Thursday, November 16, 2017

Lost Horizon

Filed under: Uncategorized — m759 @ 11:29 AM

Related material —

The following image in this journal


Wednesday, November 1, 2017

Cameron on All Saints’ Day

Filed under: Uncategorized — m759 @ 2:01 PM

"Nowdays, Halloween involves plastic figures of ghosts and bats
bought from the supermarket; this is driven by commerce and
in some people’s view is an American import. But it is clear that
this time of year was traditionally regarded as one where the barrier
between this world and the other was low, and supernatural
manifestations were to be expected."

Peter J. Cameron today.

Remarks related to another "barrier" and vértigo horizontal

See also a search for  Horizon + "Western Australia"  in this  journal.

From that search:  A sort of horizon, a "line at infinity," that is perhaps
more meaningful to most Cameron readers than the above remarks
by Borges —

Sunday, January 1, 2017

Like the Horizon

Filed under: Uncategorized — Tags: , — m759 @ 1:00 PM

(Continued from a remark by art critic Peter Schjeldahl quoted here
last  year on New Year's Day in the post "Art as Religion.")

"The unhurried curve got me. 
It was like the horizon of a world
that made a non-world of
all of the space outside it."

— Peter Schjeldahl, "Postscript: Ellsworth Kelly,"
The New Yorker , December 30, 2015

This suggests some further material from the paper 
that was quoted here yesterday on New Year's Eve —

"In teaching a course on combinatorics I have found
students doubting the existence of a finite projective
plane geometry with thirteen points on the grounds
that they could not draw it (with 'straight' lines)
on paper although they had tried to do so. Such a
lack of appreciation of the spirit of the subject is but
a consequence of the elements of formal geometry
no longer being taught in undergraduate courses.
Yet these students were demanding the best proof of
existence, namely, production of the object described."

— Derrick Breach (See his obituary from 1996.)

A related illustration of the 13-point projective plane 
from the University of Western Australia:

Projective plane of order 3

(The four points on the curve
at the right of the image are
the points on the line at infinity .)

The above image is from a post of August 7, 2012,
"The Space of Horizons."  A related image — 

Click on the above image for further remarks.

Monday, October 6, 2014

Arcs and Shards

Filed under: Uncategorized — m759 @ 10:21 AM

Ben Brantley in The New York Times  today on a Broadway opening:

“As Christopher navigates his way through an increasingly
unfamiliar landscape, both physical and emotional, the arcs
of his adventures are drawn into being.

So are the shards of sensory overload.”

Arc — See a search for Line at Infinity:

Shard — See Shard and Pythagorean Selfie:

Tuesday, August 7, 2012

The Space of Horizons

Filed under: Uncategorized — m759 @ 4:00 AM

“In the space of horizons that neither love nor hate”
— Wallace Stevens, “Things of August”

Seven years ago yesterday—

IMAGE- 3x3 grid related to Borofsky's 'Four Gods'

For some context, see Rosetta Stone as a Metaphor.

Related material from the University of Western Australia

Projective plane of order 3

(The four points on the curve
at the right of the image are
the points on the line at infinity.)

Art critic Robert Hughes,  who nearly died in Western
Australia in a 1999 car crash, actually met his death
yesterday at Calvary Hospital in the Bronx.

See also Hughes on “slow art” in this journal.

Wednesday, August 1, 2012

Elementary Finite Geometry

Filed under: Uncategorized — Tags: — m759 @ 7:16 PM

I. General finite geometry (without coordinates):

A finite affine plane of order has n^2 points.

A finite projective plane of order n  has n^2 + n + 1 

points because it is formed from an order-n finite affine 

plane by adding a line at infinity  that contains n + 1 points.


Affine plane of order 3

Projective plane of order 3

II. Galois finite geometry (with coordinates over a Galois field):

A finite projective Galois plane of order n has n^2 + n + 1

points because it is formed from a finite affine Galois 3-space

of order n with n^3 points by discarding the point (0,0,0) and 

identifying the points whose coordinates are multiples of the

(n-1) nonzero scalars.

Note: The resulting Galois plane of order n has 

(n^3-1)/(n-1)= (n^2 + n + 1) points because 

(n^2 + n + 1)(n – 1) =

(n^3 + n^2 + n – n^2 – n – 1) = (n^3 – 1) .

III. Related art:

Another version of a 1994 picture that accompanied a New Yorker
article, "Atheists with Attitude," in the issue dated May 21, 2007:

IMAGE- 'Four Gods,' by Jonathan Borofsky

The Four Gods  of Borofsky correspond to the four axes of 
  of a square and to the four points on a line at infinity 
in an order-3 projective plane as described in Part I above.

Those who prefer literature to mathematics may, if they like,
view the Borofsky work as depicting

"Blake's Four Zoas, which represent four aspects
of the Almighty God" —Wikipedia

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