Log24

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.

Sunday, February 24, 2019

Exceeding His Grasp

Filed under: General,Geometry — Tags: , , , — m759 @ 4:09 PM

See also Hudson Hawk in this  journal.

Saturday, January 26, 2019

In Memory of a Composer

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

who reportedly died early today in Paris, a tribute from
those who wrote the English lyrics for "Windmills of Your Mind" —

Legrand tribute related to 'Eddington Song'

Installasjon

Filed under: General,Geometry — Tags: , , — m759 @ 5:13 AM

'Josefine Lyche,' 'Interlock, Interlac, Interweave'

The above cryptic search result indicates that there may
soon be a new Norwegian art installation based on this page
of Eddington (via Log24) —

See also other posts tagged Kummerhenge.

Wednesday, January 23, 2019

Decorated

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

For those who prefer more elaborate decorations —

1.  A Facebook image from last August … 

2.  The Facebook glider suggests a tune from "The Thomas Crown Affair"
     (1968) that appeared in a Dec. 16, 2018 post on Christianity and
     "interlocking names"—

'The Eddington Song'

The revised lyrics describe a square space.

3.  An even more  elaborate square space:
     the Dance of the Snowflakes from
     Balanchine's version of The Nutcracker —

Sunday, December 16, 2018

Interlocking

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

http://m759.net/wordpress/?s=Eddington+Song

'The Eddington Song'

Friday, November 23, 2018

Artfield Studio

Filed under: General,Geometry — m759 @ 12:10 PM

The title is that of an new Internet domain, artfield.studio,
that is used only to store branded links  such as . . .

artfield.studio/trinity-dice

artfield.studio/dirac-and-geometry

artfield.studio/eddington

artfield.studio/kummerhenge

artfield.studio/pinterest .

Some context —

artfield.studio/patrick-blackburn.

Thursday, November 22, 2018

Geometric Incarnation

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

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

— T. S. Eliot in Four Quartets

Note also the four 4×4 arrays surrounding the central diamond
in the chi  of the chi-rho  page of the Book of Kells

From a Log24 post
of March 17, 2012

"Interlocking, interlacing, interweaving"

— Condensed version of page 141 in Eddington's
1939 Philosophy of Physical Science

Wednesday, November 21, 2018

Opening Credits

Filed under: General,Geometry — Tags: — m759 @ 7:47 PM

From Log24 on Friday, Nov. 16

This evening's New York Times  on an opening-credits designer, 
Pablo Ferro, who reportedly died at 83 on Friday, Nov. 16 —

An example of Ferro's later work in film —

Musical accompaniment from Sunday morning —

'The Eddington Song'

Tuesday, November 20, 2018

Logos

Filed under: General,Geometry — Tags: — m759 @ 12:21 PM

(Continued)

Musical accompaniment from Sunday morning

'The Eddington Song'

Update of Nov. 21 —

The reader may contrast the above Squarespace.com logo
(a rather serpentine version of the acronym SS) with a simpler logo
for a square space (the Galois window ):

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.

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 .

Friday, July 6, 2018

Something

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

"… Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness."

— T. S. Eliot, "Burnt Norton," 1936

"Read something that means something."

Advertising slogan for The New Yorker

The previous post quoted some mystic meditations of Octavio Paz
from 1974. I prefer some less mystic remarks of Eddington from
1938 (the Tanner Lectures) published by Cambridge U. Press in 1939 —

"… we have sixteen elements with which to form a group-structure" —

See as well posts tagged Dirac and Geometry.

Thursday, June 21, 2018

Models of Being

Filed under: General,Geometry — Tags: — m759 @ 11:30 AM

A Buddhist view —

"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 Christian view —

A formal view —

From a Log24 search for High Concept:

See also Galois Tesseract.

Dirac and Geometry (continued)

Filed under: General,Geometry — Tags: , — m759 @ 10:04 AM

"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.

Saturday, April 7, 2018

Sides

Filed under: General,Geometry — Tags: — m759 @ 11:47 AM

The FBI holding cube in "The Blacklist" —

" 'The Front' is not the whole story . . . ."

— Vincent Canby, New York Times  film review, 1976,
     as quoted in Wikipedia.

See also Solomon's Cube in this  journal.

IMAGE- 'Solomon's Cube'

Webpage demonstrating symmetries of 'Solomon's Cube'

Some may view the above web page as illustrating the
Glasperlenspiel  passage quoted here in Summa Mythologica 

“"I suddenly realized that in the language, or at any rate
in the spirit of the Glass Bead Game, everything actually
was all-meaningful, that every symbol and combination of
symbols led not hither and yon, not to single examples,
experiments, and proofs, but into the center, the mystery
and innermost heart of the world, into primal knowledge.
Every transition from major to minor in a sonata, every
transformation of a myth or a religious cult, every classical
or artistic formulation was, I realized in that flashing moment,
if seen with a truly meditative mind, nothing but a direct route
into the interior of the cosmic mystery, where in the alternation
between inhaling and exhaling, between heaven and earth,
between Yin and Yang, holiness is forever being created.”

A less poetic meditation on the above 4x4x4 design cube —

"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."

See also a related remark by Lévi-Strauss in 1955

"…three different readings become possible:
left to right, top to bottom, front to back."

Friday, April 6, 2018

The Thread Phantom: A Death on Pi Day*

Filed under: General — Tags: , — m759 @ 12:42 PM

The American Mathematical Society on April 4 posted a story
about a death that they said occurred on March 14  (Pi Day):

* Notes on the Title —

The Thread Part 

The Phantom Part 

"What a yarn!" — Raymond Reddington in "The Blacklist"

Fact check on the death date reported by the AMS —

But Davis's funeral-home obituary agrees with the Pi Day date.

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.

Friday, June 30, 2017

Hurriedly Put Together

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

The previous post quoted one theologian on a book
by another theologian, saying its tone "is patronizing
and its arguments are hurriedly put together."

For a more leisurely sort of argument, see a 1995* remark 
by a mathematician, Ronald Shaw, quoted here on the morning
of Tuesday, June 27, in an update at the end of the previous day's
post "Upgrading to Six" —

". . . recall the notions of Eddington (1936) . . . ."

* In "Finite Geometry, Dirac Groups and the
Table of Real Clifford Algebras
," pages 59-99 of
R. Ablamowicz and P. Lounesto (eds.),
Clifford Algebras and Spinor Structures ,
Kluwer Academic Publishers, 1995.

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.

Sunday, February 12, 2017

Colorful Tales

Filed under: General — m759 @ 1:23 PM

“Perhaps the philosophically most relevant feature of modern science
is the emergence of abstract symbolic structures as the hard core
of objectivity behind— as Eddington puts it— the colorful tale of
the subjective storyteller mind.”

— Hermann Weyl, Philosophy of  Mathematics and
    Natural Science 
, Princeton, 1949, p. 237

Harvard University Press on the late Angus Fletcher, author of
The Topological Imagination  and Colors of the Mind

From the Harvard webpage for Colors of the Mind

Angus Fletcher is one of our finest theorists of the arts,
the heir to I. A. Richards, Erich Auerbach, Northrop Frye.
This… book…  aims to open another field of study:
how thought— the act, the experience of thinking—
is represented in literature.

. . . .

Fletcher’s resources are large, and his step is sure.
The reader samples his piercing vision of Milton’s

Satan, the original Thinker,
leaving the pain of thinking
as his legacy for mankind.

A 1992 review by Vinay Dharwadker of Colors of the Mind —

See also the above word "dianoia" in The Echo in Plato's Cave.
Some context 

This post was suggested by a memorial piece today in
the Los Angeles Review of Books

A Florilegium for Angus Fletcher

By Kenneth Gross, Lindsay Waters, V. N. Alexander,
Paul Auster, Harold Bloom, Stanley Fish, K. J. Knoespel,
Mitchell Meltzer, Victoria Nelson, Joan Richardson,
Dorian Sagan, Susan Stewart, Eric Wilson, Michael Wood

Fletcher reportedly died on November 28, 2016.

"I learned from Fletcher how to apprehend
the daemonic element in poetic imagination."

— Harold Bloom in today's Los Angeles florilegium

For more on Bloom and the daemonic, see a Log24 post,
"Interpenetration," from the date of Fletcher's death.

Some backstory:  Dharwadker in this journal.

Friday, February 19, 2016

Umberto Eco, 1932-2016

Filed under: General — m759 @ 9:48 PM

See as well Eddington's "game with four letters" and "BC/AD."

Friday, February 12, 2016

The Game

Filed under: General,Geometry — Tags: — m759 @ 8:14 PM

"Right through hell
 there is a path…."
 — Malcolm Lowry,
Under the Volcano

Monday, February 8, 2016

A Game with Four Letters

Filed under: General,Geometry — Tags: — m759 @ 2:56 PM

Related material — Posts tagged Dirac and Geometry.

For an example of what Eddington calls "an open mind,"
see the 1958 letters of Nanavira Thera.
(Among the "Early Letters" in Seeking the Path ).

Monday, November 23, 2015

Dirac and Line Geometry

Filed under: General,Geometry — Tags: , , — m759 @ 2:29 AM

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

Friday, November 20, 2015

Anticommuting Dirac Matrices as Skew Lines

Filed under: General,Geometry — Tags: — m759 @ 11:45 PM

(Continued from November 13)

The work of Ron Shaw in this area, ca. 1994-1995, does not
display explicitly the correspondence between anticommutativity
in the set of Dirac matrices and skewness in a line complex of
PG(3,2), the projective 3-space over the 2-element Galois field.

Here is an explicit picture —

Anticommuting Dirac matrices as spreads of projective lines

References:  

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

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

Shaw, Ron, "Finite Geometry, Dirac Groups, and the Table of
Real Clifford Algebras," undated article at ResearchGate.net

Update of November 23:

See my post of Nov. 23 on publications by E. M. Bruins
in 1949 and 1959 on Dirac matrices and line geometry,
and on another author who gives some historical background
going back to Eddington.

Some more-recent related material from the Slovak school of
finite geometry and quantum theory —

Saniga, 'Finite Projective Spaces, Geometric Spreads of Lines and Multi-Qubits,' excerpt

The matrices underlying the Saniga paper are those of Pauli, not
those of Dirac, but these two sorts of matrices are closely related.

Thursday, April 23, 2015

Colorful Tale

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

(A sequel to yesterday's ART WARS and this
morning's De Colores )

“Perhaps the philosophically most relevant feature
of modern science is the emergence of abstract
symbolic structures as the hard core of objectivity
behind– as Eddington puts it– the colorful tale
of the subjective storyteller mind.” — Hermann Weyl
(Philosophy of  Mathematics and Natural Science ,
Princeton, 1949, p. 237)

See also Deathly Hallows.

Sunday, December 28, 2014

Cube of Ultron

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

The Blacklist “Pilot” Review

"There is an element of camp to this series though. Spader is
quite gleefully channeling Anthony Hopkins, complete with being
a well educated, elegant man locked away in a super-cell.
Speaking of that super-cell, it’s kind of ridiculous. They’ve got him
locked up in an abandoned post office warehouse on a little
platform with a chair inside  a giant metal cube that looks like
it could have been built by Tony Stark. And as Liz approaches
to talk to him, the entire front of the cube  opens and the whole
thing slides back to leave just the platform and chair. Really? 
FUCKING REALLY ? "

Kate Reilly at Geekenstein.com (Sept. 27, 2013)

Tuesday, December 2, 2014

Colorful Tale

Filed under: General — m759 @ 9:45 AM

Continued.

"Perhaps the philosophically most relevant feature
of modern science is the emergence of abstract
symbolic structures as the hard core of objectivity
behind— as Eddington puts it— the colorful tale
of the subjective storyteller mind."

— Hermann Weyl in Philosophy of Mathematics
     and Natural Science
 , Princeton, 1949, p. 237

Tom Wolfe on art theorists in The Painted Word  (1975) :

"It is important to repeat that Greenberg and Rosenberg
did not create their theories in a vacuum or simply turn up
with them one day like tablets brought down from atop
Green Mountain or Red Mountain (as B. H. Friedman once
called the two men). As tout le monde  understood, they
were not only theories but … hot news,
straight from the studios, from the scene."

The Weyl quote is a continuing theme in this journal.
The Wolfe quote appeared here on Nov. 18, 2014,
the reported date of death of Yale graduate student 
Natasha Chichilnisky-Heal.

Directions to her burial (see yesterday evening) include
a mention of "Paul Robson Street" (actually Paul
Robeson Place) near "the historic Princeton Cemetery."

This, together with the remarks by Tom Wolfe posted
here on the reported day of her death, suggests a search
for "red green black" —

The late Chichilnisky-Heal was a student of political economy.

The search colors may be interpreted, if one likes, as referring
to politics (red), economics (green), and Robeson (black).

See also Robeson in this journal.

Wednesday, February 13, 2013

Transgressing the Boundary

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

The title refers not to the 1996 Sokal hoax (which has
Boundaries , plural, in the title), but to the boundary
discussed in Monday's Penrose diamond post

"Science is a differential equation.
Religion is a boundary condition."

Alan Turing in the epigraph to the
first chapter of a book by Terence Tao

From the Tao book, page 170—

"Typically the transformed solution extends to the
boundary of the Penrose diamond and beyond…."

Transgressing the boundary between science
and religion is the topic of a 1991 paper available
at JSTOR for $29.

For the Pope on Ash Wednesday:

"Think you might have access 
to this content via your library?" —JSTOR

See also Durkheim at Harvard.

Monday, November 21, 2011

Random Reference

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

IMAGE- NY Evening Lottery Nov. 20, 2011: 245 and 0182

Joseph T. Clark, S. J., Conventional Logic and Modern Logic:
A Prelude to Transition
  (Philosophical Studies of the American
Catholic Philosophical Association, III) Woodstock, Maryland:
Woodstock College Press, 1952—

Alonzo Church, "Logic: formal, symbolic, traditional," Dictionary of Philosophy  (New York: Philosophical Library, 1942), pp. 170-182. The contents of this ambitious Dictionary are most uneven. Random reference to its pages is dangerous. But this contribution is among its best. It is condensed. But not dense. A patient and attentive study will pay big dividends in comprehension. Church knows the field and knows how to depict it. A most valuable reference.

Another book to which random reference is dangerous

http://www.log24.com/log/pix11C/111121-CosmicTrigger-245.jpg

For greater depth, see "Cassirer and Eddington on Structures,
Symmetry and Subjectivity" in Steven French's draft of
"Symmetry, Structure and the Constitution of Objects"

Saturday, September 17, 2011

The Uploading

Filed under: General — m759 @ 11:32 AM

(Continued from March 9.)

A detail from "Feist Sings 1, 2, 3, 4"—

"Uploaded by SesameStreet on Jul 18, 2008"

Those who prefer, as Weyl put it,
"
the hard core of objectivity"
to, as Eddington put it,
"the colorful tale of the subjective storyteller mind"
may consult this journal on the same day… July 18, 2008.

Friday, June 11, 2010

The Raven

Filed under: General — m759 @ 12:00 PM

"Why is a raven like a writing-desk?" — Alice's Adventures in Wonderland

Arthur Koestler, The Roots of Coincidence

"In his The Nature of the Physical World  (1928) Sir Arthur Eddington introduced his famous 'parable of the two writing desks.' One is the antique piece of furniture on which his elbows solidly rest while writing; the other is the desk as the physicist conceives it, consisting almost entirely of empty space, sheer nothingness…. Eddington concluded:

In the world of physics we watch a shadowgraph performance of familiar life. The shadow of my elbow rests on the shadow-table as the shadow-ink flows over the shadow-paper….

Though the constituents of matter could be described with great mathematical accuracy as patterns of vibrations, the question remained—  what was it that vibrated? On the one hand, these matter-waves produced physically real phenomena, such as interference patterns on a screen, or the currents in a transistor radio. On the other hand, the whole conception of matter-waves excludes by definition any medium with physical attributes as a carrier of the waves. A wave is movement; but what is that something that moves, producing the shadows on Eddington's shadow-desk? Short of calling it the grin of the Cheshire Cat, it was named the 'psi field' or 'psi function.'"

What is it that moves? Perhaps not the Cheshire Cat, but rather The Raven

Closeup, he’s blue—streaked iris blue, india-ink blue—and
black—an oily, fiery set of blacks—none of them
true—as where hate and order touch—something that cannot
become known. Stages of black but without
graduation. So there is no direction.
All of this happened, yes.

 — Jorie Graham, "The Dream of the Unified Field"

See also notes on darkness in this journal.

Saturday, July 19, 2008

Saturday July 19, 2008

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

(continued from yesterday)

Bertram Kostant, Professor Emeritus of Mathematics at MIT, on an object discussed in this week’s New Yorker:

A word about E(8). In my opinion, and shared by others, E(8) is the most magnificent ‘object’ in all of mathematics. It is like a diamond with thousands of facets. Each facet offering a different view of its unbelievable intricate internal structure.”

Hermann Weyl on the hard core of objectivity:

“Perhaps the philosophically most relevant feature of modern science is the emergence of abstract symbolic structures as the hard core of objectivity behind– as Eddington puts it– the colorful tale of the subjective storyteller mind.” (Philosophy of Mathematics and Natural Science, Princeton, 1949, p. 237)


Steven H. Cullinane on the symmetries of a 4×4 array of points:

A Structure-Endowed Entity

“A guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity S, try to determine its group of automorphisms, the group of those element-wise transformations which leave all structural relations undisturbed.  You can expect to gain a deep insight into the constitution of S in this way.”

— Hermann Weyl in Symmetry

Let us apply Weyl’s lesson to the following “structure-endowed entity.”

4x4 array of dots

What is the order of the resulting group of automorphisms?

The above group of
automorphisms plays
a role in what Weyl,
following Eddington,
  called a “colorful tale”–

The Diamond 16 Puzzle

The Diamond 16 Puzzle

This puzzle shows
that the 4×4 array can
also be viewed in
thousands of ways.

“You can make 322,560
pairs of patterns. Each
 pair pictures a different
symmetry of the underlying
16-point space.”

— Steven H. Cullinane,
July 17, 2008

For other parts of the tale,
see Ashay Dharwadker,
the Four-Color Theorem,
and Usenet Postings
.

Friday, July 18, 2008

Friday July 18, 2008

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

Hard Core

David Corfield quotes Weyl in a weblog entry, "Hierarchy and Emergence," at the n-Category Cafe this morning:

"Perhaps the philosophically most relevant feature of modern science is the emergence of abstract symbolic structures as the hard core of objectivity behind– as Eddington puts it– the colorful tale of the subjective storyteller mind." (Philosophy of Mathematics and Natural Science [Princeton, 1949], p. 237)

For the same quotation in a combinatorial context, see the foreword by A. W. Tucker, "Combinatorial Problems," to a special issue of the IBM Journal of Research and Development, November 1960 (1-page pdf).

See also yesterday's Log24 entry.
 

Tuesday, May 27, 2003

Tuesday May 27, 2003

Filed under: General,Geometry — Tags: — m759 @ 5:01 AM

Mental Health Month, Day 27:

Conspiracy Theory and
Solomon's Seal

In our journey through Mental Health Month, we have now arrived at day 27. This number, the number of lines on a non-singular cubic surface in complex projective 3-space, suggests it may be time to recall the following note (a sort of syllabus for an imaginary course) from August 1997, the month that the Mel Gibson film "Conspiracy Theory" was released.

Conspiracy Theory 101
August 13, 1997

Fiction:

(A) Masks of the Illuminati, by Robert Anton Wilson, Pocket Books, New York, 1981.  Freemasonry meets The Force (starring James Joyce and Albert Einstein).
(B) The Number of the Beast, by Robert A. Heinlein, Ballantine Books, New York, 1980.  "Pantheistic multiple solipsism" and transformation groups in n-dimensional space combine to yield "the ultimate total philosophy." (p. 438). 
(C) The Essential Blake, edited by Stanley Kunitz, MJF Books, New York, 1987.  "Fearful symmetry" in context.

Fact:

(1) The Cosmic Trigger, by Robert Anton Wilson, Falcon Press, Phoenix, 1986 (first published 1977).  Page 245 reveals that "the most comprehensive conspiracy theory," that of the physicist Sir Arthur Eddington, is remarkably similar to Heinlein's theory in (B) above.
(2) The Development of Mathematics, by E. T. Bell, 2nd. ed., McGraw-Hill, New York, 1945.  See the discussion of "Solomon's seal," a geometric configuration in complex projective 3-space.  This is as good a candidate as any for Wilson's "Holy Guardian Angel" in (A) above.
(3) Finite Projective Spaces of Three Dimensions, by J. W. P. Hirschfeld, Clarendon Press, Oxford, 1985.  Chapter 20 shows how to represent Solomon's seal in the 63-point 5-dimensional projective space over the 2-element field.  (The corresponding 6-dimensional affine space, with 64 points, is reminiscent of Heinlein's 6-dimensional space.)
 

See also China's 3,000-year-old "Book of Transformations," the I Ching, for more philosophy and lore of the affine 6-dimensional space over the binary field.

© 1997 S. H. Cullinane 

For a more up-to-date and detailed look at the mathematics mentioned above, see

Abstract Configurations
in Algebraic Geometry
,

by Igor Dolgachev.

"Art isn't easy." — Stephen Sondheim

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