Log24

Wednesday, October 19, 2016

The Crosswicks Curse Continues

Filed under: General — Tags: — m759 @ 11:29 AM

"There is  such a thing as 1906 "

Thursday, September 15, 2016

The Smallest Perfect Number/Universe

Filed under: General,Geometry — Tags: , , — m759 @ 6:29 AM

The smallest perfect number,* six, meets
"the smallest perfect universe,"** PG(3,2).

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

  * For the definition of "perfect number," see any introductory
    number-theory text that deals with the history of the subject.
** The phrase "smallest perfect universe" as a name for PG(3,2),
     the projective 3-space over the 2-element Galois field GF(2),
     was coined by math writer Burkard Polster. Cullinane's square
     model of PG(3,2) differs from the earlier tetrahedral model
     discussed by Polster.

Wednesday, May 21, 2014

The Tetrahedral Model of PG(3,2)

Filed under: General,Geometry — Tags: , — m759 @ 10:15 PM

The page of Whitehead linked to this morning
suggests a review of Polster's tetrahedral model
of the finite projective 3-space PG(3,2) over the
two-element Galois field GF(2).

The above passage from Whitehead's 1906 book suggests
that the tetrahedral model may be older than Polster thinks.

Shown at right below is a correspondence between Whitehead's
version of the tetrahedral model and my own square  model,
based on the 4×4 array I call the Galois tesseract  (at left below).

(Click to enlarge.)

Friday, June 8, 2012

Cartoon Graveyard

Filed under: General — m759 @ 2:01 AM

Whitehead and Russell, 'Logicomix' page 181

For some background, see "Cartoon Graveyard" and "Many Dimensions."

Wednesday, April 20, 2011

Romancing the Cube

Filed under: General,Geometry — m759 @ 1:00 AM

It was a dark and stormy night…

http://www.log24.com/log/pix11/110420-DarkAndStormy-Logicomix.jpg

— Page 180, Logicomix

“… the class of reflections is larger in some sense over an arbitrary field than over a characteristic zero field.”

– Julia Hartmann and Anne V. Shepler, “Jacobians of Reflection Groups

For some context, see the small cube in “A Simple Reflection Group of Order 168.”

See also the larger cube in Many Dimensions” + Whitehead in this journal (scroll down to get past the current post).

That search refers to a work by Whitehead published in 1906, the year at the top of the Logicomix  page above—

http://www.log24.com/log/pix11/110420-Whitehead1906Axioms.jpg

A related remark on axiomatics that has metaphysical overtones suitable for a dark and stormy night

“An adequate understanding of mathematical identity requires a missing theory that will account for the relationships between formal systems that describe the same items. At present, such relationships can at best be heuristically described in terms that invoke some notion of an ‘intelligent user standing outside the system.'”

— Gian-Carlo Rota, “Syntax, Semantics, and…” in Indiscrete Thoughts . See also the original 1988 article.

Tuesday, November 23, 2010

Tale

Filed under: General — Tags: , — m759 @ 8:00 PM

A reviewer says Steve Martin finds in his new novel An Object of Beauty  "a sardonic morality tale."

From this journal on the day The Cube  was published (see today's Art Object ) —

Monday February 20, 2006

m759 @ 12:00 AM

The Past Revisited

From Log24 a year ago on this date, a quote from Many Dimensions  (1931), by Charles Williams:

“Lord Arglay had a suspicion that the Stone would be purely logical.  Yes, he thought, but what, in that sense, were the rules of its pure logic?”

For the rest of the story, see the downloadable version at Project Gutenberg of Australia.

See also a post on Mathematics and Narrative from Nov. 14, 2009.

That post compares characters in Many Dimensions  to those in Logicomix

Whitehead and Russell, 'Logicomix' page 181

Saturday, November 14, 2009

Mathematics and Narrative, continued:

Filed under: General,Geometry — Tags: — m759 @ 10:10 PM

A graphic novel reviewed in the current Washington Post  features Alfred North Whitehead and Bertrand Russell–

Whitehead and Russell, 'Logicomix' page 181

Related material:

Whitehead on Fano’s finite projective three-space:

“This is proved by the consideration of a three dimensional geometry in which there are only fifteen points.”

The Axioms of Projective Geometry , Cambridge University Press, 1906

A related affine six-space:

Grey cube, 4x4x4

Further reading:

See Solomon’s Cube and the link at the end of today’s previous entry, then compare and contrast the above portraits of Whitehead and Russell with Charles Williams’s portraits of Sir Giles Tumulty and Lord Arglay in the novel Many Dimensions .

It was a dark and stormy night….

Powered by WordPress