Symmetry « Search Results

Tuesday, February 14, 2006

Tuesday February 14, 2006

Filed under: General — Tags: Modernism Rally — m759 @ 7:20 am

Elitist Valentine

“… ‘elite’ is a term of opprobrium on both sides of the Atlantic for both left and right for entirely different reasons– for the right, an ‘elitist’ is an unpatriotic, degenerate left-wing fan of the avant-garde; for the left, he is an undemocratic enemy of the people.”

— Charles Rosen, review of The Oxford History of Western Music in the Feb. 23, 2006, New York Review of Books

The first person that comes to mind as fitting both left and right descriptions is T. S. Eliot. Hence the following:

The image “http://www.log24.com/log/pix05/050503-Poets.jpg” cannot be displayed, because it contains errors.

The image “http://www.log24.com/log/pix05/050501-Quad.jpg” cannot be displayed, because it contains errors.

A Jungian on this six-line figure:

“They are the same six lines that exist in the I Ching…. Now observe the square more closely: four of the lines are of equal length, the other two are longer…. For this reason symmetry cannot be statically produced and a dance results.”

— Marie-Louise von Franz,
Number and Time (1970)

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Tuesday, February 7, 2006

Tuesday February 7, 2006

Filed under: General,Geometry — m759 @ 4:00 pm

Today’s birthdays:

E. T. Bell and G. H. Hardy.

I added a paragraph today to the diamond theorem page:

“Some of the patterns resulting from the action of G on D have been known for thousands of years. (See Jablan, Symmetry and Ornament, Ch. 2.6.) It is perhaps surprising that the patterns’ interrelationships and symmetries can be explained fully only by using mathematics discovered just recently (relative to the patterns’ age)– in particular, the theory of automorphism groups of finite geometries.”

This blend of mathematical history and mathematics proper seems not inappropriate for a birth date shared by a mathematical historian (Bell) and a pure mathematician (Hardy).

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Tuesday, January 24, 2006

Tuesday January 24, 2006

Filed under: General — Tags: Free-Floating Signs — m759 @ 7:00 am

ART WARS
for Michael Harris
(See previous entry.)

Related material:A classic book in a postmodern
(“free-floating signs”) cover —

The image “http://www.log24.com/log/pix06/060124-Symmetry2.jpg” cannot be displayed, because it contains errors.

This is my Princeton Companion
to Mathematics, from the days
when Princeton University Press
had higher scholarly standards.

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Monday, December 26, 2005

Monday December 26, 2005

Filed under: General — Tags: Hexagram 20, Xbox — m759 @ 7:00 pm

Language Game on
Boxing Day

In the box-style I Ching
Hexagram 34,
The Power of the Great,
is represented by

The image “http://www.log24.com/theory/images/Box34.gif” cannot be displayed, because it contains errors. .

Art is represented
by a box
(Hexagram 20,
Contemplation, View)

The image “http://www.log24.com/theory/images/Box20.gif” cannot be displayed, because it contains errors. .

And of course
great art
is represented by
an X in a box.
(Hexagram 2,
The Receptive)

The image “http://www.log24.com/theory/images/Box02.gif” cannot be displayed, because it contains errors. .

“… as a Chinese jar still
Moves perpetually
in its stillness”

“… at the still point,
there the dance is.”

— T. S. Eliot

A Jungian on this six-line figure:

“They are the same six lines that exist in the I Ching…. Now observe the square more closely: four of the lines are of equal length, the other two are longer…. For this reason symmetry cannot be statically produced and a dance results.”

— Marie-Louise von Franz,
Number and Time

For those who prefer
technology to poetry,
there is the Xbox 360.

(Today is day 360 of 2005.)

Comments (1)

Monday, October 31, 2005

Monday October 31, 2005

Filed under: General — Tags: Dante Time, Halloween 2005, Master Plan, Ternary.group, Ternary.space, Yale 9 — m759 @ 2:00 am

Balance

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"An asymmetrical balance is sought since it possesses more movement. This is achieved by the imaginary plotting of the character upon a nine-fold square, invented by some ingenious writer of the Tang dynasty. If the square were divided in half or in four, the result would be symmetrical, but the nine-fold square permits balanced asymmetry."

— Chiang Yee, Chinese Calligraphy, quoted in Aspen no. 10, item 8

"'Burnt Norton' opens as a meditation on time. Many comparable and contrasting views are introduced. The lines are drenched with reminiscences of Heraclitus' fragments on flux and movement…. the chief contrast around which Eliot constructs this poem is that between the view of time as a mere continuum, and the difficult paradoxical Christian view of how man lives both 'in and out of time,' how he is immersed in the flux and yet can penetrate to the eternal by apprehending timeless existence within time and above it. But even for the Christian the moments of release from the pressures of the flux are rare, though they alone redeem the sad wastage of otherwise unillumined existence. Eliot recalls one such moment of peculiar poignance, a childhood moment in the rose-garden– a symbol he has previously used, in many variants, for the birth of desire. Its implications are intricate and even ambiguous, since they raise the whole problem of how to discriminate between supernatural vision and mere illusion. Other variations here on the theme of how time is conquered are more directly apprehensible. In dwelling on the extension of time into movement, Eliot takes up an image he had used in 'Triumphal March': 'at the still point of the turning world.' This notion of 'a mathematically pure point' (as Philip Wheelwright has called it) seems to be Eliot's poetic equivalent in our cosmology for Dante's 'unmoved Mover,' another way of symbolising a timeless release from the 'outer compulsions' of the world. Still another variation is the passage on the Chinese jar in the final section. Here Eliot, in a conception comparable to Wallace Stevens' 'Anecdote of the Jar,' has suggested how art conquers time:

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

— F. O. Matthiessen, The Achievement of T.S. Eliot,
Oxford University Press, 1958, as quoted in On "Burnt Norton"

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Friday, September 2, 2005

Friday September 2, 2005

Filed under: General — m759 @ 9:57 am

Soap

Faith
Faith is an island in the setting sun
But proof, yes
Proof is the bottom line for everyone
— Paul Simon, “Proof”

This morning’s bottom line:
From Polya-Burnside Counting (pdf),
from today’s New York Times,
and from “related topics” in
article on Symmetry in Wikipedia:

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The image “http://www.log24.com/log/pix05B/050902-Burnside.jpg” cannot be displayed, because it contains errors.

R. L. Burnside

Burnside’s lemma —

The image “http://www.log24.com/log/pix05B/050902-Proof.gif” cannot be displayed, because it contains errors.

Raise your weary wings
against the rain, my baby
Wash your tangled curls
with gambler’s soap
— Paul Simon, “Proof”

Lottery numbers for
Pennsylvania, Sept. 1, 2005:

“Proof is the bottom line for everyone”–
Day = 120

“Faith is an island in the setting sun”–
Evening = 511

Saturday, June 4, 2005

Saturday June 4, 2005

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

Drama of the Diagonal

The 4×4 Square:
French Perspectives

Earendil_Silmarils:
The image “http://www.log24.com/log/pix05A/050604-Fuite1.jpg” cannot be displayed, because it contains errors.

   Les Anamorphoses:



"Pour construire un dessin en perspective,
   le peintre trace sur sa toile des repères:
   la ligne d'horizon (1),
   le point de fuite principal (2)
   où se rencontre les lignes de fuite (3)
   et le point de fuite des diagonales (4)."
   _______________________________

Serge Mehl,
   Perspective &
Géométrie Projective:

   "… la géométrie projective était souvent
   synonyme de géométrie supérieure.
   Elle s'opposait à la géométrie
   euclidienne: élémentaire…

La géométrie projective, certes supérieure
   car assez ardue, permet d'établir
   de façon élégante des résultats de
   la géométrie élémentaire."

Similarly…

Finite projective geometry
(in particular, Galois geometry)
   is certainly superior to
   the elementary geometry of
quilt-pattern symmetry
and allows us to establish
   de façon élégante
   some results of that
   elementary geometry.

Other Related Material…

   from algebra rather than
   geometry, and from a German
   rather than from the French:

"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 U. Press, 1946

Evariste Galois

Weyl also says that the profound branch
of mathematics known as Galois theory

   "… is nothing else but the
   relativity theory for the set Sigma,
   a set which, by its discrete and
finite character, is conceptually
   so much simpler than the
   infinite set of points in space
   or space-time dealt with
   by ordinary relativity theory."
— Weyl, Symmetry,
   Princeton U. Press, 1952

   Metaphor and Algebra…

"Perhaps every science must
start with metaphor
and end with algebra;
and perhaps without metaphor
there would never have been
any algebra."

— attributed, in varying forms, to
Max Black, Models and Metaphors, 1962

For metaphor and
algebra combined, see

"Symmetry invariance
in a diamond ring,"

A.M.S. abstract 79T-A37,
Notices of the
American Mathematical Society,
February 1979, pages A-193, 194 —
the original version of the 4×4 case
of the diamond theorem.

More on Max Black…

"When approaching unfamiliar territory, we often, as observed earlier, try to describe or frame the novel situation using metaphors based on relations perceived in a familiar domain, and by using our powers of association, and our ability to exploit the structural similarity, we go on to conjecture new features for consideration, often not noticed at the outset. The metaphor works, according to Max Black, by transferring the associated ideas and implications of the secondary to the primary system, and by selecting, emphasising and suppressing features of the primary in such a way that new slants on it are illuminated."

— Paul Thompson, University College, Oxford,
The Nature and Role of Intuition
in Mathematical Epistemology

A New Slant…

That intuition, metaphor (i.e., analogy), and association may lead us astray is well known. The examples of French perspective above show what might happen if someone ignorant of finite geometry were to associate the phrase "4×4 square" with the phrase "projective geometry." The results are ridiculously inappropriate, but at least the second example does, literally, illuminate "new slants"– i.e., diagonals– within the perspective drawing of the 4×4 square.

Similarly, analogy led the ancient Greeks to believe that the diagonal of a square is commensurate with the side… until someone gave them a new slant on the subject.

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Friday, May 27, 2005

Friday May 27, 2005

Filed under: General,Geometry — m759 @ 12:25 pm

Drama of the Diagonal,
Part Deux

Wednesday’s entry The Turning discussed a work by Roger Cooke. Cooke presents a

“fanciful story (based on Plato’s dialogue Meno).”

The History of Mathematics is the title of the Cooke book.

Associated Press thought for today:

“History is not, of course, a cookbook offering pretested recipes. It teaches by analogy, not by maxims. It can illuminate the consequences of actions in comparable situations, yet each generation must discover for itself what situations are in fact comparable.”
— Henry Kissinger (whose birthday is today)

For Henry Kissinger on his birthday:
a link to Geometry for Jews.

This link suggests a search for material
on the art of Sol LeWitt, which leads to
an article by Barry Cipra,
The “Sol LeWitt” Puzzle:
A Problem in 16 Squares (ps),
a discussion of a 4×4 array
of square linear designs.
Cipra says that

“If you like, there are three symmetry groups lurking within the LeWitt puzzle: the rotation/reflection group of order 8, a toroidal group of order 16, and an ‘existential’* group of order 16. The first group is the most obvious. The third, once you see it, is also obvious.”

* Jean-Paul Sartre,
Being and Nothingness,
Philosophical Library, 1956
[reference by Cipra]

For another famous group lurking near, if not within, a 4×4 array, click on Kissinger’s birthday link above.

Kissinger’s remark (above) on analogy suggests the following analogy to the previous entry’s (Drama of the Diagonal) figure:

The image “http://www.log24.com/log/pix05/021126-diagonH2.jpg” cannot be displayed, because it contains errors.

Logos Alogos II:
Horizon

This figure in turn, together with Cipra’s reference to Sartre, suggests the following excerpts (via Amazon.com)–

From Sartre’s Being and Nothingness, translated by Hazel E. Barnes, 1993 Washington Square Press reprint edition:

1.	on Page 51:
	“He makes himself known to himself from the other side of the world and he looks from the horizon toward himself to recover his inner being. Man is ‘a being of distances.'”
2.	on Page 154:
	“… impossible, for the for-itself attained by the realization of the Possible will make itself be as for-itself–that is, with another horizon of possibilities. Hence the constant disappointment which accompanies repletion, the famous: ‘Is it only this?’….”
3.	on Page 155:
	“… end of the desires. But the possible repletion appears as a non-positional correlate of the non-thetic self-consciousness on the horizon of the glass-in-the-midst-of-the-world.”
4.	on Page 158:
	“… it is in time that my possibilities appear on the horizon of the world which they make mine. If, then, human reality is itself apprehended as temporal….”
5.	on Page 180:
	“… else time is an illusion and chronology disguises a strictly logical order of deducibility. If the future is pre-outlined on the horizon of the world, this can be only by a being which is its own future; that is, which is to come….”
6.	on Page 186:
	“… It appears on the horizon to announce to me what I am from the standpoint of what I shall be.”
7.	on Page 332:
	“… the boat or the yacht to be overtaken, and the entire world (spectators, performance, etc.) which is profiled on the horizon. It is on the common ground of this co-existence that the abrupt revelation of my ‘being-unto-death’….”
8.	on Page 359:
	“… eyes as objects which manifest the look. The Other can not even be the object aimed at emptily at the horizon of my being for the Other.”
9.	on Page 392:
	“… defending and against which he was leaning as against a wail, suddenly opens fan-wise and becomes the foreground, the welcoming horizon toward which he is fleeing for refuge.”
10.	on Page 502:
	“… desires her in so far as this sleep appears on the ground of consciousness. Consciousness therefore remains always at the horizon of the desired body; it makes the meaning and the unity of the body.”

11.	on Page 506:
	“… itself body in order to appropriate the Other’s body apprehended as an organic totality in situation with consciousness on the horizon— what then is the meaning of desire?”
12.	on Page 661:
	“I was already outlining an interpretation of his reply; I transported myself already to the four corners of the horizon, ready to return from there to Pierre in order to understand him.”
13.	on Page 754:
	“Thus to the extent that I appear to myself as creating objects by the sole relation of appropriation, these objects are myself. The pen and the pipe, the clothing, the desk, the house– are myself. The totality of my possessions reflects the totality of my being. I am what I have. It is I myself which I touch in this cup, in this trinket. This mountain which I climb is myself to the extent that I conquer it; and when I am at its summit, which I have ‘achieved’ at the cost of this same effort, when I attain this magnificent view of the valley and the surrounding peaks, then I am the view; the panorama is myself dilated to the horizon, for it exists only through me, only for me.”

Illustration of the
last horizon remark:

From CIPRA – Slovenia,
the Institute for the
Protection of the Alps

For more on the horizon, being, and nothingness, see

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Wednesday, May 4, 2005

Wednesday May 4, 2005

Filed under: General,Geometry — Tags: Alperin, Eightfold Cube, May 6 2005 — m759 @ 1:00 pm

The Fano Plane
Revisualized:

or, The Eightfold Cube

Here is the usual model of the seven points and seven lines (including the circle) of the smallest finite projective plane (the Fano plane):

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Every permutation of the plane's points that preserves collinearity is a symmetry of the plane. The group of symmetries of the Fano plane is of order 168 and is isomorphic to the group PSL(2,7) = PSL(3,2) = GL(3,2). (See Cameron on linear groups (pdf).)

The above model indicates with great clarity six symmetries of the plane– those it shares with the equilateral triangle. It does not, however, indicate where the other 162 symmetries come from.

Shown below is a new model of this same projective plane, using partitions of cubes to represent points:

The cubes' partitioning planes are added in binary (1+1=0) fashion. Three partitioned cubes are collinear if and only if their partitioning planes' binary sum equals zero.

The second model is useful because it lets us generate naturally all 168 symmetries of the Fano plane by splitting a cube into a set of four parallel 1x1x2 slices in the three ways possible, then arbitrarily permuting the slices in each of the three sets of four. See examples below.

Fano plane group - generating permutations

For a proof that such permutations generate the 168 symmetries, see Binary Coordinate Systems.

(Note that this procedure, if regarded as acting on the set of eight individual subcubes of each cube in the diagram, actually generates a group of 168*8 = 1,344 permutations. But the group's action on the diagram's seven partitions of the subcubes yields only 168 distinct results. This illustrates the difference between affine and projective spaces over the binary field GF(2). In a related 2x2x2 cubic model of the affine 3-space over GF(2) whose "points" are individual subcubes, the group of eight translations is generated by interchanges of parallel 2x2x1 cube-slices. This is clearly a subgroup of the group generated by permuting 1x1x2 cube-slices. Such translations in the affine 3-space have no effect on the projective plane, since they leave each of the plane model's seven partitions– the "points" of the plane– invariant.)

To view the cubes model in a wider context, see Galois Geometry, Block Designs, and Finite-Geometry Models.

For another application of the points-as-partitions technique, see Latin-Square Geometry: Orthogonal Latin Squares as Skew Lines.

For more on the plane's symmetry group in another guise, see John Baez on Klein's Quartic Curve and the online book The Eightfold Way. For more on the mathematics of cubic models, see Solomon's Cube.

For a large downloadable folder with many other related web pages, see Notes on Finite Geometry.

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Monday, May 2, 2005

Monday May 2, 2005

Filed under: General — m759 @ 11:00 am

A Dance Results

Roger Kimball on Rosalind Krauss's
The Optical Unconscious:

"Professor Krauss even uses many of the same decorations with which she festooned earlier volumes. Bataille’s photograph of a big toe, for example, which I like to think of as her mascot, reappears. As does her favorite doodle, a little graph known as a 'Klein Group' or 'L Schema' whose sides and diagonals sport arrows pointing to corners labeled with various opposing pairs: e.g., 'ground' and 'not ground,' 'figure' and 'not figure.' Professor Krauss seems to believe that this device, lifted from the pages of structuralist theory, illuminates any number of deep mysteries: the nature of modernism, to begin with, but also the essence of gender relations, self-consciousness, perception, vision, castration anxiety, and other pressing conundrums that, as it happens, she has trouble distinguishing from the nature of modernism. Altogether, the doodle is a handy thing to have around. One is not surprised that Professor Krauss reproduces it many times in her new book."

From Drid Williams,
The Semiotics of Human Action,
Ritual, and Dance:

A Klein four-group in the context of dance

This is closely related to
Beckett's "Quad" figure

A Jungian on this six-line figure:

"They are the same six lines
that exist in the I Ching….
Now observe the square more closely:
four of the lines are of equal length,
the other two are longer….
For this reason symmetry
cannot be statically produced
and a dance results."

— Marie-Louise von Franz,
Number and Time (1970)

and to the Greimas "semiotic square":

"People have believed in the fundamental character of binary oppositions since at least classical times. For instance, in his Metaphysics Aristotle advanced as primary oppositions: form/matter, natural/unnatural, active/passive, whole/part, unity/variety, before/after and being/not-being.* But it is not in isolation that the rhetorical power of such oppositions resides, but in their articulation in relation to other oppositions. In Aristotle's Physics the four elements of earth, air, fire and water were said to be opposed in pairs. For more than two thousand years oppositional patterns based on these four elements were widely accepted as the fundamental structure underlying surface reality….

The structuralist semiotician Algirdas Greimas introduced the semiotic square (which he adapted from the 'logical square' of scholastic philosophy) as a means of analysing paired concepts more fully…."

— Daniel Chandler, Semiotics for Beginners.

* Compare Chandler's list of Aristotle's primary oppositions with Aristotle's list (also in the Metaphysics) of Pythagorean oppositions (see Midrash Jazz Quartet).

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Sunday, May 1, 2005

Sunday May 1, 2005

Filed under: General — Tags: Logos and Branding — m759 @ 1:11 pm

Logos

Harvard's Barry Mazur on
one mathematical style:

"It’s the barest, most Beckett-like vocabulary
that incorporates the theory and nothing else."

Samuel Beckett, Quad (1981):

The image “http://www.log24.com/log/pix05/050501-Quad.jpg” cannot be displayed, because it contains errors.

A Jungian on this six-line logo:

A related logo from
Columbia University's
Department of Art History
and Archaeology:

The image “http://www.log24.com/log/pix05/050501-ArtHist2.gif” cannot be displayed, because it contains errors.

Also from that department:

Rosalind Krauss,

The image “http://www.log24.com/log/pix05/050501-Krauss.jpg” cannot be displayed, because it contains errors.

Meyer Schapiro Professor
of Modern Art and Theory:

"There is no painter in the West
who can be unaware of
the symbolic power
of the cruciform shape
and the Pandora's box
of spiritual reference
that is opened
once one uses it."

"In the garden of Adding
live Even and Odd…"
— The Midrash Jazz Quartet in
City of God, by E. L. Doctorow

THE GREEK CROSS

A cross in which all the arms
are the same length.

Here, for reference, is a Greek cross
within a nine-square grid:

The image “http://www.log24.com/log/pix05/050501-GrCross.gif” cannot be displayed, because it contains errors.

Related religious meditation for
Doctorow's "Garden of Adding"…

4 + 5 = 9.

Types of Greek cross
illustrated in Wikipedia
under "cross":

The image “http://www.log24.com/log/pix05/GrCross.gif” cannot be displayed, because it contains errors.

From designboom.com:

THE BAPTISMAL CROSS

The image “http://www.log24.com/log/pix05/050501-BaptismalCross.gif” cannot be displayed, because it contains errors.

is a cross with eight arms:
a Greek cross, which is superimposed
on a Greek 'chi,' the first letter
of the Greek word for 'Christ.'
Since the number eight is symbolic
of rebirth or regeneration,
this cross is often used
as a baptismal cross.

Related material:

The image “http://www.log24.com/theory/images/Symm-axes.jpg” cannot be displayed, because it contains errors.

Fritz Leiber's "spider"
or "double cross" logo.
See Why Me? and
A Shot at Redemption.

Happy Orthodox Easter.

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Thursday, March 3, 2005

Tuesday, February 8, 2005

Tuesday February 8, 2005

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

New from the
Oscar-winning producer,
director, and screenwriter
of “A Beautiful Mind” –

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The image “http://www.log24.com/log/pix05/050208-Crowe.jpg” cannot be displayed, because it contains errors.

With apologies to Dan Brown…

“The Divine Proportion…

is an irrational number and
the positive solution
of the quadratic equation

x² – x – 1 = 0,

which is (1+Sqrt(5))/2,
about 1.618034.

The Greek letter ‘phi’
(see below for the symbol)
is sometimes used
to represent this number.”

The image “http://www.log24.com/log/pix05/050208-pentagon2.gif” cannot be displayed, because it contains errors.

— Don Cohen

For another approach to
the divine proportion, see

Best Picture.

“The rogue’s yarn that will run through much of the material is the algebraic symmetry to which the name of Galois is attached and which I wanted to introduce in as concrete and appealing a way as possible….

Apart from its intrinsic appeal, that is the reason for treating the construction of the pentagon, and our task today will be to acquire some feel for this construction. It is not easy.”

— R. P. Langlands, 1999 lecture (pdf) at the Institute for Advanced Study, Princeton, in the spirit of Hermann Weyl

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Monday, January 24, 2005

Thursday, November 11, 2004

Thursday November 11, 2004

Filed under: General — m759 @ 4:04 pm

Blasphemy?

From the profile of
psifenix at livejournal.com:

"Perhaps I shall embrace Islam.
Its standards for poetry
seem very high."

— Susan Voight in the novel
Freedom & Necessity,
by Emma Bull and Steven Brust

From psifenix
at livejournal.com
on Oct. 3, 2004:

"S₄ is the one true God,
and C₂×C₂ is
His [normal] prophet."

— psifenix

"Group theory is so beautiful. If I ever get the chance to go back in time or mess with an agrarian civilization, I'm going to try to get them to worship finite groups of small order. Why? Symmetry is nice, and you can turn groups into pretty pictures, too… Each group could represent an attribute of god, and the nonabelian attributes would be the scary and wrathful ones. The prime cyclic groups could symbolize the cycles of life and death. And the trivial group would be the origin of all things!

I'm on crack, but that's okay!"

— psifenix

"You can turn groups
into pretty pictures" —

IMAGE- 'Study of O' and 'Portrait of O' (the octahedral group)

Shown above are two ways of
picturing the octahedral group O,
also known as
the symmetric group S₄.

For further details, see
Diamond Theory.

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Friday, September 17, 2004

Friday September 17, 2004

Filed under: General — Tags: Glasperlenspiel, Strange Change — m759 @ 3:57 pm

3:57:09…
Time is a Weapon

In memory of rock star and NRA member Johnny Ramone, who died on Wednesday, Sept. 15:

“You’ve got to ask yourself a question.”
— Clint Eastwood as Dirty Harry

“At the end, when the agent pumps Neo full of lead, the agent is using a .357 Magnum. That gun only holds 9 bullets, but the agent shoots 10 shots at Neo. I don’t know where he got that gun.”

— Jesse Baumann,
The Matrix: The Magic Bullet

Manufacturer:
Ta’as Israel Industries,
Ramat Hasharon, Israel

Friday, August 01, 2003:

Fearful Meditation

Ray Price — Time

“The Max D. Barnes-penned title track, with its stark-reality lyrics, is nothing short of haunting: ‘Time is a weapon, it’s cold and it’s cruel; It knows no religion and plays by no rules; Time has no conscience when it’s all said and done; Like a beast in the jungle that devours its young.’ That’s so good, it hurts! Price’s still-amazing vocals are simply the chilling icing on the cake.”

— Lisa Berg, NashvilleCountry.com

O fearful meditation!
Where, alack,
Shall time’s best jewel
from time’s chest lie hid?

— Shakespeare, Sonnet 65

Clue: click here. This in turn leads to my March 4 entry Fearful Symmetry, which contains the following:

“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….”

— Hermann Hesse, The Glass Bead Game

“How strange the change from major to minor….”

— Cole Porter, “Every Time We Say Goodbye“

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Tuesday, September 14, 2004

Tuesday September 14, 2004

Filed under: General,Geometry — Tags: I Ching Wheel, Square Wheel — m759 @ 3:00 pm

The Square Wheel

Harmonic analysis may be based either on the circular (i.e., trigonometric) functions or on the square (i. e., Walsh) functions. George Mackey's masterly historical survey showed that the discovery of Fourier analysis, based on the circle, was of comparable importance (within mathematics) to the discovery (within general human history) of the wheel. Harmonic analysis based on square functions– the "square wheel," as it were– is also not without its importance.

For some observations of Stephen Wolfram on square-wheel analysis, see pp. 573 ff. in Wolfram's magnum opus, A New Kind of Science (Wolfram Media, May 14, 2002). Wolfram's illustration of this topic is closely related, as it happens, to a note on the symmetry of finite-geometry hyperplanes that I wrote in 1986. A web page pointing out this same symmetry in Walsh functions was archived on Oct. 30, 2001.

That web page is significant (as later versions point out) partly because it shows that just as the phrase "the circular functions" is applied to the trigonometric functions, the phrase "the square functions" might well be applied to Walsh functions– which have, in fact, properties very like those of the trig functions. For details, see Symmetry of Walsh Functions, updated today.

"While the reader may draw many a moral from our tale, I hope that the story is of interest for its own sake. Moreover, I hope that it may inspire others, participants or observers, to preserve the true and complete record of our mathematical times."

— From Error-Correcting Codes
Through Sphere Packings
To Simple Groups,
by Thomas M. Thompson,
Mathematical Association of America, 1983

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Friday, September 10, 2004

Friday September 10, 2004

Filed under: General — m759 @ 1:13 pm

Philosophy

For Samira Bellil,

who died in Paris on
Friday, Sept. 3, 2004…

From the link at
Symmetry and Change
in the Dreamtime,
Part 8, Friday,
Sept. 3, 2004,
Noon…

Under heaven
thunder rolls…

Log24 on Sept. 10, 2002—

Three songs from Sept. 10
in various preceding years–

“Good morning little schoolgirl
Good morning little schoolgirl
Can I come home with
Can I come home with you“

— Rod Stewart, Sept. 10, 1964

“Tell your mamma, girl, I can’t stay long
We got things we gotta catch up on
Mmmm, you know
You know what I’m sayin’ “

— Neil Diamond, Sept. 10, 1966

“A time of war, a time of peace
A time of love, a time of hate
A time you may embrace
A time to refrain from embracing“

— The Byrds, Sept. 10, 1965

Further verses from the Byrds
seem appropriate on this, the day
of Samira Bellil’s funeral:

To everything, turn, turn, turn,
there is a season, turn, turn, turn…

Tournante

“It’s not even called rape. They call it
a tournante, or pass-round.
The banality is deliberate:
a joint, a girl – same difference.”

… and a time to every purpose
under heaven.

“… The kind of school where teacher
Fabrice Genestal kept hearing
the word “tournante” and didn’t click
what it meant, till he and Sillam
sat the kids down in after-school
workshops, and got talking.”

— Metropolitan Police Service, London

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Sunday, September 5, 2004

Sunday September 5, 2004

Filed under: General — m759 @ 5:29 pm

A Story of Sorts

Sometimes one’s journal entries seem to be telling a story…

This was the case for Log24 entries of Tuesday through Friday last week. Unfortunately, the story they told is about as coherent as Finnegans Wake.

Anyone interested can find the story, put into chronological order and prefaced with a summary, at

Symmetry and Change
in the Dreamtime.

Comments (1)

Sunday September 5, 2004

Filed under: General — m759 @ 4:00 pm

Symmetry and Change
in the Dreamtime

Notes from the Journal
of Steven H. Cullinane

Summary:

Aug 31 2004 07:31:01 PM	Early Evening, Shining Star
Sep 01 2004 09:00:35 AM	Words and Images
Sep 01 2004 12:07:28 PM	Whale Rider
Sep 02 2004 11:11:42 AM	Heaven and Earth
Sep 02 2004 07:00:23 PM	Whale Road
Sep 03 2004 12:00:54 AM	Cinderella’s Slipper
Sep 03 2004 10:01:56 AM	Another September Morn
Sep 03 2004 12:00:25 PM	Noon
Sep 03 2004 01:13:49 PM	De Nada
Sep 03 2004 03:17:13 PM	Ite, Missa Est

Tuesday, August 31, 2004

Symmetry and Change, Part 1…

Early Evening,
Shining Star

7:31:01 PM ET

Hexagram 01
The Creative:

The Image

Heaven

The movement of heaven
is full of power.

Click on picture
for details.

The Clare Lawler Prize
for Literature goes to…

Under the Volcano,
Chapter VI:

“What have I got out of my life? Contacts with famous men… The occasion Einstein asked me the time, for instance. That summer evening…. smiles when I say I don’t know. And yet asked me. Yes: the great Jew, who has upset the whole world’s notions of time and space, once leaned down… to ask me… ragged freshman… at the first approach of the evening star, the time. And smiled again when I pointed out the clock neither of us had noticed.”

For the thoughts on time
of another famous man,
from Mexico, see the
Nobel Prize acceptance speech
of Octavio Paz,
“In Search of the Present.”

Wednesday, September 1, 2004

Symmetry and Change, Part 2…

Words and Images

9:00:35 AM ET

Hexagram 35
Progress:

The Image

Fire

Earth

The sun rises over the earth.

From Aug. 18, 2004:

“Oh, my Lolita. I have only words
to play with!” (Nabokov, Lolita)

“This is the best toy train set
a boy ever had!”
(Orson Welles, after first touring
RKO Studios, quoted in Halliwell)

“As the quotes above by Nabokov and Welles suggest, we need to be able to account for the specific functions available to narrative in each medium, for the specific elements that empirical creators will ‘play with’ in crafting their narratives.”

— Donald F. Larsson

For
James Whale
and
William French Anderson —

Words
In the Spirit of
Dave Barry’s Book of Bad Songs:

Stay for just a while…
Stay, and let me look at you.
It’s been so long, I hardly knew you.
Standing in the door…
Stay with me a while.
I only want to talk to you.
We’ve traveled halfway ’round the world
To find ourselves again.

September morn…
We danced until the night
      became a brand new day,
Two lovers playing scenes
      from some romantic play.
September morning still can
      make me feel this way.

Look at what you’ve done…
Why, you’ve become a grown-up girl…

— Neil Diamond

Images
In the Spirit of
September Morn:

The Last Day of Summer:
Photographs by Jock Sturges

“In 1990, the FBI entered Sturges’s studio and seized his work, claiming violation of child pornography laws.”

Related material:

Bill’s Diamond Theory

and

Log24 entries of
Aug. 15, 2004.

Those interested in the political implications of Diamond’s songs may enjoy Neil Performs at Kerry Fundraiser.

I personally enjoyed this site’s description of Billy Crystal’s remarks, which included “a joke about former President Clinton’s forthcoming children’s book — ‘It’s called The Little Engine That Could Because It Could.'”

“Puff, puff, woo, woo, off we go!”

Wednesday, September 1, 2004

Symmetry and Change, Part 3…

Whale Rider

12:07:28 PM

Hexagram 28
Preponderance of
the Great:

The Image

Lake

Wind

The lake rises
above the trees.

Cullinane College News:

“Congratulations to Clare Lawler, who participated very successfully in the recently held Secondary Schools Judo Championships in Wellington.”

For an explanation of this entry’s title, see the previous two entries and

Oxford Word
(Log24, July 10, 2004)

Thursday, September 2, 2004

Symmetry and Change, Part 4…

Heaven and Earth

11:11:42 AM ET

Hexagram 42
Increase:

The Image

Wind

Thunder

Wind and thunder:
the image of Increase.

“This time resembles that of
the marriage of heaven and earth”

Kylie

Finney

Well if you want to ride
you gotta ride it like you find it.
Get your ticket at the station
of the Rock Island Line. — Lonnie Donegan (d. Nov. 3) and others

The Rock Island Line’s namesake depot
in Rock Island, Illinois

“What it all boiled down to really was everybody giving everybody else a hard time for no good reason whatever… You just couldn’t march to your own music. Nowadays, you couldn’t even hear it… It was lost, the music which each person had inside himself, and which put him in step with things as they should be.”

— The Grifters, Ch. 10, 1963, by
James Myers Thompson

“The Old Man’s still an artist
with a Thompson.”
— Terry in “Miller’s Crossing”

For some of “the music which
each person had inside,”
click on the picture
with the Thompson.

It may be that Kylie is,
in her own way, an artist…
with a 357:

(Hits counter at
The Quality of Diamond
as of 11:05 AM Sept. 2, 2004)

For more on
“the marriage of heaven and earth,”
see
Plato, Pegasus, and the Evening Star.

Thursday, September 2, 2004

Symmetry and Change, Part 5…

Whale Road

7:00:23 PM

Hexagram 23
Splitting Apart:

The Image

Mountain

Earth

The mountain rests
on the earth.

“… the plot is different but the monsters, names, and manner of speaking will ring a bell.”

— Frank Pinto, Jr., review of Seamus Heaney’s translation of Beowulf

Other recommended reading, found during a search for the implications of today’s previous entry, “Hexagram 42”:

Water Wings.

This excellent meditation
on symmetry and change
comes from a site whose
home page
has the following image:

Friday, September 3, 2004

Symmetry and Change, Part 6…

Cinderella’s Slipper

12:00:54 AM ET

Hexagram 54
The Marrying Maiden:

The Image

Thunder

Lake

See

The hundredletter
thunderwords of
Finnegans Wake

“… a Thoreau-like retreat
by a nearby lake….
Both men have
a ‘touch of the poet’….
The symmetry is perfect.”

Friday, September 3, 2004

Symmetry and Change, Part 7…

Another September Morn

10:01:56 AM ET

Hexagram 56:
The Wanderer

The Image

Fire

Mountain

Fire on the mountain,
Run boys run…
Devil’s in the House of
The Rising Sun!

Friday, September 3, 2004

Symmetry and Change, Part 8…

Noon

12:00:25 PM ET

Hexagram 25
Innocence:

The Image

Heaven

Thunder

Under heaven
thunder rolls.

Friday, September 3, 2004

Symmetry and Change, Part 9…

De Nada

Helen Lane

1:13:49 PM ET

Hexagram 49
Revolution:

The Image

Lake

Fire Fire in the lake:
the image of Revolution.

“I sit now in a little room off the bar at four-thirty in the morning drinking ochas and then mescal and writing this on some Bella Vista notepaper I filched the other night…. But this is worst of all, to feel your soul dying. I wonder if it is because to-night my soul has really died that I feel at the moment something like peace. Or is it because right through hell there is a path, as Blake well knew, and though I may not take it, sometimes lately in dreams I have been able to see it? …And this is how I sometimes think of myself, as a great explorer who has discovered some extraordinary land from which he can never return to give his knowledge to the world: but the name of this land is hell. It is not Mexico of course but in the heart.”

— Malcolm Lowry, Under the Volcano

Friday, September 3, 2004

Symmetry and Change, conclusion…

Ite, Missa Est

3:17:13 PM ET

Hexagram 13
Fellowship With Men:

The Image

Heaven

Fire

Heaven together with fire.

“A pretty girl —
is like a melody —- !”

For details, see
A Mass for Lucero.

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Friday, September 3, 2004

Friday September 3, 2004

Filed under: General — m759 @ 3:17 pm

Symmetry and Change, conclusion…

Ite, Missa Est

3:17:13 PM ET

Hexagram 13
Fellowship With Men:

The Image

Heaven

Fire

Heaven together with fire.

“A pretty girl —
is like a melody —- !”

For details, see
A Mass for Lucero.

Comments Off

Friday September 3, 2004

Filed under: General — m759 @ 1:13 pm

Symmetry and Change, Part 9…

De Nada

Helen Lane

1:13:49 PM ET

Hexagram 49
Revolution:

The Image

Lake

Fire

Fire in the lake:
the image of Revolution.

"I sit now in a little room off the bar at four-thirty in the morning drinking ochas and then mescal and writing this on some Bella Vista notepaper I filched the other night…. But this is worst of all, to feel your soul dying. I wonder if it is because to-night my soul has really died that I feel at the moment something like peace. Or is it because right through hell there is a path, as Blake well knew, and though I may not take it, sometimes lately in dreams I have been able to see it? …And this is how I sometimes think of myself, as a great explorer who has discovered some extraordinary land from which he can never return to give his knowledge to the world: but the name of this land is hell. It is not Mexico of course but in the heart."

— Malcolm Lowry, Under the Volcano

Comments Off

Friday September 3, 2004

Filed under: General — m759 @ 12:00 pm

Symmetry and Change, Part 8…

Noon

12:00:25 PM ET

Hexagram 25
Innocence:

The Image

Heaven

Thunder

Under heaven
thunder rolls.

Comments Off

Friday September 3, 2004

Filed under: General — m759 @ 10:01 am

Symmetry and Change, Part 7…

Another September Morn

10:01:56 AM ET

Hexagram 56:
The Wanderer

The Image

Fire

Mountain

Fire on the mountain,
Run boys run…
Devil’s in the House of
The Rising Sun!

Comments Off

Friday September 3, 2004

Filed under: General — m759 @ 12:00 am

Symmetry and Change, Part 6…

Cinderella’s Slipper

12:00:54 AM ET

Hexagram 54
The Marrying Maiden:

The Image

Thunder

Lake

See

The hundredletter
thunderwords of
Finnegans Wake

“… a Thoreau-like retreat
by a nearby lake….
Both men have
a ‘touch of the poet’….
The symmetry is perfect.”

Comments Off

Thursday, September 2, 2004

Thursday September 2, 2004

Filed under: General — m759 @ 7:00 pm

Symmetry and Change, Part 5…

Whale Road

7:00:23 PM

Hexagram 23
Splitting Apart:

The Image

Mountain

Earth

The mountain rests
on the earth.

“… the plot is different but the monsters, names, and manner of speaking will ring a bell.”

— Frank Pinto, Jr., review of Seamus Heaney’s translation of Beowulf

Other recommended reading, found during a search for the implications of today’s previous entry, “Hexagram 42”:

Water Wings.

This excellent meditation
on symmetry and change
comes from a site whose
home page
has the following image:

Comments Off

Thursday September 2, 2004

Filed under: General — m759 @ 11:11 am

Symmetry and Change, Part 4…

Heaven and Earth

11:11:42 AM ET

Hexagram 42
Increase:

The Image

Wind

Thunder

Wind and thunder:
the image of Increase.

“This time resembles that of
the marriage of heaven and earth”

Kylie

Finney

Well if you want to ride
you gotta ride it like you find it.
Get your ticket at the station
of the Rock Island Line. — Lonnie Donegan (d. Nov. 3) and others

The Rock Island Line’s namesake depot
in Rock Island, Illinois

— The Grifters, Ch. 10, 1963, by
James Myers Thompson

“The Old Man’s still an artist
with a Thompson.”
— Terry in “Miller’s Crossing“

For some of “the music which
each person had inside,”
click on the picture
with the Thompson.

It may be that Kylie is,
in her own way, an artist…
with a 357:

(Hits counter at
The Quality of Diamond
as of 11:05 AM Sept. 2, 2004)

For more on
“the marriage of heaven and earth,”
see
Plato, Pegasus, and the Evening Star.

Comments Off

Wednesday, September 1, 2004

Wednesday September 1, 2004

Filed under: General — m759 @ 12:07 pm

Symmetry and Change, Part 3…

Whale Rider

12:07:28 PM

Hexagram 28
Preponderance of
the Great:

The Image

Lake

Wind

The lake rises
above the trees.

Cullinane College News:

“Congratulations to Clare Lawler, who participated very successfully in the recently held Secondary Schools Judo Championships in Wellington.”

For an explanation of this entry’s title, see the previous two entries and

Oxford Word
(Log24, July 10, 2004)

Comments Off

Wednesday September 1, 2004

Filed under: General — m759 @ 9:00 am

Symmetry and Change, Part 2…

Words and Images

9:00:35 AM ET

Hexagram 35
Progress:

The Image

Fire

Earth

The sun rises over the earth.

From Aug. 18, 2004:

“Oh, my Lolita. I have only words
to play with!” (Nabokov, Lolita)

“This is the best toy train set
a boy ever had!”
(Orson Welles, after first touring
RKO Studios, quoted in Halliwell)

— Donald F. Larsson

For
James Whale
and
William French Anderson —

Words
In the Spirit of
Dave Barry’s Book of Bad Songs:

September morn…
We danced until the night
      became a brand new day,
Two lovers playing scenes
      from some romantic play.
September morning still can
      make me feel this way.

Look at what you’ve done…
Why, you’ve become a grown-up girl…

— Neil Diamond

Images
In the Spirit of
September Morn:

The Last Day of Summer:
Photographs by Jock Sturges

“In 1990, the FBI entered Sturges’s studio and seized his work, claiming violation of child pornography laws.”

Related material:

Bill’s Diamond Theory

and

Log24 entries of
Aug. 15, 2004.

Those interested in the political implications of Diamond’s songs may enjoy Neil Performs at Kerry Fundraiser.

“Puff, puff, woo, woo, off we go!”

Comments Off

Tuesday, August 31, 2004

Tuesday August 31, 2004

Filed under: General — m759 @ 7:31 pm

Symmetry and Change, Part 1…

Early Evening,
Shining Star

7:31:01 PM ET

Hexagram 01
The Creative:

The Image

Heaven

The movement of heaven
is full of power.

Click on picture
for details.

The Clare Lawler Prize
for Literature goes to…

Under the Volcano,
Chapter VI:

For the thoughts on time
of another famous man,
from Mexico, see the
Nobel Prize acceptance speech
of Octavio Paz,
“In Search of the Present.”

Comments Off

Tuesday, August 10, 2004

Tuesday August 10, 2004

Filed under: General,Geometry — Tags: Change Logos, Gods and Giants, Strange Change — m759 @ 3:01 pm

Battle of Gods and Giants

In checking the quotations from Dante in the previous entry, I came across the intriguing site Gigantomachia:

"A gigantomachia or primordial battle between the gods has been retold in myth, cult, art and theory for thousands of years, from the Egyptians to Heidegger. This site will present the history of the theme. But it will do so in an attempt to raise the question of the contemporary relevance of it. Does the gigantomachia take place today? Where? When? In what relation to you and me?"

Perhaps atop the Empire State Building?

(See An Affair to Remember and Empire State Building to Honor Fay Wray.)

Perhaps in relation to what the late poet Donald Justice called "the wood within"?

Perhaps in relation to T. S. Eliot's "The Waste Land" and the Feast of the Metamorphosis?

Or perhaps not.

Perhaps at Pergamon:

Perhaps at Pergamon Press:

Invariants

"What modern painters are trying to do,
if they only knew it, is paint invariants."

— James J. Gibson in Leonardo
(Vol. 11, pp. 227-235.
Pergamon Press Ltd., 1978)

An example of invariant structure:

The three line diagrams above result from the three partitions, into pairs of 2-element sets, of the 4-element set from which the entries of the bottom colored figure are drawn. Taken as a set, these three line diagrams describe the structure of the bottom colored figure. After coordinatizing the figure in a suitable manner, we find that this set of three line diagrams is invariant under the group of 16 binary translations acting on the colored figure.

A more remarkable invariance — that of symmetry itself — is observed if we arbitrarily and repeatedly permute rows and/or columns and/or 2×2 quadrants of the colored figure above. Each resulting figure has some ordinary or color-interchange symmetry.

This sort of mathematics illustrates the invisible "form" or "idea" behind the visible two-color pattern. Hence it exemplifies, in a way, the conflict described by Plato between those who say that "real existence belongs only to that which can be handled" and those who say that "true reality consists in certain intelligible and bodiless forms."

For further details, see a section on Plato in the Gigantomachia site.

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Sunday, July 18, 2004

Sunday July 18, 2004

Filed under: General — m759 @ 1:22 am

New Web Page:

Reflections on Symmetry

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Saturday, July 10, 2004

Saturday July 10, 2004

Filed under: General — Tags: OED, Whanganui — m759 @ 3:17 pm

Oxford Word

From today's obituary in The New York Times of R. W. Burchfield, editor of A Supplement to the Oxford English Dictionary:

"Robert William Burchfield was born Jan. 27, 1923, in Wanganui, New Zealand. In 1949, after earning an undergraduate degree at Victoria University College in Wellington, he accepted a Rhodes scholarship to Oxford.

There, he read Medieval English literature with C.S. Lewis and J.R.R. Tolkien."

For more on literature and Wanganui, see my entry of Jan. 19. 2003, from which the following is taken.

Literature
and
Geography

"Literature begins
with geography."

— Attributed to
Robert Frost

The Maori Court at
the Wanganui Museum

"Cullinane College is a Catholic co-educational college, set to open in Wanganui (New Zealand) on the 29th of January, 2003."

The 29th of January will be the 40th anniversary of the death of Saint Robert Frost.

New Zealand, perhaps the most beautiful country on the planet, is noted for being the setting of the film version of Lord of the Rings, which was written by a devout Catholic, J. R. R. Tolkien.

For other New Zealand themes, see Alfred Bester's novels The Stars My Destination and The Deceivers.

The original title of The Stars My Destination was Tyger! Tyger! after Blake's poem.

For more on fearful symmetry, see the work of Marston Conder, professor of mathematics at the University of Auckland, New Zealand.

Comments (1)

Thursday, May 27, 2004

Thursday May 27, 2004

Filed under: General — Tags: Eden, Springer — m759 @ 10:10 am

Ineluctable

On the poetry of Geoffrey Hill:

"… why read him? Because of the things he writes about—war and peace and sacrifice, and the search for meaning and the truths of the heart, and for that haunting sense that, in spite of war and terror and the indifferences that make up our daily hells, there really is some grander reality, some ineluctable presence we keep touching. There remains in Hill the daunting possibility that it may actually all cohere in the end, or at least enough of it to keep us searching for more.

There is a hard edge to Hill, a strong Calvinist streak in him, and an intelligence that reminds one of Milton….."

— Paul Mariani, review in America of Geoffrey Hill's The Orchards of Syon

"Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one."

"A very short space of time through very short times of space…. Am I walking into eternity along Sandymount strand?"

— James Joyce, Ulysses, Proteus chapter

"Time has been unfolded into space."

— James O. Coplien, Bell Labs

"Pattern and symmetry are closely related."

— James O. Coplien on Symmetry Breaking

"… as the critic S. L. Goldberg puts it, 'the chapter explores the Protean transformations of matter in time . . . apprehensible only in the condition of flux . . . as object . . . and Stephen himself, as subject. In the one aspect Stephen is seeking the principles of change and the underlying substance of sensory experience; in the other, he is seeking his self among its temporal manifestations'….

— Goldberg, S.L. 'Homer and the Nightmare of History.' Modern Critical Views: James Joyce. Ed. Harold Bloom. New York: Chelsea House, 1986. 21-38."

— from the Choate site of David M. Loeb

In summary:

Coplien

Joyce

Aleph,
alpha:
nought,
nought,
one:

See also Time Fold.

(By the way, Jorn Barger seems
to have emerged from seclusion.)

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Thursday, May 20, 2004

Thursday May 20, 2004

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

Parable

"A comparison or analogy. The word is simply a transliteration of the Greek word: parabolé (literally: 'what is thrown beside' or 'juxtaposed'), a term used to designate the geometric application we call a 'parabola.'…. The basic parables are extended similes or metaphors."

— http://religion.rutgers.edu/nt/
primer/parable.html

"If one style of thought stands out as the most potent explanation of genius, it is the ability to make juxtapositions that elude mere mortals. Call it a facility with metaphor, the ability to connect the unconnected, to see relationships to which others are blind."

— Sharon Begley, "The Puzzle of Genius," Newsweek magazine, June 28, 1993, p. 50

"The poet sets one metaphor against another and hopes that the sparks set off by the juxtaposition will ignite something in the mind as well. Hopkins’ poem 'Pied Beauty' has to do with 'creation.' "

— Speaking in Parables, Ch. 2, by Sallie McFague

"The Act of Creation is, I believe, a more truly creative work than any of Koestler's novels…. According to him, the creative faculty in whatever form is owing to a circumstance which he calls 'bisociation.' And we recognize this intuitively whenever we laugh at a joke, are dazzled by a fine metaphor, are astonished and excited by a unification of styles, or 'see,' for the first time, the possibility of a significant theoretical breakthrough in a scientific inquiry. In short, one touch of genius—or bisociation—makes the whole world kin. Or so Koestler believes."

— Henry David Aiken, The Metaphysics of Arthur Koestler, New York Review of Books, Dec. 17, 1964

For further details, see

Speaking in Parables:
A Study in Metaphor and Theology

by Sallie McFague

Fortress Press, Philadelphia, 1975

Introduction
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7

"Perhaps every science must start with metaphor and end with algebra; and perhaps without metaphor there would never have been any algebra."

— attributed, in varying forms (1, 2, 3), to Max Black, Models and Metaphors, 1962

For metaphor and algebra combined, see

"Symmetry invariance in a diamond ring," A.M.S. abstract 79T-A37, Notices of the Amer. Math. Soc., February 1979, pages A-193, 194 — the original version of the 4×4 case of the diamond theorem.

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Friday, May 7, 2004

Friday May 7, 2004

Filed under: General — m759 @ 11:11 am

Religion of the Lottery:

Ground Zero Revisited

The midday New York lottery number for Thursday, May 6, 2004, the National Day of Prayer, was

000.

Since the log24 entry for the preceding day (Wednesday) was written in gratitude for a new transcription of Bach, and the log24 entry for the following day (today) has the time 11:11, signifying peace, the following seems as good a religious interpretation of yesterday’s lottery as any:

“In the Mass in B-Minor, Bach constructs a twenty-one-movement symmetry in which the Crucifixus is placed precisely between the Gratias and the Dona Nobis Pacem.“

— Timothy A. Smith,
Intentionality and Meaningfulness
in Bach’s Cyclical Works

“Nothing is random.”
— Mark Helprin,
Winter’s Tale

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Friday, February 20, 2004

Friday February 20, 2004

Filed under: General,Geometry — Tags: Finite Relativity — m759 @ 3:24 pm

Finite Relativity

Today is the 18th birthday of my note

“The Relativity Problem in Finite Geometry.”

That note begins with a quotation from 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

Here is another quotation from Weyl, on the profound branch of mathematics known as Galois theory, which he says

“… is nothing else but the relativity theory for the set Sigma, a set which, by its discrete and finite character, is conceptually so much simpler than the infinite set of points in space or space-time dealt with by ordinary relativity theory.”

— Weyl, Symmetry, Princeton University Press, 1952, p. 138

This second quotation applies equally well to the much less profound, but more accessible, part of mathematics described in Diamond Theory and in my note of Feb. 20, 1986.

Comments (1)

Friday February 20, 2004

Filed under: General,Geometry — Tags: Escher Fishes — m759 @ 12:00 am

The Da Vinci Code
and Symbology at Harvard

The protagonist of the recent bestseller The Da Vinci Code is Robert Langdon, "a professor of Religious Symbology at Harvard University." A prominent part in the novel is played by the well-known Catholic organization Opus Dei. Less well known (indeed, like Langdon, nonexistent) is the academic discipline of "symbology." (For related disciplines that do exist, click here.) What might a course in this subject at Harvard be like?

Harvard Crimson, April 10, 2003:

While Opus Dei members said that they do not refer to their practices of recruitment as "fishing," the Work’s founder does describe the process of what he calls "winning new apostles" with an aquatic metaphor.

Point #978 of The Way invokes a passage in the New Testament in which Jesus tells Peter that he will make him a "fisher of men." The point reads:

" ‘Follow me, and I will make you into fishers of men.’ Not without reason does our Lord use these words: men—like fish—have to be caught by the head. What evangelical depth there is in the ‘intellectual apostolate!’ ”

IMAGE- Escher, 'Fishes and Scales'

IMAGE- Cullinane, 'Invariance'

Exercise for Symbology 101:

Describe the symmetry
in each of the pictures above.
Show that the second picture
retains its underlying structural
symmetry under a group of
322,560 transformations.

Having reviewed yesterday's notes
on Gombrich, Gadamer, and Panofsky,
discuss the astrological meaning of
the above symbols in light of
today's date, February 20.

Extra credit:

Relate the above astrological
symbolism to the four-diamond
symbol in Jung's Aion.

Happy metaphors!

Robert Langdon

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Tuesday, November 11, 2003

Tuesday November 11, 2003

Filed under: General,Geometry — Tags: For Cameron Duncan — m759 @ 11:11 am

11:11

“Why do we remember the past
but not the future?”

— Stephen Hawking,
A Brief History of Time,
Ch. 9, “The Arrow of Time”

For another look at
the arrow of time, see

Time Fold.

Imaginary Time: The Concept

The flow of imaginary time is at right angles to that of ordinary time. “Imaginary time is a relatively simple concept that is rather difficult to visualize or conceptualize. In essence, it is another direction of time moving at right angles to ordinary time. In the image at right, the light gray lines represent ordinary time flowing from left to right – past to future. The dark gray lines depict imaginary time, moving at right angles to ordinary time.”

Is Time Quantized?

Yes.

Maybe.

We don’t really know.

Let us suppose, for the sake of argument, that time is in fact quantized and two-dimensional. Then the following picture,

from Time Fold, of “four quartets” time, of use in the study of poetry and myth, might, in fact, be of use also in theoretical physics.

In this event, last Sunday’s entry, on the symmetry group of a generic 4×4 array, might also have some physical significance.

At any rate, the Hawking quotation above suggests the following remarks from T. S. Eliot’s own brief history of time, Four Quartets:

“It seems, as one becomes older,
That the past has another pattern,
and ceases to be a mere sequence….

I sometimes wonder if that is
what Krishna meant—
Among other things—or one way
of putting the same thing:
That the future is a faded song,
a Royal Rose or a lavender spray
Of wistful regret for those who are
not yet here to regret,
Pressed between yellow leaves
of a book that has never been opened.
And the way up is the way down,
the way forward is the way back.”

Related reading:

The Wisdom of Old Age and

Poetry, Language, Thought.

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Sunday, November 9, 2003

Sunday November 9, 2003

Filed under: General,Geometry — Tags: Kummerhenge — m759 @ 5:00 pm

For Hermann Weyl's Birthday:

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

Exercise: 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 answer will, of course, depend on which aspects of the array's structure you choose to examine. It could be in the hundreds, or in the hundreds of thousands.)

Comments (1)

Thursday, November 6, 2003

Thursday November 6, 2003

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

Legacy Codes:

The Most Violent Poem

Lore of the Manhattan Project:

From The Trinity Site —

“I imagined Oppenheimer saying aloud,
‘Batter my heart, three person’d God,”
unexpectedly recalling John Donne’s ‘Holy Sonnet [14],’
and then he knew, ‘ “Trinity” will do.’
Memory has its reasons.

‘Batter my heart’ — I remember these words.
I first heard them on a fall day at Duke University in 1963.
Inside a classroom twelve of us were
seated around a long seminar table
listening to Reynolds Price recite this holy sonnet….

I remember Reynolds saying, slowly, carefully,
‘This is the most violent poem in the English language.’ ”

Related Entertainment

Today’s birthday:
director Mike Nichols

From a dead Righteous Brother:

“If you believe in forever
Then life is just a one-night stand.”

— Bobby Hatfield, found dead
in his hotel room at
7 PM EST Wednesday, Nov. 5, 2003,
before a concert scheduled at
Western Michigan University, Kalamazoo.

From a review of The Matrix Revolutions:

“You’d have to be totally blind at the end
to miss the Christian symbolism….
Trinity gets a glimpse of heaven…. And in the end…
God Put A Rainbow In The Clouds.”

Moral of the
Entertainment:

According to Chu Hsi [Zhu Xi],

“Li” is
“the principle or coherence
or order or pattern
underlying the cosmos.”

— Smith, Bol, Adler, and Wyatt,
Sung Dynasty Uses of the I Ching,
Princeton University Press, 1990

Related Non-Entertainment

Symmetry and a Trinity
(for the dotting-the-eye symbol above)

Introduction to Harmonic Analysis
(for musical and historical background)

Mathematical Proofs
(for the spirit of Western Michigan
University, Kalamazoo)

Moral of the
Non-Entertainment:

“Many kinds of entity
become easier to handle
by decomposing them into
components belonging to spaces
invariant under specified symmetries.”

— The importance of
mathematical conceptualisation
by David Corfield,
Department of History and
Philosophy of Science,
University of Cambridge

See, too,
Symmetry of Walsh Functions and
Geometry of the I Ching.

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Saturday, November 1, 2003

Saturday November 1, 2003

Filed under: General,Geometry — Tags: All Saints — m759 @ 1:05 pm

Symmetry in Diamond Theory:
Robbing Peter to Pay Paul

"Groups arise in most areas of pure and applied mathematics, usually as a set of operators or transformations of some structure. The appearance of a group generally reflects some kind of symmetry in the object under study, and such symmetry may be considered one of the fundamental notions of mathematics."

— Peter Webb

"Counter-change is sometimes known as Robbing Peter to Pay Paul."

— Helen Kelley Patchwork


Paul Robeson in King Solomon's Mines	Counterchange symmetry

For a look at the Soviet approach
to counterchange symmetry, see

The Kishinev School of Discrete Geometry.

The larger cultural context:

See War of Ideas (Oct. 24),
The Hunt for Red October (Oct. 25),
On the Left (Oct. 25), and
ART WARS for Trotsky's Birthday (Oct. 26).

Comments (2)

Sunday, August 17, 2003

Sunday August 17, 2003

Filed under: General,Geometry — Tags: Cube Coloring, Cube School — m759 @ 6:21 pm

Diamond theory is the theory of affine groups over GF(2) acting on small square and cubic arrays. In the simplest case, the symmetric group of degree 4 acts on a two-colored diamond figure like that in Plato's Meno dialogue, yielding 24 distinct patterns, each of which has some ordinary or color-interchange symmetry .

This symmetry invariance can be generalized to (at least) a group of order approximately 1.3 trillion acting on a 4x4x4 array of cubes.

The theory has applications to finite geometry and to the construction of the large Witt design underlying the Mathieu group of degree 24.

Friday, August 1, 2003

Friday August 1, 2003

Filed under: General — Tags: Beadgame Space, Glasperlenspiel, Strange Change — m759 @ 1:40 pm

Fearful Meditation

Ray Price — Time

— Lisa Berg, NashvilleCountry.com

O fearful meditation! Where, alack,
Shall time’s best jewel from time’s chest lie hid?

— Shakespeare, Sonnet 65

Clue: click here. This in turn leads to my March 4 entry Fearful Symmetry, which contains the following:

— Hermann Hesse, The Glass Bead Game

“How strange the change from major to minor….”

— Cole Porter, “Every Time We Say Goodbye“

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Sunday, June 22, 2003

Sunday June 22, 2003

Filed under: General,Geometry — m759 @ 2:28 am

The Real Hogwarts

is at no single geographical location; it is distributed throughout the planet, and it is perhaps best known (apart from its disguises in the fiction of J. K. Rowling, C. S. Lewis, Charles Williams, and other Inklings) as Christ Church. Some relevant links:

Christ Church College, Oxford

Harry Potter at Christ Church, Oxford

Christchurch, New Zealand

University of Canterbury
Physical Sciences Library:
Keeping Current with the Web:
Maths & Statistics, June 2002

Diamond Theory:
Symmetry in Binary Spaces
http://m759.freeservers.com/
The author of this site is Steven Cullinane, who has also written booklets on the subject. The web site provides detailed discussions of Diamond Theory, and is intended for college math students or mathematicians. According to Cullinane, Diamond Theory is best classified in the subject of “finite automorphism groups of algebraic, geometric, or combinatorial structures.” The site also includes links to other resources. From the NSDL Scout Report for Math, Engineering and Technology, Volume 1, No. 9, 7 June 2002, Copyright Internet Scout Project 1994-2002. http://scout.cs.wisc.edu

Christ Church, Christchurch Road,
Virginia Water, England

The Vicar of Christ Church has written a scathing attack on white-trash religion, “Christian Zionism,” that should be required reading for all prospective U. S. demagogues.

Finally, on this Sunday in June, with The New York Review of Books of July 3, 2003, headlining the religion of Scientism (Freeman Dyson reviewing Gleick’s new book on Newton), it seems fitting to provide a link to an oasis of civilisation in the home town of mathematician John Nash — Bluefield, West Virginia.

Christ Church,
Bluefield, West Virginia

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Saturday, June 7, 2003

Saturday June 7, 2003

Filed under: General — m759 @ 6:00 am

Blanche’s Waltz

For the birthday of Miss Jessica Tandy

Wien, Wien, nur du allein
Sollst stets die Stadt meiner Träume sein!
Dort, wo die alten Häuser stehn,
Dort, wo die lieblichen Mädchen gehn!
Wien, Wien, nur du allein
Sollst stets die Stadt meiner Träume sein!
Dort, wo ich glücklich und selig bin,
Ist Wien, ist Wien, mein Wien!

The web page where I found today’s midi of “Wien, Wien, nur du allein” offers a view of the pulpit of the Stephansdom in Vienna. From Hermann Weyl‘s Symmetry:

“Here (Fig. 41) is the gracefully designed staircase of the pulpit of the Stephan’s dome in Vienna; a triquetrum alternates with a swastika-like wheel.”

The closest to Weyl’s Figure 41 that I can find on the Web is located here.

Perhaps Stanley Kowalski had a lower opinion than Blanche DuBois of swastika-like wheels.

Comments (1)

Tuesday, May 27, 2003

Tuesday May 27, 2003

Filed under: General,Geometry — Tags: Solomon's Mental Health Month — 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

Comments (1)

Saturday, May 24, 2003

Saturday May 24, 2003

Filed under: General — m759 @ 1:06 am

Mental Health Month, Day 24:

The Sacred Day of
Kali, the Dark Lady

On this day, Gypsies from all over Europe gather in Provence for the sacred day of St. Sarah, also known as Kali.

Various representations of Kali exist; there is a novel about the ways men have pictured her:

From the prologue to
The Dark Lady,
by Mike Resnick.

She was old when the earth was young.

She stood atop Cemetery Ridge when Pickett made his charge, and she was there when the six hundred rode into the Valley of Death. She was at Pompeii when Mount Vesuvius blew, and she was in the forests of Siberia when the comet hit. She hunted elephant with Selous and buffalo with Cody, and she was there the night the high wire broke beneath the Flying Wallendas. She was at the fall of Troy and the Little Bighorn, and she watched Manolete and Dominguez face the brave bulls in the bloodstained arenas of Madrid….

She has no name, no past, no present, no future. She wears only black, and though she has been seen by many men, she is known to only a handful of them. You’ll see her — if you see her at all — just after you’ve taken your last breath. Then, before you exhale for the final time, she’ll appear, silent and sad-eyed, and beckon to you.

She is the Dark Lady, and this is her story.

The above is one of the best descriptions of Kali I know of in literature; another is in a short story by Fritz Leiber, “Damnation Morning.” It is not coincidental that one collection of Leiber’s writings is called “Dark Ladies.”

My journal note “Biblical Proportions” was in part inspired by Leiber.

Frank Sinatra may have pictured her as Ava Gardner. I think I saw her the night Sinatra died… hence my entries of March 31 and April 2, 2003.

It is perhaps not irrelevant that Kali is, among other things, a mother goddess, and that my entry “Raiders of the Lost Matrix” of May 20 deals with this concept and with the number 24.

The above religious symbol (see “Damnation Morning“) pictures both the axes of symmetry of the square¹ and a pattern with intriguing combinatorial properties². It also is the basis of a puzzle³ I purchased on August 29, 1997 — Judgment Day in Terminator 2. Linda Hamilton as Sarah Connor in that film is an excellent representation of the Dark Lady, both as mother figure and as Death Goddess.

Sarah Connor

Background music: “Bit by bit…” — Stephen Sondheim… See Sondheim and the Judgment Day puzzle in my entry of May 20. The Lottery Covenant.

¹ A. W. Joshi, Elements of Group Theory for Physicists, Third Edition, Wiley, 1982, p. 5

² V. K. Balakrishnan, Combinatorics, McGraw-Hill, 1995, p. 180

³ The Izzi Puzzle

Comments (3)

Wednesday, April 2, 2003

Wednesday April 2, 2003

Filed under: General,Geometry — Tags: 1982 J9, Logos and Branding, Reinhardt, Sein Feld — 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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Friday, March 21, 2003

Friday March 21, 2003

Filed under: General — Tags: Beadgame Space, Glasperlenspiel, Map Systems — m759 @ 9:29 am

ART WARS:

Readings for Bach's Birthday

Larry J. Solomon:

Symmetry as a Compositional Determinant,
Chapter VIII: New Transformations

In Solomon's work, a sequence of notes is represented as a set of positions within a Latin square:

Transformations of the Latin square correspond to transformations of the musical notes. For related material, see The Glass Bead Game, by Hermann Hesse, and Charles Cameron's sites on the Game.

Steven H. Cullinane:

Orthogonal Latin Squares as Skew Lines, and

Map Systems

Dorothy Sayers:

"The function of imaginative speech is not to prove, but to create–to discover new similarities, and to arrange them to form new entities, to build new self-consistent worlds out of the universe of undifferentiated mind-stuff." (Christian Letters to a Post-Christian World, Grand Rapids: Eerdmans, 1969, p. xiii)

— Quoted by Timothy A. Smith, "Intentionality and Meaningfulness in Bach's Cyclical Works"

Edward Sapir:

"…linguistics has also that profoundly serene and satisfying quality which inheres in mathematics and in music and which may be described as the creation out of simple elements of a self-contained universe of forms. Linguistics has neither the sweep nor the instrumental power of mathematics, nor has it the universal aesthetic appeal of music. But under its crabbed, technical, appearance there lies hidden the same classical spirit, the same freedom in restraint, which animates mathematics and music at their purest."

— "The Grammarian and his Language,"
American Mercury 1:149-155, 1924

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Thursday, March 13, 2003

Thursday March 13, 2003

Filed under: General,Geometry — m759 @ 2:45 am

Birthday Song

Today is the birthday of the late Jewish media magnate and art collector Walter H. Annenberg, whose name appears on a website that includes the following text:

Shape and Space in Geometry

“Making quilt blocks is an excellent way to explore symmetry. A quilt block is made of 16 smaller squares. Each small square consists of two triangles. Study this example of a quilt block:

This block has a certain symmetry. The right half is a mirror image of the left, and the top half is a mirror of the bottom.”

Symmetries of patterns such as the above are the subject of my 1976 monograph “ Diamond Theory,” which also deals with “shape and space in geometry,” but in a much more sophisticated way. For more on Annenberg, see my previous entry, “Daimon Theory.” For more on the historical significance of March 13, see Neil Sedaka, who also has a birthday today, in “ Jews in the News.”

Sedaka is, of course, noted for the hit tune “Happy Birthday, Sweet Sixteen,” our site music for today.

See also Geometry for Jews and related entries.

For the phrase “diamond theory” in a religious and philosophical context, see

Pilate, Truth, and Friday the Thirteenth.

“It’s quarter to three….” — Frank Sinatra

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Wednesday, March 12, 2003

Wednesday March 12, 2003

Filed under: General,Geometry — m759 @ 2:03 am

Daimon Theory

Today is allegedly the anniversary of the canonization, in 1622, of two rather important members of the Society of Jesus (Jesuits):

Ignatius Loyola…
Click here for Loyola’s legacy of strategic intelligence.

Francis Xavier…
Click here for Xavier’s legacy of strategic stupidity.

We can thank (or blame) a Jesuit (Gerard Manley Hopkins) for the poetic phrase “immortal diamond.” He may have been influenced by Plato, who has Socrates using a diamond figure in an argument for the immortality of the soul. Confusingly, Socrates also talked about his “daimon” (pronounced dye-moan). Combining these similar-sounding concepts, we have Doctor Stephen A. Diamond writing about daimons — a choice of author and topic that neatly combines the strategic intelligence of Loyola with the strategic stupidity of Xavier.

The cover illustration is perhaps not of Dr. Diamond himself.

A link between diamond theory and daimon theory is furnished by the charitable legacy of the non-practicing Jew Walter Annenberg.

For Annenberg and diamond theory, see this site on the elementary geometry of quilt blocks, which credits the Annenberg Foundation for support.

For Annenberg and daimon theory, see this site on Socrates, which has a similar Annenberg support credit.

Advanced disciples of Annenberg can learn much from the Perseus site about daimon theory. Let us pray that Abrahamic religious bigotry does not stand in their way. Less advanced disciples of Annenberg may find fulfillment in teaching children the beauty of elementary 4×4 quilt-block symmetry. Let us pray that academic bigotry does not prevent these same children, when they have grown older, from learning the deeper, and more difficult, beauties of diamond theory.

Daimon Theory

Diamond Theory

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Tuesday, March 4, 2003

Tuesday March 4, 2003

Filed under: General — Tags: Glasperlenspiel — m759 @ 9:25 pm

Fearful Symmetry

I just Googled this phrase and found the following site, which turns out to be related to my previous entry on the Bead Game and the death of John P. Thompson.

Fearful Symmetry:
The Music Master’s Lecture,

by Daniel d’Quincy.

This in turn links to an excerpt from The Glass Bead Game that includes this passage:

“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.”

It is very easy to get dangerously confused about holiness, but here are some relevant quotes:

“You will have to allow me to digress a bit in order to bring ourselves to a sufficiently elevated perspective… I warn you, it will require an attitude of playfulness on your part. Our approach will aim more at sincerity than seriousness. The attitude I’m aiming at is best expressed, I suppose, in the playing of a unique game, known by its German name as Das Glasperlenspiel, and which we may translate as the Glass Bead Game.”

— Daniel d’Quincy, Fearful Symmetry

“7:11”

— God himself said this, at least according to the previous entry and to my Jan. 28 entry, State of the Communion.

“Seven is heaven.”

— See my web page Eight is a Gate.

“An excellent example of a ‘universal’ in the sense of Charles Williams, Jung, or Plato is Hexagram 11 in China’s 3,000-year-old classic, the I Ching:

‘Heaven and earth unite:
the image of PEACE.’
(Wilhelm/Baynes translation,
Princeton University Press, 1967)”

— S. H. Cullinane, Plato, Pegasus, and the Evening Star

Thus we may associate the numbers 7 and 11 with the notions of heaven and peace; for a somewhat darker association of the time 7:11 with Kali as Time the Destroyer, see my last entry and also my previous entries

Fat Man and Dancing Girl (Feb. 18, 2003), and

Time and Eternity (Feb. 1, 2003).

Comments (1)

Tuesday, January 21, 2003

Tuesday January 21, 2003

Filed under: General — Tags: Big Sur, Cold Mountain, Single Jewel — m759 @ 5:09 pm

Diablo Ballet

Thanks to Meghan for the following:

not going, not coming,
rooted, deep and still
not reaching out, not reaching in
just resting, at the center
a single jewel, the flawless crystal drop
in the blaze of its brilliance
the way beyond.

— Shih Te (c. 730)

It turns out that Shih Te ("Foundling") was the sidekick of Han Shan ("Cold Mountain"). Here are some relevant links:

Thoughts of Robert Frost (see past two days' entries) lead to "Two Tramps in Mud Time," which in turn leads to Jack Kerouac and Gary Snyder splitting wood in The Dharma Bums.

This in turn leads, via a search on "Kerouac" and "axe," to the sentence

"There's the grace of an axe handle
as good as an Eglevsky ballet,"

in Big Sur.

Kerouac taught me when I was 16 and he is still teaching me now that I am 60.

Searching for "Eglevsky ballet" leads to this site on André Eglevsky, his work, his life, and his children. A further search leads to his daughter Marina Eglevsky, who stages dance for the Diablo Ballet.

Born to Dance

Marina Eglevsky and
the Diablo Ballet —
a rare and gifted
pas de deux

Those who feel the above is too "arty" for them may nevertheless appreciate the movie by the same name: "Born to Dance" (1936), starring Eleanor Powell and James Stewart.

In the larger metaphorical sense, of course, Powell and Eglevsky are both part of the same dance… at the "still point" described so well by Shih Te.

"just resting, at the center
a single jewel…"

"At the still point,
there the dance is."
— T. S. Eliot

From Marshall's Jewelers, Tucson —

A Diamond-Cutter Sutra:

The ideal cut is a mathematical formula for cutting diamonds to precise angles and proportions to maximize the reflection and refraction of light. In addition to these ideal proportions, the polish and symmetry of the diamond is done to the highest standards also. Only then does it qualify to receive the American Gem Society (AGS) "triple zero" rating. A "zero" rating is the most perfect rating that the AGS gives evaluating the cut, polish, and symmetry of the diamond.

When a diamond receives the "zero" rating for each of these areas, (cut, polish, and symmetry), it gets three "zeros," hence the "triple zero" rating. Because of this attention to detail, it takes up to four times longer to cut a diamond to these standards than an "average" diamond.

You may choose to compromise on color or clarity but to ensure the most brilliant diamond you should not compromise on cut….

The "triple zero" ideal cut guarantees you a magnificent balance of brilliance, sparkle, and fire.

Postscript of 1/25/03:

See also the obituary of Irene Diamond, ballet patron, for whom the New York City Ballet's "Diamond Project" is named. Diamond died on January 21, 2003, the date of the above weblog entry.

Comments (1)

Sunday, January 19, 2003

Sunday January 19, 2003

Filed under: General — Tags: Whanganui — m759 @ 4:30 pm

Literature
and
Geography

“Literature begins
with geography.”

— Attributed to
Robert Frost

The Maori Court at
the Wanganui Museum

“Cullinane College is a Catholic co-educational college, set to open in Wanganui (New Zealand) on the 29th of January, 2003.”

The 29th of January will be the 40th anniversary of the death of Saint Robert Frost.

New Zealand, perhaps the most beautiful country on the planet, is noted for being the setting of the film version of Lord of the Rings, which was written by a devout Catholic, J. R. R. Tolkien.

Here is a rather Catholic meditation on life and death in Tolkien’s work:

Frodo: “…He deserves death.”

Gandalf: “Deserves it! I daresay he does. Many that live deserve death. And some that die deserve life. Can you give it to them? Then do not be too eager to deal out death in judgement.”

Personally, I prefer Clint Eastwood’s version of this dialogue:

The Schofield Kid: “Well, I guess they had it coming.”

William Munny: “We all have it coming, Kid.”

For other New Zealand themes, see Alfred Bester’s novels The Stars My Destination and The Deceivers.

The original title of The Stars My Destination was Tyger! Tyger! after Blake’s poem.

For more on fearful symmetry, see the work of Marston Conder, professor of mathematics at the University of Auckland, New Zealand.

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Wednesday, December 4, 2002

Wednesday December 4, 2002

Filed under: General,Geometry — m759 @ 11:22 pm

Symmetry and a Trinity

From a web page titled Spectra:

"What we learn from our whole discussion and what has indeed become 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. After that you may start to investigate symmetric configurations of elements, i.e., configurations which are invariant under a certain subgroup of the group of all automorphisms . . ."

— Hermann Weyl in Symmetry, Princeton University Press, 1952, page 144

"… any color at all can be made from three different colors, in our case, red, green, and blue lights. By suitably mixing the three together we can make anything at all, as we demonstrated . . .

Further, these laws are very interesting mathematically. For those who are interested in the mathematics of the thing, it turns out as follows. Suppose that we take our three colors, which were red, green, and blue, but label them A, B, and C, and call them our primary colors. Then any color could be made by certain amounts of these three: say an amount a of color A, an amount b of color B, and an amount c of color C makes X:

X = aA + bB + cC.

Now suppose another color Y is made from the same three colors:

Y = a'A + b'B + c'C.

Then it turns out that the mixture of the two lights (it is one of the consequences of the laws that we have already mentioned) is obtained by taking the sum of the components of X and Y:

Z = X + Y = (a + a')A + (b + b')B + (c + c')C.

It is just like the mathematics of the addition of vectors, where (a, b, c ) are the components of one vector, and (a', b', c' ) are those of another vector, and the new light Z is then the "sum" of the vectors. This subject has always appealed to physicists and mathematicians."

— According to the author of the Spectra site, this is Richard Feynman in Elementary Particles and the Laws of Physics, The 1986 Dirac Memorial Lectures, by Feynman and Steven Weinberg, Cambridge University Press, 1989.

These two concepts — symmetry as invariance under a group of transformations, and complicated things as linear combinations (the technical name for Feynman's sums) of simpler things — underlie much of modern mathematics, both pure and applied.

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Tuesday, December 3, 2002

Tuesday December 3, 2002

Filed under: General,Geometry — Tags: Change Logos, Strange Change — m759 @ 1:45 pm

Symmetry, Invariance, and Objectivity

The book Invariances: The Structure of the Objective World, by Harvard philosopher Robert Nozick, was reviewed in the New York Review of Books issue dated June 27, 2002.

On page 76 of this book, published by Harvard University Press in 2001, Nozick writes:

"An objective fact is invariant under various transformations. It is this invariance that constitutes something as an objective truth…."

Compare this with Hermann Weyl's definition in his classic Symmetry (Princeton University Press, 1952, page 132):

"Objectivity means invariance with respect to the group of automorphisms."

It has finally been pointed out in the Review, by a professor at Göttingen, that Nozick's book should have included Weyl's definition.

I pointed this out on June 10, 2002.

For a survey of material on this topic, see this Google search on "nozick invariances weyl" (without the quotes).

Nozick's omitting Weyl's definition amounts to blatant plagiarism of an idea.

Of course, including Weyl's definition would have required Nozick to discuss seriously the concept of groups of automorphisms. Such a discussion would not have been compatible with the current level of philosophical discussion at Harvard, which apparently seldom rises above the level of cocktail-party chatter.

A similarly low level of discourse is found in the essay "Geometrical Creatures," by Jim Holt, also in the issue of the New York Review of Books dated December 19, 2002. Holt at least writes well, and includes (if only in parentheses) a remark that is highly relevant to the Nozick-vs.-Weyl discussion of invariance elsewhere in the Review:

"All the geometries ever imagined turn out to be variations on a single theme: how certain properties of a space remain unchanged when its points get rearranged." (p. 69)

This is perhaps suitable for intelligent but ignorant adolescents; even they, however, should be given some historical background. Holt is talking here about the Erlangen program of Felix Christian Klein, and should say so. For a more sophisticated and nuanced discussion, see this web page on Klein's Erlangen Program, apparently by Jean-Pierre Marquis, Département de Philosophie, Université de Montréal. For more by Marquis, see my later entry for today, "From the Erlangen Program to Category Theory."

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Saturday, November 9, 2002

Saturday November 9, 2002

Filed under: General,Geometry — m759 @ 4:44 am

Birthdate of Hermann Weyl

Weyl

Plato’s Diamond

Result of a Google search.

Category: Science > Math > Algebra > Group Theory

Weyl, H.: Symmetry.
Description of the book Symmetry by Weyl, H., published by Princeton University Press. … pup.princeton.edu/titles/
865.html – 7k – Nov. 8, 2002

Wednesday, November 6, 2002

Wednesday November 6, 2002

Filed under: General — m759 @ 2:22 pm

Today's birthdays: Mike Nichols and Sally Field.

Who is Sylvia?
What is she?

From A Beautiful Mind, by Sylvia Nasar:

Prologue

Where the statue stood
Of Newton with his prism and silent face,
The marble index of a mind for ever
Voyaging through strange seas of Thought, alone.
— WILLIAM WORDSWORTH

John Forbes Nash, Jr. — mathematical genius, inventor of a theory of rational behavior, visionary of the thinking machine — had been sitting with his visitor, also a mathematician, for nearly half an hour. It was late on a weekday afternoon in the spring of 1959, and, though it was only May, uncomfortably warm. Nash was slumped in an armchair in one corner of the hospital lounge, carelessly dressed in a nylon shirt that hung limply over his unbelted trousers. His powerful frame was slack as a rag doll's, his finely molded features expressionless. He had been staring dully at a spot immediately in front of the left foot of Harvard professor George Mackey, hardly moving except to brush his long dark hair away from his forehead in a fitful, repetitive motion. His visitor sat upright, oppressed by the silence, acutely conscious that the doors to the room were locked. Mackey finally could contain himself no longer. His voice was slightly querulous, but he strained to be gentle. "How could you," began Mackey, "how could you, a mathematician, a man devoted to reason and logical proof…how could you believe that extraterrestrials are sending you messages? How could you believe that you are being recruited by aliens from outer space to save the world? How could you…?"

Nash looked up at last and fixed Mackey with an unblinking stare as cool and dispassionate as that of any bird or snake. "Because," Nash said slowly in his soft, reasonable southern drawl, as if talking to himself, "the ideas I had about supernatural beings came to me the same way that my mathematical ideas did. So I took them seriously."

What I take seriously:

Introduction to Topology and Modern Analysis, by George F. Simmons, McGraw-Hill, New York, 1963

An Introduction to Abstract Harmonic Analysis, by Lynn H. Loomis, Van Nostrand, Princeton, 1953

"Harmonic Analysis as the Exploitation of Symmetry — A Historical Survey," by George W. Mackey, pp. 543-698, Bulletin of the American Mathematical Society, July 1980

Walsh Functions and Their Applications, by K. G. Beauchamp, Academic Press, New York, 1975

Walsh Series: An Introduction to Dyadic Harmonic Analysis, by F. Schipp, P. Simon, W. R. Wade, and J. Pal, Adam Hilger Ltd., 1990

The review, by W. R. Wade, of Walsh Series and Transforms (Golubov, Efimov, and Skvortsov, publ. by Kluwer, Netherlands, 1991) in the Bulletin of the American Mathematical Society, April 1992, pp. 348-359

Music courtesy of Franz Schubert.

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Tuesday, October 22, 2002

Tuesday October 22, 2002

Filed under: General,Geometry — m759 @ 1:16 am

Introduction to
Harmonic Analysis

From Dr. Mac’s Cultural Calendar for Oct. 22:

The French actress Catherine Deneuve was born on this day in Paris in 1943….
The Beach Boys released the single “Good Vibrations” on this day in 1966.

“I hear the sound of a
gentle word

On the wind that lifts
her perfume
through the air.”

— The Beach Boys

In honor of Deneuve and of George W. Mackey, author of the classic 156-page essay, “Harmonic analysis* as the exploitation of symmetry† — A historical survey” (Bulletin of the American Mathematical Society (New Series), Vol. 3, No. 1, Part 1 (July 1980), pp. 543-698), this site’s music is, for the time being, “Good Vibrations.”

For more on harmonic analysis, see “Group Representations and Harmonic Analysis from Euler to Langlands,” by Anthony W. Knapp, Part I and Part II.

* For “the simplest non-trivial model for harmonic analysis,” the Walsh functions, see F. Schipp et. al., Walsh Series: An Introduction to Dyadic Harmonic Analysis, Hilger, 1990. For Mackey’s “exploitation of symmetry” in this context, see my note Symmetry of Walsh Functions, and also the footnote below.

† “Now, it is no easy business defining what one means by the term conceptual…. I think we can say that the conceptual is usually expressible in terms of broad principles. A nice example of this comes in form of harmonic analysis, which is based on the idea, whose scope has been shown by George Mackey… to be immense, that many kinds of entity become easier to handle by decomposing them into components belonging to spaces invariant under specified symmetries.”

— The importance of mathematical conceptualisation,
by David Corfield, Department of History and Philosophy of Science, University of Cambridge

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Sunday, July 28, 2002

Sunday July 28, 2002

Filed under: General — m759 @ 2:16 pm

Memories, Dreams, Reflections

Saul Steinberg in The New York Review of Books issue dated August 15, 2002, page 32:

“The idea of reflections came to me in reading an observation by Pascal, cited in a book by W. H. Auden, who wrote an unusual kind of autobiography by collecting all the quotations he had annotated in the course of his life, which is a good way of displaying oneself, as a reflection of these quotations. Among them this observation by Pascal, which could have been made only by a mathematician….”

Pascal’s observation is that humans, animals, and plants have bilateral symmetry, but in nature at large there is only symmetry about a horizontal axis… reflections in water, nature’s mirror.

This seems related to the puzzling question of why a mirror reverses left and right, but not up and down.

The Steinberg quote is from the book Reflections and Shadows, reviewed here.

Bibliographic data on Auden’s commonplace book:

AUTHOR      Auden, W. H. (Wystan Hugh),              1907-1973. TITLE       A Certain World; a Commonplace Book   
            [selected by] W. H. Auden.
PUBLISHER   New York, Viking Press [1970]
SUBJECT     Commonplace-books.

A couple of websites on commonplace books:

Quotation Collections and

Weblets as Commonplace Books.

A classic:

The Practical Cogitator – The Thinker’s Anthology,
by Charles P. Curtis, Jr., and Ferris Greenslet,
Houghton Mifflin Company Boston, Massachusetts
c 1962 Third Edition – Revised and Enlarged

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Saturday, July 20, 2002

Saturday July 20, 2002

Filed under: General,Geometry — Tags: Change Logos, Eightfold Cube, four-set, Map Systems, Octad, Octad Generator, Strange Change — m759 @ 10:13 pm

ABSTRACT: Finite projective geometry explains the surprising symmetry properties of some simple graphic designs– found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the "Miracle Octad Generator" of R. T. Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.

We regard the four-diamond figure D above as a 4×4 array of two-color diagonally-divided square tiles.

Let G be the group of 322,560 permutations of these 16 tiles generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2×2 quadrants.

THEOREM: Every G-image of D (as at right, below) has some ordinary or color-interchange symmetry.

Example:

For an animated version, click here.

Remarks:

Some of the patterns resulting from the action of G on D have been known for thousands of years. (See Jablan, Symmetry and Ornament, Ch. 2.6.) It is perhaps surprising that the patterns' interrelationships and symmetries can be explained fully only by using mathematics discovered just recently (relative to the patterns' age)– in particular, the theory of automorphism groups of finite geometries.

Using this theory, we can summarize the patterns' properties by saying that G is isomorphic to the affine group A on the linear 4-space over GF(2) and that the 35 structures of the 840 = 35 x 24 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2).

This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets, and indicating the locations of these two-sets of tiles within the 4×4 patterns. The lines of the line diagrams may be added in a binary fashion (i.e., 1+1=0). Each three-set of line diagrams sums to zero– i.e., each diagram in a three-set is the binary sum of the other two diagrams in the set. Thus, the 35 three-sets of line diagrams correspond to the 35 three-point lines of the finite projective 3-space PG(3,2).

For example, here are the line diagrams for the figures above:

Shown below are the 15 possible line diagrams resulting from row/column/quadrant permutations. These 15 diagrams may, as noted above, be regarded as the 15 points of the projective 3-space PG(3,2).

The symmetry of the line diagrams accounts for the symmetry of the two-color patterns. (A proof shows that a 2nx2n two-color triangular half-squares pattern with such line diagrams must have a 2×2 center with a symmetry, and that this symmetry must be shared by the entire pattern.)

Among the 35 structures of the 840 4×4 arrays of tiles, orthogonality (in the sense of Latin-square orthogonality) corresponds to skewness of lines in the finite projective space PG(3,2). This was stated by the author in a 1978 note. (The note apparently had little effect. A quarter-century later, P. Govaerts, D. Jungnickel, L. Storme, and J. A. Thas wrote that skew (i.e., nonintersecting) lines in a projective space seem "at first sight not at all related" to orthogonal Latin squares.)

We can define sums and products so that the G-images of D generate an ideal (1024 patterns characterized by all horizontal or vertical "cuts" being uninterrupted) of a ring of 4096 symmetric patterns. There is an infinite family of such "diamond" rings, isomorphic to rings of matrices over GF(4).

The proof uses a decomposition technique for functions into a finite field that might be of more general use.

The underlying geometry of the 4×4 patterns is closely related to the Miracle Octad Generator of R. T. Curtis– used in the construction of the Steiner system S(5,8,24)– and hence is also related to the Leech lattice, which, as Walter Feit has remarked, "is a blown up version of S(5,8,24)."

For a movable JavaScript version of these 4×4 patterns, see The Diamond 16 Puzzle.

The above is an expanded version of Abstract 79T-A37, "Symmetry invariance in a diamond ring," by Steven H. Cullinane, Notices of the American Mathematical Society, February 1979, pages A-193, 194.

For a discussion of other cases of the theorem, click here.

Related pages:

The Diamond 16 Puzzle

Diamond Theory in 1937:
A Brief Historical Note

Notes on Finite Geometry

Geometry of the 4×4 Square

Binary Coordinate Systems

The 35 Lines of PG(3,2)

Map Systems:
Function Decomposition over a Finite Field

The Diamond Theorem–
The 2×2, the 2x2x2, the 4×4, and the 4x4x4 Cases

Diamond Theory

Latin-Square Geometry