Thursday, November 16, 2017
A Line at Infinity
Saturday, January 6, 2018
Yale News
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.
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.
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 IYou.
The other basic word is the word pair IIt;
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 IYou is different from
that in the basic word IIt.
— 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:
ichdu, iches, ichsie, dues, dusie, essie.
This is in accordance with some 1974 remarks by
MarieLouise 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
"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."
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
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
(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 nonworld 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 13point 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
Wednesday, May 13, 2015
Space
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 …
Friday, January 16, 2015
A versus PA
"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 4point affine plane A and
the 7point projective plane PA —
The circleintriangle 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)
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
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
“In the space of horizons that neither love nor hate”
— Wallace Stevens, “Things of August”
Seven years ago yesterday—
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
I. General finite geometry (without coordinates):
A finite affine plane of order n has n^2 points.
A finite projective plane of order n has n^2 + n + 1
points because it is formed from an ordern finite affine
plane by adding a line at infinity that contains n + 1 points.
Examples—


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 3space
of order n with n^3 points by discarding the point (0,0,0) and
identifying the points whose coordinates are multiples of the
(n1) nonzero scalars.
Note: The resulting Galois plane of order n has
(n^31)/(n1)= (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:
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 order3 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
"The cube has…13 axes of symmetry:
6 C_{2} (axes joining midpoints of opposite edges),
4 C_{3} (space diagonals), and
3C_{4} (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 13point Galois plane.
The geometer's 3×3×3 cube–
27 separate subcubes unconnected
by any Rubiklike mechanism–
The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
subcubes in the (Galois) 3×3×3 cube–
A closely related structure–
the finite projective plane
with 13 points and 13 lines–
A later version of the 13point plane
by Ed Pegg Jr.–
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)–
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 3space converted to a vector 3space.
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
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, 
A related visual
association of ideas —
("The association is the idea"
— Ian Lee, The Third Word War)
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.) 
See also UD's recent
A MustRead and In My Day*
as well as the five
Log24 entries ending
Sept. 20, 2002.
* Hey Hey
Thursday, December 29, 2005
Thursday December 29, 2005
Parallel Lines
Meet at Infinity
From Log24, Dec. 16, 2005: 
From today's New York Times, a man who died (like Charlie Chaplin and W. C. Fields) on Christmas Day: 
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)
The Rock Island Line's namesake depot
in Rock Island, Illinois
Sunday, November 16, 2003
Sunday November 16, 2003
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 onetoone 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
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…."
Monday, March 10, 2003
Monday March 10, 2003
ART WARS:
Art at the Vanishing Point
Two readings from The New York Times Book Review of Sunday,
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
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,
For a more honest account of the
New York art scene, see Tom Wolfe's
The Painted Word.