Log24

Best vs. Bester

The previous post ended with a reference mentioning Rosenhain.

For a recent application of Rosenhain's work, see
Desargues via Rosenhain (April 1, 2013).

From the next day, April 2, 2013:

"The proof of Desargues' theorem of projective geometry
comes as close as a proof can to the Zen ideal.
It can be summarized in two words: 'I see!' "

– Gian-Carlo Rota in Indiscrete Thoughts (1997)

Also in that book, originally from a review in Advances in Mathematics ,
Vol. 84, Number 1, Nov. 1990, p. 136:

See, too, in the Conway-Sloane book, the Galois tesseract … 
and, in this journal, Geometry for Jews and The Deceivers , by Bester.

Desargues' theorem according to a standard textbook:

"If two triangles are perspective from a point
they are perspective from a line."

The converse, from the same book:

"If two triangles are perspective from a line
they are perspective from a point."

Desargues' theorem according to Wikipedia
combines the above statements:

"Two triangles are in perspective axially  [i.e., from a line]
if and only if they are in perspective centrally  [i.e., from a point]."

A figure often used to illustrate the theorem,
the Desargues configuration , has 10 points and 10 lines,
with 3 points on each line and 3 lines on each point.

A discussion of the "if and only if" version of the theorem
in light of Galois geometry requires a larger configuration—
15 points and 20 lines, with 3 points on each line
and 4 lines on each point.

This large  Desargues configuration involves a third triangle,
needed for the proof   (though not the statement ) of the
"if and only if" version of the theorem. Labeled simply
"Desargues' Theorem," the large  configuration is the
frontispiece to Volume I (Foundations)  of Baker's 6-volume
Principles of Geometry .

Point-line incidence in this larger configuration is,
as noted in a post of April 1, 2013, described concisely
by 20 Rosenhain tetrads  (defined in 1905 by
R. W. H. T. Hudson in Kummer's Quartic Surface ).

The third triangle, within the larger configuration,
is pictured below.

Roberta Smith in 2011 on the American Folk Art Museum (see previous post):

"It could be argued that we need a museum of folk art
the way we need a museum of modern art,
to shine a very strong, undiluted light on
a very important achievement."

Some other aesthetic remarks:

"We have had a gutful of fast art and fast food.
What we need more of is slow art: art that holds time
as a vase holds water: art that grows out of modes
of perception and whose skill and doggedness
make you think and feel; art that isn't merely sensational,
that doesn't get its message across in 10 seconds,
that isn't falsely iconic, that hooks onto something
deep-running in our natures. In a word, art that is
the very opposite of mass media. For no spiritually
authentic art can beat mass media at their own game."

— Robert Hughes, speech of June 2, 2004,
     quoted here June 15, 2007.

Perhaps, as well as museums of modern art and of folk art,
we need a Museum of Slow Art. 

One possible exhibit, from this journal Monday:

The diagram on the left is from 1922.  The 20 small squares at right
that each have 4 subsquares darkened were discussed, in a different
context, in 1905. They were re-illustrated, in a new context
(Galois geometry), in 1986. The "key" square, and the combined
illustration, is from April 1, 2013. For deeper background, see
Classical Geometry in Light of Galois Geometry.

Those who prefer faster art may consult Ten Years After.

An update to Rosenhain and Göpel Tetrads in PG(3,2) 
supplies some background from
Notes on Groups and Geometry, 1978-1986,
and from a 2002 AMS Transactions  paper.

For Rosenhain and Göpel

Related material: Quotes from H. F. Baker in posts from March 2011—

A Many-Sided Theory and Remarks on Reality.