See also Clifford in this journal, in particular
The Matrix for Quantum Mystics
(Log24, St. Andrew's Day, 2017).
Sunday, January 13, 2019
The Clifford Narrative
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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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Saturday, May 17, 2025
Annals of Metadata: Archiving the Archivist
Related cinematic meditation —
A more traditional image from this journal
on the above dies natalis — April 10, 2025 —
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Tuesday, September 17, 2024
Six to Nine, My Dear Watson … Continues.
Continued from August 9, 2024.
Today 's previous post was on Ch. 6 of Selig's Geometric Fundamentals.
Ch. 9 is on Clifford Algebra.
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Wednesday, July 3, 2024
Vedic Carnival: “Hey Rubik!” *
The Moolakaprithi Cube (as opposed to Rubik's Moola Cube ) —
"The key to these connections lies in a 3 x 3 x 3 cube, which
in Vedic Physics, forms the Moolaprakriti, a key component of
the Substratum, the invisible black hole form of matter."
— viXra.org, "Clifford Clock and the Moolakaprithi Cube"
* See Wikipedia.
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Wednesday, December 6, 2023
Das Wiener Meisterstück
|
Introduction In the present article, the research work of many years is summarized in an interim report concerning the connection between logic, space, time and matter. The author always had in mind two things, namely 1. The discovery/construction of an interface between matter and mind, and 2. some entry points for the topos view that concern graphs, grade rotations and contravariant involutions in geometric Boolean lattices. In this part of the MiTopos theme I follow the historic approach to mathematical physics and remain with the Clifford algebra of the Minkowski space. It turns out that this interface is a basic morphogenetic structure inherent in both matter and thought. It resides in both oriented spaces and logic, and most surprisingly is closely linked to the symmetries of elementary particle physics.
— "MiTopos: Space Logic I," |
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Sunday, August 1, 2021
The Savage Sixteens
"Savage ('wild,' 'undomesticated') modes of thought are primary
in human mentality. They are what we all have in common."
— "The Cerebral Savage: On the Work of Claude Lévi-Strauss,"
by Clifford Geertz (Encounter, vol. 28 no. 4, April 1967, pp. 25-32)
For more Geertz and some related art, see The Kaleidoscope Puzzle,
which lets you picture twin sixteens .
"Can you imagine the mathematical possibilities?"
— Line from "Annie Hall" (1977)
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Wednesday, July 17, 2019
The Artsy Quantum Realm
|
arXiv.org > quant-ph > arXiv:1905.06914 Quantum Physics Placing Kirkman's Schoolgirls and Quantum Spin Pairs on the Fano Plane: A Rainbow of Four Primary Colors, A Harmony of Fifteen Tones J. P. Marceaux, A. R. P. Rau (Submitted on 14 May 2019) A recreational problem from nearly two centuries ago has featured prominently in recent times in the mathematics of designs, codes, and signal processing. The number 15 that is central to the problem coincidentally features in areas of physics, especially in today's field of quantum information, as the number of basic operators of two quantum spins ("qubits"). This affords a 1:1 correspondence that we exploit to use the well-known Pauli spin or Lie-Clifford algebra of those fifteen operators to provide specific constructions as posed in the recreational problem. An algorithm is set up that, working with four basic objects, generates alternative solutions or designs. The choice of four base colors or four basic chords can thus lead to color diagrams or acoustic patterns that correspond to realizations of each design. The Fano Plane of finite projective geometry involving seven points and lines and the tetrahedral three-dimensional simplex of 15 points are key objects that feature in this study. Comments:16 pages, 10 figures Subjects:Quantum Physics (quant-ph) Cite as:arXiv:1905.06914 [quant-ph] (or arXiv:1905.06914v1 [quant-ph] for this version) Submission history
From: A. R. P. Rau [view email] |
See also other posts tagged Tetrahedron vs. Square.
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Tuesday, July 16, 2019
Schoolgirl Space for Quantum Mystics
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Monday, March 11, 2019
Overarching Metanarratives
See also "Overarching + Tesseract" in this journal. From the results
of that search, some context for the "inscape" of the previous post —
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Sunday, January 13, 2019
Into the Upside Down
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Saturday, December 22, 2018
Cremona-Richmond
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 —
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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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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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Algebra
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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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Friday, June 30, 2017
Hurriedly Put Together
The previous post quoted one theologian on a book
by another theologian, saying its tone "is patronizing
and its arguments are hurriedly put together."
For a more leisurely sort of argument, see a 1995* remark
by a mathematician, Ronald Shaw, quoted here on the morning
of Tuesday, June 27, in an update at the end of the previous day's
post "Upgrading to Six" —
". . . recall the notions of Eddington (1936) . . . ."
* In "Finite Geometry, Dirac Groups and the
Table of Real Clifford Algebras," pages 59-99 of
R. Ablamowicz and P. Lounesto (eds.),
Clifford Algebras and Spinor Structures ,
Kluwer Academic Publishers, 1995.
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Monday, June 26, 2017
Upgrading to Six
This post was suggested by the previous post — Four Dots —
and by the phrase "smallest perfect" in this journal.
Related material (click to enlarge) —
Detail —
From the work of Eddington cited in 1974 by von Franz —
See also Dirac and Geometry and Kummer in this journal.
Updates from the morning of June 27 —
Ron Shaw on Eddington's triads "associated in conjugate pairs" —
For more about hyperbolic and isotropic lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.
For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.
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Tuesday, September 20, 2016
Savage Logic
From "The Cerebral Savage," by Clifford Geertz —
(Encounter, Vol. 28 No. 4 (April 1967), pp. 25-32.)
From http://www.diamondspace.net/about.html —
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Sunday, July 24, 2016
Point Omega …
In this post, "Omega" denotes a generic 4-element set.
For instance … Cullinane's
![Logo for 'Elements of Finite Geometry'](http://m759.net/wordpress/?s="Elements-logo.gif"){kind=logo}
or Schmeikal's
.
The mathematics appropriate for describing
group actions on such a set is not Schmeikal's
Clifford algebra, but rather Galois's finite fields.
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Saturday, July 23, 2016
But Seriously …
Those who want a serious approach to the mathematics
of Clifford algebras — via finite geometry, the natural setting
of the four-group of the previous post — should consult
"Finite Geometry, Dirac Groups and the Table of
Real Clifford Algebras," by Ron Shaw (1995).
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Monday, November 23, 2015
Dirac and Line Geometry
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.
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Friday, November 20, 2015
Anticommuting Dirac Matrices as Skew Lines
(Continued from November 13)
The work of Ron Shaw in this area, ca. 1994-1995, does not
display explicitly the correspondence between anticommutativity
in the set of Dirac matrices and skewness in a line complex of
PG(3,2), the projective 3-space over the 2-element Galois field.
Here is an explicit picture —
References:
Arfken, George B., Mathematical Methods for Physicists , Third Edition,
Academic Press, 1985, pages 213-214
Cullinane, Steven H., Notes on Groups and Geometry, 1978-1986
Shaw, Ron, "Finite Geometry, Dirac Groups, and the Table of
Real Clifford Algebras," undated article at ResearchGate.net
Update of November 23:
See my post of Nov. 23 on publications by E. M. Bruins
in 1949 and 1959 on Dirac matrices and line geometry,
and on another author who gives some historical background
going back to Eddington.
Some more-recent related material from the Slovak school of
finite geometry and quantum theory —
The matrices underlying the Saniga paper are those of Pauli, not
those of Dirac, but these two sorts of matrices are closely related.
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Thursday, November 19, 2015
Highlights of the Dirac-Mathieu Connection
For the connection of the title, see the post of Friday, November 13th, 2015.
For the essentials of this connection, see the following two documents —
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Sunday, November 15, 2015
The Diamond and the Cube
Anyone who clicked on the Dirac search at the end of
the previous post, "Dirac's Diamond," may wonder why the
"Solomon's Cube" post of 11 AM Sunday, March 1, 2009,
appeared in the Dirac search results, since there is no
apparent mention of Dirac in that Sunday post.
<!– See also "a linear transformation of V6… which preserves
the Klein quadric; in this way we arrive at the isomorphism of
Sym(8) withthe full orthogonal group O+(6; 2)." in "The
Classification of Flats in PG(9,2) which are External to the
Grassmannian G1,4,2 Authors: Shaw, Ron;
 Maks, Johannes; Gordon, Neil; Source: Designs,
Codes and Cryptography, Volume 34, Numbers 2-3, February
2005 , pp. 203-227; Publisher: Springer.  For more details,
see "Finite Geometry, Dirac Groups and the Table of Real
Clifford Algebras," by R. Shaw (U. of Hull), pp. 59-99 in
Clifford Algebras and Spinor Structures, by By Albert
Crumeyrolle, Rafał Abłamowicz, Pertti Lounesto,
published by Springer, 1995. –>
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Friday, November 13, 2015
A Connection between the 16 Dirac Matrices and the Large Mathieu Group
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)—
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Thursday, November 5, 2015
Lyrics
“And I slept in last night’s clothes and tomorrow’s dreams
But they’re not quite what they seem.”
— Lyrics to "Uma Thurman," by Fall Out Boy
(Sung at CMA Awards last night)
"Does an empty vessel stop making most noise
once filled only with hopes and dreams?"
— Comment at Not Even Wrong this morning
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Saturday, August 29, 2015
Studio Time
(The title is a phrase from Oslo artist Josefine Lyche's Instagram page today.)
Note that 6 PM ET is midnight in Oslo.
An image from St. Ursula's Day, 2010 —
Related material:
"Is it a genuine demolition of the walls which seem
to separate mind from mind …. ?"
— Clifford Geertz, conclusion of “The Cerebral Savage:
On the Work of Claude Lévi-Strauss“
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Sunday, June 8, 2014
Vide
"The relevance of a geometric theorem is determined by what the theorem
tells us about space, and not by the eventual difficulty of the proof."
— Gian-Carlo Rota discussing the theorem of Desargues
What space tells us about the theorem :
In the simplest case of a projective space (as opposed to a plane ),
there are 15 points and 35 lines: 15 Göpel lines and 20 Rosenhain lines.*
The theorem of Desargues in this simplest case is essentially a symmetry
within the set of 20 Rosenhain lines. The symmetry, a reflection
about the main diagonal in the square model of this space, interchanges
10 horizontally oriented (row-based) lines with 10 corresponding
vertically oriented (column-based) lines.
Vide Classical Geometry in Light of Galois Geometry.
* Update of June 9: For a more traditional nomenclature, see (for instance)
R. Shaw, 1995. The "simplest case" link above was added to point out that
the two types of lines named are derived from a natural symplectic polarity
in the space. The square model of the space, apparently first described in
notes written in October and December, 1978, makes this polarity clearly visible:
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Tuesday, April 9, 2013
Four Quartets
For the cruelest month
Click for a much larger version of the photo below.
These four Kountry Korn quartets are from the Fox Valleyaires
Men's Barbershop Chorus of Appleton, Wisconsin.
See also the fine arts here on Saturday, April 6, 2013—
The New York Times Magazine cover story
a decade ago, on Sunday, April 6, 2003:
"The artists demanded space
in tune with their aesthetic."
— "The Dia Generation,"
by Michael Kimmelman
Related material:
See Wikipedia for the difference between binary numbers
and binary coordinates from the finite Galois field GF(2).
For some background, see the relativity problem.
See also the chapter on vector spaces in Korn & Korn
(originally published by McGraw-Hill)—
.
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