Log24

Tuesday, November 7, 2017

Polarities and Correlation

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

"Read something that means something."
                — New Yorker  ad

'Knight' octad labeling by the 8 points of the projective line over GF(7) .

Thursday, December 1, 2016

Correlation/Correlative

Filed under: General — m759 @ 6:00 AM

http://m759.net/wordpress/?s="Correlation"

http://m759.net/wordpress/?tag=correlative

Related literary reference —

"The only way of expressing emotion in the form of art
is by finding an 'objective correlative'; in other words,
a set of objects, a situation, a chain of events which
shall be the formula of that particular  emotion; such that
when the external facts, which must terminate in sensory
experience, are given, the emotion is immediately evoked.
If you examine any of Shakespeare’s more successful
tragedies, you will find this exact equivalence…."

— T. S. Eliot, "Hamlet and His Problems" (1919)

Friday, August 1, 2014

The Diamond-Theorem Correlation

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

Click image for a larger, clearer version.

IMAGE- The symplectic correlation underlying Rosenhain and Göpel tetrads

Friday, September 20, 2019

Garbage-Pail Kid

Filed under: General — m759 @ 10:02 AM

In the spirit of the Linz website in the previous post
the title refers to New Yorker  writer Adam Gopnik:

Garbage-Pail Kids: Adam Bomb

Tuesday, November 7, 2017

Polarities and Correlation

Filed under: Uncategorized — Tags:  
— m759 @ 11:00 PM 

Adam Gopnik on Philip Roth

Adam Gopnik on Philip Roth and Mickey Sabbath

See also a search in this  journal for Polarity + Correlation.

Friday, March 29, 2019

Front-Row Seed

Filed under: General — Tags: — m759 @ 4:17 PM

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

Closely related material —

The Kummer 16_6 Configuration and the Nordstrom-Robinson Code

The top two cells of the Curtis "heavy brick" are also
the key to the diamond-theorem correlation.

Monday, March 11, 2019

Overarching Metanarratives

Filed under: General — m759 @ 4:15 AM

See also "Overarching + Tesseract" in this  journal. From the results
of that search, some context for the "inscape" of the previous post —

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)

Friday, November 16, 2018

The Transposed Squares

Filed under: G-Notes,General,Geometry — m759 @ 9:12 PM
 
 

I.e. (click to enlarge) —

 

Tuesday, November 6, 2018

On Mathematical Beauty

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 2:18 AM

A phrase from the previous post —
"a size-eight dame in a size-six dress" —
suggests a review . . .

See as well the diamond-theorem correlation and . . .

Why PSL(2,7) is isomorphic to GL(3.2)

Sunday, June 24, 2018

For 6/24

Filed under: General,Geometry — Tags: — m759 @ 10:12 AM

A clue to the relationship between the Kummer (16, 6)
configuration and the large Mathieu group M24

Related material —

See too the diamond-theorem correlation.

Saturday, June 9, 2018

SASTRA paper

Filed under: General,Geometry — Tags: — m759 @ 11:14 PM

Now out from behind a paywall . . .

The diamond theorem at SASTRA —

Tuesday, April 24, 2018

Dance, Music, Space

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

". . . dance, fueled by music, opens up space."

—  Alastair Macaulay in the online New York Times  today

Putting aside the unfortunate fuel metaphor, this suggests a review —

A video published on the above date —

The video has six-plus-two dancers, a more concise arrangement
than the eight-plus-two discussed by Macaulay.

Another approach to six plus two:  the diamond-theorem correlation.

Saturday, December 23, 2017

The Right Stuff

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 1:12 PM

A figure related to the general connecting theorem  of Koen Thas —

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)

See also posts tagged Dirac and Geometry in this  journal.

Those who prefer narrative to mathematics may, if they so fancy, call
the above Thas connecting theorem a "quantum tesseract theorem ."

Monday, December 11, 2017

The Diamond Theorem at SASTRA

Filed under: General,Geometry — Tags: — m759 @ 12:35 PM

The following IEEE paper is behind a paywall,
but the first page is now available for free
at deepdyve.com

For further details on the diamond theorem, see
finitegeometry.org/sc/ or the archived version at . . .

DOI

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)

Friday, December 8, 2017

Mythos and Logos

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

Part I:  Black Magician

"Schools of criticism create their own canons, elevating certain texts,
discarding others. Yet some works – Malcolm Lowry’s Under the Volcano
is one of them – lend themselves readily to all critical approaches."

— Joan Givner, review of 
A Darkness That Murmured: Essays on Malcolm Lowry and the Twentieth Century
by Frederick Asals and Paul Tiessen, eds.

The Asals-Tiessen book (U. of Toronto Press, 2000) was cited today
by Margaret Soltan (in the link below) as the source of this quotation —

"When one thinks of the general sort of snacky
under-earnest writers whose works like wind-chimes
rattle in our heads now, it is easier to forgive Lowry
his pretentious seriousness, his old-fashioned ambitions,
his Proustian plans, [his efforts] to replace the reader’s
consciousness wholly with a black magician’s."

A possible source, Perle Epstein, for the view of Lowry as black magician —

Part II:  Mythos  and Logos

Part I above suggests a review of Adam Gopnik as black magician
(a figure from Mythos ) —

Tuesday, November 7, 2017

Polarities and Correlation

Filed under: Uncategorized — Tags:  
— m759 @ 11:00 PM 

— and of an opposing figure from Logos
     Paul B. Yale, in the references below:

Thursday, November 30, 2017

The Matrix for Quantum Mystics

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 10:29 PM

Scholia on the title — See Quantum + Mystic in this journal.

The Matrix of Lévi-Strauss

"In Vol. I of Structural Anthropology , p. 209, I have shown that
this analysis alone can account for the double aspect of time
representation in all mythical systems: the narrative is both
'in time' (it consists of a succession of events) and 'beyond'
(its value is permanent)." — Claude Lévi-Strauss, 1976

I prefer the earlier, better-known, remarks on time by T. S. Eliot
in Four Quartets , and the following four quartets (from
The Matrix Meets the Grid) —

.

From a Log24 post of June 26-27, 2017:

A work of Eddington cited in 1974 by von Franz

See also Dirac and Geometry and Kummer in this journal.

Ron Shaw on Eddington's triads "associated in conjugate pairs" —

For more about hyperbolic  and isotropic  lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.

For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.

Sunday, November 26, 2017

Upper West Side Story:

Filed under: General — m759 @ 1:00 PM

The Linotype Fixer

( Sequel to "The Typewriter Fixer" * )

From The Hollywood Reporter  on Tuesday, Nov. 7, 2017 —

* "The Typewriter Fixer" refers to a typewriter repair shop
   on New York's Upper West Side —

The Hollywood Reporter 's  promotional piece  above is from
Tuesday, November 7, 2017.  For another meditation suited to
the Upper West Side, see this  journal on that date —

Tuesday, November 7, 2017

Polarities and Correlation

Filed under: Uncategorized — Tags:  
— m759 @ 11:00 PM 

Adam Gopnik on Philip Roth

Adam Gopnik on Philip Roth and Mickey Sabbath

Saturday, October 28, 2017

Lowell Brown at Vanity Fair

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 8:18 PM

A sequel to the post  CP  is for Consolation Prize  (Sept. 3, 2016)

An image from Log24 on this date last year:

A recent comment on a discussion of CP symmetry

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.

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

Saturday, April 15, 2017

Quanta Dating

Filed under: General — Tags: , — m759 @ 3:15 PM

From Quanta Magazine  —

For the Church of Synchronology

See also this  journal on July 17, 2014, and March 28, 2017.

Saturday, March 4, 2017

Hidden Figure

Filed under: General — m759 @ 3:17 AM

From this morning's New York Times  Wire

A cybersecurity-related image from Thursday evening

The diamond theorem correlation at the University of Bradford

Thursday, March 2, 2017

Raiders of the Lost Crucible Continues

Filed under: General — m759 @ 9:59 PM

Cover, 2005 paperback edition of 'Refiner's Fire,' a 1977 novel by Mark Helprin

Mariner Books paperback, 2005

See, too, this evening's A Common Space
and earlier posts on Raiders of the Lost Crucible.

Also not without relevance —

The diamond theorem correlation at the University of Bradford

Wednesday, February 15, 2017

Warp and Woof

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

Space —

Space structure —

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).

Space shuttle —

Related ethnic remarks —

As opposed to Michael  Larsen —

Funny, you don't look  Danish.

Wednesday, November 23, 2016

Yogiism

Filed under: General,Geometry — Tags: , — m759 @ 12:31 PM

From the American Mathematical Society (AMS) webpage today —

From the current AMS Notices

Related material from a post of Aug. 6, 2014

http://www.log24.com/log/pix10B/100915-SteinbergOnChevalleyGroups.jpg

(Here "five point sets" should be "five-point sets.")

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  structure* now appears in the figure
illustrating the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2).

* The phrase as used here is a deliberate 
abuse of language .  For the real definition of 
“symplectic structure,” see (for instance) 
“Symplectic Geometry,” by Ana Cannas da Silva
(article written for Handbook of Differential
Geometry 
, Vol 2.) To establish that the above
figure is indeed symplectic , see the post 
Zero System of July 31, 2014.

Friday, October 28, 2016

Diamond-Theorem Application

Filed under: General,Geometry — Tags: — m759 @ 1:06 PM
 

Abstract:

"Protection of digital content from being tapped by intruders is a crucial task in the present generation of Internet world. In this paper, we proposed an implementation of new visual secret sharing scheme for gray level images using diamond theorem correlation. A secret image has broken into 4 × 4 non overlapped blocks and patterns of diamond theorem are applied sequentially to ensure the secure image transmission. Separate diamond patterns are utilized to share the blocks of both odd and even sectors. Finally, the numerical results show that a novel secret shares are generated by using diamond theorem correlations. Histogram representations demonstrate the novelty of the proposed visual secret sharing scheme."

— "New visual secret sharing scheme for gray-level images using diamond theorem correlation pattern structure," by  V. Harish, N. Rajesh Kumar, and N. R. Raajan.

Published in: 2016 International Conference on Circuit, Power and Computing Technologies (ICCPCT).
Date of Conference: 18-19 March 2016. Publisher: IEEE.
Date Added to IEEE Xplore: 04 August 2016

Excerpts —

Related material — Posts tagged Diamond Theorem Correlation.

Friday, August 26, 2016

Wolfe vs. Chomsky

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

1.  Tom Wolfe has a new book on Chomsky, "The Kingdom of Speech."

2.  This suggests a review of a post of Aug. 11, 2014, Syntactic/Symplectic.

To paraphrase Wittgenstein, sentence 1 above is about "correlating in real life"
(cf. Crooked House and Wolfe's From Bauhaus to Our House ), and may be 
compared to sentence 2 above, which links to a sort of "correlating in
mathematics" that is a particular example of the more general sort of
mathematical correlating mentioned by Wittgenstein in 1939.

Saturday, November 21, 2015

The Zero System

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

For the title phrase, see Encyclopedia of Mathematics .
The zero system  illustrated in the previous post*
should not be confused with the cinematic Zero Theorem .

* More precisely, in the part showing the 15 lines fixed under
   a zero-system polarity in PG(3,2).  For the zero system 
   itself, see diamond-theorem correlation.

Friday, November 13, 2015

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



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

References:

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

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

Related material:

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

Background reading:

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

Thursday, October 15, 2015

Contrapuntal Interweaving

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

The title is a phrase from R. D. Laing's book The Politics of Experience .
(Published in the psychedelic year 1967. The later "contrapuntal interweaving"
below is of a less psychedelic nature.)

An illustration of the "interweaving' part of the title —
The "deep structure" of the diamond theorem:

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

The word "symplectic" from the end of last Sunday's (Oct. 11) sermon
describes the "interwoven" nature of the above illustration.

An illustration of the "contrapuntal" part of the title (click to enlarge):

The diamond-theorem correlation

 

Saturday, July 4, 2015

Context

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

Some context for yesterday's post on a symplectic polarity —

This 1986 note may or may not have inspired some remarks 
of Wolf Barth in his foreword to the 1990 reissue of Hudson's
1905 Kummer's Quartic Surface .

See also the diamond-theorem correlation.  

Wednesday, February 25, 2015

Words and Images

Filed under: General,Geometry — Tags: — m759 @ 5:30 PM

The words:  "symplectic polarity"—

The images:

The Natural Symplectic Polarity in PG(3,2)

Symmetry Invariance in a Diamond Ring

The Diamond-Theorem Correlation

Picturing the Smallest Projective 3-Space

Quilt Block Designs

Saturday, February 21, 2015

High and Low Concepts

Filed under: General,Geometry — Tags: — m759 @ 4:30 PM

Steven Pressfield on April 25, 2012:

What exactly is High Concept?

Let’s start with its opposite, low concept.
Low concept stories are personal,
idiosyncratic, ambiguous, often European. 
“Well, it’s a sensitive fable about a Swedish
sardine fisherman whose wife and daughter
find themselves conflicted over … ”

ZZZZZZZZ.

Fans of Oslo artist Josefine Lyche know she has
valiantly struggled to find a high-concept approach
to the diamond theorem. Any such approach must,
unfortunately, reckon with the following low
(i.e., not easily summarized) concept —

The Diamond Theorem Correlation:

From left to right

http://www.log24.com/log/pix14B/140824-Diamond-Theorem-Correlation-1202w.jpg

http://www.log24.com/log/pix14B/140731-Diamond-Theorem-Correlation-747w.jpg

http://www.log24.com/log/pix14B/140824-Picturing_the_Smallest-1986.gif

http://www.log24.com/log/pix14B/140806-ProjPoints.gif

For some backstory, see ProjPoints.gif and "Symplectic Polarity" in this journal.

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) —

Monday, August 11, 2014

Syntactic/Symplectic

(Continued from August 9, 2014.)

Syntactic:

Symplectic:

"Visual forms— lines, colors, proportions, etc.— are just as capable of
articulation , i.e. of complex combination, as words. But the laws that govern
this sort of articulation are altogether different from the laws of syntax that
govern language. The most radical difference is that visual forms are not
discursive 
. They do not present their constituents successively, but
simultaneously, so the relations determining a visual structure are grasped
in one act of vision."

– Susanne K. LangerPhilosophy in a New Key

For examples, see The Diamond-Theorem Correlation
in Rosenhain and Göpel Tetrads in PG(3,2).

This is a symplectic  correlation,* constructed using the following
visual structure:

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

* Defined in (for instance) Paul B. Yale, Geometry and Symmetry ,
Holden-Day, 1968, sections 6.9 and 6.10.

Wednesday, August 6, 2014

Symplectic Structure*

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

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  structure** now appears in the figure
illustrating the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2).

Some related passages from the literature:

http://www.log24.com/log/pix10B/100915-SteinbergOnChevalleyGroups.jpg

* The title is a deliberate abuse of language .
For the real definition of "symplectic structure," see (for instance)
"Symplectic Geometry," by Ana Cannas da Silva (article written for
Handbook of Differential Geometry, vol 2.) To establish that the
above figure is indeed symplectic , see the post Zero System of
July 31, 2014.

** See Steven H. Cullinane, Inscapes III, 1986

Thursday, July 31, 2014

Zero System

Filed under: General,Geometry — Tags: , — m759 @ 6:11 PM

The title phrase (not to be confused with the film 'The Zero Theorem')
means, according to the Encyclopedia of Mathematics,
a null system , and

"A null system is also called null polarity,
a symplectic polarity or a symplectic correlation….
it is a polarity such that every point lies in its own
polar hyperplane."

See Reinhold Baer, "Null Systems in Projective Space,"
Bulletin of the American Mathematical Society, Vol. 51
(1945), pp. 903-906.

An example in PG(3,2), the projective 3-space over the
two-element Galois field GF(2):

IMAGE- The natural symplectic polarity in PG(3,2), illustrating a symplectic structure

See also the 10 AM ET post of Sunday, June 8, 2014, on this topic.

Sunday, June 8, 2014

Vide

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

Some background on the large Desargues configuration

“The relevance of a geometric theorem is determined by what the theorem
tells us about space, and not by the eventual difficulty of the proof.”

— Gian-Carlo Rota discussing the theorem of Desargues

What space  tells us about the theorem :  

In the simplest case of a projective space  (as opposed to a plane ),
there are 15 points and 35 lines: 15 Göpel  lines and 20 Rosenhain  lines.*
The theorem of Desargues in this simplest case is essentially a symmetry
within the set of 20 Rosenhain lines. The symmetry, a reflection
about the main diagonal in the square model of this space, interchanges
10 horizontally oriented (row-based) lines with 10 corresponding
vertically oriented (column-based) lines.

Vide  Classical Geometry in Light of Galois Geometry.

* Update of June 9: For a more traditional nomenclature, see (for instance)
R. Shaw, 1995.  The “simplest case” link above was added to point out that
the two types of lines named are derived from a natural symplectic polarity 
in the space. The square model of the space, apparently first described in
notes written in October and December, 1978, makes this polarity clearly visible:

A coordinate-free approach to symplectic structure

Tuesday, July 9, 2013

Vril Chick

Filed under: General,Geometry — m759 @ 4:30 AM

Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo

Profile picture for "Jo Lyxe" (Josefine Lyche) at Vimeo

Compare to an image of Vril muse Maria Orsitsch.

From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —

Josefine Lyche
Born in 1973 in Bergen, Norway.
Lives and works in Oslo and Berlin.

Keywords (to help place my artwork in the
proper context): Aliens, affine geometry, affine
planes, affine spaces, automorphisms, binary
codes, block designs, classical groups, codes,
coding theory, collineations, combinatorial,
combinatorics, conjugacy classes, the Conwell
correspondence, correlations, Cullinane,
R. T. Curtis, design theory, the diamond theorem,
diamond theory, duads, duality, error correcting
codes, esoteric, exceptional groups,
extraterrestrials, finite fields, finite geometry, finite
groups, finite rings, Galois fields, generalized
quadrangles, generators, geometry, GF(2),
GF(4), the (24,12) Golay code, group actions,
group theory, Hadamard matrices, hypercube,
hyperplanes, hyperspace, incidence structures,
invariance, Karnaugh maps, Kirkman’s schoolgirls
problem, Latin squares, Leech lattice, linear
groups, linear spaces, linear transformations,
Magick, Mathieu groups, matrix theory, Meno,
Miracle Octad Generator, MOG, multiply transitive
groups, occultism, octahedron, the octahedral
group, Orsic, orthogonal arrays, outer automorphisms,
parallelisms, partial geometries,
permutation groups, PG(3,2), Plato, Platonic
solids, polarities, Polya-Burnside theorem, projective
geometry, projective planes, projective
spaces, projectivities, Pythagoras, reincarnation,
Reed-Muller codes, the relativity problem,
reverse engineering, sacred geometry, Singer
cycle, skew lines, Socrates, sporadic simple
groups, Steiner systems, Sylvester, symmetric,
symmetry, symplectic, synthemes, synthematic,
Theosophical Society tesseract, Tessla, transvections,
Venn diagrams, Vril society, Walsh
functions, Witt designs.

(See also the original catalog page.)

Clearly most of this (the non-highlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.

For some background, see (for instance) 
Conspiracy Theories and Secret Societies for Dummies .

Saturday, August 20, 2011

Castle Rock

Filed under: General,Geometry — m759 @ 6:29 PM

Happy birthday to Amy Adams
(actress from Castle Rock, Colorado)

"The metaphor for metamorphosis…" —Endgame

Related material:

"The idea that reality consists of multiple 'levels,' each mirroring all others in some fashion, is a diagnostic feature of premodern cosmologies in general…."

Scholarly paper on "Correlative Cosmologies"

"How many layers are there to human thought? Sometimes in art, just as in people’s conversations, we’re aware of only one at a time. On other occasions, though, we realize just how many layers can be in simultaneous action, and we’re given a sense of both revelation and mystery. When a choreographer responds to music— when one artist reacts in detail to another— the sensation of multilayering can affect us as an insight not just into dance but into the regions of the mind.

The triple bill by the Mark Morris Dance Group at the Rose Theater, presented on Thursday night as part of the Mostly Mozart Festival, moves from simple to complex, and from plain entertainment to an astonishingly beautiful and intricate demonstration of genius….

'Socrates' (2010), which closed the program, is a calm and objective work that has no special dance excitement and whips up no vehement audience reaction. Its beauty, however, is extraordinary. It’s possible to trace in it terms of arithmetic, geometry, dualism, epistemology and ontology, and it acts as a demonstration of art and as a reflection of life, philosophy and death."

— Alastair Macaulay in today's New York Times

SOCRATES: Let us turn off the road a little….

Libretto for Mark Morris's 'Socrates'

See also Amy Adams's new film "On the Road"
in a story from Aug. 5, 2010 as well as a different story,
Eightgate, from that same date:

A 2x4 array of squares

The above reference to "metamorphosis" may be seen,
if one likes, as a reference to the group of all projectivities
and correlations in the finite projective space PG(3,2)—
a group isomorphic to the 40,320 transformations of S8
acting on the above eight-part figure.

See also The Moore Correspondence from last year
on today's date, August 20.

For some background, see a book by Peter J. Cameron,
who has figured in several recent Log24 posts—

http://www.log24.com/log/pix11B/110820-Parallelisms60.jpg

"At the still point, there the dance is."
               — Four Quartets

Monday, November 17, 2008

Monday November 17, 2008

Filed under: General,Geometry — m759 @ 9:00 AM
Limits

From the previous entry:

“If it’s a seamless whole you want,
 pray to Apollo, who sets the limits
  within which such a work can exist.”

— Margaret Atwood,
author of Cat’s Eye

The 3x3 square

Happy birthday
to the late
Eugene Wigner

… and a belated
Merry Christmas
 to Paul Newman:

Elke Sommer, former Erlangen Gymnasium student, in 'The Prize' with Paul Newman, released Christmas Day, 1963

“The laws of nature permit us to foresee events on the basis of the knowledge of other events; the principles of invariance should permit us to establish new correlations between events, on the basis of the knowledge of established correlations between events. This is exactly what they do.”

— Eugene Wigner, Nobel Prize Lecture, December 12, 1963

Wednesday, April 23, 2008

Wednesday April 23, 2008

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

Upscale Realism

or, "Have some more
wine and cheese, Barack."

(See April 15, 5:01 AM)

  Allyn Jackson on Rebecca Goldstein
in the April 2006 AMS Notices (pdf)

"Rebecca Goldstein’s 1983 novel The Mind-Body Problem has been widely admired among mathematicians for its authentic depiction of academic life, as well as for its exploration of how philosophical issues impinge on everyday life. Her new book, Incompleteness: The Proof and Paradox of Kurt Gödel, is a volume in the 'Great Discoveries' series published by W. W. Norton….

In March 2005 the Mathematical Sciences Research Institute (MSRI) in Berkeley held a public event in which its special projects director, Robert Osserman, talked with Goldstein about her work. The conversation, which took place before an audience of about fifty people at the Commonwealth Club in San Francisco, was taped….

A member of the audience posed a question that has been on the minds of many of Goldstein’s readers: Is The Mind-Body Problem based on her own life? She did indeed study philosophy at Princeton, finishing her Ph.D. in 1976 with a thesis titled 'Reduction, Realism, and the Mind.' She said that while there are correlations between her life and the novel, the book is not autobiographical….

She… talked about the relationship between Gödel and his colleague at the Institute for Advanced Study, Albert Einstein. The two were very different: As Goldstein put it, 'Einstein was a real mensch, and Gödel was very neurotic.' Nevertheless, a friendship sprang up between the two. It was based in part, Goldstein speculated, on their both being exiles– exiles from Europe and intellectual exiles. Gödel's work was sometimes taken to mean that even mathematical truth is uncertain, she noted, while Einstein's theories of relativity were seen as implying the sweeping view that 'everything is relative.' These misinterpretations irked both men, said Goldstein. 'Einstein and Gödel were realists and did not like it when their work was put to the opposite purpose.'"


Related material:

From Log24 on
March 22 (Tuesday of
Passion Week), 2005:

 
"'What is this Stone?' Chloe asked…. 'It is told that, when the Merciful One made the worlds, first of all He created that Stone and gave it to the Divine One whom the Jews call Shekinah, and as she gazed upon it the universes arose and had being.'"

Many Dimensions,
by Charles Williams, 1931

For more on this theme
appropriate to Passion Week
Jews playing God — see

The image “http://www.log24.com/log/pix05/050322-Trio.jpg” cannot be displayed, because it contains errors.

Rebecca Goldstein
in conversation with
Bob Osserman
of the
Mathematical Sciences
Research Institute
at the
Commonwealth Club,
San Francisco,
Tuesday, March 22.

Wine and cheese
reception at 5:15 PM
(San Francisco time).

From
UPSCALE,
a website of the
physics department at
the University of Toronto:

Mirror Symmetry

Robert Fludd: Universe as mirror image of God

"The image [above]
is a depiction of
the universe as a
mirror image of God,
drawn by Robert Fludd
in the early 17th century.

The caption of the
upper triangle reads:

'That most divine and beautiful
counterpart visible below in the
flowing image of the universe.'

The caption of the
lower triangle is:

'A shadow, likeness, or
reflection of the insubstantial*
triangle visible in the image
of the universe.'"

* Sic. The original is incomprehensibilis, a technical theological term. See Dorothy Sayers on the Athanasian Creed and John 1:5.

For further iconology of the
above equilateral triangles,
see Star Wars (May 25, 2003),
Mani Padme (March 10, 2008),
Rite of Sping (March 14, 2008),
and
Art History: The Pope of Hope
(In honor of John Paul II
three days after his death
in April 2005).

Happy Shakespeare's Birthday.

Saturday, January 6, 2007

Saturday January 6, 2007

Filed under: General,Geometry — m759 @ 10:31 AM
An Epiphany
for the Birthday
of E. L. Doctorow,
Author of
City of God

(Doctorow wrote about
 New York. A city more
  closely associated with
 God is Jerusalem.)

On the morning of January 2 this year, inspired by Sambin’s “basic picture,” I considered an entry dealing with Galois lattices (pdf).  This train of thought was halted by news of the death earlier that morning of Teddy Kollek, 95, a founder of the Israeli intelligence service and six-term mayor of Jerusalem. (This led later to the entry “Damnation Morning“– a reference to the Fritz Leiber short story.)

This morning’s entry reboards the Galois train of thought.

Here are some relevant links:

Galois Connections (a French weblog entry providing an brief overview of Galois theory and an introduction to the use of Galois lattices in “formal concept analysis“)

Ontology (an introduction to formal concept analysis linked to on 3/31/06)

One motive for resuming consideration of Galois lattices today is to honor the late A. Richard Newton, a pioneer in engineering design who died at 55– also on Tuesday, Jan. 2, the date of Kollek’s death.  Today’s New York Times obituary for Newton says that “most recently, Professor Newton championed the study of synthetic biology.”

A check of syntheticbiology.org leads to a web page on– again– ontology.

For the relationship between ontology (in the semantic-web sense) and Galois lattices, see (for instance)

Knowledge Organisation and Information Retrieval Using Galois Lattices” (ps) and its references.

An epiphany within all this that Doctorow might appreciate is the following from Wikipedia, found by following a link to “upper ontology” in the syntheticbiology.org ontology page:

  • There is no self-evident way of dividing the world up into concepts.
  • There is no neutral ground that can serve as a means of translating between specialized (lower) ontologies.
  • Human language itself is already an arbitrary approximation of just one among many possible conceptual maps. To draw any necessary correlation between English words and any number of intellectual concepts we might like to represent in our ontologies is just asking for trouble.

Related material:

The intellectual concepts
mentioned by Richard Powers
at the end of tomorrow’s
New York Times Book Review.
(See the links on these concepts
in yesterday’s “Goldberg Variation.”)

See also Old School Tie.

Tuesday, October 3, 2006

Tuesday October 3, 2006

Filed under: General,Geometry — m759 @ 9:26 AM

Serious

"I don't think the 'diamond theorem' is anything serious, so I started with blitzing that."

Charles Matthews at Wikipedia, Oct. 2, 2006

"The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas."

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

Matthews yesterday deleted references to the diamond theorem and related material in the following Wikipedia articles:

Affine group‎
Reflection group‎
Symmetry in mathematics‎
Incidence structure‎
Invariant (mathematics)‎
Symmetry‎
Finite geometry‎
Group action‎
History of geometry‎

This would appear to be a fairly large complex of mathematical ideas.

See also the following "large complex" cited, following the above words of Hardy, in Diamond Theory:

Affine geometry, affine planes, affine spaces, automorphisms, binary codes, block designs, classical groups, codes, coding theory, collineations, combinatorial, combinatorics, conjugacy classes, the Conwell correspondence, correlations, design theory, duads, duality, error correcting codes, exceptional groups, finite fields, finite geometry, finite groups, finite rings, Galois fields, generalized quadrangles, generators, geometry, GF(2), GF(4), the (24,12) Golay code, group actions, group theory, Hadamard matrices, hypercube, hyperplanes, hyperspace, incidence structures, invariance, Karnaugh maps, Kirkman's schoolgirl problem, Latin squares, Leech lattice, linear groups, linear spaces, linear transformations, Mathieu groups, matrix theory, Meno, Miracle Octad Generator, MOG, multiply transitive groups, octads, the octahedral group, orthogonal arrays, outer automorphisms, parallelisms, partial geometries, permutation groups, PG(3,2), polarities, Polya-Burnside theorem, projective geometry, projective planes, projective spaces, projectivities, Reed-Muller codes, the relativity problem, Singer cycle, skew lines,  sporadic simple groups, Steiner systems, symmetric, symmetry, symplectic, synthemes, synthematic, tesseract, transvections, Walsh functions, Witt designs.

Sunday, May 25, 2003

Sunday May 25, 2003

Filed under: General,Geometry — m759 @ 9:26 PM

 STAR WARS  
opened on this date in 1977.

From the web page Amande:

Le Christ et la Vierge apparurent souvent entourés d’une auréole en forme d’amande: la mandorle.

Étymologiquement, le mot amande est une altération de amandala, qui dérive lui-même du latin classique amygdala….

L’amande a… une connotation symbolique, celle du sexe féminin. Elle figure souvent la vulve. Elle est alors en analogie avec la yoni du vocabulaire de l’hindouisme, la vulve ou la matrice, représentée par une amande ou une noix coupée en deux.

Screenshot of the online
New York Times, May 25, 2003:

Ariel the Hutt and Princess Amygdala

Introduction to Yantra

by Horia Cristescu and
Dan Bozaru 

The Triangle (TRIKONA)
The triangle (TRIKONA) is the symbol of
SHAKTI , the feminine energy or aspect of Creation. The triangle pointing down represents the YONI , the feminine sexual organ and the symbol of the supreme source of the Universe, and when the triangle is pointing upwards it signifies intense spiritual aspiration, the sublimation of one’s nature into the most subtle planes and the element of fire (AGNI TATTVA). The fire is always oriented upwards, thus the correlation with the upward triangle – SHIVA KONA. On the other hand, the downward pointing triangle signifies the element of water which always tends to flown and occupy the lowest possible position. This triangle is known as SHAKTI KONA.

The intersection of two geometric forms (lines, triangles, circles, etc.) represents forces that are even more intense than those generated by the simple forms. Such an interpenetration indicates a high level in the dynamic interaction of the correspondent energies. The empty spaces generated by such combinations are described as very efficient operational fields of the forces emanating from the central point of the YANTRA. That is why we can very often encounter representations of MANTRAS in such spaces. YANTRA and MANTRA are complementary aspects of SHIVA and their use together is much more efficient than the use of one alone.


The Six Points Star (SHATKONA)
A typical combination often found in the graphical structure of a YANTRA is the superposition of two triangles, one pointing upwards and the other downwards, forming a star with six points (SHATKONA), also known as David’s Star. This form symbolically represents the union of
PURUSHA and PRAKRITI or SHIVA-SHAKTI, without which there could be no Creation.

AMEN.

Wednesday, April 2, 2003

Wednesday April 2, 2003

Filed under: General,Geometry — Tags: — m759 @ 2:30 PM

Symmetries…. May 15, 1998

The following journal note, from the day after Sinatra died, was written before I heard of his death.  Note particularly the quote from Rilke.  Other material was suggested, in part, by Alasdair Gray’s Glasgow novel 1982 Janine.  The “Sein Feld” heading is a reference to the Seinfeld final episode, which aired May 14, 1998.  The first column contains a reference to angels — apparently Hell’s Angels — and the second column provides a somewhat more serious look at this theological topic.

Sein Feld

                        

1984 Janine

“But Angels love their own
And they’re reaching out
    for you
Janine… Oh Janine
— Kim Wilde lyric,
    Teases & Dares album,
    1984, apparently about
    a British biker girl

 

Logos means above all relation.”
— Simone Weil,
    Gateway to God,
    Glasgow, 1982

Gesang ist Dasein….
 Ein Hauch um nichts.
 Ein Wehn im Gott.
 Ein Wind
.”
— Not Heidegger but Rilke:
Sonnets to Orpheus, I, 3

Geometry and Theology

PA lottery May 14, 1998:
256
   
S8  The group of all projectivities and correlations of PG(3,2).

The above isomorphism implies the geometry of the Mathieu group M24.

“The Leech lattice is a blown-up version of
S(5,8,24).”
— W. Feit

“We have strong evidence that the creator of the universe loves symmetry.”
— Freeman Dyson

“Mackey presents eight axioms from which he deduces the [quantum] theory.”
— M. Schechter

“Theology is about words; science is about things.
— Freeman Dyson, New York Review of Books, 5/28/98

What is “256” about?

Tape purchased 12/23/97:
 

Django
Reinhardt

      Gypsy Jazz

“In the middle of 1982 Janine there are pages in which Jock McLeish is fighting with drugs and alcohol, attempting to either die or come through and get free of his fantasies. In his delirium, he hears the voice of God, which enters in small print, pushing against the larger type of his ravings.  Something God says is repeated on the first and last pages of Unlikely Stories, Mostly, complete with illustration and the words ‘Scotland 1984’ beside it. God’s statement is ‘Work as if you were in the early days of a better nation.’  It is the inherent optimism in that statement that perhaps best captures the strength of Aladair Gray’s fiction, its straightforwardness and exuberance.”
— Toby Olson, “Eros in Glasgow,” in Book World, The Washington Post, December 16, 1984

 For another look at angels, see “Winging It,” by Christopher R. Miller, The New York Times Book Review Bookend page for Sunday, May 24, 1998. May 24 is the feast day of Sara (also known by the Hindu name Kali), patron saint of Gypsies.

For another, later (July 16, 1998) reply to Dyson, from a source better known than myself, see Why Religion Matters, by Huston Smith, Harper Collins, 2001, page 66.

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