The first 12 pages of my 1976 preprint "Diamond Theory" are
now scanned and uploaded. See a slideshow.
For downloading, all 12 pages are combined in a PDF.
Monday, January 1, 2018
Diamond Theory 1976
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Wednesday, December 27, 2017
On Fiction and Mathematics
"There is always an awareness in her fiction
of the subjectivity of perception, and
the kaleidoscopic permutations
that memory can work on reality."
This is from a New York Times article subtitled
"Alice Munro, Nobel Winner, Mines the Inner Lives
of Girls and Women" …
The New York Times article was linked to by Marjorie Senechal
in a Huffington Post article of All Saints' Day 2013.
Further material on kaleidoscopic permutations —
See the Log24 post Symmetry of May 3, 2016.
For further material on mining, see Diamond-Mine:
"SEE HEAR READ" — Walt Disney Productions
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For Day 27 of December 2017
See the 27-part structure of
the 3x3x3 Galois cube
as well as Autism Sunday 2015.
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Tuesday, December 26, 2017
Raiders of the Lost Stone
Two Students of Structure
A comment on Sean Kelly's Christmas Morning column on "aliveness"
in the New York Times philosophy series The Stone —
Diana Senechal's 1999 doctoral thesis at Yale was titled
"Diabolical Structures in the Poetics of Nikolai Gogol."
Her mother, Marjorie Senechal, has written extensively on symmetry
and served as editor-in-chief of The Mathematical Intelligencer .
From a 2013 memoir by Marjorie Senechal —
|
"While I was in Holland my enterprising student assistant at Smith had found, in Soviet Physics – Crystallography, an article by N. N. Sheftal' on tetrahedral penetration twins. She gave it to me on my return. It was just what I was looking for. The twins Sheftal' described had evidently begun as (111) contact twins, with the two crystallites rotated 60o with respect to one another. As they grew, he suggested, each crystal overgrew the edges of the other and proceeded to spread across the adjacent facet. When all was said and done, they looked like they'd grown through each other, but the reality was over-and-around. Brilliant! I thought. Could I apply this to cubes? No, evidently not. Cube facets are all (100) planes. But . . . these crystals might not have been cubes in their earliest stages, when twinning occurred! I wrote a paper on "The mechanism of certain growth twins of the penetration type" and sent it to Martin Buerger, editor of Neues Jarbuch für Mineralogie. This was before the Wrinch symposium; I had never met him. Buerger rejected it by return mail, mostly on the grounds that I hadn't quoted any of Buerger's many papers on twinning. And so I learned about turf wars in twin domains. In fact I hadn't read his papers but I quickly did. I added a reference to one of them, the paper was published, and we became friends.[5]
After reading Professor Sheftal's paper I wrote to him in Moscow; a warm and encouraging correspondence ensued, and we wrote a paper together long distance.[6] Then I heard about the scientific exchanges between the Academies of Science of the USSR and USA. I applied to spend a year at the Shubnikov Institute for Crystallography, where Sheftal' worked. I would, I proposed, study crystal growth with him, and color symmetry with Koptsik. To my delight, I was accepted for an 11-month stay. Of course the children, now 11 and 14, would come too and attend Russian schools and learn Russian; they'd managed in Holland, hadn't they? Diana, my older daughter, was as delighted as I was. We had gone to Holland on a Russian boat, and she had fallen in love with the language. (Today she holds a Ph.D. in Slavic Languages and Literature from Yale.) . . . . |
Philosophy professors and those whose only interest in mathematics
is as a path to the occult may consult the Log24 posts tagged Tsimtsum.
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Monday, December 25, 2017
Every Picture Tells a Story
The movie marquee below
("Batman" and "Lethal Weapon 2")
indicates that the recent film "IT"
is set in the summer of 1989.
The marquee suggests a review. Also . . . .
"… the thing that has shown up every twenty-seven years
or so . . . . It always comes back, you see. It."
— King, Stephen. IT (p. 151). Scribner. Kindle Edition.
Note that the flashback summer in King's book,
1958… plus 27 is 1985… plus 27 is 2012.
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Saturday, December 23, 2017
The Right Stuff
A figure related to the general connecting theorem of Koen Thas —
See also posts tagged Dirac and Geometry in this journal.
Those who prefer narrative to mathematics may, if they so fancy, call
the above Thas connecting theorem a "quantum tesseract theorem ."
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Friday, December 22, 2017
Thursday, December 21, 2017
Wednesday, December 20, 2017
A Snow Ball for Clifford Irving (1930-2017)
William Grimes in The New York Times this evening —
"Clifford Irving, who perpetrated one of the biggest literary hoaxes
of the 20th century in the early 1970s when he concocted a
supposedly authorized autobiography of the billionaire Howard Hughes
based on meetings and interviews that never took place, died on Tuesday
at a hospice facility near his home in Sarasota, Fla. He was 87."
A figure reproduced here on Tuesday —
A related figure —
See too the 1973 Orson Welles film "F for Fake."
Some background on the second figure above —
posts tagged April 8-11, 2016.
Some background on the first figure above —
today's previous post, January 2018 AMS Notices.
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January 2018 AMS Notices
Update of 9:29 PM ET Dec. 20, 2017 —
See in particular, in the above Notices , the article
"Algebraic Structures on Polytopes," by Federico Ardila,
within the 2018 Joint Mathematics Meeting Lecture Sampler.
Related reading:
arXiv:1711.09102v1 [hep-th] 24 Nov 2017,
"Scattering Forms and the Positive Geometry of
Kinematics, Color and the Worldsheet," by
Nima Arkani-Hamed, Yuntao Bai, Song He, Gongwang Yan
(Submitted to the arXiv on 24 Nov. 2017).
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Tuesday, December 19, 2017
A Mythologem for Meletinsky
The word "mythologem" on page 55 of The Burning Fountain
by Philip Wheelwright, revised edition of 1968 (p. 91 in the 1954
edition), suggests a Web search for that word. It was notably often
used in the 1998 English translation of a book by Eleazar Meletinsky
first published in Russian in 1976 —
Meletinsky reportedly died on December 17, 2005.
In his memory, Log24 posts from that date are now tagged Mythologem Day.
"And we may see the meadow in December,
icy white and crystalline" — Johnny Mercer
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Monday, December 18, 2017
Mathematics and Art
From the American Mathematical Society homepage today —
From concinnitasproject.org —
"Concinnitas is the title of a portfolio of fine art prints. . . .
The portfolio draws its name from a word famously used
by the Renaissance scholar, artist, architect, and philosopher
Leon Battista Alberti (1404-1472) to connote the balance of
number, outline, and position (in essence, number, geometry,
and topology) that he believed characterize a beautiful work of art."
The favicon of the Concinnitas Project —
The structure of the Concinnitas favicon —
This structure is from page 15 of
"Diamond Theory," a 1976 preprint —
.
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Wheelwright and the Dance
The page preceding that of yesterday's post Wheelwright and the Wheel —
See also a Log24 search for
"Four Quartets" + "Four Elements".
A graphic approach to this concept:
"The Bounded Space" —
"The Fire, Air, Earth, and Water" —
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Saturday, December 16, 2017
Dagger Definitions (Review)
The previous post suggests a review of
the philosophical concept of universals —
A part of the above-mentioned 2011 "Saturday evening's post" that is
relevant to the illustration at the end of today's previous post —
Note the whatness of Singer's dagger definitions —
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Triptychs
Two readings by James Parker —
From next year’s first Atlantic issue
From last month’s Atlantic issue
“Let’s return to that hillside where Clayton exited his Mercedes.
In the gray light, he climbs the pasture. Halfway up the slope,
three horses are standing: sculpturally still, casually composed
in a perfect triptych of horsitude.”
— James Parker in The Atlantic , Nov. 2017 issue
Logos-related material
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Monday, December 11, 2017
A Diamond Metaphor
For some remarks related to the title, see Black + Algebra + Metaphor.
There is apparently no relationship between Scholze's metaphor
and my own use of the word "diamond" in finite geometry.
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The Diamond Theorem at SASTRA
The following IEEE paper is behind a paywall,
but the first page is now available for free
at deepdyve.com —
For further details on the diamond theorem, see
finitegeometry.org/sc/ or the archived version at . . .
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Sunday, December 10, 2017
Geometry
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) —
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Friday, December 8, 2017
Mythos and Logos
Part I: Black Magician
"Schools of criticism create their own canons, elevating certain texts,
discarding others. Yet some works – Malcolm Lowry’s Under the Volcano
is one of them – lend themselves readily to all critical approaches."
— Joan Givner, review of
A Darkness That Murmured: Essays on Malcolm Lowry and the Twentieth Century
by Frederick Asals and Paul Tiessen, eds.
The Asals-Tiessen book (U. of Toronto Press, 2000) was cited today
by Margaret Soltan (in the link below) as the source of this quotation —
"When one thinks of the general sort of snacky
under-earnest writers whose works like wind-chimes
rattle in our heads now, it is easier to forgive Lowry
his pretentious seriousness, his old-fashioned ambitions,
his Proustian plans, [his efforts] to replace the reader’s
consciousness wholly with a black magician’s."
A possible source, Perle Epstein, for the view of Lowry as black magician —
Part II: Mythos and Logos
Part I above suggests a review of Adam Gopnik as black magician
(a figure from Mythos ) —
|
Tuesday, November 7, 2017
Polarities and Correlation
|
— and of an opposing figure from Logos ,
Paul B. Yale, in the references below:
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Friday, December 1, 2017
The Architect and the Matrix
In memory of Yale art historian Vincent Scully, who reportedly
died at 97 last night at his home in Lynchburg, Va., some remarks
from the firm of architect John Outram and from Scully —
Update from the morning of December 2 —
The above 3×3 figure is of course not unrelated to
the 4×4 figure in The Matrix for Quantum Mystics:
.
See as well Tsimtsum in this journal.
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Thursday, November 30, 2017
The Matrix for Quantum Mystics
Scholia on the title — See Quantum + Mystic in this journal.
"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.
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Wednesday, November 29, 2017
Definitions
See also Inscape in this journal and, for a related Chapel Hill thesis,
the post Kummer and Dirac.
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Monday, November 27, 2017
The Golay Code via Witt’s Construction
Hansen, Robert Peter, "Construction and Simplicity of
the Large Mathieu Groups" (2011). Master's Theses. 4053.
http://scholarworks.sjsu.edu/etd_theses/4053.
See also The Matrix Meets the Grid (Log24, Nov. 24).
More generally, see SPLAG in this journal.
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Friday, November 24, 2017
The Matrix Meets the Grid
The Matrix —
The Grid —
Picturing the Witt Construction —
"Read something that means something." — New Yorker ad
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Thursday, November 23, 2017
Lévi-Strauss vs. Propp
| Claude Lévi-Strauss
From his “Structure and Form: 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 |
See also “Lévi-Strauss” + Formula in this journal.
Some background related to the previous post —
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Wednesday, November 22, 2017
Goethe on All Souls’ Day
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
The logo at left above is that of The Point .
The menu icon at right above is perhaps better
suited to illustrate Verwandlungslehre .
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The Prize
Bernd Sturmfels to Receive 2018
George David Birkhoff Prize in Applied Mathematics
— American Mathematical Society on
Monday, November 20th, 2017
See also Sturmfels and Birkhoff + Geometry in this journal.
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“Design is how it works” — Steve Jobs
News item from this afternoon —
The above phrase "mapping systems" suggests a review
of my own very different "map systems." From a search
for that phrase in this journal —
See also "A Four-Color Theorem: Function Decomposition
Over a Finite Field."
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Sunday, November 19, 2017
Galois Space
This is a sequel to yesterday's post Cube Space Continued.
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Review
"New to the series are the Trials of the Gods events
that pit players against Ancient Egyptian gods."
— Review of the new game Assassin's Creed: Origins
"How much story do you want?" — George Balanchine
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Saturday, November 18, 2017
Cube Space Continued
James Propp in the current Math Horizons on the eightfold cube —
For another puerile approach to the eightfold cube,
see Cube Space, 1984-2003 (Oct. 24, 2008).
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Thursday, November 16, 2017
Lost Horizon
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Sunday, November 12, 2017
Review
|
Sunday, October 29, 2017
File System… Unlocked
|
See as well Chloë Grace Moretz portraying a schoolgirl problem.
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Friday, November 10, 2017
Annals of Rarefied Scholarship
From Cambridge Core, suggested by a reference to
that website in the previous post and by the following
bibliographic data . . .
https://doi.org/10.1017/fmp.2016.5
Downloaded from https://www.cambridge.org/core
on 10 Nov 2017 at 19:06:19
See Conwell + Princeton in this journal.
Related art —
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Tuesday, November 7, 2017
Polarities and Correlation
"Read something that means something."
— New Yorker ad
.
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Sunday, November 5, 2017
Trinity
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Saturday, November 4, 2017
Seven-Cycles in an Octad
Figures from a search in this journal for Springer Knight
and from the All Souls' Day post The Trojan Pony —
For those who prefer pure abstraction to the quasi-figurative
1985 seven-cycle above, a different 7-cycle for M24 , from 1998 —
Compare and contrast with my own "knight" labeling
of a 4-row 2-column array (an M24 octad, in the system
of R. T. Curtis) by the 8 points of the projective line
over GF(7), from 2008 —
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Thursday, November 2, 2017
The Trojan Pony
From a search in this journal for Springer Knight —
Related material from Academia —
See also Log24 posts from the above "magic" date,
December 4, 2014, now tagged The Pony Argument.
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Wednesday, November 1, 2017
Many Years
From Tony Phillips's American Mathematical Society column
for November 2017 —
" It is significant that the authors chose to place this announcement
of their results not in a mathematics journal but in one aimed at a
much larger scientific audience; their writing is appropriately
expository, especially in the introduction. Nature itself ran an
assessment of the paper in their 'News and Views' section,
October 4: 'Mathematics: A pariah finds a home,' by Terry Gannon.
Gannon sets the stage, again in terms suitable for wide consumption,
and sketches out the story. He ends 'It is always difficult to gauge
the importance of a mathematical result without the hindsight
that many years brings. Nevertheless, Duncan et al. have shown us
a door. Whether it is to a new closet, house or world, we cannot yet say,
but the results are certainly unexpected, and no one will think of
the pariahs in the same way again.' "
See as well Log24 on the above date — Text and Context.
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Cameron on All Saints’ Day
"Nowdays, Halloween involves plastic figures of ghosts and bats
bought from the supermarket; this is driven by commerce and
in some people’s view is an American import. But it is clear that
this time of year was traditionally regarded as one where the barrier
between this world and the other was low, and supernatural
manifestations were to be expected."
Remarks related to another "barrier" and vértigo horizontal —
See also a search for Horizon + "Western Australia" in this journal.
From that search: A sort of horizon, a "line at infinity," that is perhaps
more meaningful to most Cameron readers than the above remarks
by Borges —
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Monday, October 30, 2017
Plato and His Modern Rivals
The previous post's Lewis Carroll cover,
modified to illustrate Plato's diamond —
See also "To Forge
a Head" (Oct. 27).
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Sunday, October 29, 2017
Rivals
The passage from Lewis Carroll's Euclid and His Modern Rivals
in the previous post suggests two illustrations —
Click the Trudeau book for related Log24 posts.
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File System… Unlocked
Logo from the above webpage —
See also the similar structure of the eightfold cube, and …
Related dialogue from the new film "Unlocked" —
1057
01:31:59,926 –> 01:32:01,301
Nice to have you back, Alice.
1058
01:32:04,009 –> 01:32:05,467
Don't be a stranger.
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Saturday, October 28, 2017
Lowell Brown at Vanity Fair
A sequel to the post CP is for Consolation Prize (Sept. 3, 2016)
An image from Log24 on this date last year:
A recent comment on a discussion of CP symmetry —
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Just the Facts
The New Yorker on the recent film "The Square" —
"It’s an aesthetic that presents,
so to speak, just the facts,
as if the facts themselves weren’t
deeply layered with living history
and crisscrossed with vectors
of divergent ideas and ideals."
— Richard Brody, Thursday, Oct. 26, 2017
For other images deeply layered and crisscrossed ,
see Geometry of the I Ching.
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Friday, October 27, 2017
To Forge a Head
The title was suggested by a 2014 Vanity Fair piece
by James Toback (Harvard '66).
"He squinted at this vision of a Qualityless world for a while,
conjured up more details, thought about it, and then squinted
some more and thought some more and then finally circled
back to where he was before.
Squareness.
That's the look. That sums it. Squareness. When you subtract
quality you get squareness. Absence of Quality is the essence
of squareness."
— Robert M. Pirsig, Zen and the Art of Motorcycle Maintenance
And when you add quality?
A related Zen joke from Final Club (June 19, 2017) —
.
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Thursday, October 26, 2017
A Center
This post was suggested by a New York Times obituary this evening —
"Tom Mathews, Promoter of Liberal Causes and Candidates, Dies at 96."
Mathews reportedly died on October 14, 2017.
"Mr. Mathews and his business partner Roger Craver 'dreamed for years
of finding the perfect citizen-candidate,' the authors wrote, 'a man or
woman of the center-left with a feel for issues, a history of independence,
a winning television manner and, most important of all, a center — a core
of beliefs more important to him or her than getting elected.'
Dream on.
From the date of Mathews's death:
Posts now tagged A Center for Krauss —
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Wednesday, October 25, 2017
The Palo Alto Edge
From Stanford — The death on October 9, 2017, of a man who
“always wanted to be at the most cutting of cutting-edge technology.”
Related material from Log24 on April 26, 2017 —
A sketch, adapted from Girl Scouts of Palo Alto —
Click the sketch for further details.
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Tuesday, October 24, 2017
Visual Insight
The most recent post in the "Visual Insight" blog of the
American Mathematical Society was by John Baez on Jan. 1, 2017 —
A visually related concept — See Solomon's Cube in this journal.
Chronologically related — Posts now tagged New Year's Day 2017.
Solomon's cube is the 4x4x4 case of the diamond theorem —
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Monday, October 23, 2017
Plan 9 Continues
Click for some background —
Another approach, for Dan Brown fans —
In the following passage, Brown claims that an eight-ray star
with arrowheads at the rays' ends is "the mathematical symbol for
entropy." Brown may have first encountered this symbol at a
questionable "Sacred Science" website. Wikipedia discusses
some even less respectable uses of the symbol.
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Saturday, October 21, 2017
Crimson Algebra
"Category theory has become the central gateway
through which to learn pure mathematics."
— David Spivak, Harvard Math Table, Oct. 24, 2017
— The New Yorker , issue of October 23, 2017
See as well posts tagged Death Warmed Over.
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Thursday, October 19, 2017
Design Grammar***
The elementary shapes at the top of the figure below mirror
the looking-glass property of the classical Lo Shu square.
The nine shapes at top left* and their looking-glass reflection
illustrate the looking-glass reflection relating two orthogonal
Latin squares over the three digits of modulo-three arithmetic.
Combining these two orthogonal Latin squares,** we have a
representation in base three of the numbers from 0 to 8.
Adding 1 to each of these numbers yields the Lo Shu square.
* The array at top left is from the cover of
Wonder Years:
Werkplaats Typografie 1998-2008.
** A well-known construction.
*** For other instances of what might be
called "design grammar" in combinatorics,
see a slide presentation by Robin Wilson.
No reference to the work of Chomsky is
intended.
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Graphic Design: Fast Forward
Typographical: »
Eightfold Cube:
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Wednesday, October 18, 2017
Three Small Grids
An earlier post today, now tagged "Three Small Magic Squares,"
suggests a review of a post from October 25 three years ago
that contains the following figure —
Fans of the October Revolution may enjoy a passage
by Rosalind Krauss on grids:
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Dürer for St. Luke’s Day
Structure of the Dürer magic square
16 3 2 13
5 10 11 8 decreased by 1 is …
9 6 7 12
4 15 14 1
15 2 1 12
4 9 10 7
8 5 6 11
3 14 13 0 .
Base 4 —
33 02 01 30
10 21 22 13
20 11 12 23
03 32 31 00 .
Two-part decomposition of base-4 array
as two (non-Latin) orthogonal arrays —
3 0 0 3 3 2 1 0
1 2 2 1 0 1 2 3
2 1 1 2 0 1 2 3
0 3 3 0 3 2 1 0 .
Base 2 –
1111 0010 0001 1100
0100 1001 1010 0111
1000 0101 0110 1011
0011 1110 1101 0000 .
Four-part decomposition of base-2 array
as four affine hyperplanes over GF(2) —
1001 1001 1100 1010
0110 1001 0011 0101
1001 0110 0011 0101
0110 0110 1100 1010 .
— Steven H. Cullinane,
October 18, 2017
See also recent related analyses of
noted 3×3 and 5×5 magic squares.
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Tuesday, October 17, 2017
Plan 9 Continues
See also Holy Field in this journal.
Some related mathematics —
Analysis of the Lo Shu structure —
Structure of the 3×3 magic square:
4 9 2
3 5 7 decreased by 1 is …
8 1 6
3 8 1
2 4 6
7 0 5
In base 3 —
10 22 01
02 11 20
21 00 12
As orthogonal Latin squares
(a well-known construction) —
1 2 0 0 2 1
0 1 2 2 1 0
2 0 1 1 0 2 .
— Steven H. Cullinane,
October 17, 2017
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The Movement of Analogy
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.
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Status Symbols
"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 by Octavio Paz, translated by Helen Lane
|
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Monday, October 16, 2017
Highway 61 Revisited
"God said to Abraham …." — Bob Dylan, "Highway 61 Revisited"
Related material —
-
"Inner Truth" in this journal
-
"On Linguistic Creation," June 25, 1999
-
"Ultra Super Magic Squares of 5×5" at
http://mathsforeurope.digibel.be/magic.htm .
See as well Charles Small, Harvard '64,
"Magic Squares over Fields" —
— and Conway-Norton-Ryba in this journal.
Some remarks on an order-five magic square over GF(52):
on the numbers 0 to 24:
22 5 18 1 14
3 11 24 7 15
9 17 0 13 21
10 23 6 19 2
16 4 12 20 8
Base-5:
42 10 33 01 24
03 21 44 12 30
14 32 00 23 41
20 43 11 34 02
31 04 22 40 13
Regarding the above digits as representing
elements of the vector 2-space over GF(5)
(or the vector 1-space over GF(52)) …
All vector row sums = (0, 0) (or 0, over GF(52)).
All vector column sums = same.
Above array as two
orthogonal Latin squares:
4 1 3 0 2 2 0 3 1 4
0 2 4 1 3 3 1 4 2 0
1 3 0 2 4 4 2 0 3 1
2 4 1 3 0 0 3 1 4 2
3 0 2 4 1 1 4 2 0 3
— Steven H. Cullinane,
October 16, 2017
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Sunday, October 15, 2017
Saturday Night Not-So-Live
Hillel Italie at AP News —
"Richard Wilbur, the Pulitzer Prize-winning poet and translator
who intrigued and delighted generations of readers and theatergoers
through his rhyming editions of Moliere and his own verse on memory,
writing and nature, died. He was 96.
Wilbur died Saturday night [Oct. 14, 2017] in Belmont, Massachusetts,
with his family by his side, according to friend and fellow poet, Dana Gioia."
Images from the post "Center" in this journal on Saturday afternoon —
.
"Things fall apart; the centre cannot hold"
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Saturday, October 14, 2017
Center
"To get inside the systems of this work,
whether LeWitt's or Judd's or Morris's,
is precisely to enter
a world without a center,
a world of substitutions and transpositions
nowhere legitimated by the revelations
of a transcendental subject. This is the strength
of this work, its seriousness, and its claim to modernity."
"The center of
the quaternion group,
Q8 = {1, −1, i, −i, j, −j, k, −k} ,
is {1, −1}."
Illustration from a post of Feb. 3, 2011 —
.
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In Principio:
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Friday, October 13, 2017
Speak, Memra
The above was suggested by a Log24 review of October 13, 2002,
which in turn suggested a Log24 search for Carousel that yielded
(from Bloomsday Lottery) —
See as well Asimov's "prime radiant," and an illustration
of the number 13 as a radiant prime …
"The Prime Radiant can be adjusted to your mind,
and all corrections and additions can be made
through mental rapport. There will be nothing to
indicate that the correction or addition is yours.
In all the history of the Plan there has been no
personalization. It is rather a creation of all of us
together. Do you understand?"
"Yes, Speaker!"
— Isaac Asimov,
Second Foundation , Ch. 8: Seldon's Plan
"Before time began, there was the Cube."
— Optimus Prime
See also Transformers in this journal.
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Thursday, October 12, 2017
Wednesday, October 11, 2017
Tuesday, October 10, 2017
Another 35-Year Wait
The title refers to today's earlier post "The 35-Year Wait."
A check of my activities 35 years ago this fall, in the autumn
of 1982, yields a formula I prefer to the nonsensical, but famous,
"canonical formula" of Claude Lévi-Strauss.
My "inscape" formula, from a note of Sept. 22, 1982 —
S = f ( f ( X ) ) .
Some mathematics from last year related to the 1982 formula —
See also Inscape in this journal and posts tagged Dirac and Geometry.
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Dueling Formulas
Continued from the previous post and from posts
now tagged Dueling Formulas —
The four-diamond formula of Jung and
the four-dot "as" of Claude Lévi-Strauss:
Simplified versions of the diamonds and the dots —
::
I prefer Jung. For those who prefer Lévi-Strauss —
First edition, Cornell University Press, 1970.
A related tale — "A Meaning, Like."
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The 35-Year Wait
From the Web this morning —
A different 35-year wait:
A monograph of August 1976 —
Thirty-five years later, in a post of August 2011, "Coordinated Steps" —
"SEE HEAR READ" — Walt Disney Productions
Some other diamond-mine productions —
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Monday, October 9, 2017
Still Point for a Dance
"At the still point of the turning world. Neither flesh nor fleshless;
Neither from nor towards; at the still point, there the dance is,
But neither arrest nor movement. And do not call it fixity,
Where past and future are gathered. Neither movement from nor towards,
Neither ascent nor decline. Except for the point, the still point,
There would be no dance, and there is only the dance."
— T. S. Eliot, Four Quartets
See also a recurrent image
from this journal —
.
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Sunday, October 8, 2017
Patterns at Oxford
See also W. Tecumseh Fitch in this journal.
From the publisher (click to enlarge) —
The above publication date, 01 September 2015, suggests a review
of posts now tagged A Mirror Darkly.
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Saturday, October 7, 2017
Four Walls
From a web page quoted here on the
Feast of St. Louis, 2003 —
|
Case 9 of Hekiganroku:
A monk asked Joshu,
Joshu said, Setcho's Verse:
Its intention concealed,
Setcho (980-1052), |
The epigraph to Lefebvre's
The Production of Space (1974, translated in 1991) —
(Adapted from a prose poem, "La Higuera ,"
in ¿Águila o Sol? (1951).)
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Broken Symmetries
From posts tagged Design Deadline —
A quotation from Lefebvre:
"… an epoch-making event so generally ignored
that we have to be reminded of it at every moment.
The fact is that around 1910 a certain space was shattered…
the space… of classical perspective and geometry…."— Page 25 of The Production of Space
(Blackwell Publishing, 1991)
This suggests, for those who prefer Harvard's past glories
to its current state, a different Raum from the Zeit 1910.
In January 1910 Annals of Mathematics , then edited at Harvard,
published George M. Conwell's "The 3-space PG (3, 2) and Its Group."
This paper, while perhaps neither epoch-making nor shattering, has
a certain beauty. For some background, see this journal on February 24, 2009.
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Wednesday, October 4, 2017
Text and Context
Text —
"A field is perhaps the simplest algebraic structure we can invent."
— Hermann Weyl, 1952
Context —
See also yesterday's Personalized Book Search.
|
Full text of Symmetry – Internet Archive — https://archive.org/details/Symmetry_482
A field is perhaps the simplest algebraic 143 structure |
From a Log24 search for Mathematics+Nutshell —
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Tuesday, October 3, 2017
Show Us Your Wall
From Monday morning's post Advanced Study —
"Mathematical research currently relies on
a complex system of mutual trust
based on reputations."
— The late Vladimir Voevodsky,
Institute for Advanced Study, Princeton,
The Institute Letter , Summer 2014, p. 8
Related news from today's online New York Times —
A heading from the above screenshot: "SHOW US YOUR WALL."
This suggests a review of a concept from Galois geometry —
(On the wall — a Galois-geometry inscape .)
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Personalized Book Search
Click to enlarge —
The quote from Hermann Weyl on which the above search is based
is from a search within this journal for Springer + Knight.
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Friday, September 29, 2017
Principles Before Personalities*
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Thursday, September 28, 2017
Core
From the New York Times Wire last night —
"Mr. Hefner … styled himself as an emblem
of the sexual revolution."
From a Log24 post on September 23 —
A different emblem related to other remarks in the above Sept. 23 post —
(On the wall — a Galois-geometry inscape .)
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Saturday, September 23, 2017
The Turn of the Year
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The Turn of the Frame
"With respect to the story's content, the frame thus acts
both as an inclusion of the exterior and as an exclusion
of the interior: it is a perturbation of the outside at the
very core of the story's inside, and as such, it is a blurring
of the very difference between inside and outside."
— Shoshana Felman on a Henry James story, p. 123 in
"Turning the Screw of Interpretation,"
Yale French Studies No. 55/56 (1977), pp. 94-207.
Published by Yale University Press.
See also the previous post and The Galois Tesseract.
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Tuesday, September 19, 2017
Friday, September 15, 2017
Space Art
Silas in "Equals" (2015) —
Ever since we were kids it's been drilled into us that …
Our purpose is to explore the universe, you know.
Outer space is where we'll find …
… the answers to why we're here and …
… and where we come from.
Related material —
See also Galois Space in this journal.
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Thursday, September 14, 2017
Found …
( A sequel to the previous post, Lost )
From a link, "A Little Boy and a Little Girl," found in a Log24
search for Andersen + Atlantic —
"A few flakes of snow were falling, and one of them, rather larger
than the rest, alighted on the edge of one of the flower boxes.
This snow-flake grew larger and larger, till at last it became
the figure of a woman, dressed in garments of white gauze,
which looked like millions of starry snow-flakes linked together.
She was fair and beautiful, but made of ice—
shining and glittering ice." — "The Snow Queen"
Related material —
Analogue of the little boy from "The Snow Queen" in "Equals" (2015) —
"Nice piece of ice." — Brendan Fraser in
"The Mummy: Tomb of the Dragon Emperor" (2008).
See also the concept that everything adds up to nothing in
"The Zero Theorem" (2013) …
… and the Conway-Norton-Ryba theorem (2017).
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Group Actions
From this journal on the above publication date —
Related material — Geometric Group Theory in this journal.
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Wednesday, September 13, 2017
Summer of 1984
The previous two posts dealt, rather indirectly, with
the notion of "cube bricks" (Cullinane, 1984) —
Group actions on partitions —
Cube Bricks 1984 —
Another mathematical remark from 1984 —
For further details, see Triangles Are Square.
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Tuesday, September 12, 2017
Think Different
The New York Times online this evening —
"Mr. Jobs, who died in 2011, loomed over Tuesday’s
nostalgic presentation. The Apple C.E.O., Tim Cook,
paid tribute, his voice cracking with emotion, Mr. Jobs’s
steeple-fingered image looming as big onstage as
Big Brother’s face in the classic Macintosh '1984' commercial."
Review —
|
Thursday, September 1, 2011
How It Works
|
See also 1984 Bricks in this journal.
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Chin Music
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Goals
"Truth and clarity remained his paramount goals…"
— Benedict Nightingale in today's online New York TImes on an
English theatre director, founder of the Royal Shakespeare Company,
who reportedly died yesterday at 86.
See also Paramount in this journal.
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Monday, September 11, 2017
New Depth
A sentence from the New York Times Wire discussed in the previous post —
"Through characters like Wolverine and Swamp Thing,
he helped bring a new depth to his art form."
For Wolverine and Swamp Thing in posts related to a different
art form — geometry — see …
- Sunday School (July 6, 2014) and
- Like Decorations in a Cartoon Graveyard (March 25, 2017).
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Saturday, September 9, 2017
How It Works
Del Toro and the History of Mathematics ,
Or: Applied Bullshit Continues
For del Toro —
For the history of mathematics —
|
Thursday, September 1, 2011
How It Works
|
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Tuesday, September 5, 2017
Annals of Critical Epistemology
"But unlike many who left the Communist Party, I turned left
rather than right, and returned—or rather turned for the first time—
to a critical examination of Marx's work. I found—and still find—
that his analysis of capitalism, which for me is the heart of his work,
provides the best starting point, the best critical tools, with which—
suitably developed—to understand contemporary capitalism.
I remind you that this year is also the sesquicentennial of the
Communist Manifesto , a document that still haunts the capitalist world."
— From "Autobiographical Reflections," a talk given on June 5, 1998, by
John Stachel at the Max Planck Institute for the History of Science in Berlin
on the occasion of a workshop honoring his 70th birthday,
"Space-Time, Quantum Entanglement and Critical Epistemology."
From a passage by Stachel quoted in the previous post —
From the source for Stachel's remarks on Weyl and coordinatization —
Note that Stachel distorted Weyl's text by replacing Weyl's word
"symbols" with the word "quantities." —
This replacement makes no sense if the coordinates in question
are drawn from a Galois field — a field not of quantities , but rather
of algebraic symbols .
"You've got to pick up every stitch… Must be the season of the witch."
— Donovan song at the end of Nicole Kidman's "To Die For"
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Florence 2001
Or: Coordinatization for Physicists
This post was suggested by the link on the word "coordinatized"
in the previous post.
I regret that Weyl's term "coordinatization" perhaps has
too many syllables for the readers of recreational mathematics —
for example, of an article on 4×4 magic squares by Conway, Norton,
and Ryba to be published today by Princeton University Press.
Insight into the deeper properties of such squares unfortunately
requires both the ability to learn what a "Galois field" is and the
ability to comprehend seven-syllable words.
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Sunday, September 3, 2017
Broomsday Revisited
Ivars Peterson in 2000 on a sort of conceptual art —
" Brill has tried out a variety of grid-scrambling transformations
to see what happens. Aesthetic sensibilities govern which
transformation to use, what size the rectangular grid should be,
and which iteration to look at, he says. 'Once a fruitful
transformation, rectangle size, and iteration number have been
found, the artist is in a position to create compelling imagery.' "
— "Scrambled Grids," August 28, 2000
Or not.
If aesthetic sensibilities lead to a 23-cycle on a 4×6 grid, the results
may not be pretty —
From "Geometry of the 4×4 Square."
See a Log24 post, Noncontinuous Groups, on Broomsday 2009.
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Saturday, September 2, 2017
A Touchstone
This post was suggested by the names* (if not the very abstruse
concepts ) in the Aug. 20, 2013, preprint "A Panoramic Overview
of Inter-universal 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.
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Friday, September 1, 2017
Krell’s Journal
See references to Krell in this journal.
IMDb > Chief Quinn (Character)
From Forbidden Planet (1956) —
Chief Engineer Quinn: I'll bet any quantum mechanic
in the service would give the rest of his life to fool around
with this gadget.
Related material from the above search for Krell —
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Thursday, August 31, 2017
Autistic Enchantments
Log24 on January 31, 2015 —
|
The 1913 puzzle from above, claiming priority —
A more sophisticated puzzle related to the previous post —
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A Conway-Norton-Ryba Theorem
In a book to be published Sept. 5 by Princeton University Press,
John Conway, Simon Norton, and Alex Ryba present the following
result on order-four magic squares —
A monograph published in 1976, “Diamond Theory,” deals with
more general 4×4 squares containing entries from the Galois fields
GF(2), GF(4), or GF(16). These squares have remarkable, if not
“magic,” symmetry properties. See excerpts in a 1977 article.
See also Magic Square and Diamond Theorem in this journal.
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Wednesday, August 30, 2017
Sunday, August 27, 2017
Black Well
The “Black” of the title refers to the previous post.
For the “Well,” see Hexagram 48.
Related material —
The Galois Tesseract and, more generally, Binary Coordinate Systems.
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Saturday, August 26, 2017
Aesthetic Distance
Naive readers may suppose that this sort of thing is
related to what has been dubbed "geometric group theory."
It is not. See posts now tagged Aesthetic Distance.
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Friday, August 25, 2017
Two Paths for the Impersonal Essay
The above is a variation on a title from last night's post By Degrees.
The Literary Path —
The Hollywood Path —
Further remarks on algebra and space —
See as well the above image in yesterday's post Maori Chess, Vol. 2.
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Thursday, August 24, 2017
Maori Chess, Vol. 2
This just in …
"Genesis Potini died of a heart attack aged 46
on the 15th August 2011."
The 15th of August in New Zealand overlapped
the 14th of August in the U.S.A.
From a Log24 post, "Sunday Review," on August 14, 2011 —
Part II (from "Marshall, Meet Bagger," July 29):
"Time for you to see the field."
For further details, see the 1985 note
"Generating the Octad Generator."
McLuhan was a Toronto Catholic philosopher.
For related views of a Montreal Catholic philosopher,
see the Saturday evening post.
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Wednesday, August 23, 2017
The Diamond Theorem in Vancouver
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Monday, August 21, 2017
Thursday, August 17, 2017
For Time Cube Fans
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Tuesday, August 15, 2017
Every Picture Tells a Story
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Sunday, August 13, 2017
Compare and Contrast
From The Atlantic , September 2017 issue, online —
"How America Lost Its Mind," by former Harvard Lampoon
writer Kurt Andersen —
The Atlantic 's embedded Google ad for "Quantum Space Elements"
is, by the way, completely unrelated to similar-sounding work on
models of space in finite geometry (cf. tsimtsum ) . . .
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Logos
In memoriam —
Zadeh is known for the unfortunate phrase "fuzzy logic."
Not-so-fuzzy related material —
“Lord Arglay had a suspicion that the Stone would be
purely logical. Yes, he thought, but what, in that sense,
were the rules of its pure logic?”
—Many Dimensions (1931), by Charles Williams
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Friday, August 11, 2017
Symmetry’s Lifeboat
A post suggested by the word tzimtzum (see Wednesday)
or tsimtsum (see this morning) —
Lifeboat from the Tsimtsum in Life of Pi —
Another sort of tsimtsum, contracting infinite space to a finite space —
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Wednesday, August 9, 2017
Implosion
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Tuesday, August 8, 2017
Wednesday, August 2, 2017
Cold and Unforgiving Erudition
"It's clear that you believe nothing will ever outrank
your cold and unforgiving erudition,
however, everything you think is based,
even at the most basic neuronal level,
on the way you connect a long line of dots."
— Mary Jo Bang, "The Game of Roles" (Summer 2016)
See as well posts tagged Dots in this journal.
Related material at a basic level —
See also History of Geometry in this journal.
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Sunday, July 30, 2017
Sermon: MS R I
From Solomon's Cube —
"Here MSRI, an acronym for Mathematical Sciences Research Institute,
is pronounced 'Misery.' See Stephen King [and] K.C. Cole . . . ."
From a manuscript by Mikhail Gromov cited yesterday in MSRI Program —
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