Log24

Sunday, December 9, 2018

Quaternions in a Small Space

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

The previous post, on the 3×3 square in ancient China,
suggests a review of group actions on that square
that include the quaternion group.

Click to enlarge

Three links from the above finitegeometry.org webpage on the
quaternion group —

Related material —

Iain Aitchison on the 'symmetric generation' of R. T. Curtis

See as well the two Log24 posts of December 1st, 2018 —

Character and In Memoriam.

Wednesday, September 19, 2018

Projective

Filed under: General — Tags: — m759 @ 6:45 AM

http://www.log24.com/log/pix18/180919-Projective_Test-search-sidebar.jpg

See also Watchmen in this journal.

Tuesday, July 31, 2018

Lexicon

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

"A blank underlies the trials of device." — Wallace Stevens

IMAGE- The ninefold square .

Thursday, November 16, 2017

Lost Horizon

Filed under: G-Notes,General,Geometry — m759 @ 11:29 AM

Related material —

The following image in this journal

  .

Monday, October 9, 2017

Still Point for a Dance

Filed under: G-Notes,General,Geometry — m759 @ 12:00 PM

"At the still point of the turning world. Neither flesh nor fleshless;
Neither from nor towards; at the still point, there the dance is,
But neither arrest nor movement. And do not call it fixity,
Where past and future are gathered. Neither movement from nor towards,
Neither ascent nor decline. Except for the point, the still point,
There would be no dance, and there is only the dance."

— T. S. Eliot, Four Quartets

See also a recurrent image
from this journal —

IMAGE- The ninefold square .

Wednesday, April 12, 2017

Contracting the Spielraum

Filed under: General,Geometry — Tags: , , — m759 @ 10:00 AM

The contraction of the title is from group actions on
the ninefold square  (with the center subsquare fixed)
to group actions on the eightfold cube.

From a post of June 4, 2014

At math.stackexchange.com on March 1-12, 2013:

Is there a geometric realization of the Quaternion group?” —

The above illustration, though neatly drawn, appeared under the
cloak of anonymity.  No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note “GL(2,3) actions on a cube” of April 5, 1985).

Tuesday, March 21, 2017

Res Ipsa

Filed under: General,Geometry — Tags: — m759 @ 7:00 PM

From The Poetic Quotidian, a journal of quotations—

See also, in this journal, New Haven + Grid.

The Ninefold Square

Sunday, December 13, 2015

Another Opening, Another Show

Filed under: General — Tags: — m759 @ 10:30 PM

Ben Brantley in tonight's online review of a show that
reportedly opened off-Broadway on Dec. 10, 2015 —

" 'Mattress' has its charms, but they do wear thin. "

See also The New York Times  on Martin Gardner Nov. 30:

A companion image from this  journal
on the "Mattress" opening date —

Images of time and eternity in a 1x4x9 black monolith

Midrash:
Vonnegut Asterisk

Tuesday, July 14, 2015

Go Set a Structure

Filed under: General — Tags: , , — m759 @ 2:45 PM

Thursday, July 9, 2015

Man and His Symbols

Filed under: General,Geometry — m759 @ 2:24 PM

(Continued)

A post of July 7, Haiku for DeLillo, had a link to posts tagged "Holy Field GF(3)."

As the smallest Galois field based on an odd prime, this structure 
clearly is of fundamental importance.  

The Galois field GF(3)

It is, however, perhaps too  small  to be visually impressive.

A larger, closely related, field, GF(9), may be pictured as a 3×3 array

hence as the traditional Chinese  Holy Field.

Marketing the Holy Field

IMAGE- The Ninefold Square, in China 'The Holy Field'

The above illustration of China's  Holy Field occurred in the context of
Log24 posts on Child Buyers.   For more on child buyers, see an excellent
condemnation today by Diane Ravitch of the U. S. Secretary of Education.

Wednesday, May 13, 2015

Space

Filed under: General,Geometry — 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

Saturday, October 18, 2014

Educational Series

Filed under: General,Geometry — m759 @ 1:01 PM

Barron’s Educational Series (click to enlarge):

The Tablet of Ahkmenrah:

IMAGE- The Tablet of Ahkmenrah, from 'Night at the Museum'

 “With the Tablet of Ahkmenrah and the Cube of Rubik,
my power will know no bounds!”
— Kahmunrah in a novelization of Night at the Museum:
Battle of the Smithsonian , Barron’s Educational Series

Another educational series (this journal):

Image-- Rosalind Krauss and The Ninefold Square

Art theorist Rosalind Krauss and The Ninefold Square

IMAGE- Elementary Galois Geometry over GF(3)

Friday, October 17, 2014

Raiders of the Inarticulate

Filed under: General — m759 @ 9:00 PM

On Walter Isaacson’s The Innovators :

“Yet as the book’s five hundred–plus pages unwind, Isaacson interrupts himself to present small bromides about what it means to innovate and what we might learn from these innovators, our presumed betters. “Innovation requires articulation,” he tells us, after explaining how the main strength of Grace Hopper, a trailblazing computer scientist for the US Navy, was her ability to speak in the languages of mathematicians, engineers, programmers, and soldiers alike. ‘One useful leadership talent is knowing when to push ahead against doubters and when to heed them,’ he offers later.

The book is peppered with these kinds of passages, which often intrude on the narrative, depriving us of moments of real emotional power.”

— Jacob Silverman in Bookforum , Sept/Oct/Nov 2014

From Isaacson’s book:

IMAGE- Bletchley Park and the Colossus computer

Related material:

In memory of T. S. Eliot…

… and in memory of Stanley Chase, producer of Colossus: The Forbin Project
and of Threepenny Opera :

Ninefold square from Colossus
(“There is another system”) —

Fourfold square introducing Brecht
in  Dreigroschen Trifft Vierfarben —

Saturday, May 31, 2014

Quaternion Group Models:

Filed under: General,Geometry — Tags: — m759 @ 10:00 AM

The ninefold square, the eightfold cube, and monkeys.

IMAGE- Actions of the unit quaternions in finite geometry, on a ninefold square and on an eightfold cube

For posts on the models above, see quaternion
in this journal. For the monkeys, see

"Nothing Is More Fun than a Hypercube of Monkeys,"
Evelyn Lamb's Scientific American  weblog, May 19, 2014:

The Scientific American  item is about the preprint
"The Quaternion Group as a Symmetry Group,"
by Vi Hart and Henry Segerman (April 26, 2014):

See also  Finite Geometry and Physical Space.

Wednesday, March 5, 2014

The Good Earth

Filed under: General — m759 @ 2:00 PM

In Beijing, it is now 3 AM on March 6,
the dies natalis  of St. Pearl Buck.

(Click to enlarge.)

IMAGE- Chiang Yee on the ninefold square and Chinese calligraphy

Holy Hub Caps

Filed under: General — m759 @ 12:00 PM

Review of Joseph Campbell's The Inner Reaches of Outer Space

The reviewer compares Campbell to "one of those guys who
builds his own church out of hub caps."

A simple hub cap — see ninefold  in this journal.

Friday, February 14, 2014

Epiphany

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

“… the object sets up a kind of 
 frame or space or field 
 within which there can be epiphany.”

Charles Taylor

A frame or space or field —

IMAGE- The ninefold square

Related material —

Star Wars (January 11, 2014),

The Lyche Gate Asterisk , from 10:31 AM ET on May 22, 2010,
the date of Martin Gardner's death —

Image-- The Case of the Lyche Gate Asterisk

— and the March 2014 issue of the
Notices of the American Mathematical Society  —

See as well Epiphany 2014 (Jan. 6) in this journal and the
March Notices  on the Shaw prize —

"Established under the auspices of Run Run Shaw
in November 2002, the prize is managed and
administered by the Shaw Prize Foundation
based in Hong Kong." 

Saturday, November 16, 2013

Raiders of the Lost Theorem

Filed under: General,Geometry — Tags: — m759 @ 11:30 AM

IMAGE- The 'atomic square' in Lee Sallows's article 'The Lost Theorem'

Yes. See

The 48 actions of GL(2,3) on a 3×3 coordinate-array A,
when matrices of that group right-multiply the elements of A,
with A =

(1,1) (1,0) (1,2)
(0,1) (0,0) (0,2)
(2,1) (2,0) (2,2)

Actions of GL(2,p) on a pxp coordinate-array have the
same sorts of symmetries, where p is any odd prime.

Note that A, regarded in the Sallows manner as a magic square,
has the constant sum (0,0) in rows, columns, both diagonals, and  
all four broken diagonals (with arithmetic modulo 3).

For a more sophisticated approach to the structure of the
ninefold square, see Coxeter + Aleph.

Thursday, November 7, 2013

Recognition

Filed under: General — m759 @ 9:00 PM

This post was suggested by

1.  Movie reviews links on the NY Times  online front page—

2.  "When Death tells a story" in this journal

3.  A search for "Recognition" in this journal
     (suggested by Terence Cave's Recognitions ,
      
mentioned in a post of October 31.)

4.  The play "Magic," by G. K. Chesterton

5.  The following posts —

The post itself  consists simply of the title,
together with the above items that suggested
the title.

Tuesday, July 16, 2013

Child Buyers

Filed under: General,Geometry — Tags: — m759 @ 10:00 PM

The title refers to a classic 1960 novel by John Hersey.

“How do you  get young people excited about space?”

— Megan Garber in The Atlantic , Aug. 16, 2012
(Italics added.) (See previous four posts.)

Allyn Jackson on “Simplicity, in Mathematics and in Art,”
in the new August 2013 issue of Notices of the American
Mathematical Society

“As conventions evolve, so do notions of simplicity.
Franks mentioned Gauss’s 1831 paper that
established the respectability of complex numbers.”

This suggests a related image by Gauss, with a
remark on simplicity—

IMAGE- Complex Grid, by Gauss

Here Gauss’s diagram is not, as may appear at first glance,
a 3×3 array of squares, but is rather a 4×4 array of discrete
points (part of an infinite plane array).

Related material that does  feature the somewhat simpler 3×3 array
of squares, not  seen as part of an infinite array—

Marketing the Holy Field

IMAGE- The Ninefold Square, in China 'The Holy Field'

Click image for the original post.

For a purely mathematical view of the holy field, see Visualizing GL(2,p).

Monday, November 5, 2012

Sitting Specially

Filed under: General,Geometry — Tags: — m759 @ 5:01 AM

Some webpages at finitegeometry.org discuss
group actions on Sylvester’s duads and synthemes.

Those pages are based on the square model of
PG(3,2) described in the 1980’s by Steven H. Cullinane.

A rival tetrahedral model of PG(3,2) was described
in the 1990’s by Burkard Polster.

Polster’s tetrahedral model appears, notably, in
a Mathematics Magazine  article from April 2009—

IMAGE- Figure from article by Alex Fink and Richard Guy on how the symmetric group of degree 5 'sits specially' in the symmetric group of degree 6

Click for a pdf of the article.

Related material:

The Religion of Cubism” (May 9, 2003) and “Art and Lies
(Nov. 16, 2008).

This  post was suggested by following the link in yesterday’s
Sunday School post  to High White Noon, and the link from
there to A Study in Art Education, which mentions the date of
Rudolf Arnheim‘s death, June 9, 2007. This journal
on that date

Cryptology

IMAGE- The ninefold square

— The Delphic Corporation

The Fink-Guy article was announced in a Mathematical
Association of America newsletter dated April 15, 2009.

Those who prefer narrative to mathematics may consult
a Log24 post from a few days earlier, “Where Entertainment is God”
(April 12, 2009), and, for some backstory, The Judas Seat
(February 16, 2007).

Tuesday, October 30, 2012

Black October

Filed under: General — m759 @ 9:00 AM

From AntiChristmas 2010

Image-- Rosalind Krauss and The Ninefold Square

Art theorist Rosalind Krauss and The Ninefold Square

Krauss is a co-founder of the art journal October .
For some backgound, see the overlapping searches
Krauss October and Matrix Architect.

Sunday, October 21, 2012

Multispeech

Filed under: General — m759 @ 9:00 PM

IMAGE- The 3x3 ('ninefold') square as Chinese 'Holy Field'

     Click image above
     for some background.

IMAGE- Snow White Imprisoned- An image for the Snow Queen

IMAGE- Charlize Theron as Ravenna: 'She drifted.'

Tuesday, October 9, 2012

Too Much Meaning

Filed under: General — m759 @ 12:00 PM

Last night's post discussed ways of draining the world of meaning.

For some tastes, poets like Dante do the opposite, supplying too much  meaning.

See a New Republic  review, dated Oct. 5, in which Harvard atheist Helen Vendler discusses Dante's

"… assertion that Beatrice herself  'was this number [nine],' since nine is the square of three, the number belonging to the Trinity. Dante’s fantastic reasoning requires pages of annotation, which Frisardi, drawing on a number of commentators, furnishes to the bewildered reader. The theological elaboration of the number nine— merely one instance of how far from our own* are Dante’s habits of thought— will convince any doubting reader that the Vita Nuova  requires annotation far beyond what its pages might seem to demand."

Related material— Ninefold in this journal, and remarks by Joseph Campbell in a post, Plan 9, from Sept. 5.

* Speak for yourself, Helen.

Monday, May 7, 2012

More on Triality

Filed under: General,Geometry — Tags: — m759 @ 4:20 PM

John Baez wrote in 1996 ("Week 91") that

"I've never quite seen anyone come right out
and admit that triality arises from the
permutations of the unit vectors i, j, and k
in 3d Euclidean space."

Baez seems to come close to doing this with a
somewhat different i , j , and kHurwitz
quaternions
— in his 2005 book review
quoted here yesterday.

See also the Log24 post of Jan. 4 on quaternions,
and the following figures. The actions on cubes
in the lower figure may be viewed as illustrating
(rather indirectly) the relationship of the quaternion
group's 24 automorphisms to the 24 rotational
symmetries of the cube.

IMAGE- Actions of the unit quaternions in finite geometry, on a ninefold square and on an eightfold cube

Sunday, April 1, 2012

The Palpatine Dimension

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

A physics quote relayed at Peter Woit's weblog today—

"The relation between 4D N=4 SYM and the 6D (2, 0) theory
is just like that between Darth Vader and the Emperor.
You see Darth Vader and you think 'Isn’t he just great?
How can anyone be greater than that? No way.'
Then you meet the Emperor."

— Arkani-Hamed

Some related material from this  weblog—

(See Big Apple and Columbia Film Theory)

http://www.log24.com/log/pix12/120108-Space_Time_Penrose_Hawking.jpg

The Meno Embedding:

http://www.log24.com/log/pix10B/101128-TheEmbedding.gif

Some related material from the Web—

IMAGE- The Penrose diamond and the Klein quadric

See also uses of the word triality  in mathematics. For instance…

A discussion of triality by Edward Witten

Triality is in some sense the last of the exceptional isomorphisms,
and the role of triality for n = 6  thus makes it plausible that n = 6
is the maximum dimension for superconformal symmetry,
though I will not give a proof here.

— "Conformal Field Theory in Four and Six Dimensions"

and a discussion by Peter J. Cameron

There are exactly two non-isomorphic ways
to partition the 4-subsets of a 9-set
into nine copies of AG( 3,2).
Both admit 2-transitive groups.

— "The Klein Quadric and Triality"

Exercise: Is Witten's triality related to Cameron's?
(For some historical background, see the triality  link from above
and Cameron's Klein Correspondence and Triality.)

Cameron applies his  triality to the pure geometry of a 9-set.
For a 9-set viewed in the context of physics, see A Beginning

From MIT Commencement Day, 2011—

A symbol related to Apollo, to nine, and to "nothing"

A minimalist favicon—

IMAGE- Generic 3x3 square as favicon

This miniature 3×3 square— http://log24.com/log/pix11A/110518-3x3favicon.ico — may, if one likes,
be viewed as the "nothing" present at the Creation. 
See Feb. 19, 2011, and Jim Holt on physics.

Happy April 1.

Saturday, March 24, 2012

The David Waltz…

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

In Turing's Cathedral

"At the still point…" — T. S. Eliot

In memory of David L. Waltz, artificial-intelligence pioneer,
who died Thursday, March 22, 2012—

  1. The Log24 post of March 22 on the square-triangle theorem
  2. The March 18 post, Square-Triangle Diamond
  3. Remarks from the BBC on linguistic embedding
    that begin as follows—
         "If we draw a large triangle and embed smaller triangles in it,
          how does it look?"—
    and include discussion of a South American "tribe called Piranha" [sic ]
  4. The result of a Cartoon Bank search suggested by no. 3 above—
    (Click image for some related material.)
  5. A suggestion from the Cartoon Bank—
    IMAGE- 'Try our new grid view.'
  6. The following from the First of May, 2010

    The Nine Divisions of Heaven–

    Image-- Routledge Encyclopedia of Taoism, Vol. I, on the Nine Heavens, 'jiutian,' ed. by Fabrizio Pregadio

    Some context–

    IMAGE- The 3x3 ('ninefold') square as Chinese 'Holy Field'

    "This pattern is a square divided into nine equal parts.
    It has been called the 'Holy Field' division and
    was used throughout Chinese history for many
    different purposes, most of which were connected
    with things religious, political, or philosophical."

    – The Magic Square: Cities in Ancient China,
    by Alfred Schinz, Edition Axel Menges, 1996, p. 71

  7. The phrase "embedding the stone" —

Monday, March 5, 2012

For the Nine Muses

Filed under: General — m759 @ 1:00 PM

From "The Talented," a post of April 26, 2011—

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

And for Josefine Lyche

Unity in Multiplicity —

Pink  in Wikipedia

Friday, January 20, 2012

The Nothing That Is

Filed under: General — m759 @ 9:00 PM

"The 'one' with whom the reader has identified himself
has now become 'the listener, who listens in the snow';
he has become the snow man, and he knows winter
with a mind of winter, knows it in its strictest reality,
stripped of all imagination and human feeling.
But at that point when he sees the winter scene
reduced to absolute fact, as the object not of the mind,
but of the perfect perceptual eye that sees
'nothing that is not there,' then the scene,
devoid of  its imaginative correspondences,
has become 'the nothing that is.'"

Robert Pack, Wallace Stevens:
An Approach to His Poetry and Thought
.
New Brunswick: Rutgers UP, 1958.

 

IMAGE- The Ninefold Square at Ninevine.net

Sunday, January 1, 2012

Sunday Shul

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

"… myths are stories, and like all narratives
they unravel through time, whereas grids
are not only spatial to start with,
they are visual structures that explicitly reject
a narrative or sequential reading of any kind."

— Rosalind Krauss in "Grids,"
October  (Summer 1979), 9: 50-64.

Counterexample—

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

The Ninefold Square

See Coxeter and the Aleph and Ayn Sof

Mathematics and Narrative, Illustrated
http://www.log24.com/log/pix11/110107-The1950Aleph-Sm.jpg

Mathematics
http://www.log24.com/log/pix11/110107-ScriptAlephSm.jpg
Narrative

Monday, May 23, 2011

The Stoner Series

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

A reader comments on yesterday afternoon's New York Times
"The Stone" column by Justin E.H. Smith—

"I did indeed appreciate Mr. Smith’s essay.
And I’m curious as to what future contributions of his,
to the Stoner series, that we can look forward to."

From August 24, 2010

Der Einsatz

Motto of Plato's Academy: 'Let no one ignorant of geometry enter'

The Ninefold Square (a 3x3 grid)

Nichts ist wie es scheint.

See also the film
"23— Nichts ist so wie es scheint."

Happy day 23 of Mental Health Month.

Sunday, January 9, 2011

Sermon

Filed under: General — Tags: — m759 @ 11:00 AM

Another  topic from today's newspaper —

IMAGE- Nicole Kidman in a Sunday supplement

Commentary —

"We're gonna need more holy water." — "Season of the Witch," a film that opened Friday

See also

This morning's post Inception and the following site:

Image- The Ninefold Favicon

Note the ninefold favicon at the above site. Some background—
The Ninth Gate  in yesterday's post and this image from last September

http://www.log24.com/log/pix10B/100902-Favicon.jpg

Saturday, January 8, 2011

True Grid (continued)

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

"Rosetta Stone" as a Metaphor
  in Mathematical Narratives

For some backgound, see Mathematics and Narrative from 2005.

Yesterday's posts on mathematics and narrative discussed some properties
of the 3×3 grid (also known as the ninefold square ).

For some other properties, see (at the college-undergraduate, or MAA, level)–
Ezra Brown, 2001, "Magic Squares, Finite Planes, and Points of Inflection on Elliptic Curves."

His conclusion:

When you are done, you will be able to arrange the points into [a] 3×3 magic square,
which resembles the one in the book [5] I was reading on elliptic curves….

This result ties together threads from finite geometry, recreational mathematics,
combinatorics, calculus, algebra, and number theory. Quite a feat!

5. Viktor Prasolov and Yuri Solvyev, Elliptic Functions and Elliptic Integrals ,
    American Mathematical Society, 1997.

Brown fails to give an important clue to the historical background of this topic —
the word Hessian . (See, however, this word in the book on elliptic functions that he cites.)

Investigation of this word yields a related essay at the graduate-student, or AMS, level–
Igor Dolgachev and Michela Artebani, 2009, "The Hesse Pencil of Plane Cubic Curves ."

From the Dolgachev-Artebani introduction–

In this paper we discuss some old and new results about the widely known Hesse
configuration
  of 9 points and 12 lines in the projective plane P2(k ): each point lies
on 4 lines and each line contains 3 points, giving an abstract configuration (123, 94).

PlanetMath.org on the Hesse configuration

http://www.log24.com/log/pix11/110108-PlanetMath.jpg

A picture of the Hesse configuration–

The image “http://www.log24.com/log/pix05B/grid3x3med.bmp” cannot be displayed, because it contains errors.

(See Visualizing GL(2,p), a note from 1985).

Related notes from this journal —

From last November —

Saturday, November 13, 2010

Story

m759 @ 10:12 PM

From the December 2010 American Mathematical Society Notices

http://www.log24.com/log/pix10B/101113-Ono.gif

Related material from this  journal—

Mathematics and Narrative and

Consolation Prize (August 19, 2010)

From 2006 —

Sunday December 10, 2006

 

 m759 @ 9:00 PM

A Miniature Rosetta Stone:

The image “http://www.log24.com/log/pix05B/grid3x3med.bmp” cannot be displayed, because it contains errors.

“Function defined form, expressed in a pure geometry
that the eye could easily grasp in its entirety.”

– J. G. Ballard on Modernism
(The Guardian , March 20, 2006)

“The greatest obstacle to discovery is not ignorance –
it is the illusion of knowledge.”

— Daniel J. Boorstin,
Librarian of Congress, quoted in Beyond Geometry

Also from 2006 —

Sunday November 26, 2006

 

m759 @ 7:26 AM

Rosalind Krauss
in "Grids," 1979:

"If we open any tract– Plastic Art and Pure Plastic Art  or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter.  They are talking about Being or Mind or Spirit.  From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete.

Or, to take a more up-to-date example…."

"He was looking at the nine engravings and at the circle,
checking strange correspondences between them."
The Club Dumas ,1993

"And it's whispered that soon if we all call the tune
Then the piper will lead us to reason."
Robert Plant ,1971

The nine engravings of The Club Dumas
(filmed as "The Ninth Gate") are perhaps more
an example of the concrete than of the universal.

An example of the universal*– or, according to Krauss,
a "staircase" to the universal– is the ninefold square:

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

"This is the garden of Apollo, the field of Reason…."
John Outram, architect    

For more on the field of reason, see
Log24, Oct. 9, 2006.

A reasonable set of "strange correspondences"
in the garden of Apollo has been provided by
Ezra Brown in a mathematical essay (pdf).

Unreason is, of course, more popular.

* The ninefold square is perhaps a "concrete universal" in the sense of Hegel:

"Two determinations found in all philosophy are the concretion of the Idea and the presence of the spirit in the same; my content must at the same time be something concrete, present. This concrete was termed Reason, and for it the more noble of those men contended with the greatest enthusiasm and warmth. Thought was raised like a standard among the nations, liberty of conviction and of conscience in me. They said to mankind, 'In this sign thou shalt conquer,' for they had before their eyes what had been done in the name of the cross alone, what had been made a matter of faith and law and religion– they saw how the sign of the cross had been degraded."

– Hegel, Lectures on the History of Philosophy ,
   "Idea of a Concrete Universal Unity"

"For every kind of vampire,
there is a kind of cross."
– Thomas Pynchon   

And from last October —

Friday, October 8, 2010

 

m759 @ 12:00 PM
 

Starting Out in the Evening
… and Finishing Up at Noon

This post was suggested by last evening's post on mathematics and narrative and by Michiko Kakutani on Vargas Llosa in this morning's New York Times .

http://www.log24.com/log/pix10B/101008-StartingOut.jpg

 

Above: Frank Langella in
"Starting Out in the Evening"

Right: Johnny Depp in
"The Ninth Gate"

http://www.log24.com/log/pix10B/101008-NinthGate.jpg

"One must proceed cautiously, for this road— of truth and falsehood in the realm of fiction— is riddled with traps and any enticing oasis is usually a mirage."

– "Is Fiction the Art of Lying?"* by Mario Vargas Llosa,
    New York Times  essay of October 7, 1984

* The Web version's title has a misprint—
   "living" instead of "lying."

"You've got to pick up every stitch…"

Friday, January 7, 2011

Coxeter and the Aleph

Filed under: General,Geometry — Tags: — m759 @ 10:31 AM

In a nutshell —

Epigraph to "The Aleph," a 1945 story by Borges:

O God! I could be bounded in a nutshell,
and count myself a King of infinite space…
— Hamlet, II, 2

http://www.log24.com/log/pix11/110107-BorgesElAleph.jpg

The story in book form, 1949

A 2006 biography of geometer H.S.M. Coxeter:

http://www.log24.com/log/pix11/110107-KingOfInfiniteSpace-Sm.jpg

The Aleph (implicit in a 1950 article by Coxeter):

http://www.log24.com/log/pix11/110107-The1950Aleph-Sm.jpg

The details:

(Click to enlarge)

http://www.log24.com/log/pix11/110107-Aleph-Sm.jpg

Related material: Group Actions, 1984-2009.

Wednesday, January 5, 2011

True Grid Example

Filed under: General,Geometry — m759 @ 9:00 PM

See today's earlier posts Ode and True Grid (continued) and, in the latter's
context of tic-tac-toe war games —  Balance, from Halloween 2005 —

IMAGE- The ninefold square

“An asymmetrical balance is sought since it possesses more movement.
This is achieved by the imaginary plotting of the character
upon a nine-fold square, invented by some ingenious writer of the Tang dynasty.
If the square were divided in half or in four, the result would be symmetrical,
but the nine-fold square permits balanced asymmetry."

Paraphrase of a passage in Chiang Yee's Chinese Calligraphy

Sunday, September 12, 2010

A Fragment of Time*

Filed under: General — m759 @ 4:30 AM

Between midnight and dawn, when the past is all deception,
The future futureless, before the morning watch
When time stops and time is never ending….

Four Quartets

June 25, 2010

Image-- Rosalind Krauss and The Ninefold Square

Art Theorist Rosalind Krauss and The Ninefold Square

* Title from a quotation in the Sunday New York Times  today
  (in a story about a June 25 death beginning on page MB1 of the New York edition)

Friday, June 25, 2010

How Deep the Darkness

Filed under: General — m759 @ 12:25 PM

Image-- Rosalind Krauss and The Ninefold Square

Art Theorist Rosalind Krauss and The Ninefold Square

Saturday, May 1, 2010

An Education

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

天鈞

 

Made famous by Ursula K. Le Guin
as the book title "Lathe of Heaven,"
this Chinese phrase, tianjun, apparently
means something more like "Scales of Heaven"–
an appropriate image for Law Day 2010.

Image--Scales (the legal symbol)

An anonymous forum user says that

"…if you switch the two characters around,
you get: 鈞天, which is one of
the nine heavens, more specifically,
the middle heaven."

This is supported by a
non-anonymous source:

"I follow A.C. Graham’s translation of
Juntian as 'Level Heaven (the innermost
of the nine divisions of heaven)';
he renders Juntian guangyue as
'the mighty music of the innermost heaven.'"

— "Music in the World of Su Shi (1037-1101):
Terminology
," by Stuart H. Sargent,
Colorado State University,
Journal of Sung-Yuan Studies 32 (2002), 39-81

The Nine Divisions of Heaven–

Image-- Routledge Encyclopedia of Taoism, Vol. I, on the Nine Heavens, 'jiutian,' ed. by Fabrizio Pregadio

Some context–

The 3x3 ('ninefold') square

"This pattern is a square divided into nine equal parts.
It has been called the 'Holy Field' division and
was used throughout Chinese history for many
different purposes, most of which were connected
with things religious, political, or philosophical."

The Magic Square: Cities in Ancient China,
by Alfred Schinz, Edition Axel Menges, 1996, p. 71

Wednesday, April 7, 2010

Veritas

Filed under: General — m759 @ 9:00 AM

Some historians consider today's date, April 7, to be the date of the Crucifixion in the Roman calendar (a solar calendar, as opposed to the Jewish lunar scheme).

Since the ninefold square has been called both a symbol of Apollo and the matrix of a cross, it will serve as an icon for today–

The 3x3 square

The image “http://www.log24.com/log/pix05B/051202-Cross.gif” cannot be displayed, because it contains errors.
Adapted from
Ad Reinhardt

Friday, February 26, 2010

Household Name

Filed under: General — m759 @ 9:00 PM

Detail from yesterday's post

Surrealistic Alarm Clock

'Pussy-Footer' alarm clock, LIFE Magazine, Oct. 10, 1949, page 122
_________________________________________________________

THE SEQUEL:

Crawdaddy article on 'Surrealistic Pillow,' the classic 1967 album by Jefferson Airplane

"…and Surrealistic Pillow  became
a household name
in the house of rock ‘n’ roll."

Denise Sullivan in Crawdaddy,
October 8, 2009

Related material:

"Which Dreamed It?"
— Title of final chapter,
Through the Looking Glass

"Go ask Alice…
I think she'll know."
— Grace Slick, 1967   

The Crawdaddy  date Oct. 8, 2009
leads to the Log24 post
Graphic Austerity.

Ninefold square with shades of gray in chessboard pattern

Clicking on those words
  in that post will lead you to…
The Logic of Dreams.

Wednesday, September 30, 2009

Wednesday September 30, 2009

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

Midnight in the Garden, Autumn 2009

 

Review:

Der Einsatz

Motto of Plato's Academy: 'Let no one ignorant of geometry enter'

The Ninefold Square (a 3x3 grid)

The New York Times Magazine
on Sunday, Sept. 20, 2009:
 

From this journal on the following day, Sept. 21:

Pearl Jam 'Backspacer' album released Sept. 20, 2009

Happy birthday, Stephen King.

Today's previous entry is based on a song, "Unthought Known,"
from the above album; the cover of the album uses the 3×3 grid
shown in Sept. 20's midnight review. For related material
on the unconscious, see June 13-15, 2005.

I know more than Apollo,
For oft when he lies sleeping
I see the stars at mortal wars
In the wounded welkin weeping.

Tom O'Bedlam's Song

Friday, February 20, 2009

Friday February 20, 2009

Filed under: General,Geometry — Tags: , , — m759 @ 11:01 PM

The Cross
of Constantine

mentioned in
this afternoon's entry
"Emblematizing the Modern"
was the object of a recent
cinematic chase sequence
(successful and inspiring)
starring Mira Sorvino
at the Metropolitan
Museum of Art.

In memory of
Dr. Hunter S. Thompson,
dead by his own hand
on this date
four years ago

Rolling Stone memorial to Dr. Hunter S. Thompson

Click for details.

There is
another sort of object
we may associate with a
different museum and with
a modern Constantine
See "Art Wars for MoMA"
(Dec. 14, 2008).

This object, modern
rather than medieval,
is the ninefold square:

The ninefold square

It may suit those who,
like Rosalind Krauss
(see "Emblematizing"),
admire the grids of modern art
but view any sort of Christian
cross with fear and loathing.

For some background that
Dr. Thompson might appreciate,
see notes on Geometry and Death
in this journal, June 1-15, 2007,
and the five Log24 entries
 ending at 9 AM Dec. 10. 2006,
which include this astute
observation by J. G. Ballard:

"Modernism's attempt to build a better world with the aid of science and technology now seems almost heroic. Bertolt Brecht, no fan of modernism, remarked that the mud, blood and carnage of the first world war trenches left its survivors longing for a future that resembled a white-tiled bathroom."

Selah.

Thursday, February 19, 2009

Thursday February 19, 2009

Filed under: General — m759 @ 7:07 AM

A Sunrise
for Sunrise

"If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World, for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit.  From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete."

Rosalind Krauss, "Grids"

Yesterday's entry featured a rather simple-minded example from Krauss of how the ninefold square (said to be a symbol of Apollo)

The 3x3 grid

may be used to create a graphic design– a Greek cross, which appears also in crossword puzzles:

Crossword-puzzle design that includes Greek-cross elements

Illustration by
Paul Rand
(born Peretz Rosenbaum)

A more sophisticated example
of the ninefold square
in graphic design:

"That old Jew
gave me this here."

— A Flag for Sunrise  

The 3x3 grid as an organizing frame for Chinese calligraphy. Example-- the character for 'sunrise'
From Paul-Rand.com

Saturday, August 2, 2008

Saturday August 2, 2008

Filed under: General,Geometry — m759 @ 2:02 PM
Geometry and Death

(continued from
June 15, 2007)

Today is the anniversary
of the 1955 death of poet
Wallace Stevens.

Related material:

A poem by Stevens,

an essay on  the
relationships between
poets and philosophers —
“Bad Blood,” by
Leonard Michaels

and

The ninefold square, a symbol of Apollo

the Log24 entries
of June 14-15, 2007
.

Sunday, July 29, 2007

Sunday July 29, 2007

Filed under: General,Geometry — m759 @ 9:00 AM
The Ninefold Square

“This translation plane is defined by
a spreadset in a 2-dimensional
vector space over the field GF(3),
consisting of the following matrices.”


 

Priv.-Doz. Dr. H. Klein,
Arbeitsgruppe Geometrie,
Mathematical Seminar of
Christian-Albrechts University

(See Log24, The Nine
and Translation Plane
for Rosh Hashanah
.)

Tuesday, May 1, 2007

Tuesday May 1, 2007

Filed under: General — m759 @ 2:45 AM
May 1, 2007
2:45 AM

I could tell you a lot,
but you gotta be
true to your code.
— Sinatra

At the still point…
— Eliot

George Tenet, At the Center of the Storm

da ist der Tanz;
Doch weder Stillstand noch Bewegung.
Und nenne es nicht Beständigkeit,
Wo Vergangenheit und Zukunft sich sammeln.

 
IMAGE- Scenes from 'Der Einsatz' with ninefold square

 
to put one's back
into something
bei etwas
Einsatz zeigen
to up the ante
den Einsatz erhöhen
to debrief den Einsatz
nachher besprechen
to be on duty
im Einsatz sein
mil.to be in action im Einsatz sein
to play for
high stakes
mit hohem
Einsatz spielen

"Nine is a very
powerful Nordic number
."
— Katherine Neville,
The Magic Circle

Happy Walpurgisnacht.
 

Sunday, November 26, 2006

Sunday November 26, 2006

Filed under: General — m759 @ 7:26 AM

Rosalind Krauss
in "Grids," 1979:

"If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World, for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter.  They are talking about Being or Mind or Spirit.  From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete.

Or, to take a more up-to-date example…."

"He was looking at
the nine engravings
and at the circle,
checking strange
correspondences
between them."
The Club Dumas,1993

"And it's whispered that soon
if we all call the tune
Then the piper will lead us
to reason."
Robert Plant,1971

The nine engravings of
The Club Dumas
(filmed as "The Ninth Gate")
are perhaps more an example
of the concrete than of the
universal.

An example of the universal*–
or, according to Krauss, a
"staircase" to the universal–
is the ninefold square:

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

"This is the garden of Apollo,
the field of Reason…."
John Outram, architect    

For more on the field
of reason, see
Log24, Oct. 9, 2006.

A reasonable set of
"strange correspondences"
in the garden of Apollo
has been provided by Ezra Brown
in a mathematical essay (pdf).

Unreason is, of course,
more popular.

* The ninefold square is perhaps a "concrete universal" in the sense of Hegel:

"Two determinations found in all philosophy are the concretion of the Idea and the presence of the spirit in the same; my content must at the same time be something concrete, present. This concrete was termed Reason, and for it the more noble of those men contended with the greatest enthusiasm and warmth. Thought was raised like a standard among the nations, liberty of conviction and of conscience in me. They said to mankind, 'In this sign thou shalt conquer,' for they had before their eyes what had been done in the name of the cross alone, what had been made a matter of faith and law and religion– they saw how the sign of the cross had been degraded."

— Hegel, Lectures on the History of Philosophy, "Idea of a Concrete Universal Unity"

"For every kind of vampire,
there is a kind of cross."
— Thomas Pynchon   
 

Saturday, September 16, 2006

Saturday September 16, 2006

Filed under: General — m759 @ 11:07 AM

Pandora's Box

Part I:
The Pandora Cross

"There is no painter in the West who can be unaware of the symbolic power of the cruciform shape and the Pandora's box of spiritual reference that is opened once one uses it."

— Rosalind Krauss in "Grids"


The image “http://www.log24.com/log/pix06A/060916-Art.jpg” cannot be displayed, because it contains errors.

(See Log24, Sept. 13)

Part II:
The Opening

Remarks by the Pope on Sept. 12,
as reported by the Vatican:

Faith, Reason, and the University:
Memories and Reflections

For the result of
the Pope's remarks, see
a transcript of
 yesterday's Google News
and the following
from BBC today:

The image “http://www.log24.com/log/pix06A/060916-Benedict16.jpg” cannot be displayed, because it contains errors.
Click to enlarge the screenshot.

Part III:
Hope

The New Yorker (issue of June 5, 2006) on the late Oriana Fallaci:

"In September [2005], she had a private audience with Pope Benedict XVI at Castel Gandolfo, his summer residence outside Rome. She had criticized John Paul II for making overtures to Muslims, and for not condemning terrorism heartily enough, but she has hopes for Joseph Ratzinger."

For further details, see yesterday's Log24.


Part IV:
The Sibyl's Song

The image “http://www.log24.com/log/pix06A/060916-MC7.GIF” cannot be displayed, because it contains errors.

— From The Magic Circle,
 a spiritual narrative
 by Katherine Neville

For more on "the long-mute voice
of the past," on "darkness beneath
the volcano," and on uncorking,
see Glory Season and Harrowing.

Related material from
Log24 on Dec. 2, 2005:

Benedict XVI, before he became Pope:

"… a purely harmonious concept of beauty is not enough…. Apollo, who for Plato's Socrates was 'the God' and the guarantor of unruffled beauty as 'the truly divine' is absolutely no longer sufficient."

A symbol of Apollo:

IMAGE- The ninefold square

and a related
Christian symbol,

The image �http://www.log24.com/log/pix05B/051202-Cross.gif� cannot be displayed, because it contains errors.

the Greek Cross
(adapted from
Ad Reinhardt).

Moral of the Pandora Cross:

"Nine is a very powerful Nordic number."
— Katherine Neville in The Magic Circle…

quoted in The Nine, a Log24 entry
for Hermann Weyl's birthday,
November 9, 2004.
 

Saturday, May 14, 2005

Saturday May 14, 2005

Filed under: General — m759 @ 9:00 PM
The Nine

"Nine is a very powerful Nordic number."
— Katherine Neville, author of The Eight,

IMAGE- Cover of 'The Magic Circle,' by Katherine Neville

in The Magic Circle

"To live is to defend a form."
("Leben, das heisst eine Form verteidigen")
attributed to Hölderlin

In defense of the nine-square grid:

Constructing a translation plane based on the ninefold square

For details on the above picture, see
Translation Plane.

Sunday, April 27, 2003

Sunday April 27, 2003

Filed under: General,Geometry — m759 @ 3:24 PM

ART WARS:

Graphical Password

From a summary of “The Design and Analysis of Graphical Passwords“:

“Results from cognitive science show that people can remember pictures much better than words….

The 5×5 grid creates a good balance between security and memorability.”

 Ian Jermyn, New York University; Alain Mayer, Fabian Monrose, Michael K. Reiter, Bell Labs, Lucent Technologies; Aviel Rubin, AT&T Labs — Research

Illustration — Warren Beatty as
a graphical password:

Town & Country,”
released April 27, 2001

Those who prefer the simplicity of a 3×3 grid are referred to my entry of Jan. 9, 2003, Balanchine’s Birthday.  For material related to the “Town & Country” theme and to Balanchine, see Leadbelly Under the Volcano (Jan. 27, 2003). (“Sometimes I live in the country, sometimes I live in town…” – Huddie Ledbetter).  Those with more sophisticated tastes may prefer the work of Stephen Ledbetter on Gershwin’s piano preludes or, in view of Warren Beatty’s architectural work in “Town & Country,” the work of Stephen R. Ledbetter on window architecture.

As noted in Balanchine’s Birthday, Apollo (of the Balanchine ballet) has been associated by an architect with the 3×3, or “ninefold” grid.  The reader who wishes a deeper meditation on the number nine, related to the “Town & Country” theme and more suited to the fact that April is Poetry Month, is referred to my note of April 27 two years ago, Nine Gates to the Temple of Poetry.

Intermediate between the simplicity of the 3×3 square and the (apparent) complexity of the 5×5 square, the 4×4 square offers an introduction to geometrical concepts that appears deceptively simple, but is in reality fiendishly complex.  See Geometry for Jews.  The moral of this megilla?

32 + 42 = 52.

But that is another story.

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