and as and is are one.
See also Four Dots in this journal.

Comments Off on The Movement of Analogy
"Status: Defunct" …
As is now its owner, who reportedly
died at 80 on Sunday, October 15, 2017.
In memoriam —
Excerpts from Log24 posts on Sunday night
and yesterday evening —
.
" … listen: there's a hell
of a good universe next door; let's go"
— e. e. cummings
Some literary background —
"At the point of convergence
the play of similarities and differences
cancels itself out in order that
identity alone may shine forth.
The illusion of motionlessness,
the play of mirrors of the one:
identity is completely empty;
it is a crystallization and
in its transparent core
the movement of analogy
begins all over once again."
— The Monkey Grammarian
by Octavio Paz, translated by Helen Lane
|
Comments Off on Status Symbols
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