Log24

Friday, March 1, 2013

Spinning in Infinity

Filed under: General — Tags: — m759 @ 11:09 PM

(Continued from Jan. 13 and Feb. 19.)

The founder of Graylock Press
died at 96 in Bethesda on Feb. 19:

For some background on the original Bethesda,
see a webpage on Angels in America.

For some background on noted Graylock authors, 
see Pavel Alexandrov.

For deeper background, see a book praised by Freeman Dyson:

"The intellectual drama will attract readers
who are interested in mystical religion
and the foundations of mathematics.
The personal drama will attract readers
who are interested in a human tragedy
with characters who met their fates with
exceptional courage."

Tuesday, February 19, 2013

Configurations

Filed under: General,Geometry — Tags: — m759 @ 12:24 PM

Yesterday's post Permanence dealt with the cube
as a symmetric model of the finite projective plane
PG(2,3), which has 13 points and 13 lines. The points
and lines of the finite geometry occur in the cube as
the 13 axes of symmetry and the 13 planes through
the center perpendicular to those axes. If the three
axes lying in  a plane that cuts the cube in a hexagon
are supplemented by the axis perpendicular  to that
plane, each plane is associated with four axes and,
dually, each axis is associated with four planes.

My web page on this topic, Cubist Geometries, was
written on February 27, 2010, and first saved to the
Internet Archive on Oct. 4, 2010

For a more recent treatment of this topic that makes
exactly the same points as the 2010 page, see p. 218
of Configurations from a Graphical Viewpoint , by
Toma┼ż Pisanski and Brigitte Servatius, published by
Springer on Sept. 23, 2012 (date from both Google
Books
and Amazon.com):

For a similar 1998 treatment of the topic, see Burkard Polster's 
A Geometrical Picture Book  (Springer, 1998), pp. 103-104.

The Pisanski-Servatius book reinforces my argument of Jan. 13, 2013,
that the 13 planes through the cube's center that are perpendicular
to the 13 axes of symmetry of the cube should be called the cube's 
symmetry planes , contradicting the usual use of of that term.

That argument concerns the interplay  between Euclidean and
Galois geometry. Pisanski and Servatius (and, in 1998, Polster)
emphasize the Euclidean square and cube as guides* to
describing the structure of a Galois space. My Jan. 13 argument
uses Galois  structures as a guide to re-describing those of Euclid .
(For a similar strategy at a much more sophisticated level,
see a recent Harvard Math Table.)

Related material:  Remarks on configurations in this journal
during the month that saw publication of the Pisanski-Servatius book.

* Earlier guides: the diamond theorem (1978), similar theorems for
  2x2x2 (1984) and 4x4x4 cubes (1983), and Visualizing GL(2,p)
  (1985). See also Spaces as Hypercubes (2012).

Sunday, January 13, 2013

Spinning in Infinity

Filed under: General,Geometry — Tags: — m759 @ 1:00 PM

A note for day 13 of 2013

How the cube's 13 symmetry planes* 
are related to the finite projective plane
of order 3, with 13 points and 13 lines—

IMAGE- How the cube's symmetry planes are related to the finite projective plane of order 3, with 13 points and 13 lines

For some background, see Cubist Geometries.

* This is not the standard terminology. Most sources count
   only the 9 planes fixed pointwise under reflections  as
   "symmetry planes." This of course obscures the connection
   with finite geometry.

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