See also "Overarching + Tesseract" in this journal. From the results
of that search, some context for the "inscape" of the previous post —
Monday, March 11, 2019
Overarching Metanarratives
Monday, March 12, 2018
“Quantum Tesseract Theorem?”
Remarks related to a recent film and a notsorecent film.
For some historical background, see Dirac and Geometry in this journal.
Also (as Thas mentions) after Saniga and Planat —
The SanigaPlanat paper was submitted on December 21, 2006.
Excerpts from this journal on that date —
"Open the pod bay doors, HAL."
Friday, September 11, 2020
In Memoriam
From the Vanderbilt University obituary of Vaughan F. R. Jones —
“During the mid1980s, while Jones was working on a problem in von Neumann algebra theory, which is related to the foundations of quantum mechanics, he discovered an unexpected link between that theory and knot theory, a mathematical field dating back to the 19th century. Specifically, he found a new mathematical expression—now known as the Jones polynomial—for distinguishing between different types of knots as well as links in threedimensional space. Jones’ discovery had been missed by topologists during the previous 60 years, and his finding contributed to his selection as a Fields Medalist. ‘Now there is an area of mathematics called said Dietmar Bisch, professor of mathematics.” [Link added.] 
Related to Jones’s work —
“Topological Quantum Information Theory” at
the website of Louis H. Kauffman —
http://homepages.math.uic.edu/~kauffman/Quanta.pdf.
Kauffman —
Saturday, March 7, 2020
The “Octad Group” as Symmetries of the 4×4 Square
From “Mathieu Moonshine and Symmetry Surfing” —
(Submitted on 29 Sep 2016, last revised 22 Jan 2018)
by Matthias R. Gaberdiel (1), Christoph A. Keller (2),
and Hynek Paul (1)
(1) Institute for Theoretical Physics, ETH Zurich
(2) Department of Mathematics, ETH Zurich
https://arxiv.org/abs/1609.09302v2 —
“This presentation of the symmetry groups G_{i} is
particularly welladapted for the symmetry surfing
philosophy. In particular it is straightforward to
combine them into an overarching symmetry group G
by combining all the generators. The resulting group is
the socalled octad group
G = (Z_{2})^{4}^{ }⋊ A_{8}_{ }.
It can be described as a maximal subgroup of M_{24}
obtained by the setwise stabilizer of a particular
‘reference octad’ in the Golay code, which we take
to be O_{9 }= {3,5,6,9,15,19,23,24} ∈ 𝒢_{24}. The octad
subgroup is of order 322560, and its index in M_{24}
is 759, which is precisely the number of
different reference octads one can choose.”
This “octad group” is in fact the symmetry group of the affine 4space over GF(2),
so described in 1979 in connection not with the Golay code but with the geometry
of the 4×4 square.* Its nature as an affine group acting on the Golay code was
known long before 1979, but its description as an affine group acting on
the 4×4 square may first have been published in connection with the
Cullinane diamond theorem and Abstract 79TA37, “Symmetry invariance in a
diamond ring,” by Steven H. Cullinane in Notices of the American Mathematical
Society , February 1979, pages A193, 194.
* The Galois tesseract .
Update of March 15, 2020 —
Conway and Sloane on the “octad group” in 1993 —
Monday, October 21, 2019
Algebra and Space… Illustrated.
Related entertainment —
Detail:
George Steiner —
"Perhaps an insane conceit."
Perhaps.
See Quantum Tesseract Theorem .
Perhaps Not.
See Dirac and Geometry .
Wednesday, October 9, 2019
The Joy of Six
Note that in the pictures below of the 15 twosubsets of a sixset,
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."
Friday, September 27, 2019
The Black List
"… Max Black, the Cornell philosopher, and others have pointed out
how 'perhaps every science must start with metaphor and end with
algebra, and perhaps without the metaphor there would never have
been any algebra' …."
— Max Black, Models and Metaphors, Cornell U. Press, 1962,
page 242, as quoted in Dramas, Fields, and Metaphors, by
Victor Witter Turner, Cornell U. Press, paperback, 1975, page 25
Metaphor —
Algebra —
The 16 Dirac matrices form six anticommuting sets of five matrices each (Arfken 1985, p. 214): 1. , , , , , 2. , , , , , 3. , , , , , 4. , , , , , 5. , , , , , 6. , , , , . SEE ALSO: Pauli Matrices REFERENCES: Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 211217, 1985. Berestetskii, V. B.; Lifshitz, E. M.; and Pitaevskii, L. P. "Algebra of Dirac Matrices." §22 in Quantum Electrodynamics, 2nd ed. Oxford, England: Pergamon Press, pp. 8084, 1982. Bethe, H. A. and Salpeter, E. Quantum Mechanics of One and TwoElectron Atoms. New York: Plenum, pp. 4748, 1977. Bjorken, J. D. and Drell, S. D. Relativistic Quantum Mechanics. New York: McGrawHill, 1964. Dirac, P. A. M. Principles of Quantum Mechanics, 4th ed. Oxford, England: Oxford University Press, 1982. Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: AddisonWesley, p. 580, 1980. Good, R. H. Jr. "Properties of Dirac Matrices." Rev. Mod. Phys. 27, 187211, 1955. Referenced on WolframAlpha: Dirac Matrices CITE THIS AS: Weisstein, Eric W. "Dirac Matrices."
From MathWorld— A Wolfram Web Resource. 
Desiring the exhilarations of changes:
The motive for metaphor, shrinking from
The weight of primary noon,
The A B C of being,
The ruddy temper, the hammer
Of red and blue, the hard sound—
Steel against intimation—the sharp flash,
The vital, arrogant, fatal, dominant X.
Friday, August 16, 2019
Nocciolo
A revision of the above diagram showing
the Galoisadditiontable structure —
Related tables from August 10 —
See "Schoolgirl Space Revisited."
Saturday, August 10, 2019
Schoolgirl Space* Revisited:
The Square "Inscape" Model of
the Generalized Quadrangle W(2)
Click image to enlarge.
* The title refers to the role of PG (3,2) in Kirkman's schoolgirl problem.
For some backstory, see my post Anticommuting Dirac Matrices as Skew Lines
and, more generally, posts tagged Dirac and Geometry.
Tuesday, July 16, 2019
Schoolgirl Space for Quantum Mystics
Sunday, July 14, 2019
Old Pathways in Science:
The Quantum Tesseract Theorem Revisited
"The secret is that the supermathematician 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 Eoperators.
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.
Saturday, December 22, 2018
CremonaRichmond
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 —
Wednesday, December 12, 2018
An Inscape for Douthat
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."
Friday, December 7, 2018
The Angel Particle
(Continued from this morning)
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
See also other Log24 posts tagged Kummerhenge.
Tuesday, November 13, 2018
Blackboard Jungle Continues.
From the 1955 film "Blackboard Jungle" —
From a trailer for the recent film version of A Wrinkle in Time —
Detail of the phrase "quantum tesseract theorem":
From the 1962 book —
"There's something phoney
in the whole setup, Meg thought.
There is definitely something rotten
in the state of Camazotz."
Related mathematics from Koen Thas that some might call a
"quantum tesseract theorem" —
Some background —
See also posts tagged Dirac and Geometry. For more
background on finite geometry, see a web page
at Thas's institution, Ghent University.
Thursday, June 21, 2018
Dirac and Geometry (continued)
"Just fancy a scale model of Being
made out of string and cardboard."
— Nanavira Thera, 1 October 1957,
on a model of Kummer's Quartic Surface
mentioned by Eddington
"… a treatise on Kummer's quartic surface."
The "supermathematician" Eddington did not see fit to mention
the title or the author of the treatise he discussed.
See Hudson + Kummer in this journal.
See also posts tagged Dirac and Geometry.
Tuesday, January 9, 2018
Koen Thas and Quantum Theory
This post supplies some background for earlier posts tagged
Quantum Tesseract Theorem.
Monday, January 8, 2018
Raiders of the Lost Theorem
The Quantum Tesseract Theorem —
Raiders —
A Wrinkle in Time
starring Storm Reid,
Reese Witherspoon,
Oprah Winfrey &
Mindy Kaling
Time Magazine December 25, 2017 – January 1, 2018
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 ."
The Patterning
Friday, December 22, 2017
Thursday, December 21, 2017
Wrinkles
TIME magazine, issue of December 25th, 2017 —
" In 2003, Hand worked with Disney to produce a madeforTV movie.
Thanks to budget constraints, among other issues, the adaptation
turned out bland and uninspiring. It disappointed audiences,
L’Engle and Hand. 'This is not the dream,' Hand recalls telling herself.
'I’m sure there were people at Disney that wished I would go away.' "
Not the dream? It was, however, the nightmare, presenting very well
the encounter in Camazotz of Charles Wallace with the Tempter.
From a trailer for the latest version —
Detail:
From the 1962 book —
"There's something phoney in the whole setup, Meg thought.
There is definitely something rotten in the state of Camazotz."
Song adapted from a 1960 musical —
"In short, there's simply not
A more congenial spot
For happyeveraftering
Than here in Camazotz!"
Sunday, December 10, 2017
Geometry
See also Symplectic in this journal.
From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens 54, 5979 (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 diamondtheorem 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) —
Tuesday, October 10, 2017
Another 35Year Wait
The title refers to today's earlier post "The 35Year 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 LéviStrauss.
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.
Tuesday, November 22, 2016
Jargon
See "sacerdotal jargon" in this journal.
For those who prefer scientific jargon —
"… open its reading to
combinational possibilities
outside its larger narrative flow.
The particulars of attention,
whether subjective or objective,
are unshackled through form,
and offered as a relational matrix …."
— Kent Johnson in a 1993 essay
For some science that is not just jargon, see …
and, also from posts tagged Dirac and Geometry …
The above line complex also illustrates an outer automorphism
of the symmetric group S_{6}. See last Thursday's post "Rotman and
the Outer Automorphism."
Friday, June 3, 2016
Bruins and van Dam
A review of some recent posts on Dirac and geometry,
each of which mentions the late physicist Hendrik van Dam:
 Kummer and Dirac (May 25)
 Framework (May 25)
 Expanding the Spielraum (May 26)
 Dorje (May 26)
The first of these posts mentions the work of E. M. Bruins.
Some earlier posts that cite Bruins:
 Anticommuting Dirac Matrices as Skew Lines (Nov. 20, 2015)
 Dirac and Line Geometry (Nov. 23, 2015)
 Einstein and Geometry (Nov. 27, 2015)
Wednesday, May 25, 2016
Kummer and Dirac
From "Projective Geometry and PTSymmetric Dirac Hamiltonian,"
Y. Jack Ng and H. van Dam,
Physics Letters B , Volume 673, Issue 3,
23 March 2009, Pages 237–239
(http://arxiv.org/abs/0901.2579v2, last revised Feb. 20, 2009)
" Studies of spin½ theories in the framework of projective geometry
have been undertaken before. See, e.g., Ref. [4]. ^{1 }"
" ^{1} These papers are rather mathematical and technical.
The authors of the first two papers discuss the Dirac equation
in terms of the PluckerKlein correspondence between lines of
a threedimensional projective space and points of a quadric
in a fivedimensional projective space. The last paper shows
that the Dirac equation bears a certain relation to Kummer’s
surface, viz., the structure of the Dirac ring of matrices is
related to that of Kummer’s 16_{6} configuration . . . ."
[4]
O. Veblen
Proc. Natl. Acad. Sci. USA , 19 (1933), p. 503
Full Text via CrossRef
E.M. Bruins
Proc. Nederl. Akad. Wetensch. , 52 (1949), p. 1135
F.C. Taylor Jr., Master thesis, University of North Carolina
at Chapel Hill (1968), unpublished
A remark of my own on the structure of Kummer’s 16_{6} configuration . . . .
See as well yesterday morning's post.
Tuesday, May 24, 2016
Rosenhain and Göpel Revisited
The authors Taormina and Wendland in the previous post
discussed some mathematics they apparently did not know was
related to a classic 1905 book by R. W. H. T. Hudson, Kummer's
Quartic Surface .
"This famous book is a prototype for the possibility
of explaining and exploring a manyfaceted topic of
research, without focussing on general definitions,
formal techniques, or even fancy machinery. In this
regard, the book still stands as a highly recommendable,
unparalleled introduction to Kummer surfaces, as a
permanent source of inspiration and, last but not least,
as an everlasting symbol of mathematical culture."
— Werner Kleinert, Mathematical Reviews ,
as quoted at Amazon.com
Some 4×4 diagrams from that book are highly relevant to the
discussion by Taormina and Wendland of the 4×4 squares within
the 1974 Miracle Octad Generator of R. T. Curtis that were later,
in 1987, described by Curtis as pictures of the vector 4space over
the twoelement Galois field GF(2).
Hudson did not think of his 4×4 diagrams as illustrating a vector space,
but he did use them to picture certain subsets of the 16 cells in each
diagram that he called Rosenhain and Göpel tetrads .
Some related work of my own (click images for related posts)—
Rosenhain tetrads as 20 of the 35 projective lines in PG(3,2)
Göpel tetrads as 15 of the 35 projective lines in PG(3,2)
Related terminology describing the Göpel tetrads above
Monday, February 8, 2016
A Game with Four Letters
Related material — Posts tagged Dirac and Geometry.
For an example of what Eddington calls "an open mind,"
see the 1958 letters of Nanavira Thera.
(Among the "Early Letters" in Seeking the Path ).
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 3952 in Oziewicz et al., eds., Spinors, Twistors,
Clifford Algebras, and Quantum Deformations:
Proceedings of the Second Max Born Symposium held
near Wrocław, Poland, September 1992 . (Springer, 2012,
originally published by Kluwer in 1993.)
For morerecent remarks on quantum geometry, see a
paper by Saniga cited in today's update to my Nov. 20 post.
Friday, November 20, 2015
Anticommuting Dirac Matrices as Skew Lines
(Continued from November 13)
The work of Ron Shaw in this area, ca. 19941995, 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 3space over the 2element Galois field.
Here is an explicit picture —
References:
Arfken, George B., Mathematical Methods for Physicists , Third Edition,
Academic Press, 1985, pages 213214
Cullinane, Steven H., Notes on Groups and Geometry, 19781986
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 morerecent 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.
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 213214
Cullinane, Steven H., Notes on Groups and Geometry, 19781986
Related material:
The 6set 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 19941995)—
Wednesday, October 21, 2015
Algebra and Space
"Perhaps an insane conceit …." Perhaps.
Related remarks on algebra and space —
"The Quality Without a Name" (Log24, August 26, 2015).
Monday, December 9, 2013
Being There
Or: The Naked Blackboard Jungle
"…it would be quite a long walk
Swiftly Mrs. Who brought her hands… together.
"Now, you see," Mrs. Whatsit said,
– A Wrinkle in Time , 
Related material: Machete Math and…
Starring the late Eleanor Parker as Swiftly Mrs. Who.
Monday, August 12, 2013
Form
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—
The Galois tesseract is the basis for a representation of the smallest
projective 3space, PG(3,2), that differs from the representation at
Wolfram Demonstrations Project. For the latter, see yesterday’s post.
The tesseract representation underlies the diamond theorem, illustrated
below in its earliest form, also from the above February 1977 article—
As noted in a more recent version, the group described by
the diamond theorem is also the group of the 35 square
patterns within the 1976 Miracle Octad Generator (MOG) of
R. T. Curtis.
Tuesday, May 28, 2013
Codes
The hypercube model of the 4space over the 2element Galois field GF(2):
The phrase Galois tesseract may be used to denote a different model
of the above 4space: the 4×4 square.
MacWilliams and Sloane discussed the Miracle Octad Generator
(MOG) of R. T. Curtis further on in their book (see below), but did not
seem to realize in 1977 that the 4×4 structures within the MOG are
based on the Galoistesseract model of the 4space over GF(2).
The thirtyfive 4×4 structures within the MOG:
Curtis himself first described these 35 square MOG patterns
combinatorially, (as his title indicated) rather than
algebraically or geometrically:
A later book coauthored by Sloane, first published in 1988,
did recognize the 4×4 MOG patterns as based on the 4×4
Galoistesseract model.
Between the 1977 and 1988 Sloane books came the diamond theorem.
Update of May 29, 2013:
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977
(the year the above MacWilliamsSloane book was first published):
Sunday, May 19, 2013
Priority Claim
From an arXiv preprint submitted July 18, 2011,
and last revised on March 11, 2013 (version 4):
"By our construction, this vector space is the dual
of our hypercube F_{2}^{4} built on I \ O_{9}. The vector space
structure of the latter, to our knowledge, is first
mentioned by Curtis in [Cur89]. Hence altogether
our proposition 2.3.4 gives a novel geometric
meaning in terms of Kummer geometry to the known
vector space structure on I \ O_{9}."
[Cur89] reference:
R. T. Curtis, "Further elementary techniques using
the miracle octad generator," Proc. Edinburgh
Math. Soc. 32 (1989), 345353 (received on
July 20, 1987).
— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group M _{24 },"
arXiv.org > hepth > arXiv:1107.3834
"First mentioned by Curtis…."
No. I claim that to the best of my knowledge, the
vector space structure was first mentioned by me,
Steven H. Cullinane, in an AMS abstract submitted
in October 1978, some nine years before the
Curtis article.
Update of the above paragraph on July 6, 2013—
No. The vector space structure was described by
The vector space structure as it occurs in a 4×4 array 
See Notes on Finite Geometry for some background.
See in particular The Galois Tesseract.
For the relationship of the 1978 abstract to Kummer
geometry, see Rosenhain and Göpel Tetrads in PG(3,2).
Wednesday, May 1, 2013
The Crosswicks Curse
"There is such a thing as a tesseract." —A novel from Crosswicks
Related material from a 1905 graduate of Princeton,
"The 3Space PG(3,2) and Its Group," is now available
at Internet Archive (1 download thus far).
The 3space paper is relevant because of the
connection of the group it describes to the
"super, overarching" group of the tesseract.
Saturday, April 13, 2013
Princeton’s Christopher Robin
The title is that of a talk (see video) given by
George Dyson at a Princeton land preservation trust,
reportedly on March 21, 2013. The talk's subtitle was
"Oswald Veblen and the Sixhundredacre Woods."
Meanwhile…
Thursday, March 21, 2013

Related material for those who prefer narrative
to mathematics:
Log24 on June 6, 2006:
The Omen :

Related material for those who prefer mathematics
to narrative:
What the Omen narrative above and the mathematics of Veblen
have in common is the number 6. Veblen, who came to
Princeton in 1905 and later helped establish the Institute,
wrote extensively on projective geometry. As the British
geometer H. F. Baker pointed out, 6 is a rather important number
in that discipline. For the connection of 6 to the Göpel tetrads
figure above from March 21, see a note from May 1986.
See also last night's Veblen and Young in Light of Galois.
"There is such a thing as a tesseract." — Madeleine L'Engle
Sunday, March 17, 2013
Back to the Present
The previous post discussed some tesseract–
related mathematics from 1905.
Returning to the present, here is some arXiv activity
in the same area from March 11, 12, and 13, 2013.
Monday, June 21, 2010
Test
From a post by Ivars Peterson, Director
of Publications and Communications at
the Mathematical Association of America,
at 19:19 UTC on June 19, 2010—
Exterior panels and detail of panel,
Michener Gallery at Blanton Museum
in Austin, Texas—
Peterson associates the fourdiamond figure
with the Pythagorean theorem.
A more relevant association is the
fourdiamond view of a tesseract shown here
on June 19 (the same date as Peterson's post)
in the "Imago Creationis" post—
This figure is relevant because of a
tesseract sculpture by Peter Forakis—
This sculpture was apparently shown in the above
building— the Blanton Museum's Michener gallery—
as part of the "Reimagining Space" exhibition,
September 28, 2008January 18, 2009.
The exhibition was organized by
Linda Dalrymple Henderson, Centennial Professor
in Art History at the University of Texas at Austin
and author of The Fourth Dimension and
NonEuclidean Geometry in Modern Art
(Princeton University Press, 1983;
new ed., MIT Press, 2009).
For the sculptor Forakis in this journal,
see "The Test" (December 20, 2009).
"There is such a thing
as a tesseract."
— A Wrinkle in TIme
Monday, May 5, 2008
Monday May 5, 2008
"And take upon's
the mystery of things
as if we were God's spies"
— King Lear
From Log24 on Aug. 19, 2003
and on Ash Wednesday, 2004:
a reviewer on
An Instance of the Fingerpost::
"Perhaps we are meant to
see the story as a cubist
retelling of the crucifixion."
From Log24 on
Michaelmas 2007:
Google searches suggested by
Sunday's PA lottery numbers
(midday 170, evening 144)
and by the above
figure of Kate Beckinsale
pointing to an instance of
the number 144 —
Related material:
Beckinsale in another film
(See At the Crossroads,
Log24, Dec. 8, 2006):
"It was only in retrospect
that the silliness
became profound."
— Review of
Faust in Copenhagen
From the conclusion of
Joan Didion's 1970 novel
Play It As It Lays —
"I know what 'nothing' means,
and keep on playing."
From Play It As It Lays,
the paperback edition of 1990
(Farrar, Straus and Giroux) —
Page 170:
"By the end of a week she was thinking constantly
170
even one microsecond she would have what she had 
"The page numbers
are generally reliable."
Friday, September 7, 2007
Friday September 7, 2007
The New York Times online,
Friday, Sept. 7, 2007:
Madeleine L’Engle,
Children’s Writer,
Is Dead
Her death, of natural causes, was announced today by her publisher, Farrar, Straus and Giroux."
Related material:
Log24 entries of
August 31—
"That is how we travel."
— A Wrinkle in Time,
Chapter 5,
"The Tesseract"
— and of
September 2
(with update of
September 5)–
"There is such a thing
as a tesseract."
— A Wrinkle in Time
Sunday, September 2, 2007
Sunday September 2, 2007
Comment at the
nCategory Cafe
Re: This Week’s Finds in Mathematical Physics (Week 251)
On Spekkens’ toy system and finite geometry
Background–
 In “Week 251” (May 5, 2007), John wrote:
“Since Spekkens’ toy system resembles a qubit, he calls it a “toy bit”. He goes on to study systems of several toy bits – and the charming combinatorial geometry I just described gets even more interesting. Alas, I don’t really understand it well: I feel there must be some mathematically elegant way to describe it all, but I don’t know what it is…. All this is fascinating. It would be nice to find the mathematical structure that underlies this toy theory, much as the category of Hilbert spaces underlies honest quantum mechanics.”  In the nCategory Cafe ( May 12, 2007, 12:26 AM, ) Matt Leifer wrote:
“It’s crucial to Spekkens’ constructions, and particularly to the analog of superposition, that the statespace is discrete. Finding a good mathematical formalism for his theory (I suspect finite fields may be the way to go) and placing it within a comprehensive framework for generalized theories would be very interesting.”  In the ncategory Cafe ( May 12, 2007, 6:25 AM) John Baez wrote:
“Spekkens and I spent an afternoon trying to think about his theory as quantum mechanics over some finite field, but failed — we almost came close to proving it couldnt’ work.”
On finite geometry:
 In “Week 234” (June 12, 2006), John wrote:
“For a pretty explanation of M_{24}… try this:
… Steven H. Cullinane, Geometry of the 4 × 4 square,
http://finitegeometry.org/sc/16/geometry.html”
The actions of permutations on a 4 × 4 square in Spekkens’ paper (quantph/0401052), and Leifer’s suggestion of the need for a “generalized framework,” suggest that finite geometry might supply such a framework. The geometry in the webpage John cited is that of the affine 4space over the twoelement field.
Related material:
Sept. 5, 2007
See also arXiv:0707.0074v1 [quantph], June 30, 2007:
A fully epistemic model for a local hidden variable emulation of quantum dynamics,
by Michael Skotiniotis, Aidan Roy, and Barry C. Sanders, Institute for Quantum Information Science, University of Calgary. Abstract: "In this article we consider an augmentation of Spekkens’ toy model for the epistemic view of quantum states [1]…."
Hypercube from the Skotiniotis paper:
Reference:
Evidence for the epistemic view of quantum states: A toy theory,
Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5 (Received 11 October 2005; revised 2 November 2006; published 19 March 2007.)
Friday, August 31, 2007
Friday August 31, 2007
"…it would be quite
a long walk
for him if he had to
walk straight across."
Swiftly Mrs. Who brought
her hands… together.
"Now, you see,"
Mrs. Whatsit said,
"he would be there,
without that long trip.
That is how we travel."
— A Wrinkle in Time,
Chapter 5,
"The Tesseract"
Related material:
To Measure the Changes,
and…

Monday, May 21, 2007
Monday May 21, 2007
A more recent royal reference:
"'Yau wants to be the king of geometry,' Michael Anderson, a geometer at Stony Brook, said. 'He believes that everything should issue from him, that he should have oversight. He doesn't like people encroaching on his territory.'" –Sylvia Nasar and David Gruber in The New Yorker, issue dated Aug. 28, 2006
Wikipedia, Cultural references to the Royal Road:
"Euclid is said to have replied to King Ptolemy's request for an easier way of learning mathematics that 'there is no royal road to geometry.' Charles S. Peirce, in his 'How to Make Our Ideas Clear' (1878), says 'There is no royal road to logic, and really valuable ideas can only be had at the price of close attention.'"
Day Without Logic
(March 8, 2007)
and
The Geometry of Logic
(March 10, 2007):
There may be
no royal roads to
geometry or logic,
but…
"There is such a thing
as a tesseract."
— Madeleine L'Engle,
A Wrinkle in Time
Monday, May 14, 2007
Monday May 14, 2007
Crossing Point
From Log24's
"Footprints for Baudrillard"–
"Was there really a cherubim
waiting at the starwatching rock…?
Was he real?
What is real?
— Madeleine L'Engle, A Wind in the Door,
Farrar, Straus and Giroux, 1973,
conclusion of Chapter Three,
"The Man in the Night"
"Oh, Euclid, I suppose."
— Madeleine L'Engle, A Wrinkle in Time,
Farrar, Straus and Giroux, 1962,
conclusion of Chapter Five,
"The Tesseract"
From Log24's
Xanga footprints,
3:00 AM today:
Texas  /431103703/item.html  5/14/2007 3:00 AM 
The link leads to a Jan. 23, 2006 entry
on what one philosopher has claimed is
"exactly that crossing point
of constraint and freedom
which is the very essence
of man's nature."
Thursday, May 3, 2007
Thursday May 3, 2007
of Links
"Some postmodern theorists like to talk about the relationship between 'intertextuality' and 'hypertextuality'; intertextuality makes each text a 'mosaic of quotations' [Kristeva, Desire in Language, Columbia U. Pr., 1980, 66] and part of a larger mosaic of texts, just as each hypertext can be a web of links and part of the whole WorldWide Web." —Wikipedia
Related material
Day Without Logic,
Introduction to Logic,
The Geometry of Logic,
Structure and Logic,
SpiderMan and Fan:
"There is such a thing
as a tesseract."
— A Wrinkle in Time
Friday, April 20, 2007
Friday April 20, 2007
"… Bush spoke and answered audience questions for nearly 90 minutes inside East Grand Rapids High School in suburban Grand Rapids….
After leaving the school, Bush's motorcade stopped at the Gerald R. Ford Presidential Museum in downtown Grand Rapids, where he stood silently for a few moments after placing a bouquet of white roses at Ford's burial site on the museum grounds. The 38th president, who grew up in Grand Rapids, died Dec. 26 at age 93."
above, see this morning's entry and
an entry that it links to —
Time's Labyrinth continued —
of March 8, 2007.
For the meaning of multispeech,
see the entries of
All Hallows' Eve, 2005:
"There is such a thing
as a tesseract."
— A Wrinkle in Time
Tuesday, April 3, 2007
Tuesday April 3, 2007
Related material:
“But what is it?”
Calvin demanded.
“We know that it’s evil,
but what is it?”
“Yyouu hhave ssaidd itt!”
Mrs. Which’s voice rang out.
“Itt iss Eevill. Itt iss thee
Ppowers of Ddarrkknesss!”
“After A Wrinkle in Time was finally published, it was pointed out to me that the villain, a naked disembodied brain, was called ‘It’ because It stands for Intellectual truth as opposed to a truth which involves the whole of us, heart as well as mind. That acronym had never occurred to me. I chose the name It intuitively, because an IT does not have a heart or soul. And I did not understand consciously at the time of writing that the intellect, when it is not informed by the heart, is evil.”
theology is about words.”
— Freeman Dyson,
New York Review of Books,
issue dated May 28, 1998
Wednesday, March 7, 2007
Wednesday March 7, 2007
Baudrillard
"Was there really a cherubim
waiting at the starwatching rock…?
Was he real?
What is real?
— Madeleine L'Engle, A Wind in the Door,
Farrar, Straus and Giroux, 1973,
conclusion of Chapter Three,
"The Man in the Night"
"Oh, Euclid, I suppose."
Farrar, Straus and Giroux, 1962,
conclusion of Chapter Five,
"The Tesseract"
In memory of the French philosopher Jean Baudrillard, who died yesterday, Tuesday, March 6, 2007.
The following Xanga footprints may be regarded as illustrating Log24 remarks of Dec. 10, 2006 on the Library of Congress, geometry, and bullshit, as well as remarks of Aug. 28, 2006 on the temporal, the eternal, and St. Augustine.
From the District of Columbia–
Xanga footprints in reverse
chronological order from
the noon hour on Tuesday,
March 6, 2007, the date
of Baudrillard's death:
District of Columbia /499111929/item.html Beijing String 
3/6/2007 12:04 PM 
District of Columbia /497993036/item.html Spellbound 
3/6/2007 12:03 PM 
District of Columbia /443606342/item.html About God, Life, Death 
3/6/2007 12:03 PM 
District of Columbia /494421586/item.html A Library of Congress Reading 
3/6/2007 12:03 PM 
District of Columbia /500434851/item.html Binary Geometry 
3/6/2007 12:03 PM 
District of Columbia /404038913/item.html Prequel on St. Cecelia's Day 
3/6/2007 12:03 PM 
Thursday, March 1, 2007
Thursday March 1, 2007
Senior Honors
From the obituary in today's New York Times of historian Arthur M. Schlesinger Jr.–
"Mr. Schlesinger, partly through his appreciation of history, fully realized his good fortune. 'I have lived through interesting times and had the luck of knowing some interesting people,' he wrote.
A huge part of his luck was his father, who guided much of his early research, and even suggested the topic for his [Harvard] senior honors: Orestes A. Brownson, a 19thcentury journalist, novelist and theologian. It was published by Little, Brown in 1938 as 'Orestes A. Brownson: A Pilgrim's Progress.'"
From The Catholic Encyclopedia:
"It is sufficient for true knowledge that it affirm as real that which is truly real."
From The Diamond Theory of Truth:
"Was there really a cherubim waiting at the starwatching rock…?
Was he real?
What is real?— Madeleine L'Engle, A Wind in the Door, Farrar, Straus and Giroux, 1973, conclusion of Chapter Three, "The Man in the Night"
"Oh, Euclid, I suppose."
— Madeleine L'Engle, A Wrinkle in Time, Farrar, Straus and Giroux, 1962, conclusion of Chapter Five, "The Tesseract"
Related material: Yesterday's first annual "Tell Your Story Day" at Harvard and yesterday's entry on Euclid.
Friday, December 8, 2006
Friday December 8, 2006
Kate Beckinsale, adapted from
poster for Underworld: Evolution
(DVD release date 6/6/6)
"Appropriating the Buttonmolder's
words to Peer Gynt, he would say,
'We'll meet at the next crossroads…
and then we'll see–
I won't say more.'"
Tuesday, October 31, 2006
Tuesday October 31, 2006
From 7/07, an art review from The New York Times:
Endgame Art?
It's Borrow, Sample and Multiply
in an Exhibition at Bard College
"The show has an endgame, endtime mood….
I would call all these strategies fear of form…. the dismissal of originality is perhaps the oldest ploy in the postmodern playbook. To call yourself an artist at all is by definition to announce a faith, however unacknowledged, in some form of originality, first for yourself, second, perhaps, for the rest of us.
Fear of form above all means fear of compression– of an artistic focus that condenses experiences, ideas and feelings into something whole, committed and visually comprehensible."
— Roberta Smith
would consider the
following "found" art an
example of originality.
It nevertheless does
"announce a faith."
"First for yourself"
Today's midday
Pennsylvania number:
707
See Log24 on 7/07
and the above review.
"Second, perhaps,
for the rest of us"
Today's evening
Pennsylvania number:
384
This number is an
example of what the
reviewer calls "compression"–
"an artistic focus that condenses
experiences, ideas and feelings
into something
whole, committed
and visually comprehensible."
"Experiences"
See (for instance)
Joan Didion's writings
(1160 pages, 2.35 pounds)
on "the shifting phantasmagoria
which is our actual experience."
"Ideas"
"Feelings"
See A Wrinkle in Time.
"Whole"
The automorphisms
of the tesseract
form a group
of order 384.
"Committed"
See the discussions of
groups of degree 16 in
R. D. Carmichael's classic
Introduction to the Theory
of Groups of Finite Order.
"Visually comprehensible"
See "Diamond Theory in 1937,"
an excerpt from which
is shown below.
The "faith" announced by
the above lottery numbers
on All Hallows' Eve is
perhaps that of the artist
Madeleine L'Engle:
Friday, May 12, 2006
Friday May 12, 2006
"Does the word 'tesseract'
mean anything to you?"
— Robert A. Heinlein in
The Number of the Beast
(1980)
My reply–
Part I:
A Wrinkle in Time, by
Madeleine L'Engle
(first published in 1962)
Part II:
Diamond Theory in 1937
and
Geometry of the 4×4 Square
Part III:
"Wells and trees were dedicated to saints. But the offerings at many wells and trees were to something other than the saint; had it not been so they would not have been, as we find they often were, forbidden. Within this double and intertwined life existed those other capacities, of which we know more now, but of which we still know little– clairvoyance, clairaudience, foresight, telepathy."
— Charles Williams, Witchcraft, Faber and Faber, London, 1941
A New Yorker profile of Madeleine L'Engle from April 2004, which I found tonight online for the first time. For a related reflection on truth, stories, and values, see Saint's Day. For a wider context, see the Log24 entries of February 115, 2003 and February 115, 2006.
Wednesday, March 29, 2006
Wednesday March 29, 2006
Note: Carmichael's reference is to
A. Emch, "Triple and multiple systems, their geometric configurations and groups," Trans. Amer. Math. Soc. 31 (1929), 25–42.
— A Wrinkle in Time
Thursday, December 5, 2002
Thursday December 5, 2002
Sacerdotal Jargon
From the website
Abstracts and Preprints in Clifford Algebra [1996, Oct 8]:
Paper: clfalg/good9601
From: David M. Goodmanson
Address: 2725 68th Avenue S.E., Mercer Island, Washington 98040
Title: A graphical representation of the Dirac Algebra
Abstract: The elements of the Dirac algebra are represented by sixteen 4×4 gamma matrices, each pair of which either commute or anticommute. This paper demonstrates a correspondence between the gamma matrices and the complete graph on six points, a correspondence that provides a visual picture of the structure of the Dirac algebra. The graph shows all commutation and anticommutation relations, and can be used to illustrate the structure of subalgebras and equivalence classes and the effect of similarity transformations….
Published: Am. J. Phys. 64, 870880 (1996)
The following is a picture of K_{6}, the complete graph on six points. It may be used to illustrate various concepts in finite geometry as well as the properties of Dirac matrices described above.
From
"The Relations between Poetry and Painting,"
by Wallace Stevens:
"The theory of poetry, that is to say, the total of the theories of poetry, often seems to become in time a mystical theology or, more simply, a mystique. The reason for this must by now be clear. The reason is the same reason why the pictures in a museum of modern art often seem to become in time a mystical aesthetic, a prodigious search of appearance, as if to find a way of saying and of establishing that all things, whether below or above appearance, are one and that it is only through reality, in which they are reflected or, it may be, joined together, that we can reach them. Under such stress, reality changes from substance to subtlety, a subtlety in which it was natural for Cézanne to say: 'I see planes bestriding each other and sometimes straight lines seem to me to fall' or 'Planes in color. . . . The colored area where shimmer the souls of the planes, in the blaze of the kindled prism, the meeting of planes in the sunlight.' The conversion of our Lumpenwelt went far beyond this. It was from the point of view of another subtlety that Klee could write: 'But he is one chosen that today comes near to the secret places where original law fosters all evolution. And what artist would not establish himself there where the organic center of all movement in time and space—which he calls the mind or heart of creation— determines every function.' Conceding that this sounds a bit like sacerdotal jargon, that is not too much to allow to those that have helped to create a new reality, a modern reality, since what has been created is nothing less."