# Log24

## Thursday, March 29, 2018

### Asymmetry: An Historical YA Fantasy

Filed under: General — Tags: — m759 @ 8:48 AM

Or:  The Discreet Charm of Stéphane

For Lisa Halliday

Supplementary images —

## Wednesday, March 28, 2018

### On Unfairly Excluding Asymmetry

Filed under: General — Tags: — m759 @ 12:28 AM

A comment on the the Diamond Theorem Facebook page



Those who enjoy asymmetry may consult the "Expert's Cube" —

For further details see the previous post.

## Thursday, September 10, 2015

### Super Overarching Symmetry

Filed under: General — m759 @ 12:48 PM

Santa Fe Institute logo (see previous post) —

## Saturday, February 18, 2012

### Symmetry

Filed under: General,Geometry — m759 @ 7:35 PM

From the current Wikipedia article "Symmetry (physics)"—

"In physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are 'unchanged', according to a particular observation. A symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is 'preserved' under some change.

A family of particular transformations may be continuous  (such as rotation of a circle) or discrete  (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group)."….

"A discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance."

Note the confusion here between continuous (or discontinuous) transformations  and "continuous" (or "discontinuous," i.e. "discrete") groups .

This confusion may impede efforts to think clearly about some pure mathematics related to current physics— in particular, about the geometry of spaces made up of individual units ("points") that are not joined together in a continuous manifold.

For an attempt to forestall such confusion, see Noncontinuous Groups.

For related material, see Erlanger and Galois as well as the opening paragraphs of Diamond Theory

Symmetry is often described as invariance under a group of transformations. An unspoken assumption about symmetry in Euclidean 3-space is that the transformations involved are continuous.

Diamond theory rejects this assumption, and in so doing reveals that Euclidean symmetry may itself  be invariant under rather interesting groups of non-continuous (and a-symmetric) transformations. (These might be called noncontinuous  groups, as opposed to so-called discontinuous  (or discrete ) symmetry groups. See Weyl's Symmetry .)

For example, the affine group A on the 4-space over the 2-element field has a natural noncontinuous and asymmetric but symmetry-preserving action on the elements of a 4×4 array. (Details)

(Version first archived on March 27, 2002)

Update of Sunday, February 19, 2012—

The abuse of language by the anonymous authors
of the above Wikipedia article occurs also in more
reputable sources. For instance—

Some transformations referred to by Brading and Castellani
and their editees as "discrete symmetries" are, in fact, as
linear transformations of continuous spaces, themselves
continuous  transformations.

This unfortunate abuse of language is at least made explicit
in a 2003 text, Mathematical Perspectives on Theoretical
Physics
(Nirmala Prakash, Imperial College Press)—

"… associated[*] with any given symmetry there always exists
a continuous or a discrete group of transformations….
A symmetry whose associated group is continuous (discrete)
is called a continuous  (discrete ) symmetry ." — Pp. 235, 236

## Tuesday, March 2, 2010

### Symmetry and Automorphisms

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

From the conclusion of Weyl's Symmetry

One example of Weyl's "structure-endowed entity" is a partition of a six-element set into three disjoint two-element sets– for instance, the partition of the six faces of a cube into three pairs of opposite faces.

The automorphism group of this faces-partition contains an order-8 subgroup that is isomorphic to the abstract group C2×C2×C2 of order eight–

The action of Klein's simple group of order 168 on the Cayley diagram of C2×C2×C2 in yesterday's post furnishes an example of Weyl's statement that

"… one may ask with respect to a given abstract group: What is the group of its automorphisms…?"

## Sunday, January 13, 2019

### Sunday the Thirteenth (Revisited)

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

For some context, see "A Riddle for Davos."

## Monday, August 27, 2018

### Children of the Six Sides

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

From the former date above —

Saturday, September 17, 2016

### A Box of Nothing

Filed under: Uncategorized — m759 @ 12:13 AM

From the latter date above —

Tuesday, October 18, 2016

### Parametrization

Filed under: Uncategorized — m759 @ 6:00 AM

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

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

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

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

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

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

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

From March 2018 —

## Tuesday, March 27, 2018

### Compare and Contrast

Filed under: General,Geometry — Tags: — m759 @ 4:28 PM

Related material on automorphism groups —

The "Eightfold Cube" structure shown above with Weyl
competes rather directly with the "Eightfold Way" sculpture
shown above with Bryant. The structure and the sculpture
each illustrate Klein's order-168 simple group.

Perhaps in part because of this competition, fans of the Mathematical
Sciences Research Institute (MSRI, pronounced "Misery') are less likely
to enjoy, and discuss, the eight-cube mathematical structure  above
than they are an eight-cube mechanical puzzle  like the one below.

Note also the earlier (2006) "Design Cube 2x2x2" webpage
illustrating graphic designs on the eightfold cube. This is visually,
if not mathematically, related to the (2010) "Expert's Cube."

## Wednesday, March 7, 2018

### Unite the Seven.

Filed under: General,Geometry — Tags: , — m759 @ 10:31 AM

Related material —

The seven points of the Fano plane within

"Before time began . . . ."

— Optimus Prime

## Wednesday, November 22, 2017

### Goethe on All Souls’ Day

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

David E. Wellbery on Goethe

From an interview published on 2 November 2017 at

as later republished in

The logo at left above is that of The Point .
The menu icon at right above is perhaps better
suited to illustrate Verwandlungslehre .

## Wednesday, October 4, 2017

### Text and Context

Filed under: G-Notes,General,Geometry — m759 @ 11:00 AM

Text —

"A field is perhaps the simplest algebraic structure we can invent."

— Hermann Weyl, 1952

Context —

 Full text of Symmetry  – Internet Archive — A field is perhaps the simplest algebraic 143 structure we can invent. Its elements are numbers. Characteristic for its structure are the operations of addition and …

From a Log24 search for Mathematics+Nutshell —

## Thursday, May 11, 2017

### In Memoriam

Filed under: General — m759 @ 9:00 PM

Update of 6:16 AM Friday, May 12, 2017 —

The phrase "smallest perfect universe" is from Burkard Polster (2001).

## Saturday, September 24, 2016

### The Seven Seals

Filed under: General,Geometry — Tags: — m759 @ 7:23 AM

From Hermann Weyl's 1952 classic Symmetry —

"Galois' ideas, which for several decades remained
a book with seven seals  but later exerted a more
and more profound influence upon the whole
development of mathematics, are contained in
a farewell letter written to a friend on the eve of
his death, which he met in a silly duel at the age of
twenty-one. This letter, if judged by the novelty and
profundity of ideas it contains, is perhaps the most
substantial piece of writing in the whole literature
of mankind."

Some Galois geometry —

See the previous post for more narrative.

## Friday, August 5, 2016

### Sleight of Post

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

From an earlier Log24 post —

### Spiegel-Spiel des Gevierts

Filed under: Uncategorized — m759 @ 12:00 PM

See Cube Symbology.

From a post of the next day, July 12, 2014 —

"So there are several different genres and tones
jostling for prominence within Lexicon :
a conspiracy thriller, an almost abstract debate
about what language can do, and an ironic
questioning of some of the things it’s currently used for."

Graham Sleight in The Washington Post
a year earlier, on July 15, 2013

For the Church of Synchronology, from Log24 on the next day —

From a post titled Circles on the date of Marc Simont's death —

See as well Verhexung  in this journal.

## Monday, April 25, 2016

### Seven Seals

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

An old version of the Wikipedia article "Group theory"
(pictured in the previous post) —

"More poetically "

From Hermann Weyl's 1952 classic Symmetry

"Galois' ideas, which for several decades remained
a book with seven seals  but later exerted a more
and more profound influence upon the whole
development of mathematics, are contained in
a farewell letter written to a friend on the eve of
his death, which he met in a silly duel at the age of
twenty-one. This letter, if judged by the novelty and
profundity of ideas it contains, is perhaps the most
substantial piece of writing in the whole literature
of mankind."

The seven seals from the previous post, with some context —

These models of projective points are drawn from the underlying
structure described (in the 4×4 case) as part of the proof of the
Cullinane diamond theorem .

## Sunday, November 30, 2014

### Investments

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

From "A Piece of the Storm," by the late poet Mark Strand —

A snowflake, a blizzard of one….

From notes to Malcolm Lowry's "La Mordida" —

## Tuesday, June 3, 2014

### Galois Matrices

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

The webpage Galois.us, on Galois matrices , has been created as
a starting point for remarks on the algebra  (as opposed to the geometry)
underlying the rings of matrices mentioned in AMS abstract 79T-A37,
Symmetry invariance in a diamond ring.”

## Monday, May 19, 2014

### Cube Space

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

A sequel to this afternoon's Rubik Quote:

"The Cube was born in 1974 as a teaching tool
to help me and my students better understand
space and 3D. The Cube challenged us to find
order in chaos."

(Click image below to enlarge.)

## Friday, January 17, 2014

### The 4×4 Relativity Problem

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

The sixteen-dot square array in yesterday’s noon post suggests
the following remarks.

“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 WeylThe Classical Groups ,
Princeton University Press, 1946, p. 16

The Galois tesseract  appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—

The 1977 matrix Q is echoed in the following from 2002—

A different representation of Cullinane’s 1977 square model of the
16-point affine geometry over the two-element Galois field GF(2)
is supplied by Conway and Sloane in Sphere Packings, Lattices and Groups
(first published in 1988) :

Here a, b, c, d   are basis vectors in the vector 4-space over GF(2).
(For a 1979 version of this vector space, see AMS Abstract 79T-A37.)

See also a 2011 publication of the Mathematical Association of America —

## Sunday, January 12, 2014

### The Feast

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

"And in the Master's chambers…" — Eagles

Related fiction — Amy + Dinner and today's previous post.

Related non-fiction — a page of pure mathematics —

## Saturday, July 13, 2013

### Circles

Filed under: General,Geometry — Tags: , — m759 @ 2:22 AM

A sort of poem
by Gauss and Weyl —

Click the circle for the context in Weyl's Symmetry .

For related remarks, see the previous post.

A literary excursus—

 Brad Leithauser in a New Yorker  post of July 11, 2013: If a poet determines that a poem should begin at point A and conclude at point D, say, the mystery of how to get there—how to pass felicitously through points B and C—strikes me as an artistic task both genuine and enlivening. There are fertile mysteries of transition, no less than of termination. And I’d like to suppose that Frost himself would recognize that any ingress into a poem is better than being locked out entirely. His little two-liner, “The Secret,” suggests as much: “We dance round in a ring and suppose / But the Secret sits in the middle and knows.” Most truly good poems might be said to contain a secret: the little sacramental miracle by which you connect, intimately, with the words of a total stranger. And whether you come at the poem frontward, or backward, or inside out—whether you approach it deliberately, word by word and line by line, or you parachute into it borne on a sudden breeze from the island of Serendip—surely isn’t the important thing. What matters is whether you achieve entrance into its inner ring, and there repose companionably beside the Secret.

One should try, of course, to avoid repose in an inner circle of Hell .

## Monday, June 10, 2013

### Galois Coordinates

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

Today's previous post on coordinate systems
suggests a look at the phrase "Galois coordinates."

A search shows that the phrase, though natural,
has apparently not been used before 2011* for solutions
to what Hermann Weyl called "the relativity problem."

A thorough historical essay on Galois coordinatization
in this sense would require more academic resources
than I have available. It would likely describe a number
of applications of Galois-field coordinates to square
(and perhaps to cubical) arrays that were studied before
1976, the date of my Diamond Theory  monograph.

But such a survey might not  find any such pre-1976
coordinatization of a 4×4 array  by the 16 elements
of the vector 4-space  over the Galois field with two
elements, GF(2).

Such coordinatizations are important because of their
close relationship to the Mathieu group 24 .

See a preprint by Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group 24 ," with its remark
denying knowledge of any such coordinatization
prior to a 1989 paper by R. T. Curtis.

Related material:

Some images related to Galois coordinates, excerpted
from a Google search today (click to enlarge)—

*  A rather abstract  2011 paper that uses the phrase
"Galois coordinates" may have some implications
for the naive form of the relativity problem
related to square and cubical arrays.

## Thursday, December 27, 2012

### Object Lesson

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

Yesterday's post on the current Museum of Modern Art exhibition
"Inventing Abstraction: 1910-1925" suggests a renewed look at
abstraction and a fundamental building block: the cube.

From a recent Harvard University Press philosophical treatise on symmetry

The treatise corrects Nozick's error of not crediting Weyl's 1952 remarks
on objectivity and symmetry, but repeats Weyl's error of not crediting
Cassirer's extensive 1910 (and later) remarks on this subject.

For greater depth see Cassirer's 1910 passage on Vorstellung :

This of course echoes Schopenhauer, as do discussions of "Will and Idea" in this journal.

For the relationship of all this to MoMA and abstraction, see Cube Space and Inside the White Cube.

"The sacramental nature of the space becomes clear…." — Brian O'Doherty

## Saturday, June 16, 2012

### Chiral Problem

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

In memory of William S. Knowles, chiral chemist, who died last Wednesday (June 13, 2012)—

Detail from the Harvard Divinity School 1910 bookplate in yesterday morning's post

"ANDOVERHARVARD THEOLOGICAL LIBRARY"

Detail from Knowles's obituary in this  morning's New York Times

William Standish Knowles was born in Taunton, Mass., on June 1, 1917. He graduated a year early from the Berkshire School, a boarding school in western Massachusetts, and was admitted to Harvard. But after being strongly advised that he was not socially mature enough for college, he did a second senior year of high school at another boarding school, Phillips Academy in Andover, N.H.

Dr. Knowles graduated from Harvard with a bachelor’s degree in chemistry in 1939….

"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 WeylThe Classical Groups, Princeton University Press, 1946, p. 16

The six congruent quaternion actions illustrated above are based on the following coordinatization of the eightfold cube

Problem: Is there a different coordinatization
that yields greater symmetry in the pictures of
quaternion group actions?

A paper written in a somewhat similar spirit—

ABSTRACT: Chiral tetrahedral molecules can be dealt [with] under the standard of quaternionic algebra. Specifically, non-commutativity of quaternions is a feature directly related to the chirality of molecules….

## Sunday, March 18, 2012

### Square-Triangle Diamond

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

The diamond shape of yesterday's noon post
is not wholly without mathematical interest …

The square-triangle theorem

"Every triangle is an n -replica" is true
if and only if n  is a square.

The 16 subdiamonds of the above figure clearly
may be mapped by an affine transformation
to 16 subsquares of a square array.

(See the diamond lattice  in Weyl's Symmetry .)

Similarly for any square n , not just 16.

There is a group of 322,560 natural transformations
that permute the centers  of the 16 subsquares
in a 16-part square array. The same group may be
viewed as permuting the centers  of the 16 subtriangles
in a 16-part triangular array.

## Wednesday, October 26, 2011

### Erlanger and Galois

Filed under: General,Geometry — m759 @ 8:00 PM

Peter J. Cameron yesterday on Galois—

"He was killed in a duel at the age of 20…. His work languished for another 14 years until Liouville published it in his Journal; soon it was recognised as the foundation stone of modern algebra, a position it has never lost."

Here Cameron is discussing Galois theory, a part of algebra. Galois is known also as the founder* of group theory, a more general subject.

Group theory is an essential part of modern geometry as well as of modern algebra—

"In der Galois'schen Theorie, wie hier, concentrirt sich das Interesse auf Gruppen von Änderungen. Die Objecte, auf welche sich die Änderungen beziehen, sind allerdings verschieden; man hat es dort mit einer endlichen Zahl discreter Elemente, hier mit der unendlichen Zahl von Elementen einer stetigen Mannigfaltigkeit zu thun."

— Felix Christian Klein, Erlanger Programm , 1872

("In the Galois theory, as in ours, the interest centres on groups of transformations. The objects to which the transformations are applied are indeed different; there we have to do with a finite number of discrete elements, here with the infinite number of elements in a continuous manifoldness." (Translated by M.W. Haskell, published in Bull. New York Math. Soc. 2, (1892-1893), 215-249))

Related material from Hermann Weyl, Symmetry , Princeton University Press, 1952 (paperback reprint of 1982, pp. 143-144)—

"A field is perhaps the simplest algebraic structure we can invent. Its elements are numbers…. Space is another example of an entity endowed with a structure. Here the elements are points…. 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  Σ try to determine is 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 Σ in this way."

For a simple example of a group acting on a field (of 8 elements) that is also a space (of 8 points), see Generating the Octad Generator and Knight Moves.

* Joseph J. Rotman, An Introduction to the Theory of Groups , 4th ed., Springer, 1994, page 2

## Tuesday, June 21, 2011

### Piracy Project

Filed under: General,Geometry — Tags: , — m759 @ 2:02 AM

Recent piracy of my work as part of a London art project suggests the following.

From http://www.trussel.com/rls/rlsgb1.htm

The 2011 Long John Silver Award for academic piracy
goes to ….

Hermann Weyl, for the remark on objectivity and invariance
in his classic work Symmetry  that skillfully pirated
the much earlier work of philosopher Ernst Cassirer.

goes to …

Richard Evan Schwartz of Brown University, for his
use, without citation, of Cullinane’s work illustrating
Weyl’s “relativity problem” in a finite-geometry context.

For further details, click on the above names.

## Monday, June 20, 2011

### The Search for Invariants

Filed under: General,Geometry — m759 @ 9:29 AM

The title of a recent contribution to a London art-related "Piracy Project" begins with the phrase "The Search for Invariants."

A search for that phrase  elsewhere yields a notable 1944* paper by Ernst Cassirer, "The Concept of Group and the Theory of Perception."

Page 20: "It is a process of objectification, the characteristic nature
and tendency of which finds expression in the formation of invariants."

Cassirer's concepts seem related to Weyl's famous remark that

“Objectivity means invariance with respect to the group of automorphisms.”
Symmetry  (Princeton University Press, 1952, page 132)

See also this journal on June 23, 2010— "Group Theory and Philosophy"— as well as some Math Forum remarks on Cassirer and Weyl.

Update of 6 to 7:50 PM June 20, 2011—

Weyl's 1952 remark seems to echo remarks in 1910 and 1921 by Cassirer.
See Cassirer in 1910 and 1921 on Objectivity.

Another source on Cassirer, invariance, and objectivity—

"… physical theories prove to be theories of invariants
with regard to certain groups of transformations and
it is exactly the invariance that secures the objectivity
of a physical theory."

— SYNTHESIS PHILOSOPHICA 42 (2/2006), pp. 233–241

A search in Weyl's Symmetry  for any reference to Ernst Cassirer yields no results.

* Published in French in 1938.

## Saturday, May 7, 2011

### Annals of Mathematics

Filed under: General — m759 @ 10:35 PM

University Diaries praised today the late Robert Nozick's pedagogical showmanship.

His scholarship was less praiseworthy. His 2001 book Invariances: The Structure of the Objective World  failed, quite incredibly, to mention Hermann Weyl's classic summary of  the connection between invariance and objectivity.  See a discussion of Nozick in The New York Review of Books  of December 19, 2002

"… one should mention, first and foremost, the mathematician Hermann Weyl who was almost obsessed by this connection. In his beautiful little book Symmetry  he tersely says, 'Objectivity means invariance with respect to the group of automorphisms….'"

See also this journal on Dec. 3, 2002, and Feb. 20, 2007.

For some context, see a search on the word stem "objectiv-" in this journal.

## Wednesday, November 10, 2010

### Scavenger Hunt

Filed under: General — m759 @ 3:09 AM

A description in Pynchon's Against the Day  of William Rowan Hamilton's October 16th, 1843, discovery of quaterions—

"The moment, of course, is timeless. No beginning, no end, no duration, the light in eternal descent, not the result of conscious thought but fallen onto Hamilton, if not from some Divine source then at least when the watchdogs of Victorian pessimism were sleeping too soundly to sense, much less frighten off, the watchful scavengers of Epiphany."

New York Lottery yesterday, on Hermann Weyl's birthday— Midday 106, Evening 865.

Here 106 suggests 1/06, the date of Epiphany, and 865 turns out to be the title number of Weyl's Symmetry  at Princeton University Press—

Symmetry and quaternions are, of course, closely related.

## Monday, June 14, 2010

### Theory of Ambiguity

Filed under: General,Geometry — m759 @ 5:01 PM

Théorie de l'Ambiguité

According to a 2008 paper by Yves André of the École Normale Supérieure  of Paris—

"Ambiguity theory was the name which Galois used
when he referred to his own theory and its future developments."

The phrase "the theory of ambiguity" occurs in the testamentary letter Galois wrote to a friend, Auguste Chevalier, on the night before Galois was shot in a duel.

Hermann Weyl in Symmetry, Princeton University Press, 1952—

"This letter, if judged by the novelty and profundity of ideas it contains, is perhaps
the most substantial piece of writing in the whole literature of mankind."

Conclusion of the Galois testamentary letter, according to
the 1897 Paris edition of Galois's collected works—

The original—

A transcription—

Évariste GALOIS, Lettre-testament, adressée à Auguste Chevalier—

Tu sais mon cher Auguste, que ces sujets ne sont pas les seuls que j'aie
explorés. Mes principales méditations, depuis quelques temps,
étaient dirigées sur l'application à l'analyse transcendante de la théorie de
l'ambiguité. Il s'agissait de voir a priori, dans une relation entre des quantités
ou fonctions transcendantes, quels échanges on pouvait faire, quelles
quantités on pouvait substituer aux quantités données, sans que la relation
put cesser d'avoir lieu. Cela fait reconnaitre de suite l'impossibilité de beaucoup
d'expressions que l'on pourrait chercher. Mais je n'ai pas le temps, et mes idées
ne sont pas encore bien développées sur ce terrain, qui est
immense.

Tu feras imprimer cette lettre dans la Revue encyclopédique.

Je me suis souvent hasardé dans ma vie à avancer des propositions dont je n'étais
pas sûr. Mais tout ce que j'ai écrit là est depuis bientôt un an dans ma
tête, et il est trop de mon intérêt de ne pas me tromper pour qu'on
me soupconne d'avoir énoncé des théorèmes dont je n'aurais pas la démonstration
complète.

Tu prieras publiquement Jacobi et Gauss de donner leur avis,
non sur la vérité, mais sur l'importance des théorèmes.

Après cela, il y aura, j'espère, des gens qui trouveront leur profit
à déchiffrer tout ce gachis.

Je t'embrasse avec effusion.

E. Galois   Le 29 Mai 1832

A translation by Dr. Louis Weisner, Hunter College of the City of New York, from A Source Book in Mathematics, by David Eugene Smith, Dover Publications, 1959–

You know, my dear Auguste, that these subjects are not the only ones I have explored. My reflections, for some time, have been directed principally to the application of the theory of ambiguity to transcendental analysis. It is desired see a priori  in a relation among quantities or transcendental functions, what transformations one may make, what quantities one may substitute for the given quantities, without the relation ceasing to be valid. This enables us to recognize at once the impossibility of many expressions which we might seek. But I have no time, and my ideas are not developed in this field, which is immense.

Print this letter in the Revue Encyclopédique.

I have often in my life ventured to advance propositions of which I was uncertain; but all that I have written here has been in my head nearly a year, and it is too much to my interest not to deceive myself that I have been suspected of announcing theorems of which I had not the complete demonstration.

Ask Jacobi or Gauss publicly to give their opinion, not as to the truth, but as to the importance of the theorems.

Subsequently there will be, I hope, some people who will find it to their profit to decipher all this mess.

J t'embrasse avec effusion.

E. Galois.   May 29, 1832.

Translation, in part, in The Unravelers: Mathematical Snapshots, by Jean Francois Dars, Annick Lesne, and Anne Papillaut (A.K. Peters, 2008)–

"You know, dear Auguste, that these subjects are not the only ones I have explored. For some time my main meditations have been directed on the application to transcendental analysis of the theory of ambiguity. The aim was to see in a relation between quantities or transcendental functions, what exchanges we could make, what quantities could be substituted to the given quantities without the relation ceasing to take place. In that way we see immediately that many expressions that we might look for are impossible. But I don't have the time and my ideas are not yet developed enough in this vast field."

Another translation, by James Dolan at the n-Category Café

"My principal meditations for some time have been directed towards the application of the theory of ambiguity to transcendental analysis. It was a question of seeing a priori in a relation between quantities or transcendent functions, what exchanges one could make, which quantities one could substitute for the given quantities without the original relation ceasing to hold. That immediately made clear the impossibility of finding many expressions that one could look for. But I do not have time and my ideas are not yet well developed on this ground which is immense."

Related material

"Renormalisation et Ambiguité Galoisienne," by Alain Connes, 2004

"La Théorie de l’Ambiguïté : De Galois aux Systèmes Dynamiques," by Jean-Pierre Ramis, 2006

"Ambiguity Theory, Old and New," preprint by Yves André, May 16, 2008,

"Ambiguity Theory," post by David Corfield at the n-Category Café, May 19, 2008

"Measuring Ambiguity," inaugural lecture at Utrecht University by Gunther Cornelissen, Jan. 16, 2009

## Saturday, May 22, 2010

### In the Details

Filed under: General — Tags: — m759 @ 10:31 AM

Today's New York Times

 Byzantine "…there were fresh questions about whether the intelligence overhaul that created the post of national intelligence director was fatally flawed, and whether Mr. Obama would move gradually to further weaken the authorities granted to the director and give additional power to individual spy agencies like the Central Intelligence Agency. Mr. Blair and each of his predecessors have lamented openly that the intelligence director does not have enough power to deliver the intended shock therapy to America’s byzantine spying apparatus."

Catch-22 in Doonesbury today—

From Log24 on Jan. 5, 2010—
Artifice of Eternity

A Medal

In memory of Byzantine scholar Ihor Sevcenko,
who died at 87 on St. Stephen's Day, 2009–

Thie above image results from a Byzantine
meditation based on a detail in the previous post

"This might be a good time to
call it a day." –Today's Doonesbury

"TOMORROW ALWAYS BELONGS TO US"
Title of an exhibition by young Nordic artists
in Sweden during the summer of 2008.

The exhibition included, notably, Josefine Lyche.

## Sunday, April 4, 2010

### URBI ET ORBI

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

URBI
(Toronto)–

Click on image for some background.

ORBI

Fearful Symmetry.

## Tuesday, March 30, 2010

### Eightfold Symmetries

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

Deconstructing Design

Reconstructing Design

The phrase “eightfold way” in today’s
previous entry has a certain
graphic resonance…

For instance, an illustration from the
Wikipedia article “Noble Eightfold Path” —

St. Joseph’s Day

Harvard students who view Christian symbols
with fear and loathing may meditate
on the above as a representation of
the Gankyil rather than of the Trinity.

## Saturday, February 27, 2010

### Cubist Geometries

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

"The cube has…13 axes of symmetry:
6 C2 (axes joining midpoints of opposite edges),
4 C3 (space diagonals), and
3C4 (axes joining opposite face centroids)."
–Wolfram MathWorld article on the cube

These 13 symmetry axes can be used to illustrate the interplay between Euclidean and Galois geometry in a cubic model of the 13-point Galois plane.

The geometer's 3×3×3 cube–
27 separate subcubes unconnected
by any Rubik-like mechanism–

The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
subcubes in the (Galois) 3×3×3 cube–

A closely related structure–
the finite projective plane
with 13 points and 13 lines–

A later version of the 13-point plane
by Ed Pegg Jr.–

A group action on the 3×3×3 cube
as illustrated by a Wolfram program
by Ed Pegg Jr. (undated, but closely
related to a March 26, 1985 note
by Steven H. Cullinane)–

The above images tell a story of sorts.
The moral of the story–

Galois projective geometries can be viewed
in the context of the larger affine geometries
from which they are derived.

The standard definition of points in a Galois projective plane is that they are lines through the (arbitrarily chosen) origin in a corresponding affine 3-space converted to a vector 3-space.

If we choose the origin as the center cube in coordinatizing the 3×3×3 cube (See Weyl's relativity problem ), then the cube's 13 axes of symmetry can, if the other 26 cubes have properly (Weyl's "objectively") chosen coordinates, illustrate nicely the 13 projective points derived from the 27 affine points in the cube model.

The 13 lines of the resulting Galois projective plane may be derived from Euclidean planes  through the cube's center point that are perpendicular to the cube's 13 Euclidean symmetry axes.

The above standard definition of points in a Galois projective plane may of course also be used in a simpler structure– the eightfold cube.

(The eightfold cube also allows a less standard way to picture projective points that is related to the symmetries of "diamond" patterns formed by group actions on graphic designs.)

See also Ed Pegg Jr. on finite geometry on May 30, 2006
at the Mathematical Association of America.

## Tuesday, January 5, 2010

### Artifice of Eternity

Filed under: General,Geometry — Tags: — m759 @ 7:59 AM

A Medal

In memory of Byzantine scholar Ihor Sevcenko,
who died at 87 on St. Stephen's Day, 2009–

 William Grimes on Sevcenko in this morning's New York Times: "Perhaps his most fascinating, if uncharacteristic, literary contribution came shortly after World War II, when he worked with Ukrainians stranded in camps in Germany for displaced persons. In April 1946 he sent a letter to Orwell, asking his permission to translate 'Animal Farm' into Ukrainian for distribution in the camps. The idea instantly appealed to Orwell, who not only refused to accept any royalties but later agreed to write a preface for the edition. It remains his most detailed, searching discussion of the book."

here in the context of an Orwellian headline from
The New York Times on Christmas morning,
the day before Sevcenko died.
That headline, at the top of the online front page,
was "Arthur Koestler, Man of Darkness."

The medal, offered as an example of brightness
to counteract the darkness of the Times, was designed
by Leibniz in honor of his discovery of binary arithmetic.
See Brightness at Noon and Brightness continued.

"By groping toward the light we are made to realize
how deep the darkness is around us."
— Arthur Koestler, The Call Girls: A Tragi-Comedy,
Random House, 1973, page 118

## Sunday, May 17, 2009

### Sunday May 17, 2009

Filed under: General,Geometry — Tags: — m759 @ 7:59 AM
Design Theory

"My work is motivated by a hope that there may be a way to recapture the ancient and medieval vision of both Beauty and purpose in a way which is relevant to our own century. I even dare to hope that the two ideas may be related, that Beauty is actually part of the meaning and purpose of life."

Hans Ludwig de Vries, "On Orthogonal Resolutions of the Classical Steiner Quadruple System SQS(16)," Designs, Codes and Cryptography Vol. 48, No. 3 (Sept. 2008) 287-292 (DOI 10.1007/s10623-008-9207-5)–

"The Reverend T. P. Kirkman knew in 1862 that there exists a group of degree 16 and order 322560 with a normal, elementary abelian, subgroup of order 16 [1, p. 108]. Frobenius identified this group in 1904 as a subgroup of the Mathieu group M24 [4, p. 570]…."

1. Biggs N.L., "T. P. Kirkman, Mathematician," Bulletin of the London Mathematical Society 13, 97–120 (1981).

4. Frobenius G., "Über die Charaktere der mehrfach transitiven Gruppen," Sitzungsber. Königl. Preuss. Akad. Wiss. zu Berlin, 558–571 (1904). Reprinted in Frobenius, Gesammelte Abhandlungen III (J.-P. Serre, editor), pp. 335–348. Springer, Berlin (1968).

Olli Pottonen, "Classification of Steiner Quadruple Systems" (Master's thesis, Helsinki, 2005)–

"The concept of group actions is very useful in the study of isomorphisms of combinatorial structures."

"Simplify, simplify."
Thoreau

"Beauty is bound up
with symmetry."
Weyl

Pottonen's thesis is
dated Nov. 16, 2005.

For some remarks on
images and theology,
see Log24 on that date.

Click on the above image
for some further details.

## Friday, August 8, 2008

### Friday August 8, 2008

Filed under: General,Geometry — m759 @ 8:08 AM

Click on image for details.

## Saturday, July 19, 2008

### Saturday July 19, 2008

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

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

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

Hermann Weyl on the hard core of objectivity:

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

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

 A Structure-Endowed Entity “A guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity S, try to determine its group of automorphisms, the group of those element-wise transformations which leave all structural relations undisturbed.  You can expect to gain a deep insight into the constitution of S in this way.” — Hermann Weyl in Symmetry Let us apply Weyl’s lesson to the following “structure-endowed entity.” What is the order of the resulting group of automorphisms?

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

The Diamond 16 Puzzle

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

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

— Steven H. Cullinane,
July 17, 2008

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

## Thursday, June 21, 2007

### Thursday June 21, 2007

Filed under: General,Geometry — Tags: — m759 @ 12:07 PM
Let No Man
Write My Epigraph

"His graceful accounts of the Bach Suites for Unaccompanied Cello illuminated the works’ structural logic as well as their inner spirituality."

Allan Kozinn on Mstislav Rostropovich in The New York Times, quoted in Log24 on April 29, 2007

"At that instant he saw, in one blaze of light, an image of unutterable conviction…. the core of life, the essential pattern whence all other things proceed, the kernel of eternity."

— Thomas Wolfe, Of Time and the River, quoted in Log24 on June 9, 2005

"… the stabiliser of an octad preserves the affine space structure on its complement, and (from the construction) induces AGL(4,2) on it. (It induces A8 on the octad, the kernel of this action being the translation group of the affine space.)"

— Peter J. Cameron, "The Geometry of the Mathieu Groups" (pdf)

"… donc Dieu existe, réponse!"

— Attributed, some say falsely,
to Leonhard Euler

what the mandala really is:
'Formation, Transformation,
Eternal Mind's eternal recreation'"

(Faust, Part Two, as
quoted by Jung in
Memories, Dreams, Reflections)

"Pauli as Mephistopheles
in a 1932 parody of
Goethe's Faust at Niels Bohr's
institute in Copenhagen.
The drawing is one of
many by George Gamow
illustrating the script."
Physics Today

"Borja dropped the mutilated book on the floor with the others. He was looking at the nine engravings and at the circle, checking strange correspondences between them.

'To meet someone' was his enigmatic answer. 'To search for the stone that the Great Architect rejected, the philosopher's stone, the basis of the philosophical work. The stone of power. The devil likes metamorphoses, Corso.'"

The Club Dumas, basis for the Roman Polanski film "The Ninth Gate" (See 12/24/05.)

with the concept of automorphism."

The Innermost Kernel
(previous entry)

And from
"Symmetry in Mathematics
and Mathematics of Symmetry
"
(pdf), by Peter J. Cameron,
a paper presented at the
International Symmetry Conference,
Edinburgh, Jan. 14-17, 2007,
we have

The Epigraph–

(Here "whatever" should
of course be "whenever.")

Also from the
Cameron paper:

 Local or global? Among other (mostly more vague) definitions of symmetry, the dictionary will typically list two, something like this: • exact correspondence of parts; • remaining unchanged by transformation. Mathematicians typically consider the second, global, notion, but what about the first, local, notion, and what is the relationship between them?  A structure M is homogeneous if every isomorphism between finite substructures of M can be extended to an automorphism of M; in other words, "any local symmetry is global."

Some Log24 entries
related to the above politically
(women in mathematics)–

and mathematically–

This entry cites
Alice Devillers of Brussels–

"The aim of this thesis
is to classify certain structures
which are, from a certain
point of view, as homogeneous
as possible, that is which have
as many symmetries as possible."

"There is such a thing
as a tesseract."

## Tuesday, February 20, 2007

### Tuesday February 20, 2007

Filed under: General,Geometry — m759 @ 7:09 AM
Symmetry

Today is the 21st birthday of my note “The Relativity Problem in Finite Geometry.”

Some relevant quotations:

“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

Describing the branch of mathematics known as Galois theory, Weyl says that it

“… 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

Weyl’s set Sigma is a finite set of complex numbers.   Some other sets with “discrete and finite character” are those of 4, 8, 16, or 64 points, arranged in squares and cubes.  For illustrations, see Finite Geometry of the Square and Cube.  What Weyl calls “the relativity problem” for these sets involves fixing “objectively” a class of equivalent coordinatizations.  For what Weyl’s “objectively” means, see the article “Symmetry and Symmetry  Breaking,” by Katherine Brading and Elena Castellani, in the Stanford Encyclopedia of Philosophy:

“The old and natural idea that what is objective should not depend upon the particular perspective under which it is taken into consideration is thus reformulated in the following group-theoretical terms: what is objective is what is invariant with respect to the transformation group of reference frames, or, quoting Hermann Weyl (1952, p. 132), ‘objectivity means invariance with respect to the group of automorphisms [of space-time].‘[22]

22. The significance of the notion of invariance and its group-theoretic treatment for the issue of objectivity is explored in Born (1953), for example. For more recent discussions see Kosso (2003) and Earman (2002, Sections 6 and 7).

References:

Born, M., 1953, “Physical Reality,” Philosophical Quarterly, 3, 139-149. Reprinted in E. Castellani (ed.), Interpreting Bodies: Classical and Quantum Objects in Modern Physics, Princeton, NJ: Princeton University Press, 1998, pp. 155-167.

Earman, J., 2002, “Laws, Symmetry, and Symmetry Breaking; Invariance, Conservation Principles, and Objectivity,’ PSA 2002, Proceedings of the Biennial Meeting of the Philosophy of Science Association 2002, forthcoming [Abstract/Preprint available online]

Kosso, P., 2003, “Symmetry, objectivity, and design,” in K. Brading and E. Castellani (eds.), Symmetries in Physics: Philosophical Reflections, Cambridge: Cambridge University Press, pp. 410-421.

Weyl, H., 1952, Symmetry, Princeton, NJ: Princeton University Press.

Archives Henri Poincaré (research unit UMR 7117, at Université Nancy 2, of the CNRS)–

Minkowski, Mathematicians, and the Mathematical Theory of Relativity,” by Scott Walter, in The Expanding Worlds of General Relativity (Einstein Studies, volume 7), H. Goenner, J. Renn, J. Ritter and T. Sauer, editors, Boston/Basel: Birkhäuser, 1999, pp. 45-86–

“Developing his ideas before Göttingen mathematicians in April 1909, Klein pointed out that the new theory based on the Lorentz group (which he preferred to call ‘Invariantentheorie’) could have come from pure mathematics (1910: 19). He felt that the new theory was anticipated by the ideas on geometry and groups that he had introduced in 1872, otherwise known as the Erlangen program (see Gray 1989: 229).”

References:

Gray, Jeremy J. (1989). Ideas of Space. 2d ed. Oxford: Oxford University Press.

Klein, Felix. (1910). “Über die geometrischen Grundlagen der Lorentzgruppe.” Jahresbericht der deutschen Mathematiker-Vereinigung 19: 281-300. [Reprinted: Physikalische Zeitschrift 12 (1911): 17-27].

Related material: A pathetically garbled version of the above concepts was published in 2001 by Harvard University Press.  See Invariances: The Structure of the Objective World, by Robert Nozick.

## Monday, November 20, 2006

### Monday November 20, 2006

Filed under: General — Tags: — m759 @ 7:20 PM

Triumphs

Yesterday's link to a Log24 entry for the Feast of the Triumph of the Cross led to the following figure:

(Based on Weyl's Symmetry)

Today, an entry in the The New Criterion's weblog tells of Hilton Kramer's new collection of essays on art, The Triumph of Modernism.

From a Booklist review:

Kramer "celebrates the revelations of modern art, defining modernism as nothing less than 'the discipline of truthfulness, the rigor of honesty.'"

Further background: Kramer opposes

"willed frivolity and politicized vulgarization as fashionable enemies of high culture as represented in the recent past by the integrity of modernism."

Perhaps Kramer would agree that such integrity is exemplified by "Two Giants" of modernism described by Roberta Smith in The New York Times recently (Nov. 3– birthdate of A. B. Coble, an artist of a different kind). She is reviewing an exhibit, ''Albers and Moholy-Nagy: From the Bauhaus to the New World,'' that continues through Jan. 21 at the Whitney Museum of American Art,

945

This instance of the number 945 as an "artists' signature" is perhaps more impressive than the instances cited in yesterday's Log24 entry, Signature.

## Thursday, October 19, 2006

### Thursday October 19, 2006

Filed under: General,Geometry — m759 @ 7:59 AM
For Sir Thomas Browne

(Born Oct. 19, 1605,
died  Oct. 19, 1682)

Browne is noted for
Hydriotaphia (Urne-Buriall)
and The Garden of Cyrus.

Related material:

## Thursday, September 14, 2006

### Thursday September 14, 2006

Filed under: General — Tags: — m759 @ 7:11 PM
Today is the Feast of the
Triumph of the Cross

(Based on Weyl's Symmetry)

and the birthday of
an expert on primitive roots,

Happy birthday.

Click on pictures
for further details.

## Thursday, July 20, 2006

### Thursday July 20, 2006

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

Those who clicked on Rieff’s concept in the previous entry will know about the book that Rieff titled Sacred Order/Social Order: My Life among the Deathworks.

That entry, from Tuesday, July 18, was titled “Sacred Order,” and gave as an example the following figure:

(Based on Weyl’s Symmetry)

For the use of this same figure to represent a theatrical concept–

“It’s like stringing beads on a necklace. By the time the play ends, you have the whole necklace.”

— see Ursprache Revisited (June 9, 2006).

Of course, the figure also includes a cross– or “deathwork”– of sorts.  These incidental social properties of the figure (which is purely mathematical in origin) make it a suitable memorial for a theatre critic who died on the date of the previous entry– July 18– and for whom the American Theatre Wing’s design awards, the Henry Hewes Awards, are named.

“The annual awards honor designers… recognizing not only the traditional design categories of sets, costumes and lighting, but also ‘Notable Effects,’ which encompasses sound, music, video, puppets and other creative elements.” —BroadwayWorld.com

For more on life among the deathworks, see an excellent review of the Rieff book mentioned above.

## Monday, June 26, 2006

### Monday June 26, 2006

Filed under: General — Tags: — m759 @ 9:29 AM

In memory of
Mary Martin McLaughlin,
a scholar of Heloise and Abelard.
McLaughlin died on June 8, 2006.

"Following the parade, a speech is given by Charles Williams, based on his book The Place of the Lion. Williams explains the true meaning of the word 'realism' in both philosophy and theology. His guard of honor, bayonets gleaming, is led by William of Ockham."

A review by John D. Burlinson of Charles Williams's novel The Place of the Lion:

"… a little extra reading regarding Abelard's take on 'universals' might add a little extra spice– since Abelard is the subject of the heroine's … doctoral dissertation. I'd suggest the article 'The Medieval Problem of Universals' in the online Stanford Encyclopedia of Philosophy."

Michael L. Czapkay, a student of philosophical theology at Oxford:

"The development of logic in the schools and universities of western Europe between the eleventh and fifteenth centuries constituted a significant contribution to the history of philosophy. But no less significant was the influence of this development of logic on medieval theology. It provided the necessary conceptual apparatus for the systematization of theology. Abelard, Ockham, and Thomas Aquinas are paradigm cases of the extent to which logic played an active role in the systematic formulation of Christian theology. In fact, at certain points, for instance in modal logic, logical concepts were intimately related to theological problems, such as God's knowledge of future contingent truths."

The Medieval Problem of Universals, by Fordham's Gyula Klima, 2004:

"… for Abelard, a status is an object of the divine mind, whereby God preconceives the state of his creation from eternity."

Status Symbol

(based on Weyl's Symmetry):

"… for then we would know

the mind of God"
Stephen Hawking, 1988

For further details,
click on the picture.

## Tuesday, June 6, 2006

### Tuesday June 6, 2006

Filed under: General — Tags: — m759 @ 5:01 AM
D-Day Morning,
62 Years Later

Review: ART WARS
on Sept. 12, 2002:

Und was fur ein Bild des Christentums
ist dabei herausgekommen?

(Pentecost was Sunday, June 4, 2006.
The following Monday was formerly a
French public holiday.)

This morning's meditation:

Sous Rature

"… words must be written
sous rature, or 'under erasure.'"

The above Bild, based
on Weyl's Symmetry,
might be titled
Rature sous Rature.

## Sunday, June 4, 2006

### Sunday June 4, 2006

Filed under: General — m759 @ 1:00 PM
Images
and Words
for Baccalaureate
Day at Princeton

From Hermann Weyl’s
Symmetry,
Princeton University
Press, page 140

cover of Alan Watts’s
The Spirit of Zen

Romani flag, courtesy of

myspace.com/RomArmando

Related material:

“The Scholar Gypsy”
in The Oxford Book
of English Prose
, 1923,
edited by
Sir Arthur Quiller-Couch

This is available online:

From The Vanity of Dogmatizing,
by Joseph Glanvill
(London, printed by E.C. for
Henry Eversden at the Grey-Hound
in St.Pauls-Church-Yard,  1661)

Pages 195-201:

Now that this strange power of the Imagination is no Impossibility; the wonderful signatures in the Foetus caus’d by the Imagination of the Mother, is no contemptible Item. The sympathies of laughing & gaping together, are resolv’d into this Principle: and I see not why the phancy of one man may not determine the cogitation of another rightly qualified, as easily as his bodily motion. This influence seems to be no more unreasonable, then [sic] that of one string of a Lute upon another; when a stroak on it causeth a proportionable motion in the sympathizing confort, which is distant from it and not sensibly touched. Now if this notion be strictly verifiable; ’twill yeeld us a good account of how Angels inject thoughts into our minds, and know our cogitations: and here we may see the source of some kinds of fascination. If we are prejudic’d against the speculation, because we cannot conceive the manner of so strange an operation; we shall indeed receive no help from the common Philosophy: But yet the Hypothesis of a Mundane soul, lately reviv’d by that incomparable Platonist and Cartesian, Dr. H. More, will handsomely relieve us. Or if any would rather have a Mechanical account; I think it may probably be made out some such way as follow. Imagination is inward Sense. To Sense is required a motion of certain Filaments of the Brain; and consequently in Imagination there’s the like: they only differing in this, that the motion of the one proceeds immediately from external objects; but that of the other hath its immediate rise within us. Now then, when any part of the Brain is stringly agitated; that, which is next and most capable to receive the motive Impress, must in like manner be moved. Now we cannot conceive any thing more capable of motion, then the fluid matter, that’s interspers’d among all bodies, and contiguous to them. So then, the agitated parts of the Brain begetting a motion in the proxime Aether; it is propagated through the liquid medium, as we see the motion is which is caus’d by a stone thrown into the water. Now, when the thus moved matter meets with anything like that, from which it received its primary impress; it will proportionably move it, as it is in Musical strings tuned Unisons. And thus the motion being convey’d, from the Brain of one man to the Phancy of another; it is there receiv’d from the instrument of conveyance, the subtil matter; and the same kind of strings being moved, and much of whay after the same manner as in the first Imaginant; the Soul is awaken’d to the same apprehensions, as were they that caus’d them. I pretend not to any exactness or infallibility in this account, fore-seeing many scruples that must be removed to make it perfect: ‘Tis only a hint of the possibility of mechanically solving the Phaenomenon; though very likely it may require many other circumstances completely to make it out. But ’tis not my business here to follow it: I leave it therefore to receive accomplishment from maturer Inventions.

## Monday, April 10, 2006

### Monday April 10, 2006

Filed under: General,Geometry — Tags: — m759 @ 9:20 PM
Club
continued
"What other colleges call fraternities,
Princeton calls Eating Clubs."

Illustrated below:
The Restaurant Quarré in Berlin,
with a view of the Brandenburg Gate.

Related etymology:
OF. quarré square, F. carré,
Webster's Revised
Unabridged Dictionary, 1913

Related material:

(1) A symbol of symmetry
Hermann Weyl:

Source —
Timothy A. Smith on
Bach's Fugue No. 21,
the Well-Tempered
Clavier, Book II
(pdf or Shockwave)

"It is of course an act of chutzpah,
some would say almost heresy,
to challenge Bach so explicitly
on his own turf."

(3) The five Log24 entries
culminating on Pi Day,
March 14, 2006

(4) The following event at the
Harvard University
mathematics department
on March 14, 2006, also
featuring Noam D. Elkies:

"At 3:14 p.m., six contestants began
exactly three minutes and 14 seconds
to eat as much pie as they could.

'Five, four, pi, three, two, one,'
Elkies counted down as the
contestants shoved the last
mouthful of pie
into their mouths…."

Noam D. Elkies

## Sunday, March 19, 2006

### Sunday March 19, 2006

Filed under: General — m759 @ 6:09 PM
St. Joseph’s Day

Cut Numbers and
In the Hand of Dante,
both by Nick Tosches,

and Symmetry,
by Hermann Weyl:

Related material:
Kernel of Eternity
(a Log24 entry of June 9, 2005)
and the comment on that entry
by ItAlIaNoBoI.

## Friday, March 17, 2006

### Friday March 17, 2006

Filed under: General,Geometry — m759 @ 2:28 AM
George W. Mackey,
Harvard mathematician,

Mackey was born, according to Wikipedia, on Feb. 1, 1916.  He died, according to Harvard University, on the night of March 14-15, 2006.  He was the author of, notably, “Harmonic Analysis as the Exploitation of Symmetry — A Historical Survey,” pp. 543-698 in Bulletin of the American Mathematical Society (New Series), Vol. 3, No. 1, July 1980.  This is available in a hardcover book published in 1992 by the A.M.S., The Scope and History of Commutative and Noncommutative Harmonic Analysis. (370 pages, ISBN 0-8218-9903-1).  A paperback edition of this book will apparently be published this month by Oxford University Press (ISBN 978-0-8218-3790-7).

From Oxford U.P.–

Contents

• Introduction
• Harmonic analysis as the exploitation of symmetry: A historical survey
• Herman Weyl and the application of group theory to quantum mechanics
• The significance of invariant measures for harmonic analysis
• Weyl’s program and modern physics
• Induced representations and the applications of harmonic analysis
• Von Neumann and the early days of ergodic theory
• Final remarks

Related material:
Log24, Oct. 22, 2002.
Women’s history month continues.

## Saturday, June 4, 2005

### Saturday June 4, 2005

Filed under: General,Geometry — m759 @ 7:00 PM
Drama of the Diagonal

The 4×4 Square:
French Perspectives

Earendil_Silmarils:

“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
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.

## 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” –

“The Divine Proportion

is an irrational number and
the positive solution

x2 – x – 1 = 0,

which is (1+Sqrt(5))/2,

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

For another approach to
the divine proportion, see

“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

## Friday, February 20, 2004

### Friday February 20, 2004

Filed under: General,Geometry — Tags: — m759 @ 3:24 PM

Finite Relativity

Today is the 18th birthday of my note

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.

## Sunday, November 9, 2003

### Sunday November 9, 2003

Filed under: General,Geometry — Tags: — 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."

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.)

## 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.

## 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.

## Tuesday, December 3, 2002

### Tuesday December 3, 2002

Filed under: General,Geometry — Tags: — 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."

## Saturday, November 9, 2002

### Saturday November 9, 2002

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

Birthdate of Hermann Weyl

 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 Sponsored Link Symmetry PuzzleNew free online puzzle illustrates the mathematics of symmetry.m759.freeservers.com/puzzle.html

Quotation from Weyl’s Symmetry:

Symmetry is a vast subject, significant in art and nature. Mathematics lies at its root, and it would be hard to find a better one on which to demonstrate the working of the mathematical intellect.”

In honor of Princeton University, of Sylvia Nasar (see entries of Nov, 6), of the Presbyterian Church (see entry of Nov. 8), and of Professor Weyl (whose work partly inspired the website Diamond Theory), this site’s background music is now Pink Floyd’s

Updates of Friday, November 15, 2002:

In order to clarify the meaning of “Shine” and “Crazy” in the above, consult the following —