Log24

Friday, August 16, 2013

Six-Set Geometry

Filed under: General,Geometry — Tags: — m759 @ 5:24 AM

From April 23, 2013, in
​"Classical Geometry in Light of Galois Geometry"—

Click above image for some background from 1986.

Related material on six-set geometry from the classical literature—

Baker, H. F., "Note II: On the Hexagrammum Mysticum  of Pascal,"
in Principles of Geometry , Vol. II, Camb. U. Press, 1930, pp. 219-236  

Richmond, H. W., "The Figure Formed from Six Points in Space of Four Dimensions,"
Mathematische Annalen  (1900), Volume 53, Issue 1-2, pp 161-176

Richmond, H. W., "On the Figure of Six Points in Space of Four Dimensions," 
Quarterly Journal of Pure and Applied Mathematics , Vol. 31 (1900), pp. 125-160

Monday, October 26, 2020

Set + Design

Filed under: General — Tags: — m759 @ 9:41 PM

In memoriam . . .

http://m759.net/wordpress/?s=Set+Design .

Wednesday, September 23, 2020

Geometry of Even Subsets

Filed under: General — Tags: , — m759 @ 12:06 AM

Various posts here on the geometry underlying the Mathieu group M24
are now tagged with the phrase “Geometry of Even Subsets.”

For example, a post with this diagram . . .

Thursday, April 30, 2020

Walpurgisnacht Geometry

Filed under: General — Tags: — m759 @ 11:59 PM

A version more explicitly connected to finite geometry

For the six synthematic totals , see The Joy of Six.

Sunday, December 8, 2019

Geometry of 6 and 8

Filed under: General — Tags: , , — m759 @ 4:03 AM

Just as
the finite space PG(3,2) is
the geometry of the 6-set, so is
the finite space PG(5,2)
the geometry of the 8-set.*

Selah.

* Consider, for the 6-set, the 32
(16, modulo complementation)
0-, 2-, 4-, and 6-subsets,
and, for the 8-set, the 128
(64, modulo complementation)
0-, 2-, 4-, 6-, and 8-subsets.

Update of 11:02 AM ET the same day:

See also Eightfold Geometry, a note from 2010.

Wednesday, October 9, 2019

The Joy of Six

Note that in the pictures below of the 15 two-subsets of a six-set,
the symbols 1 through 6 in Hudson's square array of 1905 occupy the
same positions as the anticommuting Dirac matrices in Arfken's 1985
square array. Similarly occupying these positions are the skew lines
within a generalized quadrangle (a line complex) inside PG(3,2).

Anticommuting Dirac matrices as spreads of projective lines

Related narrative The "Quantum Tesseract Theorem."

Saturday, September 21, 2019

Six Dots

Filed under: General — Tags: — m759 @ 8:11 AM

See other posts now tagged Six Dots.

Related narrative:  Raiders of the Lost Ring .

Tuesday, July 2, 2019

Depth Psychology Meets Inscape Geometry

Filed under: General — m759 @ 3:00 AM

An illustration from the previous post may be interpreted
as an attempt to unbokeh  an inscape

The 15 lines above are Euclidean  lines based on pairs within a six-set
For examples of Galois  lines so based, see Six-Set Geometry:

Monday, April 29, 2019

Geometry Review

Filed under: General — m759 @ 9:02 PM

See also Good Friday 2003 … Continued .

Monday, February 25, 2019

The Deep Six

Filed under: General,Geometry — Tags: , , , , — m759 @ 11:00 AM

". . . this notion of ‘depth’ is an elusive one
even for a mathematician who can recognize it. . . ."

— G. H. Hardy, A Mathematician's Apology

See Six-Set in this journal.

Monday, February 18, 2019

The Joy of Six

Filed under: General,Geometry — Tags: , , — m759 @ 3:00 PM


__________________________________________________________________________

See also the previous post.

I prefer the work of Josefine Lyche on the smallest perfect number/universe.

Context —

Lyche's Lynx760 installations and Vigeland's nearby Norwegian  clusterfuck.

Monday, August 27, 2018

Children of the Six Sides

Filed under: General,Geometry — Tags: — m759 @ 11:32 AM

http://www.log24.com/log/pix18/180827-Terminator-3-tx-arrival-publ-160917.jpg

http://www.log24.com/log/pix18/180827-Terminator-3-tx-arrival-publ-161018.jpg

From the former date above —

Saturday, September 17, 2016

A Box of Nothing

Filed under: Uncategorized — m759 @ 12:13 AM

(Continued)

"And six sides to bounce it all off of.

From the latter date above —

Tuesday, October 18, 2016

Parametrization

Filed under: Uncategorized — m759 @ 6:00 AM

The term "parametrization," as discussed in Wikipedia, seems useful for describing labelings that are not, at least at first glance, of a vector-space  nature.

Examples: The labelings of a 4×4 array by a blank space plus the 15 two-subsets of a six-set (Hudson, 1905) or by a blank plus the 5 elements and the 10 two-subsets of a five-set (derived in 2014 from a 1906 page by Whitehead), or by a blank plus the 15 line diagrams of the diamond theorem.

Thus "parametrization" is apparently more general than the word "coodinatization" used by Hermann Weyl —

“This is the relativity problem:  to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.”

— Hermann Weyl, The Classical Groups , Princeton University Press, 1946, p. 16

Note, however, that Weyl's definition of "coordinatization" is not limited to vector-space  coordinates. He describes it as simply a mapping to a set of reproducible symbols

(But Weyl does imply that these symbols should, like vector-space coordinates, admit a group of transformations among themselves that can be used to describe transformations of the point-space being coordinatized.)

From March 2018 —

http://www.log24.com/log/pix18/180827-MIT-Rubik-Robot.jpg

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 "super-mathematician" Eddington did not see fit to mention
the title or the author of the treatise he discussed.

See Hudson + Kummer in this  journal.

See also posts tagged Dirac and Geometry.

Sunday, December 10, 2017

Geometry

Google search result for Plato + Statesman + interlacing + interweaving

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….”

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven)

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) —

Anticommuting Dirac matrices as spreads of projective lines

Ron Shaw on the 15 lines of the classical generalized quadrangle W(2), a general linear complex in PG(3,2)

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.

Thursday, May 25, 2017

The Story of Six Continues

Filed under: General,Geometry — Tags: , — m759 @ 6:00 AM

A post of March 22, 2017, was titled "The Story of Six."

Related material from that date —

"I meant… a larger map." — Number Six in "The Prisoner"

Friday, April 14, 2017

Hudson and Finite Geometry

Filed under: General,Geometry — Tags: , — m759 @ 3:00 AM

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

The above four-element sets of black subsquares of a 4×4 square array 
are 15 of the 60 Göpel tetrads , and 20 of the 80 Rosenhain tetrads , defined
by R. W. H. T. Hudson in his 1905 classic Kummer's Quartic Surface .

Hudson did not  view these 35 tetrads as planes through the origin in a finite
affine 4-space (or, equivalently, as lines in the corresponding finite projective
3-space).

In order to view them in this way, one can view the tetrads as derived,
via the 15 two-element subsets of a six-element set, from the 16 elements
of the binary Galois affine space pictured above at top left.

This space is formed by taking symmetric-difference (Galois binary)
sums of the 15 two-element subsets, and identifying any resulting four-
element (or, summing three disjoint two-element subsets, six-element)
subsets with their complements.  This process was described in my note
"The 2-subsets of a 6-set are the points of a PG(3,2)" of May 26, 1986.

The space was later described in the following —

IMAGE- Dolgachev and Keum, coordinatization of the 4x4 array in 'Birational Automorphisms of Quartic Hessian Surfaces,' AMS Transactions, 2002

Wednesday, March 22, 2017

The Story of Six

Filed under: General,Geometry — Tags: , , — m759 @ 8:01 PM

On a psychotherapist who died at 86 on Monday —

"He studied mathematics and statistics at the Courant Institute,
a part of New York University — he would later write   a
mathematical fable, Numberland  (1987)."

The New York Times  online this evening
 


 

From Publishers Weekly

This wry parable by a psychotherapist contains one basic message: though death is inevitable, each moment in life is to be cherished. In the orderly but sterile kingdom of Numberland, digits live together harmoniously under a rigid president called The Professor. Their stable society is held intact by the firm conviction that they are immortal: When has a number ever died? This placid universe is plunged into chaos when the inquisitive hero SIX crosses over into the human world and converses with a young mathematician. This supposedly impossible transition convinces the ruling hierarchy that if SIX can talk to a mortal, then the rest of the numbers are, after all, mortal. The digits conclude that any effort or achievement is pointless in the face of inevitable death, and the cipher society breaks down completely. The solution? Banish SIX to the farthest corners of kingdom. Weinberg (The Heart of Psychotherapy ) uses his fable to gently satirize the military, academics, politicians and, above all, psychiatrists. But his tale is basically inspirational; a triumphant SIX miraculously returns from exile and quells the turmoil by showing his fellow digits that knowledge of one's mortality should enrich all other experiences and that death ultimately provides a frame for the magnificent picture that is life. 

Copyright 1987 Reed Business Information, Inc.

See also The Prisoner in this journal.

Friday, December 23, 2016

Memory, History, Geometry

Filed under: General,Geometry — Tags: — m759 @ 6:48 PM

(Continued)

Code Blue

Update of 7:04 PM ET —

The source of the 404 message in the browsing history above
was the footnote below:

Friday, December 16, 2016

Memory, History, Geometry

Filed under: General,Geometry — Tags: — m759 @ 9:48 AM

These are Rothko's Swamps .

See a Log24 search for related meditations.

For all three topics combined, see Coxeter —

" There is a pleasantly discursive treatment 
of Pontius Pilate’s unanswered question
‘What is truth?’ "

— Coxeter, 1987, introduction to Trudeau’s
     The Non-Euclidean Revolution

Update of 10 AM ET —  Related material, with an elementary example:

Posts tagged "Defining Form." The example —

IMAGE- Triangular models of the 4-point affine plane A and 7-point projective plane PA

Thursday, March 24, 2016

The Nervous Set*

Filed under: General — Tags: — m759 @ 12:00 PM

The previous post suggests a review of the saying
"There is  such a thing as a 4-set."

* Title of a 1959 musical

Monday, December 14, 2015

Dirac and Geometry

Filed under: General,Geometry — Tags: , — m759 @ 10:30 AM

(Continued)

See a post by Peter Woit from Sept. 24, 2005 — Dirac's Hidden Geometry.

The connection, if any, with recent Log24 posts on Dirac and Geometry
is not immediately apparent.  Some related remarks from a novel —

From Broken Symmetries by Paul Preuss
(first published by Simon and Schuster in 1983) —

"He pondered the source of her fascination with the occult, which sooner or later seemed to entangle a lot of thoughtful people who were not already mired in establishmentarian science or religion. It was  the religious impulse, at base. Even reason itself could function as a religion, he supposed— but only for those of severely limited imagination. 

He’d toyed with 'psi' himself, written a couple of papers now much quoted by crackpots, to his chagrin. The reason he and so many other theoretical physicists were suckers for the stuff was easy to understand— for two-thirds of a century an enigma had rested at the heart of theoretical physics, a contradiction, a hard kernel of paradox. Quantum theory was inextricable from the uncertainty relations. 

The classical fox knows many things, but the quantum-mechanical hedgehog knows only one big thing— at a time. 'Complementarity,' Bohr had called it, a rubbery notion the great professor had stretched to include numerous pairs of opposites. Peter Slater was willing to call it absurdity, and unlike some of his older colleagues who, following in Einstein’s footsteps, demanded causal explanations for everything (at least in principle), Peter had never thirsted after 'hidden variables' to explain what could not be pictured. Mathematical relationships were enough to satisfy him, mere formal relationships which existed at all times, everywhere, at once. It was a thin nectar, but he was convinced it was the nectar of the gods. 

The psychic investigators, on the other hand, demanded to know how  the mind and the psychical world were related. Through ectoplasm, perhaps? Some fifth force of nature? Extra dimensions of spacetime? All these naive explanations were on a par with the assumption that psi is propagated by a species of nonlocal hidden variables, the favored explanation of sophisticates; ignotum per ignotius

'In this connection one should particularly remember that the human language permits the construction of sentences which do not involve any consequences and which therefore have no content at all…' The words were Heisenberg’s, lecturing in 1929 on the irreducible ambiguity of the uncertainty relations. They reminded Peter of Evan Harris Walker’s ingenious theory of the psi force, a theory that assigned psi both positive and negative values in such a way that the mere presence of a skeptic in the near vicinity of a sensitive psychic investigation could force null results. Neat, Dr. Walker, thought Peter Slater— neat, and totally without content. 

One had to be willing to tolerate ambiguity; one had to be willing to be crazy. Heisenberg himself was only human— he’d persuasively woven ambiguity into the fabric of the universe itself, but in that same set of 1929 lectures he’d rejected Dirac’s then-new wave equations with the remark, 'Here spontaneous transitions may occur to the states of negative energy; as these have never been observed, the theory is certainly wrong.' It was a reasonable conclusion, and that was its fault, for Dirac’s equations suggested the existence of antimatter: the first antiparticles, whose existence might never have been suspected without Dirac’s crazy results, were found less than three years later. 

Those so-called crazy psychics were too sane, that was their problem— they were too stubborn to admit that the universe was already more bizarre than anything they could imagine in their wildest dreams of wizardry."

Particularly relevant

"Mathematical relationships were enough to satisfy him,
mere formal relationships which existed at all times,
everywhere, at once."

Some related pure  mathematics

Anticommuting Dirac matrices as spreads of projective lines

Tuesday, December 1, 2015

Pascal’s Finite Geometry

Filed under: General,Geometry — Tags: — m759 @ 12:01 AM

See a search for "large Desargues configuration" in this journal.

The 6 Jan. 2015 preprint "Danzer's Configuration Revisited," 
by Boben, Gévay, and Pisanski, places this configuration,
which they call the Cayley-Salmon configuration , in the 
interesting context of Pascal's Hexagrammum Mysticum .

They show how the Cayley-Salmon configuration is, in a sense,
dual to something they call the Steiner-Plücker configuration .

This duality appears implicitly in my note of April 26, 1986,
"Picturing the smallest projective 3-space." The six-sets at
the bottom of that note, together with Figures 3 and 4
of Boben et. al. , indicate how this works.

The duality was, as they note, previously described in 1898.

Related material on six-set geometry from the classical literature—

Baker, H. F., "Note II: On the Hexagrammum Mysticum  of Pascal,"
in Principles of Geometry , Vol. II, Camb. U. Press, 1930, pp. 219-236  

Richmond, H. W., "The Figure Formed from Six Points in Space of Four Dimensions,"
Mathematische Annalen  (1900), Volume 53, Issue 1-2, pp 161-176

Richmond, H. W., "On the Figure of Six Points in Space of Four Dimensions," 
Quarterly Journal of Pure and Applied Mathematics , Vol. 31 (1900), pp. 125-160

Related material on six-set geometry from a more recent source —

Cullinane, Steven H., "Classical Geometry in Light of Galois Geometry," webpage

Friday, November 27, 2015

Einstein and Geometry

Filed under: General,Geometry — Tags: — m759 @ 2:01 PM

(A Prequel to Dirac and Geometry)

"So Einstein went back to the blackboard.
And on Nov. 25, 1915, he set down
the equation that rules the universe.
As compact and mysterious as a Viking rune,
it describes space-time as a kind of sagging mattress…."

— Dennis Overbye in The New York Times  online,
     November 24, 2015

Some pure  mathematics I prefer to the sagging Viking mattress —

Readings closely related to the above passage —

Thomas Hawkins, "From General Relativity to Group Representations:
the Background to Weyl's Papers of 1925-26
," in Matériaux pour
l'histoire des mathématiques au XXe siècle:
Actes du colloque
à la mémoire de Jean Dieudonné
, Nice, 1996  (Soc. Math.
de France, Paris, 1998), pp. 69-100.

The 19th-century algebraic theory of invariants is discussed
as what Weitzenböck called a guide "through the thicket
of formulas of general relativity."

Wallace Givens, "Tensor Coordinates of Linear Spaces," in
Annals of Mathematics  Second Series, Vol. 38, No. 2, April 1937, 
pp. 355-385.

Tensors (also used by Einstein in 1915) are related to 
the theory of line complexes in three-dimensional
projective space and to the matrices used by Dirac
in his 1928 work on quantum mechanics.

For those who prefer metaphors to mathematics —

"We acknowledge a theorem's beauty
when we see how the theorem 'fits' in its place,
how it sheds light around itself, like a Lichtung ,
a clearing in the woods." 
— Gian-Carlo Rota, Indiscrete Thoughts ,
Birkhäuser Boston, 1997, page 132

Rota fails to cite the source of his metaphor.
It is Heidegger's 1964 essay, "The End of Philosophy
and the Task of Thinking" —

"The forest clearing [ Lichtung ] is experienced
in contrast to dense forest, called Dickung  
in our older language." 
— Heidegger's Basic Writings 
edited by David Farrell Krell, 
Harper Collins paperback, 1993, page 441

Monday, November 23, 2015

Dirac and Line Geometry

Some background for my post of Nov. 20,
"Anticommuting Dirac Matrices as Skew Lines" —

First page of 'Configurations in Quantum Mechanics,' by E.M. Bruins, 1959

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

Tuesday, July 14, 2015

Go Set a Structure

Filed under: General — Tags: , , — m759 @ 2:45 PM

Saturday, January 24, 2015

Now We Are Six (continued)

Filed under: General — Tags: — m759 @ 11:01 AM

Wikipedia on a 1953 novel by Theodore Sturgeon:

"The novel concerns the coming together of
six extraordinary people with strange powers…."

Review of the novel by Ted Gioia at Conceptual Fiction :

"If there is a flaw to More Than Human , it comes from
the writer’s desire to achieve more than fiction. If you 
think that sci-fi books don’t pay attention to deep
inner meanings, you will be surprised by the conclusion
to this work, in which Sturgeon reaches for something
bigger than a story of this scale can deliver."

Background: Sturgeon's novella Baby is Three  (1952).

See also  6!  in this journal.

Saturday, December 6, 2014

Six-Point Theology

Filed under: General,Geometry — m759 @ 12:00 PM

On the feast of Saint Nicholas

See also the six  posts on this year's feast of Saint Andrew
and the following from the University  of St. Andrews —

Friday, April 25, 2014

Quilt Geometry

Filed under: General,Geometry — Tags: , — m759 @ 7:55 PM

or: The Dead Hand Shot

Library Thing book list: 'An Awkward Lie' and 'A Piece of Justice'

See also Tumbling Blocks Quilt and Springtime for Vishnu.

Tuesday, April 23, 2013

The Six-Set

Filed under: General,Geometry — Tags: , — m759 @ 3:00 AM

The configurations recently discussed in
Classical Geometry in Light of Galois Geometry
are not unrelated to the 27 "Solomon's Seal Lines
extensively studied in the 19th century.

See, in particular—

IMAGE- Archibald Henderson on six-set geometry (1911)

The following figures supply the connection of Henderson's six-set
to the Galois geometry previously discussed in "Classical Geometry…"—

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

Saturday, November 10, 2012

Battlefield Geometry

Filed under: General,Geometry — Tags: — m759 @ 5:24 AM

(Continued)

Click to enlarge.

Related material from Wikipedia— Baseball metaphors for sex.

"Build it…"

Friday, November 9, 2012

Battlefield Geometry

Filed under: General,Geometry — Tags: — m759 @ 3:28 PM

(Continued from Sept. 11, 2007)

CIA Director David Petraeus resigns, cites extramarital affair

Trouble with the curve?

Wednesday, April 28, 2010

Eightfold Geometry

Filed under: General,Geometry — Tags: , , — m759 @ 11:07 AM

Image-- The 35 partitions of an 8-set into two 4-sets

Image-- Analysis of structure of the 35 partitions of an 8-set into two 4-sets

Image-- Miracle Octad Generator of R.T. Curtis

Related web pages:

Miracle Octad Generator,
Generating the Octad Generator,
Geometry of the 4×4 Square

Related folklore:

“It is commonly known that there is a bijection between the 35 unordered triples of a 7-set [i.e., the 35 partitions of an 8-set into two 4-sets] and the 35 lines of PG(3,2) such that lines intersect if and only if the corresponding triples have exactly one element in common.” –“Generalized Polygons and Semipartial Geometries,” by F. De Clerck, J. A. Thas, and H. Van Maldeghem, April 1996 minicourse, example 5 on page 6

The Miracle Octad Generator may be regarded as illustrating the folklore.

Update of August 20, 2010–

For facts rather than folklore about the above bijection, see The Moore Correspondence.

Thursday, April 22, 2010

Mere Geometry

Filed under: General,Geometry — Tags: — m759 @ 1:00 PM

Image-- semeion estin ou meros outhen

Image-- Euclid's definition of 'point'

Stanford Encyclopedia of Philosophy

Mereology (from the Greek μερος, ‘part’) is the theory of parthood relations: of the relations of part to whole and the relations of part to part within a whole. Its roots can be traced back to the early days of philosophy, beginning with the Presocratics….”

A non-Euclidean* approach to parts–

Image-- examples from Galois affine geometry

Corresponding non-Euclidean*
projective points —

Image-- The smallest Galois geometries

Richard J. Trudeau in The Non-Euclidean Revolution, chapter on “Geometry and the Diamond Theory of Truth”–

“… Plato and Kant, and most of the philosophers and scientists in the 2200-year interval between them, did share the following general presumptions:

(1) Diamonds– informative, certain truths about the world– exist.
(2) The theorems of Euclidean geometry are diamonds.

Presumption (1) is what I referred to earlier as the ‘Diamond Theory’ of truth. It is far, far older than deductive geometry.”

Trudeau’s book was published in 1987. The non-Euclidean* figures above illustrate concepts from a 1976 monograph, also called “Diamond Theory.”

Although non-Euclidean,* the theorems of the 1976 “Diamond Theory” are also, in Trudeau’s terminology, diamonds.

* “Non-Euclidean” here means merely “other than  Euclidean.” No violation of Euclid’s parallel postulate is implied.

Tuesday, October 27, 2020

“To Illustrate My Last Remark”*

Filed under: General — Tags: — m759 @ 9:32 AM

* Song lyric, soundtrack album of
Midnight in the Garden of Good and Evil

Monday, October 26, 2020

Annals of Artspeak

Filed under: General — Tags: — m759 @ 9:14 PM

See also LeWitt in this journal.

Wednesday, April 8, 2020

For LA Boulevardiers

Filed under: General — Tags: , — m759 @ 10:15 PM

A screenshot from 10:07 PM EDT —

See also this journal on Sunset Boulevard.

Friday, February 7, 2020

Correspondences

Filed under: General — Tags: , , — m759 @ 1:05 PM

The 15  2-subsets of a 6-set correspond to the 15 points of PG(3,2).
(Cullinane, 1986*)

The 35  3-subsets of a 7-set correspond to the 35 lines of PG(3,2).
(Conwell, 1910)

The 56  3-subsets of an 8-set correspond to the 56 spreads of PG(3,2).
(Seidel, 1970)

Each correspondence above may have been investigated earlier than
indicated by the above dates , which are the earliest I know of.

See also Correspondences in this journal.

* The above 1986 construction of PG(3,2) from a 6-set also appeared
in the work of other authors in 1994 and 2002 . . .

Addendum at 5:09 PM suggested by an obituary today for Stephen Joyce:

See as well the word correspondences  in
James Joyce and the Hermetic Tradition,” by William York Tindall
(Journal of the History of Ideas , Jan. 1954).

Sunday, January 26, 2020

Harmonic-Analysis Building Blocks

Filed under: General — Tags: — m759 @ 1:14 PM

See also The Eightfold Cube.

Duke Blocks

Filed under: General — Tags: — m759 @ 12:38 AM

The Wall Street Journal  Jan. 24 on a Duke University professor —

"Dr. Daubechies is best known for her work on mathematical structures
called wavelets; her discoveries have been so influential, in fact, that
these are referred to in the field as Daubechies wavelets. She describes
them as 'mathematical building blocks' that can be used to extract the 
essential elements of images or signals without losing their quality—
in effect, a new universal language for scientists and researchers."

See also this  journal on January 20-21, and …

Tuesday, January 21, 2020

Eye-in-the-Pyramid Points

Filed under: General — Tags: — m759 @ 2:06 PM

"it remains only to choose a pleasing arrangement of {1, 2, … 7}
to label the eye-in-the-pyramid points.
there are, as it’ll turn out, 168 of ’em that’ll work."

— Comment at a weblog on November 27, 2010.

See also Log24 on that date.

The 11/27/2010 comment was on a post dated November 23, 2010.

See also Log24 on that  date.

Precisely.

Monday, January 20, 2020

Dyadic Harmonic Analysis:

Filed under: General — Tags: — m759 @ 8:26 PM

The Fourfold Square and Eightfold Cube

Related material:  A Google image search for "field dream" + log24.

Saturday, December 28, 2019

Caballo Blanco

Filed under: General — Tags: , , , — m759 @ 9:02 AM

The key  is the cocktail that begins the proceedings.”

– Brian Harley, Mate in Two Moves

http://www.log24.com/log/pix18/180809-The_EIght-and-coordinates-for-PSL(2,7)-actions-500w.jpg

“Just as these lines that merge to form a key
Are as chess squares . . . .” — Katherine Neville, The Eight

“The complete projective group of collineations and dualities of the
[projective] 3-space is shown to be of order [in modern notation] 8! ….
To every transformation of the 3-space there corresponds
a transformation of the [projective] 5-space. In the 5-space, there are
determined 8 sets of 7 points each, ‘heptads’ ….”

— George M. Conwell, “The 3-space PG (3, 2) and Its Group,”
The Annals of Mathematics , Second Series, Vol. 11, No. 2 (Jan., 1910),
pp. 60-76.

“It must be remarked that these 8 heptads are the key  to an elegant proof….”

— Philippe Cara, “RWPRI Geometries for the Alternating Group A8,” in
Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference
(July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis,
James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97.

Sunday, December 22, 2019

M24 from the Eightfold Cube

Filed under: General — Tags: , — m759 @ 12:01 PM

Exercise:  Use the Guitart 7-cycles below to relate the 56 triples
in an 8-set (such as the eightfold cube) to the 56 triangles in
a well-known Klein-quartic hyperbolic-plane tiling. Then use
the correspondence of the triples with the 56 spreads of PG(3,2)
to construct M24.

Click image below to download a Guitart PowerPoint presentation.

See as well earlier posts also tagged Triangles, Spreads, Mathieu.

Friday, December 20, 2019

Triangles, Spreads, Mathieu…

Filed under: General — Tags: , — m759 @ 1:38 AM

Continued.

An addendum for the post “Triangles, Spreads, Mathieu” of Oct. 29:

Wednesday, December 11, 2019

Miracle Octad Generator Structure

Filed under: General — Tags: , , — m759 @ 11:44 PM

Miracle Octad Generator — Analysis of Structure

(Adapted from Eightfold Geometry, a note of April 28, 2010.
See also the recent post Geometry of 6 and 8.)

Friday, November 22, 2019

Triangles, Spreads, Mathieu …

Filed under: General — Tags: , — m759 @ 4:39 PM

Continued from October 29, 2019.

More illustrations (click to enlarge) —

Thursday, October 31, 2019

56 Triangles

Filed under: General — Tags: , — m759 @ 8:09 AM

The post “Triangles, Spreads, Mathieu” of October 29 has been
updated with an illustration from the Curtis Miracle Octad Generator.

Related material — A search in this journal for “56 Triangles.”

Tuesday, October 29, 2019

Triangles, Spreads, Mathieu

Filed under: General — Tags: , — m759 @ 8:04 PM

There are many approaches to constructing the Mathieu
group M24. The exercise below sketches an approach that
may or may not be new.

Exercise:

It is well-known that

 There are 56 triangles in an 8-set.
There are 56 spreads in PG(3,2).
The alternating group An is generated by 3-cycles.
The alternating group Ais isomorphic to GL(4,2).

Use the above facts, along with the correspondence
described below, to construct M24.

Some background —

A Log24 post of May 19, 2013, cites

Peter J. Cameron in a 1976 Cambridge U. Press
book — Parallelisms of Complete Designs .
See the proof of Theorem 3A.13 on pp. 59 and 60.

See also a Google search for “56 triangles” “56 spreads” Mathieu.

Update of October 31, 2019 — A related illustration —

Update of November 2, 2019 —

See also p. 284 of Geometry and Combinatorics:
Selected Works of J. J. Seidel
  (Academic Press, 1991).
That page is from a paper published in 1970.

Update of December 20, 2019 —

Monday, October 21, 2019

Algebra and Space… Illustrated.

Filed under: General — Tags: , — m759 @ 4:26 PM

Related entertainment —

Detail:

   George Steiner

"Perhaps an insane conceit."

 

Perhaps.

 

See Quantum Tesseract Theorem .

 

Perhaps Not.

 

 See Dirac and Geometry .

Friday, August 16, 2019

Nocciolo

Filed under: General — Tags: , , — m759 @ 10:45 AM

(Continued)

IMAGE- 'Nocciolo': A 'kernel' for Pascal's Hexagrammum Mysticum: The 15 2-subsets of a 6-set as points in a Galois geometry.

A revision of the above diagram showing
the Galois-addition-table structure —

Related tables from August 10

See "Schoolgirl Space Revisited."

Wednesday, July 31, 2019

The Epstein Chronicles, or:  Z is for Zorro

Filed under: General — Tags: , — m759 @ 8:54 PM

Zorro Ranch

The geometry of the 15 point-pairs in the previous post suggests a review:

From "Exploring Schoolgirl Space," July 8 —

The date  in the previous post — Oct. 9, 2018 — also suggests a review
of posts from that date now tagged Gen-Z:

Sunday, July 14, 2019

Old Pathways in Science:

Filed under: General — Tags: , — m759 @ 12:37 PM

The Quantum Tesseract Theorem Revisited

From page 274 — 

"The secret  is that the super-mathematician expresses by the anticommutation
of  his operators the property which the geometer conceives as  perpendicularity
of displacements.  That is why on p. 269 we singled out a pentad of anticommuting
operators, foreseeing that they would have an immediate application in describing
the property of perpendicular directions without using the traditional picture of space.
They express the property of perpendicularity without the picture of perpendicularity.

Thus far we have touched only the fringe of the structure of our set of sixteen E-operators.
Only by entering deeply into the theory of electrons could I show the whole structure
coming into evidence."

A related illustration, from posts tagged Dirac and Geometry —

Anticommuting Dirac matrices as spreads of projective lines

Compare and contrast Eddington's use of the word "perpendicular"
with a later use of the word by Saniga and Planat.

Friday, June 21, 2019

Cube Tales for Solstice Day

Filed under: General — Tags: , — m759 @ 3:45 PM

See also "Six-Set" in this journal
and "Cube Geometry Continues."

 
 

Friday, May 10, 2019

I Ching g6

Filed under: General — Tags: , , , — m759 @ 12:25 PM

For fans of Resonance Science

When the men on the chessboard
get up and tell you where to go ….”

Desperately Seeking Resonance

Filed under: General — Tags: , , — m759 @ 10:46 AM

Continues

Also from Fall Equinox 2018 — Looney Tune for Physicists

Thursday, May 9, 2019

Blade and Chalice at the Museum

Filed under: General — Tags: — m759 @ 9:24 PM

(For other posts on the continuing triumph of entertainment
over truth, see a Log24 search for "Night at the Museum.")

See also yesterday's post When the Men and today's previous post.

Defense Against the Dark Arts

Filed under: General — Tags: , , — m759 @ 7:00 PM

F. Lanier Graham chess set (king-queen arrangement by the Wachowskis)

Wednesday, May 8, 2019

When the Men

Filed under: General — Tags: — m759 @ 6:33 PM

In Memoriam . . .

"When the men on the chessboard
get up and tell you where to go …."

"The I Ching encodes the geometry of the fabric of spacetime."

Sure it does.
 

Wednesday, May 1, 2019

For the First of May

Filed under: General — Tags: — m759 @ 9:00 AM

"The purpose of mathematics cannot be derived from an activity 
inferior to it but from a higher sphere of human activity, namely,
religion."

 Igor Shafarevitch in 1973

"The hint half guessed, the gift half understood, is Incarnation."

— T. S. Eliot in Four Quartets

See also Ultron Cube.

Thursday, February 28, 2019

Fooling

Filed under: General — m759 @ 10:12 AM

Galois (i.e., finite) fields described as 'deep modern algebra'

IMAGE- History of Mathematics in a Nutshell

The two books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.

Note: There is no Galois (i.e., finite) field with six elements, but
the theory  of finite fields underlies applications of six-set geometry.

Wednesday, February 27, 2019

Construction of PG(3,2) from K6

Filed under: General,Geometry — Tags: , , — m759 @ 11:38 AM

From this journal on April 23, 2013

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

From this journal in 2003

From Wikipedia on Groundhog Day, 2019

Tuesday, February 26, 2019

Citation

Filed under: General — Tags: , , , — m759 @ 12:00 PM

Some related material in this journal — See a search for k6.gif.

Some related material from Harvard —

Elkies's  "15 simple transpositions" clearly correspond to the 15 edges of
the complete graph K6 and to the 15  2-subsets of a 6-set.

For the connection to PG(3,2), see Finite Geometry of the Square and Cube.

The following "manifestation" of the 2-subsets of a 6-set might serve as
the desired Wikipedia citation —

See also the above 1986 construction of PG(3,2) from a 6-set
in the work of other authors in 1994 and 2002 . . .

IMAGE- Dolgachev and Keum, coordinatization of the 4x4 array in 'Birational Automorphisms of Quartic Hessian Surfaces,' AMS Transactions, 2002

Monday, February 25, 2019

“Far from the shallow now”

Filed under: General,Geometry — Tags: , , , — m759 @ 12:06 AM

See posts tagged depth.

See as well Eddington Song and the previous post.

Wednesday, February 13, 2019

April 18, 2003 (Good Friday), Continued

Filed under: General,Geometry — Tags: , — m759 @ 11:03 AM

"The purpose of mathematics cannot be derived from an activity 
inferior to it but from a higher sphere of human activity, namely,
religion."

Igor Shafarevitch, 1973 remark published as above in 1982.

"Perhaps."

— Steven H. Cullinane, February 13, 2019

From Log24 on Good Friday, April 18, 2003

. . . What, indeed, is truth?  I doubt that the best answer can be learned from either the Communist sympathizers of MIT or the “Red Mass” leftists of Georgetown.  For a better starting point than either of these institutions, see my note of April 6, 2001, Wag the Dogma.

See, too, In Principio Erat Verbum , which notes that “numbers go to heaven who know no more of God on earth than, as it were, of sun in forest gloom.”

Since today is the anniversary of the death of MIT mathematics professor Gian-Carlo Rota, an example of “sun in forest gloom” seems the best answer to Pilate’s question on this holy day.  See

The Shining of May 29.

“Examples are the stained glass windows
of knowledge.” — Vladimir Nabokov

AGEOMETRETOS MEDEIS EISITO

Motto of Plato’s Academy


 The Exorcist, 1973

Detail from an image linked to in the above footnote —

"And the darkness comprehended it not."

Id est :

A Good Friday, 2003, article by 
a student of Shafarevitch

" there are 25 planes in W . . . . Of course,
replacing {a,b,c} by the complementary set
does not change the plane. . . ."

Of course.

See. however, Six-Set Geometry in this  journal.

Saturday, December 22, 2018

Cremona-Richmond

Filed under: General,Geometry — Tags: , — m759 @ 12:34 PM

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" —

Detail of Feb. 20, 1986, note by Steven H. Cullinane on Weyl's 'relativity problem'

Kibler's 2008 'Variations on a theme' illustrated.

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."

Thursday, December 6, 2018

The Mathieu Cube of Iain Aitchison

This journal ten years ago today —

Surprise Package

Santa and a cube
From a talk by a Melbourne mathematician on March 9, 2018 —

The Mathieu group cube of Iain Aitchison (2018, Hiroshima)

The source — Talk II below —

Search Results

pdf of talk I  (March 8, 2018)

www.math.sci.hiroshima-u.ac.jp/branched/…/Aitchison-Hiroshima-2018-Talk1-2.pdf

Iain Aitchison. Hiroshima  University March 2018 … Immediate: Talk given last year at Hiroshima  (originally Caltech 2010).

pdf of talk II  (March 9, 2018)  (with model for M24)

www.math.sci.hiroshima-u.ac.jp/branched/files/…/Aitchison-Hiroshima-2-2018.pdf

Iain Aitchison. Hiroshima  University March 2018. (IRA: Hiroshima  03-2018). Highly symmetric objects II.

Abstract

www.math.sci.hiroshima-u.ac.jp/branched/files/2018/abstract/Aitchison.txt

Iain AITCHISON  Title: Construction of highly symmetric Riemann surfaces , related manifolds, and some exceptional objects, I, II Abstract: Since antiquity, some …

Related material — 

The 56 triangles of  the eightfold cube . . .

The Eightfold Cube: The Beauty of Klein's Simple Group

   Image from Christmas Day 2005.

Sunday, November 18, 2018

Space Music

Filed under: General,Geometry — Tags: , , — m759 @ 9:27 AM

'The Eddington Song,' based on 'The Philosophy of Physical Science,' p. 141 (1939)

Update of Nov. 19 —

"Design is how it works." — Steve Jobs

See also www.cullinane.design.

Sunday, October 21, 2018

For Connoisseurs of Bad Art

Filed under: General — Tags: , — m759 @ 7:00 PM

Yesterday afternoon's post "Study in Blue and Pink" featured 
an image related to the "Blade and Chalice" of Dan Brown 

Requiem for a comics character known as "The Blue Blade" —

 

"We all float down here."

About the corresponding "Pink Chalice," the less said the better.

Saturday, October 20, 2018

Study in Blue and Pink

Filed under: General,Geometry — Tags: , — m759 @ 3:00 PM

Related Log24 posts — See Blade + Chalice.

Tuesday, September 25, 2018

Trinity

Filed under: General — Tags: , , — m759 @ 10:10 AM

See some posts related to three names
associated with Trinity College, Cambridge —

Atiyah + Shaw + Eddington .

Saturday, September 1, 2018

Ron Shaw — D. 21 June 2016

The date of Ron Shaw's 2016 death appears to be June 21:

http://www.log24.com/log/pix18/180901-Ron_Shaw-d_21_June_2016-LMS-500w.jpg

All other Internet sources I have seen omit the June 21 date.

This  journal on that date —

http://www.log24.com/log/pix18/180901-The_Central_Structure-21_June_2016.jpg

Sunday, July 1, 2018

Deutsche Ordnung

The title is from a phrase spoken, notably, by Yul Brynner
to Christopher Plummer in the 1966 film “Triple Cross.”

Related structures —

Greg Egan’s animated image of the Klein quartic —

For a smaller tetrahedral arrangement, within the Steiner quadruple
system of order 8 modeled by the eightfold cube, see a book chapter
by Michael Huber of Tübingen

Steiner quadruple system in eightfold cube

For further details, see the June 29 post Triangles in the Eightfold Cube.

See also, from an April 2013 philosophical conference:

Abstract for a talk at the City University of New York:

The Experience of Meaning
Jan Zwicky, University of Victoria
09:00-09:40 Friday, April 5, 2013

Once the question of truth is settled, and often prior to it, what we value in a mathematical proof or conjecture is what we value in a work of lyric art: potency of meaning. An absence of clutter is a feature of such artifacts: they possess a resonant clarity that allows their meaning to break on our inner eye like light. But this absence of clutter is not tantamount to ‘being simple’: consider Eliot’s Four Quartets  or Mozart’s late symphonies. Some truths are complex, and they are simplified  at the cost of distortion, at the cost of ceasing to be  truths. Nonetheless, it’s often possible to express a complex truth in a way that precipitates a powerful experience of meaning. It is that experience we seek — not simplicity per se , but the flash of insight, the sense we’ve seen into the heart of things. I’ll first try to say something about what is involved in such recognitions; and then something about why an absence of clutter matters to them.

For the talk itself, see a YouTube video.

The conference talks also appear in a book.

The book begins with an epigraph by Hilbert

Friday, June 29, 2018

Triangles in the Eightfold Cube

Filed under: General,Geometry — Tags: , — m759 @ 9:10 PM

From a post of July 25, 2008, “56 Triangles,” on the Klein quartic
and the eightfold cube

Baez’s discussion says that the Klein quartic’s 56 triangles
can be partitioned into 7 eight-triangle Egan ‘cubes’ that
correspond to the 7 points of the Fano plane in such a way
that automorphisms of the Klein quartic correspond to
automorphisms of the Fano plane. Show that the
56 triangles within the eightfold cube can also be partitioned
into 7 eight-triangle sets that correspond to the 7 points of the
Fano plane in such a way that (affine) transformations of the
eightfold cube induce (projective) automorphisms of the Fano plane.”

Related material from 1975 —

More recently

Thursday, March 29, 2018

“Before Creation Itself . . .”

Filed under: General,Geometry — Tags: , , , — m759 @ 10:13 AM

From the Diamond Theorem Facebook page —

A question three hours ago at that page

“Is this Time Cube?”

Notes toward an answer —

And from Six-Set Geometry in this journal . . .

Monday, March 12, 2018

“Quantum Tesseract Theorem?”

Filed under: General,Geometry — Tags: , , — m759 @ 11:00 AM

Remarks related to a recent film and a not-so-recent film.

For some historical background, see Dirac and Geometry in this journal.

Also (as Thas mentions) after Saniga and Planat —

The Saniga-Planat paper was submitted on December 21, 2006.

Excerpts from this  journal on that date —

A Halmos tombstone and the tale of HAL and the pod bay doors

     "Open the pod bay doors, HAL."

Thursday, March 1, 2018

The Movement of Analogy: Hume vs. Paz

Filed under: General — Tags: , — m759 @ 11:00 AM

Hume, from posts tagged "four-set" in this journal —

"The mind is a kind of theatre, where several perceptions
successively make their appearance; pass, repass, glide away,
and mingle in an infinite variety of postures and situations.
There is properly no simplicity in it at one time, nor identity
in different, whatever natural propension we may have
to imagine that simplicity and identity."

Paz, from a search for Paz + Identity in this journal —

"At the point of convergence
the play of similarities and differences
cancels itself out in order that 
identity alone may shine forth
The illusion of motionlessness,
the play of mirrors of the one: 
identity is completely empty;
it is a crystallization and
in its transparent core
the movement of analogy 
begins all over once again."

— The Monkey Grammarian 

by Octavio Paz, translated by Helen Lane 

Saturday, February 17, 2018

The Binary Revolution

Michael Atiyah on the late Ron Shaw

Phrases by Atiyah related to the importance in mathematics
of the two-element Galois field GF(2) —

  • “The digital revolution based on the 2 symbols (0,1)”
  • “The algebra of George Boole”
  • “Binary codes”
  • “Dirac’s spinors, with their up/down dichotomy”

These phrases are from the year-end review of Trinity College,
Cambridge, Trinity Annual Record 2017 .

I prefer other, purely geometric, reasons for the importance of GF(2) —

  • The 2×2 square
  • The 2x2x2 cube
  • The 4×4 square
  • The 4x4x4 cube

See Finite Geometry of the Square and Cube.

See also today’s earlier post God’s Dice and Atiyah on the theology of
(Boolean) algebra vs. (Galois) geometry:

God’s Dice

Filed under: General,Geometry — Tags: , — m759 @ 10:45 AM

On a Trinity classmate of Ian Macdonald (see previous post)—

Atiyah's eulogy of Shaw in Trinity Annual Record 2017 
is on pages 137 through 146.  The conclusion —

 

Friday, February 2, 2018

For Plato’s Cave

Filed under: General,Geometry — Tags: , , — m759 @ 12:06 PM

"Plato's allegory of the cave describes prisoners,
inhabiting the cave since childhood, immobile,
facing an interior wall. A large fire burns behind
the prisoners, and as people pass this fire their
shadows are cast upon the cave's wall, and
these shadows of the activity being played out
behind the prisoner become the only version of
reality that the prisoner knows."

— From the Occupy Space gallery in Ireland

IMAGE- Patrick McGoohan as 'The Prisoner,' with lapel button that says '6.'

Monday, October 30, 2017

For Devil’s Night

Filed under: General — Tags: — m759 @ 11:25 PM

Location,  Location,  Location

From a Los Angeles Times  piece on Epiphany (Jan. 6), 1988 —

“Some 30 paces east of the spooky old Chateau Marmont is
the intersection of Selma and Sunset Boulevard.” . . . .
“Though it is not much of an intersection, the owner of
the liquor store on that corner might resent that you have
slotted his parking lot in the Twilight Zone. . . .
And directly across Sunset from Selma looking south is
where the infamous Garden of Allah used to stand. . . .”

Tuesday, October 10, 2017

Another 35-Year Wait

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 9:00 PM

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.

The Lévi-Strauss formula

My "inscape" formula, from a note of Sept. 22, 1982 —

S = f ( f ( X ) ) .

Some mathematics from last year related to the 1982 formula —

Koen Thas, 'Unextendible Mututally Unbiased Bases' (2016)

See also Inscape in this  journal and posts tagged Dirac and Geometry.

Tuesday, September 12, 2017

Goals

Filed under: General,Geometry — Tags: — m759 @ 12:18 PM

"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.

Monday, September 11, 2017

New Depth

Filed under: General,Geometry — Tags: — m759 @ 9:48 PM

A sentence from the New York Times Wire  discussed in the previous post

NYT Wire on Len Wein: 'Through characters like Wolverine and Swamp Thing, he helped bring a new depth to his art form.'

"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 …

Wednesday, August 9, 2017

War Game: 8/09

Filed under: General — Tags: — m759 @ 1:20 PM

"Fire and Fury  is one of the most popular
historical military miniatures wargames . . . .

For the new player it is an easy game
to learn and enjoy." 

fireandfury.com

Thursday, May 25, 2017

Presidential Address of November 19, 1976

Filed under: General — Tags: — m759 @ 2:14 PM

Deep Problems in the Faculty Lounge

Filed under: General — Tags: — m759 @ 7:14 AM

Tuesday, May 2, 2017

Image Albums

Filed under: General,Geometry — Tags: — m759 @ 1:05 PM

Pinterest boards uploaded to the new m759.net/piwigo

Diamond Theorem 

Diamond Theorem Correlation

Miracle Octad Generator

The Eightfold Cube

Six-Set Geometry

Diamond Theory Cover

Update of May 2 —

Four-Color Decomposition

Binary Galois Spaces

The Galois Tesseract

Update of May 3 —

Desargues via Galois

The Tetrahedral Model

Solomon's Cube

Update of May 8 —

Art Space board created at Pinterest

Wednesday, April 26, 2017

A Tale Unfolded

Filed under: General,Geometry — Tags: , , — m759 @ 2:00 AM

A sketch, adapted tonight from Girl Scouts of Palo Alto

From the April 14 noon post High Concept

From the April 14 3 AM post Hudson and Finite Geometry

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

From the April 24 evening post The Trials of Device

Pentagon with pentagram    

Note that Hudson's 1905 "unfolding" of even and odd puts even on top of
the square array, but my own 2013 unfolding above puts even at its left.

Sunday, March 26, 2017

Seagram Studies

Filed under: General,Geometry — Tags: — m759 @ 9:00 PM

From a search in this journal for Seagram + Tradition

Related art:  Saturday afternoon's Twin Pillars of Symmetry.

Saturday, March 25, 2017

Twin Pillars of Symmetry

Filed under: General,Geometry — Tags: — m759 @ 1:00 PM

The phrase "twin pillars" in a New York Times  Fashion & Style
article today suggests a look at another pair of pillars —

This pair, from the realm of memory, history, and geometry disparaged
by the late painter Mark Rothko, might be viewed by Rothko
as  "parodies of ideas (which are ghosts)." (See the previous post.)

For a relationship between a 3-dimensional simplex and the {4, 3, 3},
see my note from May 21, 2014, on the tetrahedron and the tesseract.

Like Decorations in a Cartoon Graveyard

Filed under: General,Geometry — Tags: — m759 @ 4:00 AM

Continued from April 11, 2016, and from

A tribute to Rothko suggested by the previous post

For the idea  of Rothko's obstacles, see Hexagram 39 in this journal.

Wednesday, March 8, 2017

Inscapes

Filed under: General,Geometry — Tags: — m759 @ 6:42 PM

"The particulars of attention,
whether subjective or objective,
are unshackled through form,
and offered as a relational matrix …."

— Kent Johnson in a 1993 essay

Illustration

Commentary

The 16 Dirac matrices form six anticommuting sets of five matrices each (Arfken 1985, p. 214):

1. alpha_1alpha_2alpha_3alpha_4alpha_5,

2. y_1y_2y_3y_4y_5,

3. delta_1delta_2delta_3rho_1rho_2,

4. alpha_1y_1delta_1sigma_2sigma_3,

5. alpha_2y_2delta_2sigma_1sigma_3,

6. alpha_3y_3delta_3sigma_1sigma_2.

SEE ALSO:  Pauli Matrices

REFERENCES:

Arfken, G. Mathematical Methods for Physicists, 3rd ed.  Orlando, FL: Academic Press, pp. 211-217, 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. 80-84, 1982.

Bethe, H. A. and Salpeter, E. Quantum Mechanics of One- and Two-Electron Atoms.  New York: Plenum, pp. 47-48, 1977.

Bjorken, J. D. and Drell, S. D. Relativistic Quantum Mechanics.  New York: McGraw-Hill, 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: Addison-Wesley, p. 580, 1980.

Good, R. H. Jr. "Properties of Dirac Matrices." Rev. Mod. Phys. 27, 187-211, 1955.

Referenced on Wolfram|Alpha:  Dirac Matrices

CITE THIS AS:

Weisstein, Eric W.  "Dirac Matrices."

From MathWorld— A Wolfram Web Resource. 
http://mathworld.wolfram.com/DiracMatrices.html

Friday, February 24, 2017

For Your Consideration

Filed under: General — Tags: , — m759 @ 12:25 AM

Hollywood, from the Alto Nido Apartments
to Sunset Boulevard —

From Alto Nido Apts. to Sunset Boulevard: Aerial view including Los Angeles Film School

See also the Jan. 31 post "Sunset Passion."

Friday, February 3, 2017

Raiders of the Lost Chalice

Filed under: General — Tags: — m759 @ 9:30 AM

Personally, I prefer
the religious symbolism
of Hudson Hawk .

Sunday, December 25, 2016

Credit Where Due

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM

See also Robert M. Pirsig in this journal on Dec. 26, 2012.

Saturday, December 24, 2016

Early X Piece

Filed under: General,Geometry — Tags: — m759 @ 11:00 AM

In memory of an American artist whose work resembles that of
the Soviet constructivist Karl Ioganson (c. 1890-1929).

The American artist reportedly died on Thursday, Dec. 22, 2016.

"In fact, the (re-)discovery of this novel structural principle was made in 1948-49 by a young American artist whom Koleichuk also mentions, Kenneth Snelson. In the summer of 1948, Snelson had gone to study with Joseph Albers who was then teaching at Black Mountain College. . . . One of the first works he made upon his return home was Early X Piece  which he dates to December 1948 . . . . "

— "In the Laboratory of Constructivism:
      Karl Ioganson's Cold Structures,"
      by Maria GoughOCTOBER  Magazine, MIT,
      Issue 84, Spring 1998, pp. 91-117

The word "constructivism" also refers to a philosophy of mathematics.
See a Log24 post, "Constructivist Witness,"  of 1 AM ET on the above
date of death.

Thursday, December 22, 2016

The Laugh-Hospital

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM

Constructivism in mathematics and the laughing academy

See also, from the above publication date, Hudson's Inscape.
The inscape is illustrated in posts now tagged Laughing Academy.

Constructivist Witness

Filed under: General,Geometry — Tags: — m759 @ 1:00 AM

The title refers to a philosophy of mathematics.

For those who prefer metaphor Folk Etymology.

See also Stages of Math at Princeton's  
Institute for Advanced Study in March 2013 —

— and in this journal starting in August 2014.

Monday, December 19, 2016

ART WARS

Filed under: General,Geometry — Tags: , — m759 @ 10:25 PM

See also all posts now tagged Memory, History, Geometry.

Tetrahedral Cayley-Salmon Model

Filed under: General,Geometry — Tags: , — m759 @ 9:38 AM

The figure below is one approach to the exercise
posted here on December 10, 2016.

Tetrahedral model (minus six lines) of the large Desargues configuration

Some background from earlier posts —


IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

Click the image below to enlarge it.

Polster's tetrahedral model of the small Desargues configuration

Sunday, December 18, 2016

Two Models of the Small Desargues Configuration

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM

Click image to enlarge.

Polster's tetrahedral model of the small Desargues configuration

See also the large  Desargues configuration in this journal.

Saturday, December 17, 2016

Tetrahedral Death Star

Filed under: General,Geometry — Tags: , — m759 @ 10:00 PM

Continuing the "Memory, History, Geometry" theme
from yesterday

See Tetrahedral,  Oblivion,  and Tetrahedral Oblivion.

IMAGE- From 'Oblivion' (2013), the Mother Ship

"Welcome home, Jack."

Friday, December 16, 2016

Read Something That Means Something

Filed under: General,Geometry — Tags: — m759 @ 5:29 PM

Rothko’s Swamps

Filed under: General,Geometry — Tags: — m759 @ 2:45 AM

"… you don’t write off an aging loved one
just because he or she becomes cranky."

— Peter Schjeldahl on Rothko in The New Yorker ,
issue dated December 19 & 26, 2016, page 27

He was cranky in his forties too —

See Rothko + Swamp in this journal.

Related attitude —

From Subway Art for Times Square Church , Nov. 7

Tuesday, December 13, 2016

The Thirteenth Novel

Filed under: General,Geometry — Tags: — m759 @ 4:00 PM

John Updike on Don DeLillo's thirteenth novel, Cosmopolis

" DeLillo’s post-Christian search for 'an order at some deep level'
has brought him to global computerization:
'the zero-oneness of the world, the digital imperative . . . . ' "

The New Yorker , issue dated March 31, 2003

On that date ….

Related remark —

" There is a pleasantly discursive treatment 
of Pontius Pilate’s unanswered question
‘What is truth?’ "

— Coxeter, 1987, introduction to Trudeau’s
     The Non-Euclidean Revolution

Saturday, December 10, 2016

Folk Etymology

Images from Burkard Polster's Geometrical Picture Book

See as well in this journal the large  Desargues configuration, with
15 points and 20 lines instead of 10 points and 10 lines as above.

Exercise:  Can the large Desargues configuration be formed
by adding 5 points and 10 lines to the above Polster model
of the small configuration in such a way as to preserve
the small-configuration model's striking symmetry?  
(Note: The related figure below from May 21, 2014, is not
necessarily very helpful. Try the Wolfram Demonstrations
model
, which requires a free player download.)

Labeling the Tetrahedral Model (Click to enlarge) —

Related folk etymology (see point a  above) —

Related literature —

The concept  of "fire in the center" at The New Yorker , 
issue dated December 12, 2016, on pages 38-39 in the
poem by Marsha de la O titled "A Natural History of Light."

Cézanne's Greetings.

Wednesday, December 7, 2016

Spreads and Conwell’s Heptads

Filed under: General,Geometry — Tags: — m759 @ 7:11 PM

For a concise historical summary of the interplay between
the geometry of an 8-set and that of a 16-set that is
involved in the the Miracle Octad Generator approach
to the large Mathieu group M24, see Section 2 of 

Alan R. Prince
A near projective plane of order 6 (pp. 97-105)
Innovations in Incidence Geometry
Volume 13 (Spring/Fall 2013).

This interplay, notably discussed by Conwell and
by Edge, involves spreads and Conwell’s heptads .

Update, morning of the following day (7:07 ET) — related material:

See also “56 spreads” in this  journal.

Tuesday, October 18, 2016

Parametrization

Filed under: General,Geometry — Tags: — m759 @ 6:00 AM

The term "parametrization," as discussed in Wikipedia,
seems useful for describing labelings that are not, at least
at first glance, of a vector-space  nature.

Examples: The labelings of a 4×4 array by a blank space
plus the 15 two-subsets of a six-set (Hudson, 1905) or by a
blank plus the 5 elements and the 10 two-subsets of a five-set
(derived in 2014 from a 1906 page by Whitehead), or by 
a blank plus the 15 line diagrams of the diamond theorem.

Thus "parametrization" is apparently more general than
the word "coodinatization" used by Hermann Weyl —

“This is the relativity problem:  to fix objectively
a class of equivalent coordinatizations and to
ascertain the group of transformations S
mediating between them.”

— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16

Note, however, that Weyl's definition of "coordinatization"
is not limited to vector-space  coordinates. He describes it
as simply a mapping to a set of reproducible symbols

(But Weyl does imply that these symbols should, like vector-space 
coordinates, admit a group of transformations among themselves
that can be used to describe transformations of the point-space
being coordinatized.)

Thursday, September 15, 2016

The Smallest Perfect Number/Universe

Filed under: General,Geometry — Tags: , , — m759 @ 6:29 AM

The smallest perfect number,* six, meets
"the smallest perfect universe,"** PG(3,2).

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

  * For the definition of "perfect number," see any introductory
    number-theory text that deals with the history of the subject.
** The phrase "smallest perfect universe" as a name for PG(3,2),
     the projective 3-space over the 2-element Galois field GF(2),
     was coined by math writer Burkard Polster. Cullinane's square
     model of PG(3,2) differs from the earlier tetrahedral model
     discussed by Polster.

Tuesday, September 13, 2016

Parametrizing the 4×4 Array

Filed under: General,Geometry — Tags: , , , — m759 @ 10:00 PM

The previous post discussed the parametrization of 
the 4×4 array as a vector 4-space over the 2-element 
Galois field GF(2).

The 4×4 array may also be parametrized by the symbol
0  along with the fifteen 2-subsets of a 6-set, as in Hudson's
1905 classic Kummer's Quartic Surface

Hudson in 1905:

These two ways of parametrizing the 4×4 array — as a finite space
and as an array of 2-element sets —  were related to one another
by Cullinane in 1986 in describing, in connection with the Curtis
"Miracle Octad Generator,"  what turned out to be 15 of Hudson's
1905 "Göpel tetrads":

A recap by Cullinane in 2013:

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

Click images for further details.

Monday, September 12, 2016

The Kummer Lattice

Filed under: General,Geometry — Tags: , , , — m759 @ 2:00 PM

The previous post quoted Tom Wolfe on Chomsky's use of
the word "array." 

An example of particular interest is the 4×4  array
(whether of dots or of unit squares) —

      .

Some context for the 4×4 array —

The following definition indicates that the 4×4 array, when
suitably coordinatized, underlies the Kummer lattice .

Further background on the Kummer lattice:

Alice Garbagnati and Alessandra Sarti, 
"Kummer Surfaces and K3 surfaces
with $(Z/2Z)^4$ symplectic action." 
To appear in Rocky Mountain J. Math.

The above article is written from the viewpoint of traditional
algebraic geometry. For a less traditional view of the underlying
affine 4-space from finite  geometry, see the website
Finite Geometry of the Square and Cube.

Some further context

"To our knowledge, the relation of the Golay code
to the Kummer lattice is a new observation."

— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of
Kummer surfaces in the Mathieu group M24 
"

As noted earlier, Taormina and Wendland seem not to be aware of
R. W. H. T. Hudson's use of the (uncoordinatized*) 4×4 array in his
1905 book Kummer's Quartic Surface.  The array was coordinatized,
i.e. given a "vector space structure," by Cullinane eight years prior to
the cited remarks of Curtis.

* Update of Sept. 14: "Uncoordinatized," but parametrized  by 0 and
the 15 two-subsets of a six-set. See the post of Sept. 13.

Friday, September 9, 2016

Notation

Filed under: General — Tags: , — m759 @ 10:00 PM

The New York Times  interviews Alan Moore

"A version of this article appears in print on September 11, 2016,
on page BR9 of the Sunday Book Review …."

"What genres do you prefer? And which do you avoid?"

"To be honest, having worked in genre for so long, I’m happiest
when I’m outside it altogether, or perhaps more accurately,
when I can conjure multiple genres all at once, in accordance
with my theory (now available, I believe, as a greeting card and
fridge magnet) that human life as we experience it is a
simultaneous multiplicity of genres. I put it much more elegantly
on the magnet."

Tuesday, August 9, 2016

Gestalt

Filed under: General,Geometry — Tags: , — m759 @ 9:29 PM

See "Smallest Perfect" and "We Are Six."

Tuesday, July 12, 2016

Group Elements and Skew Lines

Filed under: General,Geometry — Tags: — m759 @ 12:00 AM

The following passage by Igor Dolgachev (Good Friday, 2003
seems somewhat relevant (via its connection to Kummer's 166 )
to previous remarks here on Dirac matrices and geometry

Note related remarks from E. M. Bruins in 1959 —

First page of 'Configurations in Quantum Mechanics,' by E.M. Bruins, 1959

Monday, June 20, 2016

Plan 9 Continues

Filed under: General — Tags: — m759 @ 11:00 AM

See …

At the Still Point … (February 12, 2008)

For Balanchine's Birthday (January 9, 2007)

Go Set a Structure (Various dates)

and …

Friday, June 3, 2016

Bruins and van Dam

Filed under: General,Geometry — Tags: , — m759 @ 8:00 AM

A review of some recent posts on Dirac and geometry,
each of which mentions the late physicist Hendrik van Dam:

The first of these posts mentions the work of E. M. Bruins.
Some earlier posts that cite Bruins:

Monday, May 30, 2016

Perfect Universe

Filed under: General,Geometry — Tags: — m759 @ 7:00 PM

(A sequel to the previous post, Perfect Number)

Since antiquity,  six has been known as
"the smallest perfect number." The word "perfect"
here means that a number is the sum of its 
proper divisors — in the case of six: 1, 2, and 3.

The properties of a six-element set (a "6-set") 
divided into three 2-sets and divided into two 3-sets
are those of what Burkard Polster, using the same 
adjective in a different sense, has called 
"the smallest perfect universe" — PG(3,2), the projective
3-dimensional space over the 2-element Galois field.

A Google search for the phrase "smallest perfect universe"
suggests a turnaround in meaning , if not in finance, 
that might please Yahoo CEO Marissa Mayer on her birthday —

The semantic  turnaround here in the meaning  of "perfect"
is accompanied by a model  turnaround in the picture  of PG(3,2) as
Polster's tetrahedral  model is replaced by Cullinane's square  model.

Further background from the previous post —

See also Kirkman's Schoolgirl Problem.

Perfect Number

Filed under: General,Geometry — Tags: — m759 @ 10:00 AM

"Ageometretos me eisito."—
"Let no one ignorant of geometry enter."—
Said to be a saying of Plato, part of the
seal of the American Mathematical Society—

For the birthday of Marissa Mayer, who turns 41 today —

VOGUE Magazine,
AUGUST 16, 2013 12:01 AM
by JACOB WEISBERG —

"As she works to reverse the fortunes of a failing Silicon Valley
giant, Yahoo’s Marissa Mayer has fueled a national debate
about the office life, motherhood, and what it takes to be the
CEO of the moment.

'I really like even numbers, and
I like heavily divisible numbers.
Twelve is my lucky number—
I just love how divisible it is.
I don’t like odd numbers, and
I really don’t like primes.
When I turned 37,
I put on a strong face, but
I was not looking forward to 37.
But 37 turned out to be a pretty amazing year.
Especially considering that
36 is divisible by twelve!'

A few things may strike you while listening to Marissa Mayer
deliver this riff . . . . "

Yes, they may.

A smaller number for Marissa's meditations:

Six has been known since antiquity as the first "perfect" number.
Why it was so called is of little interest to anyone but historians
of number theory  (a discipline that is not, as Wikipedia notes, 
to be confused with numerology .)

What part geometry , on the other hand, played in Marissa's education,
I do not know.

Here, for what it's worth, is a figure from a review of posts in this journal
on the key role played by the number six in geometry —

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