Log24

Friday, December 1, 2017

The Architect and the Matrix

Filed under: General,Geometry — m759 @ 1:00 PM

In memory of Yale art historian Vincent Scully, who reportedly
died at 97 last night at his home in Lynchburg, Va., some remarks
from the firm of architect John Outram and from Scully —

Update from the morning of December 2 —

The above 3×3 figure is of course not unrelated to
the 4×4 figure in The Matrix for Quantum Mystics:

 .

See as well Tsimtsum in this journal.

Harold Bloom on tsimtsum as sublimation

Thursday, June 21, 2018

Cavell’s Matrix

Filed under: General — Tags: , — m759 @ 3:00 AM

From an obituary for Stanley Cavell, Harvard philosopher
who reportedly died at 91 on Tuesday,  June 19:

The London Review of Books  weblog yesterday —

"Michael Wood reviewed [Cavell’s] 
Philosophy the Day after Tomorrow  in 2005:

'The ordinary slips away from us. If we ignore it, we lose it.
If we look at it closely, it becomes extraordinary, the way
words or names become strange if we keep staring at them.
The very notion turns into a baffling riddle.' "

See also, in this  journal, Tuesday morning's Ici vient M. Jordan  and
this  morning's previous post.

Update of 3:24 AM from my RSS feed —

Friday, November 24, 2017

The Matrix Meets the Grid

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

The Matrix

  The Grid —

  Picturing the Witt Construction

     "Read something that means something." — New Yorker  ad

Sunday, March 5, 2017

The Omega Matrix

Filed under: General,Geometry — Tags: — m759 @ 5:00 PM

Richard Evan Schwartz on
the mathematics of the 4×4 square

See also Priority in this journal.

Monday, June 15, 2015

Omega Matrix

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

See that phrase in this journal.

See also last night's post.

The Greek letter Ω is customarily used to
denote a set that is acted upon by a group.
If the group is the affine group of 322,560
transformations of the four-dimensional
affine space over the two-element Galois
field, the appropriate Ω is the 4×4 grid above.

See the Cullinane diamond theorem.

Sunday, August 3, 2014

The Omega Matrix

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

Shown below is the matrix Omega from notes of Richard Evan Schwartz.
See also earlier versions (1976-1979) by Steven H. Cullinane.

IMAGE- The matrix Omega from notes of Richard Evan Schwartz. See also earlier versions (1977-1979) by Steven H. Cullinane.

Backstory:  The Schwartz Notes (June 1, 2011), and Schwartz on
the American Mathematical Society's current home page:

(Click to enlarge.)

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.

Saturday, September 21, 2019

Annals of Random Fandom

Filed under: General — Tags: , — m759 @ 5:46 PM

For Dan Brown fans …

… and, for fans of The Matrixanother tale
from the above death date: May 16, 2019 —

An illustration from the above
Miracle Octad Generator post:

Related mathematics — Tetrahedron vs. Square.

Thursday, February 7, 2019

Geometry of the 4×4 Square: The Kummer Configuration

Filed under: General — Tags: , — m759 @ 12:00 AM

From the series of posts tagged Kummerhenge

A Wikipedia article relating the above 4×4 square to the work of Kummer —

A somewhat more interesting aspect of the geometry of the 4×4 square
is its relationship to the 4×6 grid underlying the Miracle Octad Generator
(MOG) of R. T. Curtis.  Hudson's 1905 classic Kummer's Quartic Surface
deals with the Kummer properties above and also foreshadows, without
explicitly describing, the finite-geometry properties of the 4×4 square as
a finite affine 4-space — properties that are of use in studying the Mathieu
group M24  with the aid of the MOG.

Thursday, July 12, 2018

Kummerhenge Illustrated

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

      

“… the utterly real thing in writing is the only thing that counts…."

— Maxwell Perkins to Ernest Hemingway, Aug. 30, 1935

"Omega is as real  as we need it to be."

— Burt Lancaster in "The Osterman Weekend"

Thursday, June 21, 2018

Kummerhenge

Filed under: General,Geometry — Tags: , — m759 @ 2:19 AM

See also the Omega Matrix in this  journal.

Tuesday, June 19, 2018

Ici vient M. Jordan

Filed under: General,Geometry — Tags: , — m759 @ 2:13 AM

NY Times correction, online June 16, about 'Here Comes Mr. Jordan' and 'Heaven Can Wait'

See also this  journal on Saturday morning, June 16.

Monday, November 27, 2017

The Golay Code via Witt’s Construction

Filed under: General,Geometry — Tags: — m759 @ 2:08 PM

Reference

Hansen, Robert Peter, "Construction and Simplicity of
the Large Mathieu Groups" (2011). Master's Theses. 4053. 
http://scholarworks.sjsu.edu/etd_theses/4053.

See also The Matrix Meets the Grid (Log24, Nov. 24).
More generally, see SPLAG in this journal.

Tuesday, March 7, 2017

Signature Backdrop

Filed under: General — Tags: — m759 @ 9:00 PM

"The Bitter End’s signature stage backdrop —
a bare 150-year-old brick wall — helped distinguish it from
other popular bohemian hangouts like the Village Gate  
and the Village Vanguard. It appeared on the cover of
Peter, Paul and Mary’s first album."

The New York Times  this evening on a Sunday death

Commentary

“Looking carefully at Golay’s code is like staring into the sun.”

— Richard Evan Schwartz

See also Schwartz in "The Omega Matrix," a post of 5 PM ET Sunday:

Tuesday, May 24, 2016

Rosenhain and Göpel Revisited

The authors Taormina and Wendland in the previous post
discussed some mathematics they apparently did not know was
related to a classic 1905 book by R. W. H. T. Hudson, Kummer's
Quartic Surface
.

"This famous book is a prototype for the possibility
of explaining and exploring a many-faceted topic of
research, without focussing on general definitions,
formal techniques, or even fancy machinery. In this
regard, the book still stands as a highly recommendable,
unparalleled introduction to Kummer surfaces, as a
permanent source of inspiration and, last but not least, 
as an everlasting symbol of mathematical culture."

— Werner Kleinert, Mathematical Reviews ,
     as quoted at Amazon.com

Some 4×4 diagrams from that book are highly relevant to the
discussion by Taormina and Wendland of the 4×4 squares within
the 1974 Miracle Octad Generator of R. T. Curtis that were later,
in 1987, described by Curtis as pictures of the vector 4-space over
the two-element Galois field GF(2).

Hudson did not think of his 4×4 diagrams as illustrating a vector space,
but he did use them to picture certain subsets of the 16 cells in each
diagram that he called Rosenhain and Göpel tetrads .

Some related work of my own (click images for related posts)—

Rosenhain tetrads as 20 of the 35 projective lines in PG(3,2)

IMAGE- Desargues's theorem in light of Galois geometry

Göpel tetrads as 15 of the 35 projective lines in PG(3,2)

Anticommuting Dirac matrices as spreads of projective lines

Related terminology describing the Göpel tetrads above

Ron Shaw on symplectic geometry and a linear complex in PG(3,2)

Monday, June 15, 2015

Slow Art

Filed under: General — Tags: — m759 @ 2:03 PM

Slowness is sometimes in the eye of the beholder.

See this journal on Slow Art Day 2015.

Related material: Epistemic States in this journal.

Wednesday, August 13, 2014

Stranger than Dreams*

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

Illustration from a discussion of a symplectic structure 
in a 4×4 array quoted here on January 17, 2014 —

See symplectic structure in this journal.

* The final words of Point Omega , a 2010 novel by Don DeLillo.
See also Omega Matrix in this journal.

Thursday, August 7, 2014

The Omega Mystery

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

See a post,  The Omega Matrix, from the date of her death.

Related material:

"When Death tells a story, you really have to listen."
— Cover of The Book Thief

A scene from the film of the above book —

“Looking carefully at Golay’s code is like staring into the sun.”

— Richard Evan Schwartz

Some context — "Mathematics, Magic, and Mystery" —
See posts tagged April Awareness 2014.

Tuesday, August 5, 2014

The Omega Story

Filed under: General — Tags: — m759 @ 1:00 AM
 

"We tell ourselves stories in order to live…. We interpret what we see, select the most workable of multiple choices. We live entirely, especially if we are writers, by the imposition of a narrative line upon disparate images, by the 'ideas' with which we have learned to freeze the shifting phantasmagoria which is our actual experience."

Joan Didion

See also a post from May 4, 2011 (the date, according to a Google
search, of untitled notes regarding a matrix called Omega).

Monday, August 4, 2014

A Wrinkle in Space

Filed under: General,Geometry — Tags: , — m759 @ 10:30 AM

"There is  such a thing as a tesseract." — Madeleine L'Engle

An approach via the Omega Matrix:

http://www.log24.com/log/pix10A/100619-TesseractAnd4x4.gif

See, too, Rosenhain and Göpel as The Shadow Guests .

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…"

Monday, October 9, 2006

Monday October 9, 2006

Filed under: General,Geometry — m759 @ 9:00 AM
 
ART WARS:
To Apollo
 
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

To Apollo (10/09/02)
Art Wars: Apollo and Dionysus
(10/09/02)
Balanchine's Birthday
(01/09/03)

Art Theory for Yom Kippur
(10/05/03)

A Form
(05/22/04)
Ineluctable
(05/27/04)

A Form, continued
(06/05/04)
Parallelisms
(06/06/04)
Ado
(06/25/04)

Deep Game
(06/26/04)
Gameplayers of Zen
(06/27/04)
And So To Bed
(06/29/04)
Translation Plane for Rosh Hashanah
(09/15/04)
Derrida Dead
(10/09/04)
The Nine
(11/09/04)
From Tate to Plato
(11/19/04)
Art History
(05/11/05)
A Miniature Rosetta Stone
(08/06/05)
High Concept
(8/23/05) 
High Concept, Continued
(8/24/05)
Analogical Train of Thought
(8/25/05)
Today's Sermon: Magical Thinking
(10/09/05)
Balance
(10/31/05)
Matrix
(11/01/05)
Seven is Heaven, Eight is a Gate
(11/12/05)
Nine is a Vine
(11/12/05)
Apollo and Christ
(12/02/05)
Hamilton's Whirligig
(01/05/06)
Cross
(01/06/06)
On Beauty
(01/26/06)
Sunday Morning
(01/29/06)
Centre
(01/29/06)
New Haven
(01/29/06) 
Washington Ballet
(02/05/06)
Catholic Schools Sermon
(02/05/06)
The Logic of Apollo
(02/05/06)
Game Boy
(08/06/06)
Art Wars Continued: The Krauss Cross
(09/13/06)
Art Wars Continued: Pandora's Box
(09/16/06)
The Pope in Plato's Cave
(09/16/06)
Today's Birthdays
(09/26/06)
Symbology 101
(09/26/06)

Saturday, July 30, 2005

Saturday July 30, 2005

Filed under: General — Tags: , — m759 @ 11:21 AM

Born today: Laurence Fishburne

Matrix

"The nine-fold square has centre, periphery, axes and diagonals. But all are present only in their bare essentials. It is also a sequence of eight triads. Four pass through the centre and four do not. This is the garden of Apollo, the field of Reason, sheltered by the Gate from the turmoil of the Delta, with its endless cycles of erasure and reinscription. This is the Temple of Solomon, as inscribed, for example, by a nine-fold compartmentation to provide the ground plan of Yale…."

Architects John Outram Associates
    on work at Rice University

Yale Daily News, Jan. 11, 2001:  

    "When New Haven was founded, the city was laid out into a grid of nine squares surrounded by a great wilderness.
    Last year History of Art Professor Emeritus Vincent Scully said the original town plan reflected a feeling that the new city should be sacred.
    Scully said the colony's founders thought of their new Puritan settlement as a 'nine-square paradise on Earth, heaven on earth, New Haven, New Jerusalem.'"

"Real and unreal are two in one:
    New Haven
 Before and after one arrives…."

 — Wallace Stevens,
    "An Ordinary Evening in New Haven,' XXVIII
 

Related material:
 Log24 entries on
St. Peter's Day, 2004

Wednesday, November 5, 2003

Wednesday November 5, 2003

Filed under: General — Tags: , — m759 @ 2:23 PM

Game Over

 "Everything that has a beginning
     has an end."

— The Matrix Revolutions

Matrix, by Knots, Inc., 1979.

"Easy to master — A lifetime to enjoy!"

The object for 2 players (8-adult)
is to be the first to form a line
consisting of 4 different
colored chips.

Imagist Poem

Digital 'tears in the rain'

Image suggesting the 'Go chip' in 'Wild Palms'

(Recall the Go-chip
in Wild Palms.)

Sunday, May 25, 2003

Sunday May 25, 2003

Filed under: General,Geometry — Tags: — m759 @ 7:11 PM

ART WARS

Mental Health Month, Day 25:

Matrix of the Death God

Having dealt yesterday with the Death Goddess Sarah, we turn today to the Death God Abraham.  (See Jacques Derrida, The Gift of Death, University of Chicago Press, 1996.)  For a lengthy list of pictures of this damned homicidal lunatic about to murder his son, see The Text This Week.

 

See, too, The Matrix of Abraham, illustrated below.  This is taken from a book by R. M. Abraham, Diversions and Pastimes, published by Constable and Company, London, in 1933.

The Matrix of Abraham

A summary of the religious import of the above from Princeton University Press:

“Moslems of the Middle Ages were fascinated by pandiagonal squares with 1 in the center…. The Moslems thought of the central 1 as being symbolic of the unity of Allah.  Indeed, they were so awed by that symbol that they often left blank the central cell on which the 1 should be positioned.”

— Clifford A. Pickover, The Zen of Magic Squares, Circles, and Stars, Princeton U. Press, 2002, pp. 71-72

Other appearances of this religious icon on the Web:

On Linguistic Creation

Picasso’s Birthday

A less religious approach to the icon may be found on page 393 of R. D. Carmichael’s Introduction to the Theory of Groups of Finite Order (Ginn, Boston, 1937, reprinted by Dover, 1956).

This matrix did not originate with Abraham but, unlike Neo, I have not yet found its Architect.

Friday, January 10, 2003

Friday January 10, 2003

Filed under: General,Geometry — m759 @ 8:15 PM

Story

“How much story do you want?” 
— George Balanchine

While researching yesterday’s entry on Balanchine, Apollo, and the nine Muses, I came across this architect’s remarks, partially quoted yesterday and continued here:

“The icon that I use for this element is the nine-fold square…. This is the garden of Apollo, the field of Reason….  This is the Temple of Solomon, as inscribed, for example, by a nine-fold compartmentation to provide the ground plan of Yale, as described to me by Professor Hersey.”

Duncanology Part 3

Checking this out yesterday, I came across the following at a Yale University Art Gallery site:

“This exhibition of nine boldly colored, asymmetrically designed quilts selected from a private collection will be displayed in the Matrix Gallery….

With the guidance of Professor Maude Southwell Wahlman, author of ‘Signs and Symbols: African Images in African American Quilts,’ the collector has explored and gathered examples….”

Exploring and gathering examples myself today, I received a book in the mail — W. M. Spackman’s On the Decay of Humanism (Rutgers University Press, 1967) — and picked up a second-hand book at a sale — Barbara Michaels’s Stitches in Time (Harper Collins Publishers, 1995).

The Spackman book includes the following poem at the end:

In sandarac etui for sepulchre
  lies the cered body of a poisoned queen;
     and in her mouth and hair, and at her feet,
     and in the grey folds of her winding-sheet,
  there sifts a dreamy powder, smooth and green,
the magic of an idle sorcerer,
  an ancient spell, cast when the shroud was spun.
     In death her hands clasp amourously a bowl
     that still contains the fragments of her soul,
  a tale of Beauty sought, and Beauty won,
his false lips kissed, and Beauty dead for her.

— Alexander B. Griswold, Princeton ’28, in the
    Nassau Literary Magazine of December 1925

From a synopsis of Michaels‘s Stitches in Time:

“Michaels follows Rachel, a graduate student studying women’s crafts–weaving, spinning, quilting, embroidery–and the superstitions connected with them. Linking all important rites of passage to the garments created as markers of these occasions leads Rachel to her theory: in societies in which magic was practiced, the garment was meant to protect its wearer. She gains evidence that her theory is valid when an evil antique bridal quilt enters her life.”

Although Stitches in Time is about a quilt — stitched, not spun — Griswold’s line

“an ancient spell, cast when the shroud was spun” 

is very closely related to the evil spell in Michaels’s book. 

The above events display a certain synchronicity that Wallace Stevens might appreciate, especially in light of the following remark in a review of Stitches in Time:

“…the premise is too outlandish for even the suspension of disbelief….” (Publishers Weekly, 4/24/95)

Stevens might reply,

The very man despising honest quilts
Lies quilted to his poll in his despite.

— “The Comedian as the Letter C,” Part V

Finally, those who prefer stories to the more formal qualities of pure dance (ballet) pure mathematics (see previous entry), pure (instrumental) music, and pure (abstract, as in quilt designs) art, can consult the oeuvre of Jodie Foster — as in my 

Pearl Harbor Day entry on Buddhism.

An art historian named Griswold — perhaps that very same Griswold quoted above — might have a thing or two to say to Jodie on her recent film “Anna and the King.”  In the April, 1957, issue of The Journal of the Siam Society, Alexander B. Griswold takes issue with Broadway’s and Hollywood’s “grotesque caricature” of Siamese society, and ultimately with Anna herself:

“The real fault lies in the two books they ultimately spring from — The English Governess at the Court of Siam and The Romance of the Harem — both written by Mrs. Anna Leonowens.”

Is a puzzlement.

See also The Diamond 16 Puzzle for some quilt designs.

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