Log24

Thursday, November 16, 2017

A Line at Infinity

Filed under: Geometry — m759 @ 12:00 PM

Saturday, January 6, 2018

Yale News

Filed under: Uncategorized — m759 @ 5:24 AM

The Yale of the title is not the university, but rather the
mathematician Paul B. Yale. Yale's illustration of the Fano
plane is below.

IMAGE- Triangular models of the 4-point affine plane A and 7-point projective plane PA

A different illustration from a mathematician named Greenberg —

This illustration of the ominous phrase "line at infinity"
may serve as a sort of Deathly Hallows  for Greenberg.
According to the AMS website yesterday, he died on
December 12, 2017:

A search of this  journal for Greenberg yields no mention of
the dead mathematician, but does yield some remarks
on art that are pehaps less bleak than the above illustration.

For instance —

Art adapted from the Google search screen. Discuss.

Sunday, December 31, 2017

Ich, Du, etc., etc.

Filed under: Uncategorized — m759 @ 9:00 AM

Recent posts involving the English pronoun IT referred to
classic tales of horror by Madeleine L'Engle and Stephen King.

Those posts suggest some further remarks by Martin Buber

THE WORLD IS TWOFOLD for man
     in accordance with his twofold attitude.
The attitude of man is twofold
     in accordance with the two basic words he can speak.
The basic words are not single words but word pairs.
One basic word is the word pair I-You.
The other basic word is the word pair I-It;
     but this basic word is not changed when
     He or She takes the place of It.
Thus the I of man is also twofold.
For the I of the basic word I-You is different from
     that in the basic word I-It.

— Buber, Martin. I and Thou, Trans. Kaufmann
     (p. 53). Kindle Edition. 

Four German pronouns from the above passage
by Martin Buber lead to six pronoun pairs:

ich-du, ich-es, ich-sie, du-es, du-sie, es-sie.

This is in accordance with some 1974 remarks by
Marie-Louise von Franz

The following passage by Buber may confuse readers of
L'Engle and King with its use, in translation, of "it" instead of
the original German "sie" ("she," corresponding to "die Welt") —

Here, for comparison, are the original German and the translation.

As for "that you in which the lines of relation, though parallel,
intersect," and "intimations of eternity," see Log24 posts on
the concept "line at infinity" as well as "Lost Horizon."

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 —

Wednesday, February 22, 2017

Horizon

Filed under: Uncategorized — m759 @ 11:00 PM

A memorable phrase —

"the transcendental horizon of the ‘I’."

For some backstory, see a Google search for

Marion + transcendental + horizon.

For a perhaps more intelligible horizon, see

Line at Infinity  in this journal.

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.

Friday, February 26, 2016

Literacy Test

Filed under: Uncategorized — Tags: , — m759 @ 3:13 PM
 
כֵּֽאלֹהִ֔ים

Being Interpreted:

9 + 4 = 13.

Midrash

Line at Infinity .

Wednesday, May 13, 2015

Space

Filed under: Uncategorized — Tags: — m759 @ 2:00 PM

Notes on space for day 13 of May, 2015 —

The 13 symmetry axes of the cube may be viewed as
the 13 points of the Galois projective space PG(2,3).
This space (a plane) may also be viewed as the nine points
of the Galois affine space AG(2,3) plus the four points on
an added "line at infinity."

Related poetic material:

The ninefold square and Apollo, as well as 

http://www.log24.com/log/pix11A/110426-ApolloAndDionysus.jpg

Friday, January 16, 2015

A versus PA

Filed under: Uncategorized — Tags: — m759 @ 8:48 PM

"Reality is the beginning not the end,
Naked Alpha, not the hierophant Omega,
of dense investiture, with luminous vassals."

— “An Ordinary Evening in New Haven” VI

From the series of posts tagged "Defining Form" —

The 4-point affine plane A  and
the 7-point projective plane PA  —

IMAGE- Triangular models of the 4-point affine plane A and 7-point projective plane PA

The circle-in-triangle of Yale's Figure 30b (PA ) may,
if one likes, be seen as having an occult meaning.

For the mathematical  meaning of the circle in PA
see a search for "line at infinity."

A different, cubic, model of PA  is perhaps more perspicuous.

Monday, December 29, 2014

Dodecahedron Model of PG(2,5)

Filed under: Uncategorized — Tags: , — m759 @ 2:28 PM

Recent posts tagged Sagan Dodecahedron 
mention an association between that Platonic
solid and the 5×5 grid. That grid, when extended
by the six points on a "line at infinity," yields
the 31 points of the finite projective plane of
order five.  

For details of how the dodecahedron serves as
a model of this projective plane (PG(2,5)), see
Polster's A Geometrical Picture Book , p. 120:

For associations of the grid with magic rather than
with Plato, see a search for 5×5 in this journal.

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.

Examples—

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 
symmetry
  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

Saturday, February 27, 2010

Cubist Geometries

Filed under: Uncategorized — Tags: , — m759 @ 2:01 PM

"The cube has…13 axes of symmetry:
  6 C2 (axes joining midpoints of opposite edges),
4 C3 (space diagonals), and
3C4 (axes joining opposite face centroids)."
–Wolfram MathWorld article on the cube

These 13 symmetry axes can be used to illustrate the interplay between Euclidean and Galois geometry in a cubic model of the 13-point Galois plane.

The geometer's 3×3×3 cube–
27 separate subcubes unconnected
by any Rubik-like mechanism–

The 3x3x3 geometer's cube, with coordinates

The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
  subcubes in the (Galois) 3×3×3 cube–

The 13 symmetry axes of the cube

A closely related structure–
the finite projective plane
with 13 points and 13 lines–

Oxley's 2004 drawing of the 13-point projective plane

A later version of the 13-point plane
by Ed Pegg Jr.–

Ed Pegg Jr.'s 2007 drawing of the 13-point projective plane

A group action on the 3×3×3 cube
as illustrated by a Wolfram program
by Ed Pegg Jr. (undated, but closely
related to a March 26, 1985 note
by Steven H. Cullinane)–

Ed Pegg Jr.'s program at Wolfram demonstrating concepts of a 1985 note by Cullinane

The above images tell a story of sorts.
The moral of the story–

Galois projective geometries can be viewed
in the context of the larger affine geometries
from which they are derived.

The standard definition of points in a Galois projective plane is that they are lines through the (arbitrarily chosen) origin in a corresponding affine 3-space converted to a vector 3-space.

If we choose the origin as the center cube in coordinatizing the 3×3×3 cube (See Weyl's relativity problem ), then the cube's 13 axes of symmetry can, if the other 26 cubes have properly (Weyl's "objectively") chosen coordinates, illustrate nicely the 13 projective points derived from the 27 affine points in the cube model.

The 13 lines of the resulting Galois projective plane may be derived from Euclidean planes  through the cube's center point that are perpendicular to the cube's 13 Euclidean symmetry axes.

The above standard definition of points in a Galois projective plane may of course also be used in a simpler structure– the eightfold cube.

(The eightfold cube also allows a less standard way to picture projective points that is related to the symmetries of "diamond" patterns formed by group actions on graphic designs.)

See also Ed Pegg Jr. on finite geometry on May 30, 2006
at the Mathematical Association of America.

Thursday, August 28, 2008

Thursday August 28, 2008

Filed under: Uncategorized — m759 @ 5:24 AM
Associations
for the writer
known as UD

 

"Have liberty not as
     the air within a grave
Or down a well. Breathe freedom,
     oh, my native,
In the space of horizons
     that neither love nor hate."

— Wallace Stevens,
   "Things of August"

Remarks on physics, with apparently unrelated cartoon, New Yorker, Oct. 2, 2006

A related visual  
association of ideas —

("The association is the idea"
— Ian Lee, The Third Word War)

From UD Jewelry:

For  fishing enthusiasts: hook pendant from UD Jewelry

by John Braheny

"Hook" is the term you'll hear most often in the business and craft of commercial songwriting. (Well, maybe not as much as "Sorry, we can't use your song," but it's possible that the more you hear about hooks now, the less you'll hear "we can't use it" later.)

The hook has been described as "the part(s) you remember after the song is over," "the part that reaches out and grabs you," "the part you can't stop singing (even when you hate it)" and "the catchy repeated chorus…."

See also UD's recent
A Must-Read and In My Day*
as well as the five
Log24 entries ending
Sept. 20, 2002.

More seriously:
 
The date of The New Yorker issue quoted above is also the anniversary of the birth of Wallace Stevens and the date of death of mathematician Paul R. Halmos.
 
Stevens's "space of horizons" may, if one likes, be interpreted as a reference to projective geometry. Despite the bleak physicist's view of mathematics quoted above, this discipline is– thanks to Blaise Pascal— not totally lacking in literary and spiritual associations.

* Hey Hey

Thursday, December 29, 2005

Thursday December 29, 2005

Filed under: Uncategorized — Tags: — m759 @ 3:31 PM

Parallel Lines
Meet at Infinity

 

From Log24,
Dec. 16, 2005:

The image “http://www.log24.com/log/pix05B/051229-WhistleStop.jpg” cannot be displayed, because it contains errors.

From today's
New York Times,
a man who died
(like Charlie Chaplin
and W. C. Fields)
on Christmas Day:

The image “http://www.log24.com/log/pix05B/051229-DawsonClip.jpg” cannot be displayed, because it contains errors.

 

From Log24, Dec. 6, 2002,
Santa Versus the Volcano:

Well if you want to ride
you gotta ride it like you find it.
Get your ticket at the station
of the Rock Island Line.

Lonnie Donegan   
(d. Nov. 3, 2002)

and others

The Rock Island Line's namesake depot 
in Rock Island, Illinois

Sunday, November 16, 2003

Sunday November 16, 2003

Filed under: Uncategorized — Tags: , — m759 @ 7:59 PM

Russell Crowe as Santa's Helper

From The Age, Nov. 17, 2003:

"Russell Crowe's period naval epic has been relegated to second place at the US box office by an elf raised by Santa's helpers at the North Pole."

From A Midsummer Night's Dream:

"The lunatic,¹ the lover,² and the poet³
  Are of imagination all compact."

1

2

3

In acceping a British Film Award for his work in A Beautiful Mind, Crowe said that

"Richard Harris, one of the finest of this profession, recently brought to my attention the verse of Patrick Kavanagh:

'To be a poet and not know the trade,
To be a lover and repel all women,
Twin ironies by which
    great saints are made,
The agonising
    pincer jaws of heaven.' "

A theological image both more pleasant and more in keeping with the mathematical background of A Beautiful Mind is the following:

This picture, from a site titled Strange and Complex, illustrates a one-to-one correspondence between the points of the complex plane and all the points of the sphere except for the North Pole.

To complete the correspondence (to, in Shakespeare's words, make the sphere's image "all compact"), we may adjoin a "point at infinity" to the plane — the image, under the revised correspondence, of the North Pole.

For related poetry, see Stevens's "A Primitive Like an Orb."

For more on the point at infinity, see the conclusion of Midsummer Eve's Dream.

For Crowe's role as Santa's helper, consider how he has helped make known the poetry of Patrick Kavanagh, and see Kavanagh's "Advent":

O after Christmas we'll have
    no need to go searching….

… Christ comes with a January flower.

i.e. Christ Mass… as, for instance, performed by the six Jesuits who were murdered in El Salvador on this date in 1989.
  

Thursday, June 26, 2003

Thursday June 26, 2003

Filed under: Uncategorized — Tags: , — m759 @ 5:00 AM

ART WARS:
Art at the Vanishing Point

From the web page Art Wars:

"For more on the 'vanishing point,'
or 'point at infinity,' see
Midsummer Eve's Dream."

On Midsummer Eve, June 23, 2003, minimalist artist Fred Sandback killed himself.

Sandback is discussed in The Dia Generation, an April 6, 2003, New York Times Magazine article that is itself discussed at the Art Wars page.

Sandback, who majored in philosophy at Yale, once said that

"Fact and illusion are equivalents."

Two other references that may be relevant:

The Medium is
the Rear View Mirror
,

which deals with McLuhan's book Through the Vanishing Point, and a work I cited on Midsummer Eve  …

Chapter 5 of Through the Looking Glass:

" 'What is it you want to buy?' the Sheep said at last, looking up for a moment from her knitting.

'I don't quite know yet,' Alice said very gently.  'I should like to look all round me first, if I might.'

'You may look in front of you, and on both sides, if you like,' said the Sheep; 'but you ca'n't look all round you — unless you've got eyes at the back of your head.'

But these, as it happened, Alice had not got: so she contented herself with turning round, looking at the shelves as she came to them.

The shop seemed to be full of all manner of curious things — but the oddest part of it all was that, whenever she looked hard at any shelf, to make out exactly what it had on it, that particular shelf was always quite, empty, though the others round it were crowded as full as they could hold.

'Things flow about so here!' she said at last in a plaintive tone…."

 "When Alice went
     through the vanishing point
"
 

Monday, March 10, 2003

Monday March 10, 2003

Filed under: Uncategorized — Tags: , — m759 @ 5:45 AM

ART WARS:

Art at the Vanishing Point

Two readings from The New York Times Book Review of Sunday,

March 9,

2003 are relevant to our recurring "art wars" theme.  The essay on Dante by Judith Shulevitz on page 31 recalls his "point at which all times are present."  (See my March 7 entry.)  On page 12 there is a review of a novel about the alleged "high culture" of the New York art world.  The novel is centered on Leo Hertzberg, a fictional Columbia University art historian.  From Janet Burroway's review of What I Loved, by Siri Hustvedt:

"…the 'zeros' who inhabit the book… dramatize its speculations about the self…. the spectator who is 'the true vanishing point, the pinprick in the canvas.'''

Here is a canvas by Richard McGuire for April Fools' Day 1995, illustrating such a spectator.

For more on the "vanishing point," or "point at infinity," see

"Midsummer Eve's Dream."

Connoisseurs of ArtSpeak may appreciate Burroway's summary of Hustvedt's prose: "…her real canvas is philosophical, and here she explores the nature of identity in a structure of crystalline complexity."

For another "structure of crystalline
complexity," see my March 6 entry,

"Geometry for Jews."

For a more honest account of the
New York art scene, see Tom Wolfe's
 
The Painted Word.
 

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