Log24

"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.

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.”

See also related historical remarks by Weyl and by Atiyah.

“You’ve got to pick up every stitch…”
— Donovan, song on closing credits of  To Die For

“…’Supersymmetry’ was originally written
specifically for Her ….” — Pitchfork

“Eventually we see snow particles….”
— Her  screenplay by Spike Jonze

This journal on January 24, 2006:

Context:  See Free-Floating Signs.

Backstory:  Digital Member and  Uneven Break.

Related material: 

For code  in a different sense, but related to the first passage above,
see Diamond Theory Roullete, a webpage by Radamés Ajna.

For "the method of the kaleidoscope" mentioned in the second
passage above, see both the Ajna page and a webpage of my own,
Kaleidoscope Puzzle.

(A sequel to yesterday's Raiders of the Lost Music Box)

See, in this book, "Walsh Functions: A Digital Fourier Series,"
by Benjamin Jacoby (BYTE , September 1977).  Some context:
Symmetry of Walsh Functions.

"I don't like odd numbers, and I really don't like primes."

— Marissa Mayer

See Cube Symmetry Axes in this journal.

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

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

Related non-fiction — a page of pure mathematics —

From the obituary of a Bletchley Park
codebreaker who reportedly died on
Armistice Day (Monday, Nov. 11)—

"The main flaw of the Enigma machine,
seen by the inventors as a security-enhancing
measure, was that it would never encipher
a letter as itself…."

Update of 9 PM ET Nov. 13—

"The rogue’s yarn that will run through much of
the material is the algebraic symmetry to which
the name of Galois is attached…."

— Robert P. Langlands,
     Institute for Advanced Study, Princeton

"All the turmoil, all the emotions of the scenes
have been digested by the mind into
a grave intellectual whole.  It is as though
Bach had written the 1812 Overture."

— Aldous Huxley, "The Best Picture," 1925

Online biography of author Cormac McCarthy—

"… he left America on the liner Sylvania, intending to visit
the home of his Irish ancestors (a King Cormac McCarthy
built Blarney Castle)." 

Two Years Ago:

Blarney in The Harvard Crimson—

Melissa C. Wong, illustration for "Atlas to the Text,"
by Nicholas T. Rinehart:

Thirty Years Ago:

Non-Blarney from a rural outpost—

Illustration for the generalized diamond theorem,
by Steven H. Cullinane: 

See also Barry's Lexicon .

(Continued from December 30, 2012)

"And let us finally, then, observe the
parallel progress of the formations of thought
across the species of psychical onomatopoeia
of the primitives, and elementary symmetries
and contrasts, to the ideas of substances,
to metaphors, the faltering beginnings of logic,
formalisms, entities, metaphysical existences."

— Paul Valéry, Introduction to the Method of
    Leonardo da Vinci

But first, a word from our sponsor…

Brought to you by two uploads, each from Sept. 11, 2012—

Symmetry and Hierarchy and the above VINCI Genius commercial.

Surreal requiem for the late Jonathan Winters:

"They 'burn, burn, burn like fabulous yellow roman candles
exploding like spiders across the stars,'
as Jack Kerouac once wrote. It was such a powerful
image that Wal-Mart sells it as a jigsaw puzzle."

— "When the Village Was the Vanguard,"
       by Henry Allen, in today's Wall Street Journal

See also Damnation Morning and the picture in
yesterday evening's remarks on art:

The New Yorker  on Cubism:

"The style wasn’t new, exactly— or even really a style,
in its purest instances— though it would spawn no end
of novelties in art and design. Rather, it stripped naked
certain characteristics of all pictures. Looking at a Cubist
work, you are forced to see how you see. This may be
gruelling, a gymnasium workout for eye and mind.
It pays off in sophistication."

— Online "Culture Desk" weblog, posted today by Peter Schjeldahl

Non-style from 1911:

See also Cube Symmetry Planes  in this  journal.

A comment at The New Yorker  related to Schjeldahl's phrase "stripped naked"—

"Conceptualism is the least seductive modern-art movement."

POSTED 4/11/2013, 3:54:37 PM BY CHRISKELLEY

(The "conceptualism" link was added to the quoted comment.)

The previous post discussed some tesseract–
related mathematics from 1905.

Returning to the present, here is some arXiv activity
in the same area from March 11, 12, and 13, 2013.

Inscribed hexagon (1984)

The well-known fact that a regular hexagon
may be inscribed in a cube was the basis
in 1984 for two ways of coloring the faces
of a cube that serve to illustrate some graphic
aspects of embodied Galois geometry—

Inscribed hexagon (2013)

A redefinition of the term "symmetry plane"
also uses the well-known inscription
of a regular hexagon in the cube—

Related material

"Here is another way to present the deep question 1984  raises…."

— "The Quest for Permanent Novelty," by Michael W. Clune,
     The Chronicle of Higher Education , Feb. 11, 2013

“What we do may be small, but it has a certain character of permanence.”

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

George Steiner, Real Presences , first published in 1989—

The inception of critical thought, of a philosophic anthropology,
is contained in the archaic Greek definition of man as a
'language-animal'….

Richard Powers, The Gold Bug Variations , first published in 1991—

Botkin, whatever her gifts as a conversationist, is almost as old
as the rediscovery of Mendel. The other extreme in age,
Joe Lovering, beat a time-honored path out of pure math
into muddy population statistics. Ressler has seen the guy
potting about in the lab, although exactly what the excitable kid
does is anybody's guess. He looks decidedly gumfooted holding
any equipment more corporeal than a chi-square. Stuart takes
him to the Y for lunch, part of a court-your-resources campaign.
He has the sub, Levering the congealed mac and cheese.
Hardly are they seated when Joe whips out a napkin and begins
sketching proofs. He argues that the genetic code, as an
algorithmic formal system, is subject to Gödel's Incompleteness
Theorem. "That would mean the symbolic language of the code
can't be both consistent and complete. Wouldn't that be a kick
in the head?"

Kid talk, competitive showing off, intellectual fantasy.
But Ressler knows what Joe is driving at. He's toyed with similar
ideas, cast in less abstruse terms. We are the by-product of the
mechanism in there. So it must be more ingenious than us.
Anything complex enough to create consciousness may be too
complex for consciousness to understand. Yet the ultimate paradox
is Lovering, crouched over his table napkin, using proofs to
demonstrate proof's limits. Lovering laughs off recursion and takes
up another tack: the key is to find some formal symmetry folded
in this four-base chaos. Stuart distrusts this approach even more.
He picks up the tab for their two untouched lunches, thanking
Lovering politely for the insight.

Edith Piaf—

Non, rien de rien…

See last midnight's post and Theme and Variations.

"The key is to find some formal symmetry…."

Raiders of the Lost Trunk, or:

Stars in the Attic

More »

See also A Glass for Klugman :

Context: Poetry and Truth,  Eternal Recreation,  
              Solid Symmetry, and Stevens's Rock.

This post continues the April 9 post
commemorating Élie Cartan's birthday.

That post mentioned triality .
Here is John Baez reviewing
On Quaternions and Octonions:
Their Geometry, Arithmetic, and Symmetry
by John H. Conway and Derek A. Smith
(A.K. Peters, Ltd., 2003)—

"In this context, triality manifests itself
as the symmetry that cyclically permutes
the Hurwitz integers  i , j ,  and k ."

Related material— Quaternion Acts in this journal
as well as Finite Geometry and Physical Space.

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

Another book to which random reference is dangerous—

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

The hypercube has 192 rotational symmetries.
Its full symmetry group, including reflections,
is of order 384.

See (for instance) Coxeter—

Related material—

The rotational symmetry groups of the Platonic solids
(from April 25, 2011)—

— and the figure in yesterday evening's post on the hypercube—

(Animation source: MIQEL.com)

Clearly hypercube rotations of this sort carry any
of the eight 3D subcubes to the central subcube
of a central projection of the hypercube—

The 24 rotational symmeties of that subcube induce
24 rigid rotations of the entire hypercube. Hence,
as in the logic of the Platonic symmetry groups
illustrated above, the hypercube has 8 × 24 = 192
rotational symmetries.