Log24

Monday, March 12, 2018

“Quantum Tesseract Theorem?”

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

Remarks related to a recent film and a not-so-recent film.

For some historical background, see Dirac and Geometry in this journal.

Also (as Thas mentions) after Saniga and Planat —

The Saniga-Planat paper was submitted on December 21, 2006.

Excerpts from this  journal on that date —

A Halmos tombstone and the tale of HAL and the pod bay doors

     "Open the pod bay doors, HAL."

Wednesday, May 11, 2016

Jewel of Odin

Filed under: General — m759 @ 1:00 PM

The tesseract in last night's post Game Theory
suggests a search in Log24 for "Jewel of Odin."

See also Trinkets.

Wednesday, October 10, 2012

Subtitle for Odin’s Day

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

The subtitle of Jack Kerouac's novel Doctor Sax
is Faust Part Three.

Related material—

Types of Ambiguity— Galois Meets Doctor Faustus
(this journal, December 14, 2010).

See also tesseracts of Odin and of Galois.

Wednesday, September 19, 2012

For Odin’s Day*

Filed under: General — m759 @ 12:00 PM

(Mathematics and Narrative, continued)

"My dad has a great expression,"
Steve Sabol told USA TODAY Sports
last year. "He always says, 'Tell me
a fact, and I'll learn. Tell me the truth,
and I believe. But tell me a story,
and it will live in my heart forever.' "

Fact—

Sabol died yesterday.

Truth—

An art gallery in Oslo is exhibiting a tesseract.

Story—

The Jewel of Odin's Treasure Room

(Click to enlarge.)

* I.e., Wednesday. For some apt Nordic spirit,
   see Odin's Day 2012 Trailer.

Wednesday, August 31, 2011

Odin’s Day

Filed under: General — m759 @ 12:00 PM

Today is Wednesday.

O.E. Wodnesdæg  "Woden's day," a Gmc. loan-translation of L. dies Mercurii  "day of Mercury" (cf. O.N. Oðinsdagr , Swed. Onsdag , O.Fris. Wonsdei , M.Du. Wudensdach ). For Woden , see Odin  . — Online Etymology Dictionary

http://www.log24.com/log/pix11B/110831-HopkinsAsOdin.jpg

Above: Anthony Hopkins as Odin in the 2011 film "Thor"

Hugo Weaving as Johann Schmidt in the related 2011 film "Captain America"—

"The Tesseract* was the jewel of Odin's treasure room."

http://www.log24.com/log/pix11B/110831-JohannSchmidt.jpg

Weaving also played Agent Smith in The Matrix Trilogy.

The figure at the top in the circle of 13** "Thor" characters above is Agent Coulson.

"I think I'm lucky that they found out they need somebody who's connected to the real world to help bring these characters all together."

— Clark Gregg, who plays Agent Coulson in "Thor," at UGO.com

For another circle of 13, see the Crystal Skull film implicitly referenced in the Bright Star link from Abel Prize (Friday, Aug. 26, 2011)—

http://www.log24.com/log/pix11B/110831-BrightStar.jpg

Today's New York Times  has a quote about a former mathematician who died on that day (Friday, Aug. 26, 2011)—

"He treated it like a puzzle."

Sometimes that's the best you can do.

* See also tesseract  in this journal.

** For a different arrangement of 13 things, see the cube's 13 axes in this journal.

Tuesday, January 9, 2018

Koen Thas and Quantum Theory

Filed under: General — Tags: — m759 @ 9:23 AM

'General Quantum Theory,' by Koen Thas, Dec. 13, 2017, preprint

This post supplies some background for earlier posts tagged
Quantum Tesseract Theorem.

Monday, January 8, 2018

Raiders of the Lost Theorem

Filed under: General,Geometry — Tags: — m759 @ 11:15 PM
 

The Quantum Tesseract Theorem 

 


 

Raiders —

A Wrinkle in Time
starring Storm Reid,
Reese Witherspoon,
Oprah Winfrey &
Mindy Kaling

 

Time Magazine  December 25, 2017 – January 1, 2018

Saturday, December 23, 2017

The Right Stuff

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 1:12 PM

A figure related to the general connecting theorem  of Koen Thas —

Anticommuting Dirac matrices as spreads of projective lines

Ron Shaw on the 15 lines of the classical generalized quadrangle W(2), a general linear complex in PG(3,2)

See also posts tagged Dirac and Geometry in this  journal.

Those who prefer narrative to mathematics may, if they so fancy, call
the above Thas connecting theorem a "quantum tesseract theorem ."

The Patterning

Filed under: General — Tags: — m759 @ 12:25 PM

See a Log24 search for "Patterning Windows."

Related material (Click for context) —

.

IT Girl (for Sweet Home Alabama)

Filed under: General — Tags: — m759 @ 3:35 AM

Friday, December 22, 2017

IT

Filed under: General,Geometry — Tags: — m759 @ 4:08 PM

Movie marquee on Camazotz, from the 2003 film of 'A Wrinkle in Time'

From a Log24 post of October 10, 2017

Koen Thas, 'Unextendible Mututally Unbiased Bases' (Sept. 2016)

Related material from May 25, 2016 —

Thursday, December 21, 2017

Wrinkles

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

TIME magazine, issue of December 25th, 2017 —

" In 2003, Hand worked with Disney to produce a made-for-TV movie.
Thanks to budget constraints, among other issues, the adaptation
turned out bland and uninspiring. It disappointed audiences,
L’Engle and Hand. 'This is not the dream,' Hand recalls telling herself.
'I’m sure there were people at Disney that wished I would go away.' "

Not the dream?  It was, however, the nightmare, presenting very well
the encounter in Camazotz of Charles Wallace with the Tempter.

From a trailer for the latest version —

Detail:

From the 1962 book —

"There's something phoney in the whole setup, Meg thought.
There is definitely something rotten in the state of Camazotz."

Song adapted from a 1960 musical —

"In short, there's simply not
A more congenial spot
For happy-ever-aftering
Than here in Camazotz!"

Sunday, December 15, 2013

Sermon

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

Odin's Jewel

Jim Holt, the author of remarks in yesterday's
Saturday evening post

"It turns out that the Kyoto school of Buddhism
makes Heidegger seem like Rush Limbaugh—
it’s so rarified, I’ve never been able to
understand it at all. I’ve been knocking my head
against it for years."

Vanity Fair Daily , July 16, 2012

Backstory Odin + Jewel in this journal.

See also Odin on the Kyoto school —

For another version of Odin's jewel, see Log24
on the date— July 16, 2012— that Holt's Vanity Fair
remarks were published. Scroll to the bottom of the
"Mapping Problem continued" post for an instance of
the Galois tesseract —

IMAGE- The Galois tesseract as a four-dimensional vector space, from a diagram by Conway and Sloane in 'Sphere Packings, Lattices, and Groups'

Monday, December 9, 2013

Being There

Filed under: General — Tags: — m759 @ 9:16 PM

Or: The Naked Blackboard Jungle

"…it would be quite a long walk
for him if he had to walk straight across."

The image “http://www.log24.com/log/pix07A/070831-Ant1.gif” cannot be displayed, because it contains errors.

Swiftly Mrs. Who brought her hands… together.

"Now, you see," Mrs. Whatsit said,
"he would be  there, without that long trip.
That is how we travel."

The image “http://www.log24.com/log/pix07A/070831-Ant2.gif” cannot be displayed, because it contains errors.

– A Wrinkle in Time 
Chapter 5, "The Tesseract"

Related material: Machete Math and

Starring the late Eleanor Parker as Swiftly Mrs. Who.

Monday, August 12, 2013

Form

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

The Galois tesseract  appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

The Galois tesseract is the basis for a representation of the smallest 
projective 3-space, PG(3,2), that differs from the representation at
Wolfram Demonstrations Project. For the latter, see yesterday's post.

The tesseract representation underlies the diamond theorem, illustrated
below in its earliest form, also from the above February 1977 article—

IMAGE- Steven H. Cullinane, diamond theorem, from 'Diamond Theory,' Computer Graphics and Art, Vol. 2 No. 1, Feb. 1977, pp. 5-7

As noted in a more recent version, the group described by
the diamond theorem is also the group of the 35 square
patterns within the 1976 Miracle Octad Generator  (MOG) of
R. T. Curtis.

Tuesday, July 9, 2013

Vril Chick

Filed under: General,Geometry — m759 @ 4:30 AM

Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo

Profile picture for "Jo Lyxe" (Josefine Lyche) at Vimeo

Compare to an image of Vril muse Maria Orsitsch.

From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —

Josefine Lyche
Born in 1973 in Bergen, Norway.
Lives and works in Oslo and Berlin.

Keywords (to help place my artwork in the
proper context): Aliens, affine geometry, affine
planes, affine spaces, automorphisms, binary
codes, block designs, classical groups, codes,
coding theory, collineations, combinatorial,
combinatorics, conjugacy classes, the Conwell
correspondence, correlations, Cullinane,
R. T. Curtis, design theory, the diamond theorem,
diamond theory, duads, duality, error correcting
codes, esoteric, exceptional groups,
extraterrestrials, finite fields, finite geometry, finite
groups, finite rings, Galois fields, generalized
quadrangles, generators, geometry, GF(2),
GF(4), the (24,12) Golay code, group actions,
group theory, Hadamard matrices, hypercube,
hyperplanes, hyperspace, incidence structures,
invariance, Karnaugh maps, Kirkman’s schoolgirls
problem, Latin squares, Leech lattice, linear
groups, linear spaces, linear transformations,
Magick, Mathieu groups, matrix theory, Meno,
Miracle Octad Generator, MOG, multiply transitive
groups, occultism, octahedron, the octahedral
group, Orsic, orthogonal arrays, outer automorphisms,
parallelisms, partial geometries,
permutation groups, PG(3,2), Plato, Platonic
solids, polarities, Polya-Burnside theorem, projective
geometry, projective planes, projective
spaces, projectivities, Pythagoras, reincarnation,
Reed-Muller codes, the relativity problem,
reverse engineering, sacred geometry, Singer
cycle, skew lines, Socrates, sporadic simple
groups, Steiner systems, Sylvester, symmetric,
symmetry, symplectic, synthemes, synthematic,
Theosophical Society tesseract, Tessla, transvections,
Venn diagrams, Vril society, Walsh
functions, Witt designs.

(See also the original catalog page.)

Clearly most of this (the non-highlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.

For some background, see (for instance) 
Conspiracy Theories and Secret Societies for Dummies .

Tuesday, May 28, 2013

Codes

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

The hypercube  model of the 4-space over the 2-element Galois field GF(2):

IMAGE- A hyperspace model of the 4D vector space over GF(2)

The phrase Galois tesseract  may be used to denote a different model
of the above 4-space: the 4×4 square.

MacWilliams and Sloane discussed the Miracle Octad Generator
(MOG) of R. T. Curtis further on in their book (see below), but did not
seem to realize in 1977 that the 4×4 structures within the MOG are
based on the Galois-tesseract model of the 4-space over GF(2).

IMAGE- Octads within the Curtis MOG, which uses a 4x4-array model of the 4D vector space over GF(2)

The thirty-five 4×4 structures within the MOG:

IMAGE- The 35 square patterns within the Curtis MOG

Curtis himself first described these 35 square MOG patterns
combinatorially, (as his title indicated) rather than
algebraically or geometrically:

IMAGE- R. T. Curtis's combinatorial construction of 4x4 patterns within the Miracle Octad Generator

A later book co-authored by Sloane, first published in 1988,
did  recognize the 4×4 MOG patterns as based on the 4×4
Galois-tesseract model.

Between the 1977 and 1988 Sloane books came the diamond theorem.

Update of May 29, 2013:

The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977
(the year the above MacWilliams-Sloane book was first published):

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

Saturday, April 13, 2013

Princeton’s Christopher Robin

The title is that of a talk (see video) given by
George Dyson at a Princeton land preservation trust,
reportedly on March 21, 2013.  The talk's subtitle was
"Oswald Veblen and the Six-hundred-acre Woods."

Meanwhile

Thursday, March 21, 2013

Geometry of Göpel Tetrads (continued)

m759 @ 7:00 PM

An update to Rosenhain and Göpel Tetrads in PG(3,2)
supplies some background from
Notes on Groups and Geometry, 1978-1986,
and from a 2002 AMS Transactions  paper.

IMAGE- Göpel tetrads in an inscape, April 1986

Related material for those who prefer narrative
to mathematics:

Log24 on June 6, 2006:

 

The Omen:


Now we are 
 

6!

Related material for those who prefer mathematics
to narrative:

What the Omen narrative above and the mathematics of Veblen
have in common is the number 6. Veblen, who came to
Princeton in 1905 and later helped establish the Institute,
wrote extensively on projective geometry.  As the British
geometer H. F. Baker pointed out,  6 is a rather important number
in that discipline.  For the connection of 6 to the Göpel tetrads
figure above from March 21, see a note from May 1986.

See also last night's Veblen and Young in Light of Galois.

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

Thursday, January 24, 2013

Moondance

Filed under: General,Geometry — Tags: — m759 @ 9:01 PM

The title was suggested by an ad for a film that opens
at 10 PM EST today: "Hansel & Gretel: Witch Hunters."

Related material: Grimm Day 2012, as well as
Amy Adams in Raiders of the Lost Tesseract
and in a Film School Rejects page today.

See also some Norwegian art in
Trish Mayo's Photostream today and in
Omega Point (Log24, Oct. 15, 2012)—

Monday, October 15, 2012

Omega Point

m759 @ 2:00 PM 

For Sergeant-Major America—

IMAGE- Art exhibition with 'Omega Point' and geometric figures related to tesseract, along with movie 'Captain America' figure

The image is from posts of Feb. 20, 2011,
and Jan. 27, 2012.

This instance of the omega point is for 
a sergeant major who died at 92 on Wednesday,
October 10, 2012.

See also posts on that date in this journal—

Midnight,  Ambiguation,  Subtitle for Odin's Day,
 and Melancholia, Depression, Ambiguity.

Monday, October 15, 2012

Omega Point

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

For Sergeant-Major America—

IMAGE- Art exhibition with 'Omega Point' and geometric figures related to tesseract, along with movie 'Captain America' figure

The image is from posts of Feb. 20, 2011, and Jan. 27, 2012.

This instance of the omega point is for a sergeant major
who died at 92 on Wednesday, October 10, 2012.

See also posts on that date in this journal—

Midnight,  Ambiguation,  Subtitle for Odin's Day,  and
Melancholia, Depression, Ambiguity.

Wednesday, September 19, 2012

Art Wars (continued)

Filed under: General,Geometry — m759 @ 8:00 PM

Today's previous post, "For Odin's Day," discussed 
a mathematical object, the tesseract, from a strictly
narrative point of view.

In honor of George Balanchine, Odin might yield the
floor this evening to Apollo.

From a piece in today's online New York Times  titled
"How a God Finds Art (the Abridged Version)"—

"… the newness at the heart of this story,
in which art is happening for the first time…."

Some related art

IMAGE- Figure from Plato's Meno in version by Benjamin Jowett, Master of Balliol College, Oxford

and, more recently

This more recent figure is from Ian Stewart's 1996 revision 
of a 1941 classic, What Is Mathematics? , by Richard Courant
and Herbert Robbins.

Apollo might discuss with Socrates how the confused slave boy
of Plato's Meno  would react to Stewart's remark that

"The number of copies required to double an
 object's size depends on its dimension."

Apollo might also note an application of Socrates' Meno  diagram
to the tesseract of this afternoon's Odin post

.

Thursday, May 10, 2012

For Thor’s Day

Filed under: General — m759 @ 7:11 PM

Today's previous post was "Midnight in Oslo (continued)."

The link "a 4-element set" in "Midnight"
was to a more elaborate structure in a post titled "Tesseract."

In memory of an Oslo "hero of midnight"
(a phrase quoted here last September 1)—

A search for material that is more entertaining
Odin 's Tesseract.

See also a related Hollywood story in The Washington Post .

Friday, January 27, 2012

Mathematics and Narrative (continued)

Filed under: General,Geometry — Tags: — m759 @ 1:44 AM

Princeton University Press on a book it will publish in March—

Circles Disturbed: The Interplay of Mathematics and Narrative

"Circles Disturbed  brings together important thinkers in mathematics, history, and philosophy to explore the relationship between mathematics and narrative. The book's title recalls the last words of the great Greek mathematician Archimedes before he was slain by a Roman soldier— 'Don't disturb my circles'— words that seem to refer to two radically different concerns: that of the practical person living in the concrete world of reality, and that of the theoretician lost in a world of abstraction. Stories and theorems are, in a sense, the natural languages of these two worlds–stories representing the way we act and interact, and theorems giving us pure thought, distilled from the hustle and bustle of reality. Yet, though the voices of stories and theorems seem totally different, they share profound connections and similarities."

Timeline of the Marvel Cinematic Universe — Norway, March 1942

"The Red Skull finds the Tesseract, a cube of strange power,
said to be the jewel of Odin’s treasure room, in Tonsberg Norway.
 (Captain America: The First Avenger)"

Tesseracts Disturbed — (Click to enlarge)

Detail of Tesseracts Disturbed —

Narrative of the detail—

See Tesseract in this journal and Norway, May 2010

The Oslo Version and Annals of Conceptual Art.

"Oh, what a tangled web we weave…"

Sunday, September 11, 2011

First Lady

Filed under: General — m759 @ 7:59 AM

Betty Skelton, "the First Lady of Firsts," died on the last day of August.

http://www.log24.com/log/pix11B/110911-NYTobitsSm.jpg

From this  journal on August thirty-first—

"The Tesseract was the jewel of Odin's treasure room."

http://www.log24.com/log/pix11B/110831-JohannSchmidt.jpg

Hugo Weaving also played Agent Smith
in The Matrix Trilogy .

For Cynthia Zarin, biographer of Madeleine L'Engle

"There is  such a thing as a tesseract."
A Wrinkle in Time

Thursday, September 1, 2011

Boundary (continued*)

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

It is now midnight. Yesterday was Odin's Day. Today is Thor's Day.

From a weblog post on Captain America and Thor

"While all this [Captain America] is happening an SS officer, Johann Schmidt (Hugo Weaving), has found a religious artefact called the Tesseract which Schmidt describes as 'the jewel of Odin’s treasure room,' linking it in with the Thor storyline."

That's Entertainment  weblog, August 14, 2011

From Wallace Stevens, "An Ordinary Evening in New Haven," Canto III—

The point of vision and desire are the same.
It is to the hero of midnight that we pray
On a hill of stones to make beau mont thereof.

Captain America opened in the United States on Friday, July 22, 2011.

Thor opened in the United States on Friday, May 6, 2011.

"There is  such a thing as a tesseract." —A Wrinkle in Time

* Continued from August 30.

Wednesday, July 6, 2011

Nordstrom-Robinson Automorphisms

Filed under: General,Geometry — Tags: , — m759 @ 1:01 AM

A 2008 statement on the order of the automorphism group of the Nordstrom-Robinson code—

"The Nordstrom-Robinson code has an unusually large group of automorphisms (of order 8! = 40,320) and is optimal in many respects. It can be found inside the binary Golay code."

— Jürgen Bierbrauer and Jessica Fridrich, preprint of "Constructing Good Covering Codes for Applications in Steganography," Transactions on Data Hiding and Multimedia Security III, Springer Lecture Notes in Computer Science, 2008, Volume 4920/2008, 1-22

A statement by Bierbrauer from 2004 has an error that doubles the above figure—

The automorphism group of the binary Golay code G is the simple Mathieu group M24 of order |M24| = 24 × 23 × 22 × 21 × 20 × 48 in its 5-transitive action on the 24 coordinates. As M24 is transitive on octads, the stabilizer of an octad has order |M24|/759 [=322,560]. The stabilizer of NR has index 8 in this group. It follows that NR admits an automorphism group of order |M24| / (759 × 8 ) = [?] 16 × 7! [=80,640]. This is a huge symmetry group. Its structure can be inferred from the embedding in G as well. The automorphism group of NR is a semidirect product of an elementary abelian group of order 16 and the alternating group A7.

— Jürgen Bierbrauer, "Nordstrom-Robinson Code and A7-Geometry," preprint dated April 14, 2004, published in Finite Fields and Their Applications , Volume 13, Issue 1, January 2007, Pages 158-170

The error is corrected (though not detected) later in the same 2004 paper—

In fact the symmetry group of the octacode is a semidirect product of an elementary abelian group of order 16 and the simple group GL(3, 2) of order 168. This constitutes a large automorphism group (of order 2688), but the automorphism group of NR is larger yet as we saw earlier (order 40,320).

For some background, see a well-known construction of the code from the Miracle Octad Generator of R.T. Curtis—

Click to enlarge:

IMAGE - The 112 hexads of the Nordstrom-Robinson code

For some context, see the group of order 322,560 in Geometry of the 4×4 Square.

Monday, June 21, 2010

Test

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

From a post by Ivars Peterson, Director
of Publications and Communications at
the Mathematical Association of America,
at 19:19 UTC on June 19, 2010—

Exterior panels and detail of panel,
Michener Gallery at Blanton Museum
in Austin, Texas—

http://www.log24.com/log/pix10A/100621-MichenerGalleryPanel.jpg

Peterson associates the four-diamond figure
with the Pythagorean theorem.

A more relevant association is the
four-diamond view of a tesseract shown here
on June 19 (the same date as Peterson's post)
in the "Imago Creationis" post—

Image-- The Four-Diamond Tesseract

This figure is relevant because of a
tesseract sculpture by Peter Forakis—

http://www.log24.com/log/pix09A/091220-ForakisHypercube.jpg

This sculpture was apparently shown in the above
building— the Blanton Museum's Michener gallery—
as part of the "Reimagining Space" exhibition,
September 28, 2008-January 18, 2009.

The exhibition was organized by
Linda Dalrymple Henderson, Centennial Professor
in Art History at the University of Texas at Austin
and author of The Fourth Dimension and
Non-Euclidean Geometry in Modern Art
(Princeton University Press, 1983;
new ed., MIT Press, 2009).

For the sculptor Forakis in this journal,
see "The Test" (December 20, 2009).

"There is  such a thing
as a tesseract."
A Wrinkle in TIme   

Monday, May 5, 2008

Monday May 5, 2008

Filed under: General — Tags: , — m759 @ 11:07 AM
Lottery Sermon

"And take upon's
the mystery of things
 as if we were God's spies"
King Lear  

PA Lottery Sunday, May 4, 2008: mid-day 170, evening 144

From Log24 on Aug. 19, 2003
and on Ash Wednesday, 2004:
a reviewer on
An Instance of the Fingerpost::

"Perhaps we are meant to
see the story as a cubist
   retelling of the crucifixion."

From Log24 on
Michaelmas 2007:

Kate Beckinsale (in 'Pearl Harbor') pointing to an instance of the number 144

Google searches suggested by
Sunday's PA lottery numbers
(mid-day 170, evening 144)
and by the above
figure of Kate Beckinsale
pointing to an instance of
the number 144 —

Click to enlarge:

Search for the meaning of 170 and 144, the PA lottery numbers of Sunday, May 4, 2008

Related material:

Beckinsale in another film
(See At the Crossroads,
Log24, Dec. 8, 2006):

"For every kind of vampire,
there is a kind of cross."
Gravity's Rainbow  
 
Kate Beckinsale in Underworld: Evolution

 

Kate Beckinsale, adapted from
poster for Underworld: Evolution
(DVD release date 6/6/6)
 
There is such a thing
as a tesseract.

"It was only in retrospect
that the silliness
became profound."

— Review of  
Faust in Copenhagen

From the conclusion of
Joan Didion's 1970 novel
  Play It As It Lays

Cover of 'Play It As It Lays'

"I know what 'nothing' means,
and keep on playing."

From Play It As It Lays,
the paperback edition of 1990
  (Farrar, Straus and Giroux) —

Page 170:

"By the end of a week she was thinking constantly
about where her body stopped and the air began,
about the exact point in space and time that was the
difference between Maria and other. She had the sense
that if she could get that in her mind and hold it for

170  

even one micro-second she would have what she had
come to get."

"The page numbers
are generally reliable."

Michaelmas 2007   

Friday, September 7, 2007

Friday September 7, 2007

Filed under: General — Tags: — m759 @ 2:02 PM

The New York Times online,
Friday, Sept. 7, 2007:

Madeleine L’Engle,
Children’s Writer,
Is Dead

"Madeleine L’Engle, who in writing more than 60 books, including childhood fables, religious meditations and science fiction, weaved emotional tapestries transcending genre and generation, died Thursday [Sept. 6, 2007] in Connecticut. She was 88.

Her death, of natural causes, was announced today by her publisher, Farrar, Straus and Giroux."

More >>

Related material:

Log24 entries of
August 31

"That is how we travel."

The image “http://www.log24.com/log/pix07A/070831-Ant2.gif” cannot be displayed, because it contains errors.

A Wrinkle in Time,
Chapter 5,
"The Tesseract"

— and of 
September 2
(with update of
 September 5)–

"There is such a thing
as a tesseract."
A Wrinkle in Time  

Sunday, September 2, 2007

Sunday September 2, 2007

Filed under: General,Geometry — Tags: — m759 @ 5:11 PM

Comment at the
n-Category Cafe

Re: This Week’s Finds in Mathematical Physics (Week 251)

On Spekkens’ toy system and finite geometry

Background–

  • In “Week 251” (May 5, 2007), John wrote:
    “Since Spekkens’ toy system resembles a qubit, he calls it a “toy bit”. He goes on to study systems of several toy bits – and the charming combinatorial geometry I just described gets even more interesting. Alas, I don’t really understand it well: I feel there must be some mathematically elegant way to describe it all, but I don’t know what it is…. All this is fascinating. It would be nice to find the mathematical structure that underlies this toy theory, much as the category of Hilbert spaces underlies honest quantum mechanics.”
  • In the n-Category Cafe ( May 12, 2007, 12:26 AM, ) Matt Leifer wrote:
    “It’s crucial to Spekkens’ constructions, and particularly to the analog of superposition, that the state-space is discrete. Finding a good mathematical formalism for his theory (I suspect finite fields may be the way to go) and placing it within a comprehensive framework for generalized theories would be very interesting.”
  • In the n-category Cafe ( May 12, 2007, 6:25 AM) John Baez wrote:
    “Spekkens and I spent an afternoon trying to think about his theory as quantum mechanics over some finite field, but failed — we almost came close to proving it couldnt’ work.”

On finite geometry:

The actions of permutations on a 4 × 4 square in Spekkens’ paper (quant-ph/0401052), and Leifer’s suggestion of the need for a “generalized framework,” suggest that finite geometry might supply such a framework. The geometry in the webpage John cited is that of the affine 4-space over the two-element field.

Related material:

Update of
Sept. 5, 2007

See also arXiv:0707.0074v1 [quant-ph], June 30, 2007:

A fully epistemic model for a local hidden variable emulation of quantum dynamics,

by Michael Skotiniotis, Aidan Roy, and Barry C. Sanders, Institute for Quantum Information Science, University of Calgary. Abstract: "In this article we consider an augmentation of Spekkens’ toy model for the epistemic view of quantum states [1]…."
 

Skotiniotis et al. note that the group actions on the 4×4 square described in Spekkens' paper [1] may be viewed (as in Geometry of the 4×4 Square and Geometry of Logic) in the context of a hypercube, or tesseract, a structure in which adjacency is isomorphic to adjacency in the 4 × 4 square (on a torus).

Hypercube from the Skotiniotis paper:

Hypercube

Reference:

[1] Robert W. Spekkens, Phys. Rev. A 75, 032110 (2007),

Evidence for the epistemic view of quantum states: A toy theory
,

Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5 (Received 11 October 2005; revised 2 November 2006; published 19 March 2007.)

"There is such a thing
as a tesseract."
A Wrinkle in Time  
 

Friday, August 31, 2007

Friday August 31, 2007

Filed under: General — Tags: — m759 @ 10:10 PM
Being There

"…it would be quite
a long walk
for him if he had to
walk straight across."

The image “http://www.log24.com/log/pix07A/070831-Ant1.gif” cannot be displayed, because it contains errors.

Swiftly Mrs. Who brought
her hands… together.

"Now, you see,"
Mrs. Whatsit said,
"he would be there,
without that long trip.
That is how we travel."

The image “http://www.log24.com/log/pix07A/070831-Ant2.gif” cannot be displayed, because it contains errors.

A Wrinkle in Time,
Chapter 5,
"The Tesseract"

Related material:


To Measure the Changes
,

Serious Numbers,

and…

 
The image “http://www.log24.com/log/pix06A/061017-Gump2A.jpg” cannot be displayed, because it contains errors.

Balls of Fury
 

Monday, May 21, 2007

Monday May 21, 2007

Filed under: General,Geometry — Tags: — m759 @ 4:00 PM
No Royal Roads
Illustration from a
1980 article at JSTOR:

Coxeter as King of Geometry

A more recent royal reference:

"'Yau wants to be the king of geometry,' Michael Anderson, a geometer at Stony Brook, said. 'He believes that everything should issue from him, that he should have oversight. He doesn't like people encroaching on his territory.'" –Sylvia Nasar and David Gruber in The New Yorker, issue dated Aug. 28, 2006

Wikipedia, Cultural references to the Royal Road:

"Euclid is said to have replied to King Ptolemy's request for an easier way of learning mathematics that 'there is no royal road to geometry.' Charles S. Peirce, in his 'How to Make Our Ideas Clear' (1878), says 'There is no royal road to logic, and really valuable ideas can only be had at the price of close attention.'"

Related material:

Day Without Logic
(March 8, 2007)

and
The Geometry of Logic
(March 10, 2007)
:

The image “http://www.log24.com/log/pix07/070521-Tesseract.gif” cannot be displayed, because it contains errors.

There may be
no royal roads to
geometry or logic,
but…

"There is such a thing
as a tesseract."
— Madeleine L'Engle, 
A Wrinkle in Time

Monday, May 14, 2007

Monday May 14, 2007

Filed under: General — Tags: — m759 @ 3:09 AM

Crossing Point

From Log24's
"Footprints for Baudrillard"–

"Was there really a cherubim
waiting at the star-watching rock…?
Was he real?
What is real?

— Madeleine L'Engle, A Wind in the Door,
Farrar, Straus and Giroux, 1973,
conclusion of Chapter Three,
"The Man in the Night"

"Oh, Euclid, I suppose."

— Madeleine L'Engle, A Wrinkle in Time,
Farrar, Straus and Giroux, 1962,
conclusion of Chapter Five,

"The Tesseract"

From Log24's
Xanga footprints,
3:00 AM today:

 

Texas /431103703/item.html 5/14/2007 3:00 AM

The link leads to a Jan. 23, 2006 entry
on what one philosopher has claimed is
"exactly that crossing point
of constraint and freedom
which is the very essence
of man's nature."

Thursday, May 3, 2007

Thursday May 3, 2007

Filed under: General,Geometry — Tags: — m759 @ 3:00 PM
A Web
of Links

"Some postmodern theorists like to talk about the relationship between 'intertextuality' and 'hypertextuality'; intertextuality makes each text a 'mosaic of quotations' [Kristeva, Desire in Language, Columbia U. Pr., 1980, 66] and part of a larger mosaic of texts, just as each hypertext can be a web of links and part of the whole World-Wide Web." —Wikipedia
 

The image “http://www.log24.com/log/pix07/070503-Tiffany.jpg” cannot be displayed, because it contains errors.

Related material

Day Without Logic,
Introduction to Logic,
The Geometry of Logic,
Structure and Logic,
Spider-Man and Fan:

The image “http://www.log24.com/log/pix07/070503-Devillers.jpg” cannot be displayed, because it contains errors.

"There is such a thing
as a tesseract."
A Wrinkle in Time  
 

Friday, April 20, 2007

Friday April 20, 2007

Filed under: General — Tags: — m759 @ 10:31 PM
Speech

In Grand Rapids today

"… Bush spoke and answered audience questions for nearly 90 minutes inside East Grand Rapids High School in suburban Grand Rapids….

After leaving the school, Bush's motorcade stopped at the Gerald R. Ford Presidential Museum in downtown Grand Rapids, where he stood silently for a few moments after placing a bouquet of white roses at Ford's burial site on the museum grounds. The 38th president, who grew up in Grand Rapids, died Dec. 26 at age 93."

Multispeech

Mich. Lottery Apr. 20, 2007: Day 019, Night 001

 

For the meaning of the lottery icons
above, see this morning's entry and
an entry that it links to —
Time's Labyrinth continued
of March 8, 2007.

For the meaning of multispeech,
see the entries of
All Hallows' Eve, 2005:

Tesseract on the cover of The Gameplayers of Zan
 
"There is such a thing
as a tesseract."
A Wrinkle in Time 
 

Tuesday, April 3, 2007

Tuesday April 3, 2007

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

Mathematics Awareness Month

 
Related material:

"But what is it?"
Calvin demanded.
"We know that it's evil,
but what is it?"

"Yyouu hhave ssaidd itt!"
Mrs. Which's voice rang out.
"Itt iss Eevill. Itt iss thee
Ppowers of Ddarrkknesss!"

A Wrinkle in Time

AMS Notices cover, April 2007

"After A Wrinkle in Time was finally published, it was pointed out to me that the villain, a naked disembodied brain, was called 'It' because It stands for Intellectual truth as opposed to a truth which involves the whole of us, heart as well as mind.  That acronym had never occurred to me.  I chose the name It intuitively, because an IT does not have a heart or soul.  And I did not understand consciously at the time of writing that the intellect, when it is not informed by the heart, is evil."

See also
"Darkness Visible"
in ART WARS.
 
"When all is said and done,
science is about things and
theology is about words."
— Freeman Dyson,
New York Review of Books,
issue dated May 28, 1998

"Does the word 'tesseract'
mean anything to you?"
 

Wednesday, March 7, 2007

Wednesday March 7, 2007

Filed under: General,Geometry — Tags: — m759 @ 8:35 AM
Footprints for
Baudrillard

The image “http://www.log24.com/log/pix07/070307-Baudrillard.jpg” cannot be displayed, because it contains errors.

"Was there really a cherubim
waiting at the star-watching rock…?
Was he real?
What is real?

 

— Madeleine L'Engle, A Wind in the Door,
Farrar, Straus and Giroux, 1973,
conclusion of Chapter Three,
"The Man in the Night"

 

"Oh, Euclid, I suppose."

— Madeleine L'Engle, A Wrinkle in Time,
Farrar, Straus and Giroux, 1962,
conclusion of Chapter Five,
"The Tesseract"

In memory of the French philosopher Jean Baudrillard, who died yesterday, Tuesday, March 6, 2007. 

The following Xanga footprints may be regarded as illustrating Log24 remarks of Dec. 10, 2006 on the Library of Congress, geometry, and bullshit, as well as remarks of Aug. 28, 2006 on the temporal, the eternal, and St. Augustine.

From the District of Columbia–
Xanga footprints in reverse
chronological order from
the noon hour on Tuesday,
March 6, 2007, the date
of Baudrillard's death:

District of Columbia
/499111929/item.html
Beijing String
3/6/2007
12:04 PM
District of Columbia
/497993036/item.html
Spellbound
3/6/2007
12:03 PM
District of Columbia
/443606342/item.html
About God, Life, Death
3/6/2007
12:03 PM
District of Columbia
/494421586/item.html
A Library of Congress Reading
3/6/2007
12:03 PM
District of Columbia
/500434851/item.html
Binary Geometry
3/6/2007
12:03 PM
District of Columbia
/404038913/item.html
Prequel on St. Cecelia's Day
3/6/2007
12:03 PM

Thursday, March 1, 2007

Thursday March 1, 2007

Filed under: General,Geometry — Tags: — m759 @ 6:29 AM

Senior Honors

Notes in Memory of
a Father, a Son, and a Holy Ghost

From the obituary in today's New York Times of historian Arthur M. Schlesinger Jr.–

"Mr. Schlesinger, partly through his appreciation of history, fully realized his good fortune. 'I have lived through interesting times and had the luck of knowing some interesting people,' he wrote.

A huge part of his luck was his father, who guided much of his early research, and even suggested the topic for his [Harvard] senior honors: Orestes A. Brownson, a 19th-century journalist, novelist and theologian. It was published by Little, Brown in 1938 as 'Orestes A. Brownson: A Pilgrim's Progress.'"

Douglas Martin

From The Catholic Encyclopedia:

"It is sufficient for true knowledge that it affirm as real that which is truly real."

Article on Ontologism

From The Diamond Theory of Truth:

"Was there really a cherubim waiting at the star-watching rock…?
Was he real?
What is real?

— Madeleine L'Engle, A Wind in the Door, Farrar, Straus and Giroux, 1973, conclusion of Chapter Three, "The Man in the Night"

"Oh, Euclid, I suppose."

— Madeleine L'Engle, A Wrinkle in Time, Farrar, Straus and Giroux, 1962, conclusion of Chapter Five, "The Tesseract"

Related material: Yesterday's first annual "Tell Your Story Day" at Harvard and yesterday's entry on Euclid.

Friday, December 8, 2006

Friday December 8, 2006

Filed under: General — Tags: — m759 @ 9:00 AM
An Instance
of the Fingerpost
The image “http://www.log24.com/log/pix06B/061208-Date.jpg” cannot be displayed, because it contains errors.
 
"CRUCIAL (from Lat. crux, a cross),
that which has the form of a cross…
 From Francis Bacon's expression
instantia crucis (taken, as he says, from
the finger-post or crux at cross-roads)"
 
Encyclopaedia Britannica,
the classic 11th edition (1911)
 
"For every kind of vampire,
there is a kind of cross."
Gravity's Rainbow  
 
The image “http://www.log24.com/log/pix06A/060614-EvolutionBegins2.jpg” cannot be displayed, because it contains errors.

Kate Beckinsale, adapted from
poster for Underworld: Evolution
(DVD release date 6/6/6)

 
There is such a thing
as a tesseract.
A Wrinkle in Time  
 
Related material:
 
The tesseract on the cover of
The Gameplayers of Zan
(All Hallows' Eve, 2005), and
 
A Last Stitch in Time…or
A Map of the Map
of Kierkegaard's World:

"Appropriating the Button-molder's
words to Peer Gynt, he would say,
'We'll meet at the next crossroads…
and then we'll see–
I won't say more.'"

Tuesday, October 31, 2006

Tuesday October 31, 2006

Filed under: General,Geometry — Tags: , , — m759 @ 11:00 PM
To Announce a Faith

From 7/07, an art review from The New York Times:

Endgame Art?
It's Borrow, Sample and Multiply
in an Exhibition at Bard College

"The show has an endgame, end-time mood….

I would call all these strategies fear of form…. the dismissal of originality is perhaps the oldest ploy in the postmodern playbook. To call yourself an artist at all is by definition to announce a faith, however unacknowledged, in some form of originality, first for yourself, second, perhaps, for the rest of us.

Fear of form above all means fear of compression– of an artistic focus that condenses experiences, ideas and feelings into something whole, committed and visually comprehensible."

— Roberta Smith

It is doubtful that Smith
 would consider the
following "found" art an
example of originality.

It nevertheless does
"announce a faith."


The image “http://www.log24.com/log/pix06A/061031-PAlottery2.jpg” cannot be displayed, because it contains errors.


"First for yourself"

Today's mid-day
Pennsylvania number:
707

See Log24 on 7/07
and the above review.
 

"Second, perhaps,
for the rest of us"

Today's evening
Pennsylvania number:
384

This number is an
example of what the
reviewer calls "compression"–

"an artistic focus that condenses
 experiences, ideas and feelings
into something
whole, committed
 and visually comprehensible."

"Experiences"

See (for instance)

Joan Didion's writings
(1160 pages, 2.35 pounds)
on "the shifting phantasmagoria
which is our actual experience."

"Ideas"

See Plato.

"Feelings"

See A Wrinkle in Time.

"Whole"

The automorphisms
of the tesseract
form a group
of order 384.

"Committed"

See the discussions of
groups of degree 16 in
R. D. Carmichael's classic
Introduction to the Theory
of Groups of Finite Order
.

"Visually comprehensible"

See "Diamond Theory in 1937,"
an excerpt from which
is shown below.

The image “http://www.log24.com/theory/images/Carmichael440abbrev.gif” cannot be displayed, because it contains errors.

The "faith" announced by
the above lottery numbers
on All Hallows' Eve is
perhaps that of the artist
Madeleine L'Engle:

"There is such a thing
as a tesseract.
"

Tuesday, October 3, 2006

Tuesday October 3, 2006

Filed under: General,Geometry — m759 @ 9:26 AM

Serious

"I don't think the 'diamond theorem' is anything serious, so I started with blitzing that."

Charles Matthews at Wikipedia, Oct. 2, 2006

"The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas."

— G. H. Hardy, A Mathematician's Apology

Matthews yesterday deleted references to the diamond theorem and related material in the following Wikipedia articles:

Affine group‎
Reflection group‎
Symmetry in mathematics‎
Incidence structure‎
Invariant (mathematics)‎
Symmetry‎
Finite geometry‎
Group action‎
History of geometry‎

This would appear to be a fairly large complex of mathematical ideas.

See also the following "large complex" cited, following the above words of Hardy, in Diamond Theory:

Affine geometry, affine planes, affine spaces, automorphisms, binary codes, block designs, classical groups, codes, coding theory, collineations, combinatorial, combinatorics, conjugacy classes, the Conwell correspondence, correlations, design theory, duads, duality, error correcting codes, exceptional groups, finite fields, finite geometry, finite groups, finite rings, Galois fields, generalized quadrangles, generators, geometry, GF(2), GF(4), the (24,12) Golay code, group actions, group theory, Hadamard matrices, hypercube, hyperplanes, hyperspace, incidence structures, invariance, Karnaugh maps, Kirkman's schoolgirl problem, Latin squares, Leech lattice, linear groups, linear spaces, linear transformations, Mathieu groups, matrix theory, Meno, Miracle Octad Generator, MOG, multiply transitive groups, octads, the octahedral group, orthogonal arrays, outer automorphisms, parallelisms, partial geometries, permutation groups, PG(3,2), polarities, Polya-Burnside theorem, projective geometry, projective planes, projective spaces, projectivities, Reed-Muller codes, the relativity problem, Singer cycle, skew lines,  sporadic simple groups, Steiner systems, symmetric, symmetry, symplectic, synthemes, synthematic, tesseract, transvections, Walsh functions, Witt designs.

Friday, May 12, 2006

Friday May 12, 2006

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

"Does the word 'tesseract'
mean anything to you?"
— Robert A. Heinlein in
The Number of the Beast
(1980)

My reply–

Part I:

The image “http://www.log24.com/log/pix06A/WrinkleInTime1A.jpg” cannot be displayed, because it contains errors.

A Wrinkle in Time, by
Madeleine L'Engle
(first published in 1962)

Part II:

Diamond Theory in 1937
and
Geometry of the 4×4 Square

Part III:

Catholic Schools Sermon

Conclusion:
 

"Wells and trees were dedicated to saints.  But the offerings at many wells and trees were to something other than the saint; had it not been so they would not have been, as we find they often were, forbidden.  Within this double and intertwined life existed those other capacities, of which we know more now, but of which we still know little– clairvoyance, clairaudience, foresight, telepathy."

— Charles Williams, Witchcraft, Faber and Faber, London, 1941

Related material:

A New Yorker profile of Madeleine L'Engle from April 2004, which I found tonight online for the first time.  For a related reflection on truth, stories, and values, see Saint's Day.  For a wider context, see the Log24 entries of February 1-15, 2003 and February 1-15, 2006.
 

Wednesday, March 29, 2006

Wednesday March 29, 2006

Filed under: General,Geometry — Tags: , , — m759 @ 8:00 PM

The image “http://www.log24.com/theory/images/Carmichael440.gif” cannot be displayed, because it contains errors.
Note: Carmichael's reference is to
A. Emch, "Triple and multiple systems, their geometric configurations and groups," Trans. Amer. Math. Soc. 31 (1929), 25–42.

"There is such a thing as a tesseract."
A Wrinkle in Time

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