Log24

Thursday, April 26, 2018

Defining Form

Filed under: General — m759 @ 7:00 PM

Images related to the previous post

Wednesday, August 1, 2012

Defining Form

Filed under: General — m759 @ 11:01 AM

Continued from July 29 in memory of filmmaker Chris Marker,
who reportedly* died on that date at 91 at his home in Paris.

See Slides and Chantingand Where Madness Lies.

See also Sherrill Grace on Malcolm Lowry.

* Washington Post. Other sources say Marker died on July 30.

 These notably occur in Marker's masterpiece
     La Jetée (review with spoilers).

Sunday, July 29, 2012

Defining Form

Filed under: General,Geometry — Tags: — m759 @ 2:00 PM

IMAGE- Hyperplanes (square and triangular) in PG(3,2), and coordinates for AG(4,2)

Background: Square-Triangle Theorem.

For a more literary approach, see "Defining Form" in this journal
and a bibliography from the University of Zaragoza.

Saturday, January 14, 2012

Defining Form (continued)

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

Detail of Sylvie Donmoyer picture discussed
here on January 10

http://www.log24.com/log/pix12/120114-Donmoyer-Still-Life-CubeDetail.jpg

The "13" tile may refer to the 13 symmetry axes
in the 3x3x3 Galois cube, or the corresponding
13 planes through the center in that cube. (See
this morning's post and Cubist Geometries.)

Tuesday, January 10, 2012

Defining Form

Filed under: General,Geometry — Tags: , — m759 @ 9:00 AM

(Continued from Epiphany and from yesterday.)

Detail from the current American Mathematical Society homepage

http://www.log24.com/log/pix12/120110-AMS_page-Detail.jpg

Further detail, with a comparison to Dürer's magic square—

http://www.log24.com/log/pix12/120110-Donmoyer-Still-Life-Detail.jpg http://www.log24.com/log/pix12/120110-DurerSquare.jpg

The three interpenetrating planes in the foreground of Donmoyer's picture
provide a clue to the structure of the the magic square array behind them.

Group the 16 elements of Donmoyer's array into four 4-sets corresponding to the
four rows of Dürer's square, and apply the 4-color decomposition theorem.
Note the symmetry of the set of 3 line diagrams that result.

Now consider the 4-sets 1-4, 5-8, 9-12, and 13-16, and note that these
occupy the same positions in the Donmoyer square that 4-sets of
like elements occupy in the diamond-puzzle figure below—

http://www.log24.com/log/pix12/120110-DiamondPuzzleFigure.jpg

Thus the Donmoyer array also enjoys the structural  symmetry,
invariant under 322,560 transformations, of the diamond-puzzle figure.

Just as the decomposition theorem's interpenetrating lines  explain the structure
of a 4×4 square , the foreground's interpenetrating planes  explain the structure
of a 2x2x2 cube .

For an application to theology, recall that interpenetration  is a technical term
in that field, and see the following post from last year—

Saturday, June 25, 2011

 

Theology for Antichristmas

— m759 @ 12:00 PM

Hypostasis (philosophy)

"… the formula 'Three Hypostases  in one Ousia '
came to be everywhere accepted as an epitome
of the orthodox doctrine of the Holy Trinity.
This consensus, however, was not achieved
without some confusion…." —Wikipedia

http://www.log24.com/log/pix11A/110625-CubeHypostases.gif

Ousia

Click for further details:

http://www.log24.com/log/pix11A/110625-ProjectiveTrinitySm.jpg

 

Friday, January 6, 2012

Defining Form

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

IMAGE- MLA session, 'Defining Form,' chaired by Colleen Rosenfeld of Pomona College

Some related resources from Malcolm Lowry

"…his eyes ranged the Consul's books disposed quite neatly… on high shelves around the walls: Dogme et Ritual de la Haute Magie , Serpent and Siva Worship in Central America , there were two long shelves of this, together with the rusty leather bindings and frayed edges of the numerous cabbalistic and alchemical books, though some of them looked fairly new, like the Goetia of the Lemegaton of Solomon the King , probably they were treasures, but the rest were a heterogeneous collection…."

Under the Volcano , Chapter VI

— and from Matilde Marcolli

Seven books on analytical psychology

See also Marcolli in this morning's previous post, The Garden Path.

For the relevance of alchemy to form, see Alchemy in this journal.

Saturday, July 21, 2018

Building-Block Theory

Filed under: General,Geometry — m759 @ 10:56 AM

(A sequel to yesterday's Geometry for Jews)

From Dr/ Yau's own website

From this journal on the above UCI posting  date — April 6, 2018 —

From this journal on the above lecture  date — April 26, 2018 —
illustrations in a post titled Defining Form

For the relevance of the above material to building blocks,
see Eightfold Cube in this journal.

Friday, December 16, 2016

Memory, History, Geometry

Filed under: General,Geometry — Tags: — m759 @ 9:48 AM

These are Rothko's Swamps .

See a Log24 search for related meditations.

For all three topics combined, see Coxeter —

" There is a pleasantly discursive treatment 
of Pontius Pilate’s unanswered question
‘What is truth?’ "

— Coxeter, 1987, introduction to Trudeau’s
     The Non-Euclidean Revolution

Update of 10 AM ET —  Related material, with an elementary example:

Posts tagged "Defining Form." The example —

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

Friday, January 16, 2015

A versus PA

Filed under: General,Geometry — 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.

Wednesday, October 31, 2012

The Malfunctioning TARDIS

Filed under: General,Geometry — m759 @ 11:01 AM

(Continued from previous TARDIS posts)

Summary: A review of some  posts from last August is suggested by the death,
reportedly during the dark hours early on October 30, of artist Lebbeus Woods.

An (initially unauthorized) appearance of his work in the 1995 film
Twelve Monkeys 

 … suggests a review of three posts from last August.

Wednesday, August 1, 2012

Defining Form

Filed under: Uncategorized — m759 @ 11:01 AM 

Continued from July 29 in memory of filmmaker Chris Marker,
who reportedly* died on that date at 91 at his home in Paris.

See Slides and Chantingand Where Madness Lies.

See also Sherrill Grace on Malcolm Lowry.

Washington PostOther sources say Marker died on July 30.

 These notably occur in Marker's masterpiece
     La Jetée  (review with spoilers).

 

Wednesday, August 1, 2012

Triple Feature

Filed under: Uncategorized — m759 @ 11:11 PM

IMAGE- Triple Feature: 'Twelve Monkeys,' Reagan National Airport on July 31, 2012, and 'Die Hard 2'

For related material, see this morning's post Defining Form.

 

Sunday, August 12, 2012

Doctor Who

Filed under: Uncategorized — m759 @ 2:00 PM

On Robert A. Heinlein's novel Glory Road

"Glory Road  (1963) included the foldbox , a hyperdimensional packing case that was bigger inside than outside. It is unclear if Glory Road  was influenced by the debut of the science fiction television series Doctor Who  on the BBC that same year. In Doctor Who , the main character pilots a time machine called a TARDIS, which is built with technology which makes it 'dimensionally transcendental,' that is, bigger inside than out."

— Todd, Tesseract article at exampleproblems.com

From the same exampleproblems.com article—

"The connection pattern of the tesseract's vertices is the same as that of a 4×4 square array drawn on a torus; each cell (representing a vertex of the tesseract) is adjacent to exactly four other cells. See geometry of the 4×4 square."

For further details, see today's new page on vertex adjacency at finitegeometry.org.

 

"It was a dark and stormy night."— A Wrinkle in Time

Tuesday, September 25, 2012

Another Day

Filed under: General,Geometry — m759 @ 11:01 AM

Verlyn Klinkenborg in yesterday's online New York Times

"Even metaphors — the best ones anyway —
are literal-minded. But that’s a story for another day."

Another day: May 18, 2010—

Part I: At Pomona College

"Writer-in-Residence Verlyn Klinkenborg '74
Writes Essay on Graduation for New York Times"
— Pomona College news item, May 18, 2010, by
   Laura Tiffany

Part II: In this journal



Note that the geometric diamond in the screenshot above
is not blue but black.

See also Pomona College under the topic Defining Form
in this journal.

Wednesday, August 1, 2012

Triple Feature

Filed under: General — m759 @ 11:11 PM

IMAGE- Triple Feature: 'Twelve Monkeys,' Reagan National Airport on July 31, 2012, and 'Die Hard 2'

For related material, see this morning's post Defining Form.

Sunday, July 29, 2012

The Galois Tesseract

Filed under: General,Geometry — Tags: , — m759 @ 11:00 PM

(Continued)

The three parts of the figure in today's earlier post "Defining Form"—

IMAGE- Hyperplanes (square and triangular) in PG(3,2), and coordinates for AG(4,2)

— share the same vector-space structure:

   0     c     d   c + d
   a   a + c   a + d a + c + d
   b   b + c   b + d b + c + d
a + b a + b + c a + b + d   a + b + 
  c + d

   (This vector-space a b c d  diagram is from  Chapter 11 of 
    Sphere Packings, Lattices and Groups , by John Horton
    Conway and N. J. A. Sloane, first published by Springer
    in 1988.)

The fact that any  4×4 array embodies such a structure was implicit in
the diamond theorem (February 1979). Any 4×4 array, regarded as
a model of the finite geometry AG(4, 2), may be called a Galois tesseract.
(So called because of the Galois geometry involved, and because the
16 cells of a 4×4 array with opposite edges identified have the same
adjacency pattern as the 16 vertices of a tesseract (see, for instance,
Coxeter's 1950 "Self-Dual Configurations and Regular Graphs," figures
5 and 6).)

A 1982 discussion of a more abstract form of AG(4, 2):

Source:

The above 1982 remarks by Brouwer may or may not have influenced
the drawing of the above 1988 Conway-Sloane diagram.

Saturday, January 14, 2012

Damnation Morning*

Filed under: General,Geometry — Tags: , — m759 @ 5:24 AM

(Continued)

The following is adapted from a 2011 post

IMAGE- Galois vs. Rubik

* The title, that of a Fritz Leiber story, is suggested by
   the above picture of the symmetry axes of the square.
   Click "Continued" above for further details. See also
   last Wednesday's Cuber.

Tuesday, December 27, 2011

Getting with the Program

Filed under: General,Geometry — Tags: — m759 @ 4:28 AM

Stanley Fish in The New York Times  yesterday evening—

IMAGE- Stanley Fish, 'The Old Order Changeth,' Boxing Day, 2011

From the MLA program Fish discussed—

IMAGE- MLA session, 'Defining Form,' chaired by Colleen Rosenfeld of Pomona College

Above: An MLA session, "Defining Form," led
by Colleen Rosenfeld of Pomona College

An example from Pomona College in 1968—

IMAGE- Triangular models of small affine and projective finite geometries

The same underlying geometries (i.e., "form") may be modeled with
a square figure and a cubical figure rather than with the triangular
figures of 1968 shown above.

See Finite Geometry of the Square and Cube.

Those who prefer a literary approach to form may enjoy the recent post As Is.
(For some context, see Game of Shadows.)

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