Log24

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

—  Alastair Macaulay in the online New York Times  today

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

A video published on the above date —

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

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

Space —

Space structure —

From Gotay and Isenberg, “The Symplectization of Science,”
Gazette des Mathématiciens  54, 59-79 (1992):

“… what is the origin of the unusual name ‘symplectic’? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure ‘line complex group’ the ‘symplectic group.’
… the adjective ‘symplectic’ means ‘plaited together’ or ‘woven.’
This is wonderfully apt….”

The above symplectic  figure appears in remarks on
the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2).

Space shuttle —

Related ethnic remarks —

… As opposed to Michael  Larsen —

Funny, you don't look  Danish.

Material related to the previous post, "Symmetry" —

This is the group of "8 rigid motions
generated by reflections in midplanes"
of "Solomon's Cube."

Material from this journal on May 1, the date of Golomb's death —

"Weitere Informationen zu diesem Themenkreis
finden sich unter http://​www.​encyclopediaofma​th.​org/
​index.​php/​Cullinane_​diamond_​theorem und
http://​finitegeometry.​org/​sc/​gen/​coord.​html ."

"Little emblems of eternity"
— Phrase by Oliver Sacks in today's
New York Times  Sunday Review

Some other emblems —

Note the color-interchange
symmetry of each emblem
under 180-degree rotation.

Click an emblem for
some background.

Continued from All Hallows' Eve, 2014.

Last year's Halloween post displayed the
Dürer print Knight, Death, and the Devil 
(illustrated below on the cover of the book
Film and Phenomenology  by Allan Casebier).

Cover illustration: Knight, Death, and the Devil, 
by Albrecht Dürer

Some mathematics related to a different Dürer print —

The 1976 monograph "Diamond Theory" was an example
of "programmed art" in the sense established by, for
instance, Karl Gerstner. The images were produced 
according to strict rules, and were in this sense 
"programmed," but were drawn by hand.

Now an actual computer program has been written,
based on the Diamond Theory excerpts published
in the Feb. 1977 issue of Computer Graphics and Art
(Vol. 2, No. 1, pp. 5-7), that produces copies of some of
these images (and a few malformed images not  in
Diamond Theory).

See Isaac Gierard's program at GitHub—

https://github.com/matthewepler/ReCode_Project/
blob/dda7b23c5ad505340b468d9bd707fd284e6c48bf/
isaac_gierard/StevenHCullinane_DiamondTheory/
StevenHCullinane_DiamondTheory.pde

As the suffix indicates, this program is in the
Processing Development Environment language.

It produces the following sketch:

The rationale for selecting and arranging these particular images is not clear,
and some of the images suffer from defects (exercise: which ones?), but the 
overall effect of the sketch is pleasing.

For some background for the program, see The ReCode Project.

It is good to learn that the Processing language is well-adapted to making the 
images in such sketches. The overall structure of the sketch gives, however,
no clue to the underlying theory  in "Diamond Theory."

For some related remarks, see Theory (Sept. 30, 2012).

* For the title, see Darko Fritz, "Notions of the Program in 1960s Art."

In Turing's Cathedral

"At the still point…" — T. S. Eliot

In memory of David L. Waltz, artificial-intelligence pioneer,
who died Thursday, March 22, 2012—

1. The Log24 post of March 22 on the square-triangle theorem
2. The March 18 post, Square-Triangle Diamond
3. Remarks from the BBC on linguistic embedding
   that begin as follows—
        "If we draw a large triangle and embed smaller triangles in it,
         how does it look?"—
   and include discussion of a South American "tribe called Piranha" [sic ]
4. The result of a Cartoon Bank search suggested by no. 3 above—
   (Click image for some related material.)
   
5. A suggestion from the Cartoon Bank—
   

6. The following from the First of May, 2010—

   The Nine Divisions of Heaven–
   

   

   Some context–

   

   "This pattern is a square divided into nine equal parts.
   It has been called the 'Holy Field' division and
   was used throughout Chinese history for many
   different purposes, most of which were connected
   with things religious, political, or philosophical."

   – The Magic Square: Cities in Ancient China,
   by Alfred Schinz, Edition Axel Menges, 1996, p. 71

7. The phrase "embedding the stone" —
   • in an article on buying jewelry,
   • in this journal yesterday, and
   • in the Bible.

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

[* Associated how?]

For the title, see Palm Sunday.

"There is a pleasantly discursive treatment of
Pontius Pilate's unanswered question 'What is truth?'" — H. S. M. Coxeter, 1987

From this date (April 22) last year—

Richard J. Trudeau in The Non-Euclidean Revolution , chapter on "Geometry and the Diamond Theory of Truth"– "… Plato and Kant, and most of the philosophers and scientists in the 2200-year interval between them, did share the following general presumptions: (1) Diamonds– informative, certain truths about the world– exist. (2) The theorems of Euclidean geometry are diamonds. Presumption (1) is what I referred to earlier as the 'Diamond Theory' of truth. It is far, far older than deductive geometry." Trudeau's book was published in 1987. The non-Euclidean* figures above illustrate concepts from a 1976 monograph, also called "Diamond Theory." Although non-Euclidean,* the theorems of the 1976 "Diamond Theory" are also, in Trudeau's terminology, diamonds. * "Non-Euclidean" here means merely "other than  Euclidean." No violation of Euclid's parallel postulate is implied.

Trudeau comes to reject what he calls the "Diamond Theory" of truth. The trouble with his argument is the phrase "about the world."

Geometry, a part of pure mathematics, is not  about the world. See G. H. Hardy, A Mathematician's Apology .