Annals of Geometry:

The Bottom Line

From the geometry page
at cut-the-knot.org:



Related material:
this date three years ago

Sunday, March 30, 2008  3:14 PM

Philosophy Wars continued:

Friday, March 28, 2008  2:14 PM

A Yahrzeit for Virginia Woolf:

Friday, March 28, 2008  2:02 AM

Mr. Holland's Week continues...

Judgment at Hollywood

"Judgment at Nuremberg" screenwriter
Abby Mann in this morning's
New York Times obituaries:



Click image to enlarge.

Related material:

The Log24 entries of
March 15 - 25.

(Tuesday, March 25, was the
date of Mann's death.)

"Philosophers ponder the idea
of identity: what it is to give
something a name on Monday
and have it respond to
  that name on Friday...."

-- Bernard Holland in
   The New York Times

See also the five Log24
entries ending on Feb. 25.

Thursday, March 27, 2008  3:29 PM

ART WARS continued:

H.S.M. Coxeter's classic
Introduction to Geometry (2nd ed.):

Thursday, March 27, 2008  2:45 AM

STAR WARS continued:

Click image to enlarge.

The "Boy's Life" illustration is of an Arthur C. Clarke story, "Against the Fall of Night." This, according to the review quoted below, was Clarke's first story, begun in 1936 and first published in 1948. The title is from a poem by A. E. Housman, "Smooth Between Sea and Land." See Log24 on the Feast of St. Mark, 2003.

From a book review by Christopher B. Jones:

"Against the Fall of Night describes well how it often takes youth to bring forth change. The older mind becomes locked in a routine, or blocks out things because it has been told that it shouldn't think or talk about them. But the young mind is ever the explorer, seeking out knowledge without the taboos placed on it by a rigid society. Alvin is a breath of fresh air in the don't-look-over-the-wall society of Diaspar.

Myths play a big role, and an interesting religious overtone pervades the story with a long since departed being whose origins are unknown and who played an important part in Earth's past. Parallels to Jesus can easily be drawn, and the forecast shown for the longevity of religions in general seems to me to be rather accurate....

Finally, when Alvin uncovers part of the truth he has been looking for, he learns of the dangers and stagnation that can befall a xenophobic society. There are still a few such societies in the world today, and this characteristic almost always comes with negative effects-- even if it has been cultivated with the intention to protect."

An example of such a xenophobic society is furnished by the Hadassah ad currently running in the New York Times obituaries section: "Who will say Kaddish in Israel?"

Another example:

Tom Stoppard, in the London Times of Sunday, March 16, 2008, on the social unrest of forty years ago in 1968--

"Altering the psyche was supposed to change the social structure but, as a Marxist, Max knows it really works the other way: changing the social structure is the only way to change the psyche. The idea that 'make love, not war' is a more practical slogan than 'workers of the world unite' is as airy-fairy as the I Ching."

Airy-fairy, Jewey-phooey.

Clarke's 1948 story was the basis of his 1956 novel, The City and the Stars. In memory of the star Richard Widmark, here are two illustrations from St. Mark's Day, 2003:

Housman asks the reader
to tell him of runes to grave
or bastions to design
"against the fall of night."

Here, as examples, are
one rune and one bastion.

Tuesday, March 25, 2008  9:00 AM

A Midrash for Julie:

"... they were performing for God. Now God can mean whatever you want it to mean. But for me, I understood it so totally. The detail....

They did it from the inside to the outside. And from the outside to the in. And that profoundly moved me then. It was... it was the most important thing that I ever experienced."

-- Julie Taymor,
"Skewed Mirrors" interview

Here is some further commentary on the words of that entry--

On the phrase "Within You Without You"-- the title of a song by George Harrison:

"Bernard’s understanding of reality connects to this idea of 'flow': he sees reality as a product of consciousness. He rejects the idea of an 'outer' world of unchanging objects and an 'inner' world of the mind and ideas. Rather, our minds are part of the world, and vice versa."

-- Adrien Ardoin, SparkNote on
    Virginia Woolf's The Waves

On "Death and the Apple Tree"-- the title of the previous entry-- in The Waves:

"The apple tree Neville is looking at as he overhears the servants at the school discussing a local murder becomes inextricably linked to his knowledge of death. Neville finds himself unable to pass the tree, seeing it as glimmering and lovely, yet sinister and 'implacable.' When he learns that Percival is dead, he feels he is face to face once again with 'the tree which I cannot pass.' Eventually, Neville turns away from the natural world to art, which exists outside of time and can therefore transcend death. The fruit of the tree appears only in Neville’s room on his embroidered curtain, a symbol itself of nature turned into artifice. The apple tree image also echoes the apple tree from the Book of Genesis in the Bible, the fruit of which led Adam and Eve to knowledge and, therefore, expulsion from Eden."

-- Adrien Ardoin, op. cit.

Monday, March 24, 2008  3:00 PM

For the gardener's wife:

Sunday, March 23, 2008  11:00 AM

ART WARS continued:

A sequel to
The Crimson Passion

Easter Egg


Click on image
for further details.

Sunday, March 16, 2008  1:20 AM

A Play for Kristen:

Real Numbers

(continued from
March 7, 2008)



A search for the evening
 number, 332, in Log24
yields a rather famous
 line from Sophocles...

Sophocles, Antigone,
edited by Mark Griffith,
Cambridge University Press,
1999:

"Many things are formidable (deina) and none is more formidable (deinoteron) than man."

-- Antigone, lines 332-333, in Valdis Leinieks, The Plays of Sophokles, John Benjamins Publishing Company, 1982, p. 62

Related material:

The Log24 Pi Day 
mantra from 
 Roger Zelazny --
"center loosens,  
forms again elsewhere."

Saturday, March 15, 2008  1:00 PM

Happy Ides of March

Friday, March 14, 2008  12:00 AM

ART WARS continued:

Rite of Spring

From the online 
Harvard Crimson --



Related material:

A figure from  
Monday's entry --



-- and  
June 30, 2007's
Annals of Theology,
with a link to a film:
The Center of the World.

The center referred
to in that film is the
same generic "center"
displayed at Harvard
and in the above
mandorla: not the
Harvard Women's
Center, but rather
the women's center.

See also Yeats --
"the centre cannot hold,"

Stevens --
"the center of resemblance,"

and Zelazny --
"center loosens,
forms again elsewhere."

Monday, March 10, 2008  9:00 PM

Happy Birthday, Sharon Stone

Mani Padme
(Jewel in the Lotus):
 
Part I

"Raiders of the Lost Stone"
(March 10, 2006)

Part II

"Raiders of the Lost..."
(Feb. 17, 2006)

Part III

The Further
Adventures of
Tony Rome
(March 7, 2008)

Parts I and II above
may be summarized by
the famous phrase
"jewel in the lotus"--
which, some say, has
a sexual meaning--
and by the diagram



For discussions
of this structure
in Western thought,
see
the ovato tondo
and
Last to the Lost.

Sunday, March 9, 2008  1:00 AM

Religious Symbolism:

Identity



Click for context.

Related material:

Is Nothing Sacred?

Pictures of Nothing

Art Wars: Epiphany

Tilting at Whirligigs

Down the Up Staircase

Saturday, March 8, 2008  1:00 PM

Imaginary Numbers:

of literary "signature passages" --



An answer:

"The whirligig of time"
-- Shakespeare, Twelfth Night

and

Log24, Twelfth Night, 2006:

Hamilton's Whirligigs



Click image to enlarge.

Related material:

Rotation in the complex plane.

The plane was discovered
in the late 1700's by Wessel:

Caspar Wessel by J.J. O'Connor  and E.F. Robertson: "Wessel's paper [in Danish] was not noticed by the mathematical community until 1895... A French translation... was published in 1897 but an English translation of this most remarkable work was not published until 1999 (exactly 200 years after it was first published).... We have called Wessel's work remarkable, and indeed although the credit has gone to Argand, many historians of mathematics feel that Wessel's contribution was [1]:- ... superior to and more modern in spirit to Argand's. In the [1] article the approaches by Argand and Wessel are compared and contrasted. Of course Wessel was a surveyor and his paper was motivated by his surveying and cartography work:- Wessel's development proceeded rather directly from geometric problems, through geometric-intuitive reasoning, to an algebraic formula. Argand began with algebraic quantities and sought a geometric representation for them. ... Wessel's initial formulation was remarkably clear, direct, concise and modern. It is regrettable that it was not appreciated for nearly a century and hence did not have the influence it merited. However more is claimed for Wessel's single mathematical paper than the first geometric interpretation of complex numbers. In [3] Crowe credits Wessel with being the first person to add vectors. Again this shows the depth of Wessel's thinking but again, as the paper was unnoticed it had no influence on mathematical development despite appearing in the Memoirs of the Royal Danish Academy which by any standard was a major source of publications.... 1. ... Biography in Dictionary of Scientific Biography (New York 1970-1990). 3. M.J. Crowe, A History of Vector Analysis (Notre Dame, 1967)."

Friday, March 7, 2008  8:28 PM

Real Numbers, continued:

"We keep coming back
 and coming back
 To the real: to the hotel
            instead of the hymns...."

    -- Wallace Stevens,  
    "An Ordinary Evening
   in New Haven"

Friday, March 7, 2008  12:12 PM

Can the Center Hold?

Setter of Riddles

"Philosophers ponder
the idea of identity:
what it is to give
something a name
on Monday
and have it respond
to that name
   on Friday...."

-- Bernard Holland in
   The New York Times

2:45 AM Monday:



Related material:

A Kaddish for Raymond

Obituary of Paul Raymond
in today's New York Times:

"He left school at 15 and had his first experience of show business as a mind reader and clairvoyant. He developed his interest in the commercial exploitation of sex when the manager of a theater where he performed said he would hire him only if his two female assistants appeared seminaked on the stage."

Scarlett Johansson
and Natalie Portman

See also
Scarlett Johansson in Vanity Fair,
Natalie Portman in Hotel Chevalier.

Ad for Hadassah
on Monday, the day
that Raymond's death
was announced:

"Who will say Kaddish?"



Portman, of course.

Thursday, March 6, 2008  12:00 PM

Annals of Geometry:

This note is prompted by the March 4 death of Richard D. Anderson, writer on geometry, President (1981-82) of the Mathematical Association of America (MAA), and member of the MAA's Icosahedron Society.

Royal Road

"The historical road
from the Platonic solids
to the finite simple groups
is well known."

-- Steven H. Cullinane,
November 2000,
Symmetry from Plato to
the Four-Color Conjecture

Euclid is said to have remarked that "there is no royal road to geometry." The road to the end of the four-color conjecture may, however, be viewed as a royal road from geometry to the wasteland of mathematical recreations.* (See, for instance, Ch. VIII, "Map-Colouring Problems," in Mathematical Recreations and Essays, by W. W. Rouse Ball and H. S. M. Coxeter.) That road ended in 1976 at the AMS-MAA summer meeting in Toronto-- home of H. S. M. Coxeter, a.k.a. "the king of geometry."

Picture from a 1980 article
in a journal of
The Mathematical Association
of America:

See also Log24, May 21, 2007.

A different road-- from Plato to the finite simple groups-- is, as I noted in November 2000, well known. But new roadside attractions continue to appear. One such attraction is the role played by a Platonic solid-- the icosahedron-- in design theory, coding theory, and the construction of the sporadic simple group M₂₄.

"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."

-- "Block Designs," by Andries E. Brouwer (Ch. 14 (pp. 693-746) of Handbook of Combinatorics, Vol. I, MIT Press, 1995, edited by Ronald L. Graham, Martin Grötschel, and László Lovász, Section 16 (p. 716))

This Steiner system is closely connected to M₂₄ and to the extended binary Golay code. Brouwer gives an elegant construction of that code (and therefore of  M₂₄):

"Let N be the adjacency matrix of the icosahedron (points: 12 vertices, adjacent: joined by an edge). Then the rows of the 12×24 matrix (I  J-N) generate the extended binary Golay code." [Here I is the identity matrix and J is the matrix of all 1's.]

-- Op. cit., p. 719

Related material:

Finite Geometry of
the Square and Cube
and
Jewel in the Crown

"There is a pleasantly discursive
treatment of Pontius Pilate's
unanswered question
'What is truth?'"
-- H. S. M. Coxeter, 1987,
introduction to Trudeau's
"story theory" of truth

Those who prefer stories to truth
may consult the Log24 entries
 of March 1, 2, 3, 4, and 5.

They may also consult
the poet Rubén Darío:

... Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte.

* For a road out of this wasteland, back to geometry, see The Kaleidoscope Puzzle and Reflection Groups in Finite Geometry.

Wednesday, March 5, 2008  1:09 PM

Vocabulary Lesson--

(Context: March 2-4)

For CENTRAL
Central Intelligence:

"God does not play dice."
-- Paraphrase of a remark
by Albert Einstein

Another Nobel Prize winner,
Isaac Bashevis Singer--

"a God who speaks in deeds,
not in words, and whose
vocabulary is the Cosmos"

From "The Escapist:
The Reality of Fantasy Games"--

Dungeons & Dragons Dice

From today's New York Times:

A Kaddish for Gygax:

Rebecca Goldstein
   on first encountering Plato--

"I was reading Durant's section on Plato, struggling to understand his theory of the ideal Forms that lay in inviolable perfection out beyond the phantasmagoria. (That was the first, and I think the last, time that I encountered that word.)"

Related material:

For more on the word
"phantasmagoria," see
Log24 on Dec. 12, 2004
and on Sept. 23, 2006.

For phantasmagoria in action,
see Dungeons & Dragons
and Singer's (and others')
Jewish fiction.

For non-phantasmagoria,
see (for instance) the Elements
of Euclid, which culminates
in the construction of the
Platonic solids illustrated above.

See also Geometry for Jews.

Tuesday, March 4, 2008  1:00 PM

ART WARS continued:

... And for a
    Swiftly Tilting
       Shadowed Planet ...


John Trever, Albuquerque Journal, 2/29/08

The pen's point:

Log24, Dec. 11, 2006 SINGER, ISAAC: "Are Children the Ultimate Literary Critics?" -- Top of the News 29 (Nov. 1972): 32-36. "Sets forth his own aims in writing for children and laments 'slice of life' and chaos in children's literature. Maintains that children like good plots, logic, and clarity, and that they have a concern for 'so-called eternal questions.'" -- An Annotated Listing of Criticism by Linnea Hendrickson "She returned the smile, then looked across the room to her youngest brother, Charles Wallace, and to their father, who were deep in concentration, bent over the model they were building of a tesseract: the square squared, and squared again: a construction of the dimension of time." -- A Swiftly Tilting Planet, by Madeleine L'Engle For "the dimension of time," see A Fold in Time, Time Fold, and Diamond Theory in 1937.  A Swiftly Tilting Planet is a fantasy for children set partly in Vespugia, a fictional country bordered by Chile and Argentina.

Monday, March 3, 2008  11:09 AM

For a Shadowed Planet:

CENTRAL Central Intelligence
 
A Wrinkle in Time:

"Holding hands, they crossed the square. The huge CENTRAL Central Intelligence Building had only one door, but it was an enormous one, at least two stories high and wider than a room, made of a dull, bronzelike material.

'Do we just knock?' Meg giggled."

Associated Press story
by Tom Raum:

Detail:



"Do we just knock?"

"Click PLAY."
_______________________

Related material:

Der Einsatz

Monday, March 3, 2008  2:45 AM

Quarter to Three, continued:

So Set 'Em Up, Joe...

Sunday, March 2, 2008  10:31 PM

All Hallows' Eve continued:

Practical Magic

Halloween 2005:

"They don't understand
what it is to be awake,
To be living
on several planes at once
Though one cannot speak
with several voices at once."

-- T. S. Eliot,
The Family Reunion

Several voices:

Margaret Wertheim in today's
Los Angeles Times and at
The Pearly Gates of Cyberspace,

Linda Dalrymple Henderson, and

Madeleine L'Engle and husband.

From Wertheim's Pearly Gates:


"There is such a thing
as a tesseract."

-- Madeleine L'Engle   

Sunday, March 2, 2008  11:00 AM

Women's History* continued:

The Well Wrought
Closure
 


Click on image
for details.

Jonathan D. Culler
in Deconstruction:


 


"The alternative to 'crystalline closure'
is not, then, an endless and chaotic
'repetition and proliferation,' but a
*structured relationship of significance."

-- The Old New Criticism and Its Critics,
by R. V. Young, Professor of English
at North Carolina State University

Saturday, March 1, 2008  11:02 AM

Women's History Month

1997’s surprise best-seller, Cold Mountain, is the first novel of North Carolina native and travel writer, Charles Frazier. Two ancient Greek authors shape and drive the novel, set in the post-war Southern Appalachians of 1865. Homer's Odyssey frames the novel: the hero Inman undergoes epic adventures after the war, has his own Penelope waiting, and travels back to a land as remote as any island, Cold Mountain, North Carolina. But fragments of an anonymous philosopher who can be identified as Heraclitus alienate Inman from the Homeric world around him and determine his fate. Ada, his Penelope, also casts off her shroud of tradition: impatient with the 'glorious war,' no longer content to wait, Ada plunges into the new business of living. And just as the archaic, post-Homeric Greek world produced new ways of living and thought, as exemplified by Heraclitus, so too does the post-bellum world of Cold Mountain, as exemplified by Inman and Ada; their struggle, and the novel's tension, speak to and about all those caught between two worlds, epic and philosophic, whether driven by love or strife."