Wednesday, February 28, 2007 7:59 AM

Parts of a Whole:

Elements
of Geometry

The title of Euclid's Elements is, in Greek, Stoicheia.

From Lectures on the Science of Language, by Max Muller, fellow of All Souls College, Oxford. New York: Charles Scribner's Sons, 1890, pp. 88-90 --

Stoicheia

"The question is, why were the elements, or the component primary parts of things, called stoicheia by the Greeks? It is a word which has had a long history, and has passed from Greece to almost every part of the civilized world, and deserves, therefore, some attention at the hand of the etymological genealogist.

Stoichos, from which stoicheion, means a row or file, like stix and stiches in Homer. The suffix eios is the same as the Latin eius, and expresses what belongs to or has the quality of something. Therefore, as stoichos means a row, stoicheion would be what belongs to or constitutes a row....

Hence stoichos presupposes a root stich, and this root would account in Greek for the following derivations:--

stix, gen. stichos, a row, a line of soldiers
stichos, a row, a line; distich, a couplet
steicho, estichon, to march in order, step by step; to mount
stoichos, a row, a file; stoichein, to march in a line

In German, the same root yields steigen, to step, to mount, and in Sanskrit we find stigh, to mount....

Stoicheia are the degrees or steps from one end to the other, the constituent parts of a whole, forming a complete series, whether as hours, or letters, or numbers, or parts of speech, or physical elements, provided always that such elements are held together by a systematic order."

Example:

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The Miracle Octad Generator of R. T. Curtis

For the geometry of these stoicheia, see
The Smallest Perfect Universe and
Finite Geometry of the Square and Cube.

Tuesday, February 27, 2007 9:25 PM

Harvard Design, continued

Suggested by today's
New York Times story
on a Harvard student's
research on pattern in
Islamic art --

and in memory of
George Sadek --

From Log24 in July 2005:

Intersections

A Trinity Sunday sermon
quotes T. S. Eliot:

"... to apprehend
The point of intersection of the timeless
With time, is an occupation for the saint."

See also The Diamond Project.

Related material:

"... an alphabet
By which to spell out holy doom and end,
A bee for the remembering of happiness."

-- Wallace Stevens,
"The Owl in the Sarcophagus"

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Some context for these figures:
The Diamond Theory of Truth

Tuesday, February 27, 2007 7:59 AM

ART WARS: Time and Chance

Continued from 2/06:

The Poetics of Space

Log24 yesterday:

"Imprimatur.
+John Cardinal Farley,
Archbishop of New York"

Tom Hanks as Robert Langdon in The Da Vinci Code

Tom Hanks as Robert Langdon
in "The Da Vinci Code"

"... and by '+' I mean
artistic vision."

New York State Lottery
yesterday, Feb. 26, 2007:

Mid-day 206
Evening 888

For more on the artistic
significance of 206,
see 2/06.

For more on the artistic
significance of 888, see
St. Bonaventure on the
Trinity at math16.com.

A trinity:

Click on picture for further details.

Monday, February 26, 2007 9:29 AM

From the Academy:

Synaxis

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Monica Almeida/The New York Times
Martin Scorsese won the best-director
Oscar last night for "The Departed."
From left, Francis Ford Coppola, Scorsese,
George Lucas and Steven Spielberg.

"Synaxis (synaxis from synago) means gathering, assembly, reunion. It is exactly equivalent to the Latin collecta (from colligere), and corresponds to synagogue (synagoge), the place of reunion."

-- The Catholic Encyclopedia, Volume XIV. Published 1912. New York: Robert Appleton Company. Nihil Obstat, July 1, 1912. Remy Lafort, S.T.D., Censor. Imprimatur. +John Cardinal Farley, Archbishop of New York

Related material:
Yesterday's entries.

Sunday, February 25, 2007 10:31 AM

For Oscar Night

Between Two Worlds

Nicolas Cage as Ghost Rider

"I'm the only one who can
walk in both worlds.
I'm T. S. Eliot."

Four Quartets:

I caught the sudden look of some dead master
Whom I had known, forgotten, half recalled
Both one and many; in the brown baked features
The eyes of a familiar compound ghost
Both intimate and unidentifiable.
So I assumed a double part, and cried
And heard another's voice cry: 'What! are you here?'
Although we were not. I was still the same,
Knowing myself yet being someone other—
And he a face still forming; yet the words sufficed
To compel the recognition they preceded.
And so, compliant to the common wind,
Too strange to each other for misunderstanding,
In concord at this intersection time
Of meeting nowhere, no before and after,
We trod the pavement in a dead patrol.
I said: 'The wonder that I feel is easy,
Yet ease is cause of wonder. Therefore speak:
I may not comprehend, may not remember.'
And he: 'I am not eager to rehearse
My thoughts and theory which you have forgotten.
These things have served their purpose: let them be.
So with your own, and pray they be forgiven
By others, as I pray you to forgive
Both bad and good. Last season's fruit is eaten
And the fullfed beast shall kick the empty pail.
For last year's words belong to last year's language
And next year's words await another voice.
But, as the passage now presents no hindrance
To the spirit unappeased and peregrine
Between two worlds become much like each other,
So I find words I never thought to speak
In streets I never thought I should revisit
When I left my body on a distant shore.
Since our concern was speech, and speech impelled us
To purify the dialect of the tribe
And urge the mind to aftersight and foresight,
Let me disclose the gifts reserved for age
To set a crown upon your lifetime's effort.
First, the cold friction of expiring sense
Without enchantment, offering no promise
But bitter tastelessness of shadow fruit
As body and soul begin to fall asunder.
Second, the conscious impotence of rage
At human folly, and the laceration
Of laughter at what ceases to amuse.
And last, the rending pain of re-enactment
Of all that you have done, and been; the shame
Of motives late revealed, and the awareness
Of things ill done and done to others' harm
Which once you took for exercise of virtue.
Then fools' approval stings, and honour stains.
From wrong to wrong the exasperated spirit
Proceeds, unless restored by that refining fire
Where you must move in measure, like a dancer.'

Sunday, February 25, 2007 10:30 AM

Hollywood Sermon

Devil's Night in Hollywood
Revisited

On the night of October 30-31, 1993, also known as Devil's Night, there was a full Hunter's Moon and the Pennsylvania Lottery number was 666.
-- Steven H. Cullinane, 03/20/01

"Mystery surrounds the death of young actor River Phoenix.... The actor... was declared dead at 1:51 a.m. PT Sunday [Oct. 31, 1993]. Phoenix died about 50 minutes after collapsing in front of the Viper Room, a new club on the Sunset Strip...."
-- Karen Thomas, USA Today, Monday, November 1, 1993

Related material:

The five Log24 entries
ending on Yom Kippur, 2006.

Tuesday, February 20, 2007 10:15 AM

For Mardi Gras:

Anniversary

On this, the second anniversary of Hunter Thompson's death, two Xanga footprints from Texas furnish appropriate links:

Texas /514659186/item.html 2/20/2007 7:47 AM

Texas /534740724/item.html 2/20/2007 9:39 AM.

The first link is to Highway 1 Revisited (8/1/06).

The second link is to Serious (10/3/06).
(See also today's previous entry.)

Related material:

The Crimson Passion: A Drama at Mardi Gras.

Tuesday, February 20, 2007 7:09 AM

Philosophy and...

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.

Dear Dan Brown, All Eyes Are on You

"The much-borrowed Brown formula involves some very specific things. The name of a great artist, artifact or historical figure must be in the book’s story, not to mention on its cover. The narrative must start in the present day with a bizarre killing, then use that killing as a reason to investigate the past. And the past must yield a secret so big, so stunning, so saber-rattling that all of civilization may be changed by it. Probably not for the better.

This formula is neatly summarized...."

Cover illustration
for
The Judas Seat:

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Norton Anthology of
Children's Literature

The Narrative:

Princeton Scholar
and Bible Translator
Dies at 93

The Secret:

Part I

"Little 'Jack' Horner was actually Thomas Horner, steward to the Abbot of Glastonbury during the reign of King Henry VIII.... Always keen to raise fresh funds, Henry had shown a interest in Glastonbury (and other abbeys). Hoping to appease the royal appetite, the nervous Abbot, Richard Whiting, allegedly sent Thomas Horner to the King with a special gift. This was a pie containing the title deeds to twelve manor houses in the hope that these would deflect the King from acquiring Glastonbury Abbey. On his way to London, the not so loyal courier Horner apparently stuck his thumb into the pie and extracted the deeds for Mells Manor, a plum piece of real estate. The attempted bribe failed and the dissolution of the monasteries (including Glastonbury) went ahead from 1536 to 1540. Richard Whiting was subsequently executed, but the Horner family kept the house, so the moral of this one is: treachery and greed pay off, but bribery is a bad idea." --Chris Roberts, Heavy Words Lightly Thrown: The Reason Behind the Rhyme

Part II

"The Grail Table has thirteen seats, one of which is kept vacant in memory of Judas Iscariot who betrayed Christ." --Symbolism of King Arthur's Round Table

"In medieval romance, the grail was said to have been brought to Glastonbury in Britain by Joseph of Arimathea and his followers. In the time of Arthur, the quest for the Grail was the highest spiritual pursuit." --The Camelot Project

Part III

The Log24 entry for the date--
February 13, 2007--
of the above Bible scholar's death,

and the three entries preceding it:

"And what the dead had no speech for, when living, they can tell you, being dead: the communication of the dead is tongued with fire beyond the language of the living."

-- T. S. Eliot, Four Quartets