Log24

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'  "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

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.

“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.

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.

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.

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, 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 . . .

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.'

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"

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.

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.

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.

Tuesday, August 9, 2016

Gestalt

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

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

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 —

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 13, 2015

A Connection between the 16 Dirac Matrices and the Large Mathieu Group

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



Note that the six anticommuting sets of Dirac matrices listed by Arfken
correspond exactly to the six spreads in the above complex of 15 projective
lines of PG(3,2) fixed under a symplectic polarity (the diamond theorem
correlation
 
). As I noted in 1986, this correlation underlies the Miracle
Octad Generator of R. T. Curtis, hence also the large Mathieu group.

References:

Arfken, George B., Mathematical Methods for Physicists , Third Edition,
Academic Press, 1985, pages 213-214

Cullinane, Steven H., Notes on Groups and Geometry, 1978-1986

Related material:

The 6-set in my 1986 note above also appears in a 1996 paper on
the sixteen Dirac matrices by David M. Goodmanson —

Background reading:

Ron Shaw on finite geometry, Clifford algebras, and Dirac groups 
(undated compilation of publications from roughly 1994-1995)—

Sunday, August 24, 2014

Symplectic Structure…

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

In the Miracle Octad Generator (MOG):

The above details from a one-page note of April 26, 1986, refer to the
Miracle Octad Generator of R. T. Curtis, as it was published in 1976:

http://www.log24.com/log/pix10A/100514-Curtis1976MOG.jpg

From R. T. Curtis (1976). A new combinatorial approach to M24,
Mathematical Proceedings of the Cambridge Philosophical Society ,
79, pp 25-42. doi:10.1017/S0305004100052075.

The 1986 note assumed that the reader would be able to supply, from the
MOG itself, the missing top row of each heavy brick.

Note that the interchange of the two squares in the top row of each
heavy brick induces the diamond-theorem correlation.

Note also that the 20 pictured 3-subsets of a 6-set in the 1986 note
occur as paired complements  in two pictures, each showing 10 of the
3-subsets.

This pair of pictures corresponds to the 20 Rosenhain tetrads  among
the 35 lines of PG(3,2), while the picture showing the 2-subsets
corresponds to the 15 Göpel tetrads  among the 35 lines.

See Rosenhain and Göpel tetrads in PG(3,2). Some further background:

Wednesday, August 13, 2014

Symplectic Structure continued

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

Some background for the part of the 2002 paper by Dolgachev and Keum
quoted here on January 17, 2014 —

Related material in this journal (click image for posts) —

Saturday, September 21, 2013

Geometric Incarnation

The  Kummer 166  configuration  is the configuration of sixteen
6-sets within a 4×4 square array of points in which each 6-set
is determined by one of the 16 points of the array and
consists of the 3 other points in that point's row and the
3 other points in that point's column.

See Configurations and Squares.

The Wikipedia article Kummer surface  uses a rather poetic
phrase* to describe the relationship of the 166 to a number
of other mathematical concepts — "geometric incarnation."

Geometric Incarnation in the Galois Tesseract

Related material from finitegeometry.org —

IMAGE- 4x4 Geometry: Rosenhain and Göpel Tetrads and the Kummer Configuration

* Apparently from David Lehavi on March 18, 2007, at Citizendium .

Wednesday, August 21, 2013

The 21

Filed under: General,Geometry — Tags: — m759 @ 8:28 PM

A useful article on finite geometry,
"21 – 6 = 15: A Connection between Two Distinguished Geometries,"
by Albrecht Beutelspacher, American Mathematical Monthly ,
Vol. 93, No. 1, January 1986, pp. 29-41, is available for purchase
at JSTOR.

This article is related to the geometry of the six-set.
For some background, see remarks from 1986 at finitegeometry.org.

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

Sunday, April 28, 2013

The Octad Generator

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

… And the history of geometry  
Desargues, Pascal, Brianchon and Galois
in the light of complete n-points in space.

(Rewritten for clarity at about 10 AM ET April 29, with quote from Dowling added.
Updated with a reference to a Veblen and Young exercise (on p. 53) on April 30.)

Veblen and Young, Projective Geometry, Vol. I ,
Ginn and Company, 1910, page 39:

"The Desargues configuration. A very important configuration
is obtained by taking the plane section of a complete space five-point."

Each of figures 14 and 15 above has 15 points and 20 lines.
The Desargues configuration within each figure is denoted by
10 white points and 10 solid lines, with 3 points on each line and
3 lines on each point. Black  points and dashed  lines indicate the
complete space five-point and lines connecting it to the plane section
containing the Desargues configuration.

In a 1915 University of Chicago doctoral thesis, Archibald Henderson
used a complete space six -point to construct a configuration of
15 points and 20 lines in the context not of Desargues '  theorem, but
rather of Brianchon 's theorem and of the Pascal  hexagram.
Henderson's 1915 configuration is, it turns out, isomorphic to that of
the 15 points and 20 lines in the configuration constructed via a
complete space five -point five years earlier by Veblen and Young.
(See, in Veblen and Young's 1910 Vol. I, exercise 11, page 53:
"A plane section of a 6-point in space can  be considered as
3 triangles perspective in pairs from 3 collinear points with
corresponding sides meeting in 3 collinear points." This is the
large  Desargues configuration. See Classical Geometry in Light of 
Galois Geometry
.)

For this large  Desargues configuration see April 19.
For Henderson's complete six –point, see The Six-Set (April 23).
That post ends with figures relating the large  Desargues configuration
to the Galois  geometry PG(3,2) that underlies the Curtis
Miracle Octad Generator  and the large Mathieu group M24 —

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

See also Note on the MOG Correspondence from April 25, 2013.

That correspondence was also discussed in a note 28 years ago, on this date in 1985.

Thursday, April 25, 2013

Note on the MOG Correspondence

Filed under: General,Geometry — Tags: , — m759 @ 4:15 PM

In light of the April 23 post "The Six-Set,"
the caption at the bottom of a note of April 26, 1986
seems of interest:

"The R. T. Curtis correspondence between the 35 lines and the
2-subsets and 3-subsets of a 6-set. This underlies M24."

A related note from today:

IMAGE- Three-sets in the Curtis MOG

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

Monday, September 4, 2006

Monday September 4, 2006

Filed under: General,Geometry — Tags: , — m759 @ 7:20 PM
In a Nutshell:
 
The Seed

"The symmetric group S6 of permutations of 6 objects is the only symmetric group with an outer automorphism….

This outer automorphism can be regarded as the seed from which grow about half of the sporadic simple groups…."

Noam Elkies, February 2006

This "seed" may be pictured as

The outer automorphism of a six-set in action

group actions on a linear complex

within what Burkard Polster has called "the smallest perfect universe"– PG(3,2), the projective 3-space over the 2-element field.

Related material: yesterday's entry for Sylvester's birthday.

Wednesday, September 3, 2003

Wednesday September 3, 2003

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

Reciprocity

From my entry of Sept. 1, 2003:

"…the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity….

… E. M. Forster famously advised his readers, 'Only connect.' 'Reciprocity' would be Michael Kruger's succinct philosophy, with all that the word implies."

— William Boyd, review of Himmelfarb, New York Times Book Review, October 30, 1994

Last year's entry on this date: 

Today's birthday:
James Joseph Sylvester

"Mathematics is the music of reason."
— J. J. Sylvester

Sylvester, a nineteenth-century mathematician, coined the phrase "synthematic totals" to describe some structures based on 6-element sets that R. T. Curtis has called "rather unwieldy objects." See Curtis's abstract, Symmetric Generation of Finite Groups, John Baez's essay, Some Thoughts on the Number 6, and my website, Diamond Theory.

The picture above is of the complete graph K6  Six points with an edge connecting every pair of points… Fifteen edges in all.

Diamond theory describes how the 15 two-element subsets of a six-element set (represented by edges in the picture above) may be arranged as 15 of the 16 parts of a 4×4 array, and how such an array relates to group-theoretic concepts, including Sylvester's synthematic totals as they relate to constructions of the Mathieu group M24.

If diamond theory illustrates any general philosophical principle, it is probably the interplay of opposites….  "Reciprocity" in the sense of Lao Tzu.  See

Reciprocity and Reversal in Lao Tzu.

For a sense of "reciprocity" more closely related to Michael Kruger's alleged philosophy, see the Confucian concept of Shu (Analects 15:23 or 24) described in

Shu: Reciprocity.

Kruger's novel is in part about a Jew: the quintessential Jewish symbol, the star of David, embedded in the K6 graph above, expresses the reciprocity of male and female, as my May 2003 archives illustrate.  The star of David also appears as part of a graphic design for cubes that illustrate the concepts of diamond theory:

Click on the design for details.

Those who prefer a Jewish approach to physics can find the star of David, in the form of K6, applied to the sixteen 4×4 Dirac matrices, in

A Graphical Representation
of the Dirac Algebra
.

The star of David also appears, if only as a heuristic arrangement, in a note that shows generating partitions of the affine group on 64 points arranged in two opposing triplets.

Having thus, as the New York Times advises, paid tribute to a Jewish symbol, we may note, in closing, a much more sophisticated and subtle concept of reciprocity due to Euler, Legendre, and Gauss.  See

The Jewel of Arithmetic and

The Golden Theorem.

Thursday, December 5, 2002

Thursday December 5, 2002

Filed under: G-Notes,General,Geometry — Tags: , , , — m759 @ 3:17 AM

Sacerdotal Jargon

From the website

Abstracts and Preprints in Clifford Algebra [1996, Oct 8]:

Paper:  clf-alg/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, 870-880 (1996)


The following is a picture of K6, 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.

The complete graph on a six-set


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."

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