Log24

Sunday, December 3, 2023

Red Skull and the Tesseract Box

Filed under: General — Tags: — m759 @ 8:48 pm

Related art by Basquiat

Click on the above image for an Instagram description of its source.
See also related artistic remarks in this  journal on the date of that
Instagram description — October 17, 2022.

Saturday, October 24, 2020

The Galois Tesseract

Filed under: General — Tags: — m759 @ 9:32 am

Stanley E. Payne and J. A. Thas in 1983* (previous post) —

“… a 4×4 grid together with
the affine lines on it is AG(2,4).”

Payne and Thas of course use their own definition
of affine lines on a grid.

Actually, a 4×4 grid together with the affine lines on it
is, viewed in a different way, not AG(2,4) but rather AG(4,2).

For AG(4,2) in the proper context, see
Affine Groups on Small Binary Spaces and
The Galois Tesseract.

* And 26 years later,  in 2009.

Monday, October 15, 2018

Tesserae for a Tesseract

Filed under: General — Tags: , — m759 @ 8:22 pm

The source —

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."

Saturday, May 20, 2017

van Lint and Wilson Meet the Galois Tesseract*

Filed under: General,Geometry — Tags: — m759 @ 12:12 am

Click image to enlarge.

The above 35 projective lines, within a 4×4 array —


The above 15 projective planes, within a 4×4 array (in white) —

* See Galois Tesseract  in this journal.

Thursday, May 11, 2017

Reopening the Tesseract

Filed under: General,Geometry — Tags: , — m759 @ 11:27 am

Dialogue from the film "Interstellar" —

Cooper: Did it work?

TARS: I think it might have.

Cooper: How do you know?

TARS: Because the bulk beings
            are closing the tesseract.

Related material — "Bulk apperception"
in this journal, and

Wednesday, December 28, 2016

Rosetta Tesseracts

Filed under: General — m759 @ 9:00 am

Click to enlarge.

Related material Folk Etymology  (Dec. 10, 2016).

Tuesday, March 24, 2015

Brouwer on the Galois Tesseract

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

Yesterday's post suggests a review of the following —

Andries Brouwer, preprint, 1982:

"The Witt designs, Golay codes and Mathieu groups"
(unpublished as of 2013)

Pages 8-9:

Substructures of S(5, 8, 24)

An octad is a block of S(5, 8, 24).

Theorem 5.1

Let B0 be a fixed octad. The 30 octads disjoint from B0
form a self-complementary 3-(16,8,3) design, namely 

the design of the points and affine hyperplanes in AG(4, 2),
the 4-dimensional affine space over F2.

Proof….

… (iv) We have AG(4, 2).

(Proof: invoke your favorite characterization of AG(4, 2) 
or PG(3, 2), say 
Dembowski-Wagner or Veblen & Young. 

An explicit construction of the vector space is also easy….)

Related material:  Posts tagged Priority.

Tuesday, December 10, 2013

Wittgenstein’s Tesseract

Filed under: General,Geometry — m759 @ 5:14 pm

See also last night's "Pink Champagne on Ice" post.
The "ice" in that post's title refers to the white lines
forming a tesseract in the book cover's background—
"icy white and crystalline," as Johnny Mercer put it.
(A Tune for Josefine, Nov. 25.)

See also the tag Diamond Theory tesseract in this journal.

Saturday, July 6, 2013

The People’s Tesseract*

Filed under: General,Geometry — Tags: , — m759 @ 9:57 am

From Andries Brouwer

Image related, very loosely, to Falstaff's 'green fields'

* Related material:  Yesterday's evening post and The People's Cube
  (By the way, any  4×4 array is a tesseract .)

Thursday, August 16, 2012

Raiders of the Lost Tesseract

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

(Continued from August 13. See also Coxeter Graveyard.)

Coxeter exhuming Geometry

Here the tombstone says
"GEOMETRY… 600 BC — 1900 AD… R.I.P."

In the geometry of Plato illustrated below,
"the figure of eight [square] feet" is not , at this point
in the dialogue, the diamond in Jowett's picture.

An 1892 figure by Jowett illustrating Plato's Meno

Jowett's picture is nonetheless of interest for
its resemblance to a figure drawn some decades later
by the Toronto geometer H. S. M. Coxeter.

A similar 1950 figure by Coxeter illustrating a tesseract

For a less scholarly, but equally confusing, view of the number 8,
see The Eight , a novel by Katherine Neville.

Monday, August 13, 2012

Raiders of the Lost Tesseract

Filed under: General,Geometry — Tags: — m759 @ 3:33 pm

(An episode of Mathematics and Narrative )

A report on the August 9th opening of Sondheim's Into the Woods

Amy Adams… explained why she decided to take on the role of the Baker’s Wife.

“It’s the ‘Be careful what you wish’ part,” she said. “Since having a child, I’m really aware that we’re all under a social responsibility to understand the consequences of our actions.” —Amanda Gordon at businessweek.com

Related material—

Amy Adams in Sunshine Cleaning  "quickly learns the rules and ropes of her unlikely new market. (For instance, there are products out there specially formulated for cleaning up a 'decomp.')" —David Savage at Cinema Retro

Compare and contrast…

1.  The following item from Walpurgisnacht 2012

IMAGE- Excerpt from 'Unified Approach to Functional Decompositions of Switching Functions,' by Marek A. Perkowski et al., 1995

2.  The six partitions of a tesseract's 16 vertices 
       into four parallel faces in Diamond Theory in 1937

Sunday, July 29, 2012

The Galois Tesseract

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

(Continued)

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

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

— share the same vector-space structure:

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

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

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

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

Source:

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

Monday, June 4, 2012

Cube to Tesseract

Filed under: General — m759 @ 10:30 am

Yesterday's post Child's Play displayed a cube formed
by a Hasse diagram of the 8 subsets of a 3-set.*

This suggests a review of a post from last January

IMAGE- Tesseract (i.e., hypercube) formed by a Hasse diagram of the 16 subsets of a 4-element set

* See a comment on yesterday's post relating it to earlier,
  very similar, remarks by Margaret Masterman.
  I was unaware yesterday that those remarks exist. 

Tuesday, January 31, 2012

Tesseract

Filed under: General,Geometry — Tags: , — m759 @ 12:21 pm
 

"… a finite set with  elements
is sometimes called an n-set …."

Tesseract formed from a 4-set—

IMAGE- Tesseract.


The same 16 subsets or points can
be arranged in a 4×4 array that has,
when the array's opposite edges are
joined together, the same adjacencies
as those of the above tesseract.

"There is  such a thing as a 4-set."
— Saying adapted from a novel   

Update of August 12, 2012:

Figures like the above, with adjacent vertices differing in only one coordinate,
appear in a 1950 paper of H. S. M. Coxeter—

Saturday, September 3, 2011

The Galois Tesseract (continued)

A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
two-thirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79T-A37, Notices , Feb. 1979).

Here is some supporting material—

http://www.log24.com/log/pix11B/110903-Carmichael-Conway-Curtis.jpg

The passage from Carmichael above emphasizes the importance of
the 4×4 square within the MOG.

The passage from Conway and Sloane, in a book whose first edition
was published in 1988, makes explicit the structure of the MOG's
4×4 square as the affine 4-space over the 2-element Galois field.

The passage from Curtis (1974, published in 1976) describes 35 sets
of four "special tetrads" within the 4×4 square of the MOG. These
correspond to the 35 sets of four parallel 4-point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 "special tetrads" rather by the parity
of their intersections with the square's rows and columns.

The affine structure appears in the 1979 abstract mentioned above—

IMAGE- An AMS abstract from 1979 showing how the affine group AGL(4,2) of 322,560 transformations acts on a 4x4 square

The "35 structures" of the abstract were listed, with an application to
Latin-square orthogonality, in a note from December 1978

IMAGE- Projective-space structure and Latin-square orthogonality in a set of 35 square arrays

See also a 1987 article by R. T. Curtis—

Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract:

“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”

(Received July 20 1987)

Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353

* For instance:

Algebraic structure in the 4x4 square, by Cullinane (1985) and Curtis (1987)

Update of Sept. 4— This post is now a page at finitegeometry.org.

Thursday, September 1, 2011

The Galois Tesseract

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

Click to enlarge

IMAGE- The Galois Tesseract, 1979-1999

IMAGE- Review of Conway and Sloane's 'Sphere Packings...' by Rota

Wednesday, August 7, 2024

Purely Coincidental  Spatial Studies

Filed under: General — Tags: — m759 @ 5:47 am

See also the previous post.

Update of 1 (and 1:35) PM EDT the same day —

Some Spatial Studies vocabulary:  http://m759.net/wordpress/?s=Dyadic.

Tuesday, April 23, 2024

Golden Keys

Filed under: General — Tags: — m759 @ 7:55 pm

“Chess problems are the
hymn-tunes of mathematics.”
— G. H. Hardy,
A Mathematician’s Apology

The image “http://www.log24.com/log/pix04A/041010-Hardy2.jpg” cannot be displayed, because it contains errors.
The image “http://www.log24.com/log/pix04A/041010-Mate2.jpg” cannot be displayed, because it contains errors.

Bogart and Lorre in 'Casablanca' with chessboard and cocktail

The key is the cocktail that begins the proceedings.”

– Brian Harley, Mate in Two Moves

"I named this script ocode and chmod 755'd it to make it executable…"

Software forum post on the OCR program Tesseract

Garfield, Dec. 4, 2008:  Mouse's Xmas bulb-lighting

From the author of
The Pearly Gates of Cyberspace:

"Like so many other heroes
 who have seen the light
 of a higher order…."

Monday, January 22, 2024

Box Geometry

Filed under: General — Tags: , , — m759 @ 4:33 am

From a search in this journal for "Box of Nothing" —

"And the Führer digs for trinkets in the desert."

 

From a related search, for "Ghent Links" —

Related art —


More seriously See Box759.

Monday, January 15, 2024

Saturday, November 18, 2023

“Don’t solicit for your sister,* it’s not nice.” — Tom Lehrer

Filed under: General — Tags: — m759 @ 11:36 am

André Weil to his sister:

From this journal at 1:51 AM  ET Thursday, September 8, 2022

"The pleasure comes from the illusion" . . .

Exercise:

Compare and contrast the following structure with the three
"bricks" of the R. T. Curtis Miracle Octad Generator (MOG).

http://www.log24.com/log/pix11B/110805-The24.jpg

Note that the 4-row-2-column "brick" at left is quite 
different from the other two bricks, which together
show chevron variations within a Galois tesseract —

.

Further Weil remarks . . .

A Slew of Prayers

"The pleasure comes from the illusion
and the far from clear meaning;
once the illusion is dissipated,
and knowledge obtained, one becomes
indifferent at the same time;
at least in the Gitâ there is a slew of prayers
(slokas) on the subject, each one more final
than the previous ones."

*

Tuesday, October 3, 2023

Tuesday… Belgium.

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

The inscape  in the previous post suggests a review of
work by the Belgian mathematician Koen Thas on what
might be called the "quantum tesseract theorem."

Friday, July 28, 2023

Requiem for an Eagle

Filed under: General — Tags: — m759 @ 12:13 am

" 'Hotel California,' with its mysterious, allegorical lyrics,
became among the band’s best-known recordings.
It topped the Billboard Hot 100 in 1977 and won
a Grammy Award for record of the year in 1978."

The New York Times  tonight  on an Eagles founding
member who reportedly died on Wednesday, July 26.

"Mirrors on the ceiling, pink champagne on ice . . . ."

U. of California edition of Wittgenstein's 'Zettel'-- pink cover, white tesseract in background

See as well a search in this  journal for Zettel .

Monday, June 19, 2023

The Original Portable Door

Filed under: General — Tags: , , — m759 @ 3:26 pm

"… a cardboard tube, more or less the same length as
the inner core of a toilet roll, but thicker. He frowned,
took the roll out, laid it on the desk and poked up it
with the butt end of a pencil. Something slid out.
It looked like a rolled-up black plastic dustbin liner;
but when he unfolded it, he recognised it as the funny
sheet thing he’d found in the strongroom and briefly
described as an Acme Portable Door, before losing
his nerve and changing it to something less facetious." 

— Holt, Tom. The Portable Door . Orbit. Kindle Edition. 

According to goodreads.com, the Holt book was
"first published March 6, 2003."

Compare and contrast the "portable door" as a literary device
with the "tesseract" in A Wrinkle in Time  (1962).

See also this  journal on March 6, 2003.

Friday, April 28, 2023

The Small Space Model

Filed under: General — Tags: , , , — m759 @ 6:28 pm

From the previous post, "The Large Language Model,"
a passage from Wikipedia —

"… sometimes large models undergo a 'discontinuous phase shift' 
where the model suddenly acquires substantial abilities not seen
in smaller models. These are known as 'emergent abilities,' and
have been the subject of substantial study." — Wikipedia

Compare and contrast 
this with the change undergone by a "small space model,"
that of the finite affine 4-space A  with 16 points (a Galois tesseract ), 
when it is augmented by an eight-point "octad." The 30 eight-point
hyperplanes of A  then have a natural extension within the new
24-point set to 759 eight-point octads, and the 322,560 affine
automorphisms of the space expand to the 244,823,040 Mathieu
automorphisms of the 759-octad set — a (5, 8, 24) Steiner system.

For a visual analogue of the enlarged 24-point space and some remarks
on analogy by Simone Weil's brother, a mathematician, see this journal
on September 8 and 9, 2022.

Saturday, April 1, 2023

For this weekend’s Bombay Beach Biennale

Filed under: General — Tags: , — m759 @ 8:32 pm

From this  journal on the above artist's dies natalis

Tuesday, March 28, 2023

Onestone Parable

Filed under: General — Tags: , — m759 @ 12:45 pm

“You’re literally looking for like a one in a million thing.
You filter out the 999,999 of the boring ones, then
you’ve got something that’s weird, and then that’s worth
further exploration.”

— Quote from a mathematics story today at Gizmodo

A different "one in a million" mathematics story —

On Steiner Quadruple Systems of Order 16.

See also Galois Tesseract.

Sunday, February 19, 2023

Hoarding Space

Filed under: General — Tags: , — m759 @ 1:53 pm

See also the star of the previous post in . . .

Quantum Tesseract Theorem .

Friday, January 13, 2023

Dürer, Music, and Doktor Faustus

Filed under: General — Tags: — m759 @ 1:49 am

The previous post and Mann's Doktor Faustus in this journal
suggest a look at . . .

Thursday, December 22, 2022

Welcome to the Desert of the Real

Filed under: General — Tags: , — m759 @ 10:58 am

IMAGE- Aug. 5, 2005- Galois tesseract, Shakespeherian Rag, Sir Alec Guinness

Thursday, November 24, 2022

The Drum Machine

Filed under: General — Tags: , , , — m759 @ 2:18 pm

"A struggling music producer sells his soul to a 1970s drum machine."

— Summary of a short film by Kevin Ignatius, "Hook Man."

The music producer pawns his current drum device 
and acquires a demonic 1970s machine.


Artistic symbolism —

The 16-pad device at left may be viewed by enthusiasts of ekphrasis
as a Galois tesseract, and the machine at right as the voice of
Hal Foster, an art theorist who graduated from Princeton in 1977.

For an example of Foster's prose style, see
the current London Review of Books.

Friday, November 18, 2022

Datura Date

Filed under: General — m759 @ 10:00 pm

YouTube: Georgia O'Keeffe at Tate Modern

There was no Log24 post for the above Jacky Klein date,
July 6, 2016, but the posts of that month contain a flashback
to an earlier July 6 —

See as well . . .

… and some Log24 posts from the above Dec. 1, 2014, date 
now tagged Red Dawn.

Monday, November 14, 2022

Plot Structure

Filed under: General — Tags: , — m759 @ 2:21 pm

From Peter J. Cameron's weblog today

"It happens sometimes that researchers working in different fields
study the same thing, give it different names, and don’t realise that
there is further work on the subject somewhere else…."

Cameron's example of a theorem connecting work on 
the same thing in different fields —

"Theorem  A partition Δ is equitable for a graph Γ if and only if
the projection matrix onto the subspace of functions constant
on parts of Δ commutes with the adjacency matrix of Γ."

A phrase from Cameron's remarks today —

"Thus we have to consider 'plot structure'…."

For more remarks on different fields and plot structure , see
"Quantum Tesseract Theorem" in this  weblog.

Saturday, October 15, 2022

A Box of Nothing . . .

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

Continues.

"And the Führer digs for trinkets in the desert."

Sunday, September 25, 2022

Woodwork

Filed under: General — Tags: — m759 @ 11:21 pm

"The actor Nick Offerman, himself an accomplished woodworker 
and a member of Ms. Hiller’s legion of admirers, called her an
'Obi-Wan Kenobi level master.'"

The New York Times  this evening, obituary by Clay Risen
for Nancy Hiller.

Related woodwork note —

"There and back again."

Thursday, September 8, 2022

Analogy in Mathematics: Chevron Variations

André Weil in 1940 on analogy in mathematics —

. "Once it is possible to translate any particular proof from one theory to another, then the analogy has ceased to be productive for this purpose; it would cease to be at all productive if at one point we had a meaningful and natural way of deriving both theories from a single one. In this sense, around 1820, mathematicians (Gauss, Abel, Galois, Jacobi) permitted themselves, with anguish and delight, to be guided by the analogy between the division of the circle (Gauss’s problem) and the division of elliptic functions. Today, we can easily show that both problems have a place in the theory of abelian equations; we have the theory (I am speaking of a purely algebraic theory, so it is not a matter of number theory in this case) of abelian extensions. Gone is the analogy: gone are the two theories, their conflicts and their delicious reciprocal reflections, their furtive caresses, their inexplicable quarrels; alas, all is just one theory, whose majestic beauty can no longer excite us. Nothing is more fecund than these slightly adulterous relationships; nothing gives greater pleasure to the connoisseur, whether he participates in it, or even if he is an historian contemplating it retrospectively, accompanied, nevertheless, by a touch of melancholy. The pleasure comes from the illusion and the far from clear meaning; once the illusion is dissipated, and knowledge obtained, one becomes indifferent at the same time; at least in the Gitâ there is a slew of prayers (slokas) on the subject, each one more final than the previous ones."

"The pleasure comes from the illusion" . . .

Exercise:

Compare and contrast the following structure with the three
"bricks" of the R. T. Curtis Miracle Octad Generator (MOG).

http://www.log24.com/log/pix11B/110805-The24.jpg

Note that the 4-row-2-column "brick" at left is quite 
different from the other two bricks, which together
show chevron variations within a Galois tesseract —

Saturday, September 3, 2022

1984 Revisited

Filed under: General — m759 @ 2:46 pm

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168

Related material

Note the three quadruplets of parallel edges  in the 1984 figure above.

Further Reading

The above Gates article appeared earlier, in the June 2010 issue of
Physics World , with bigger illustrations. For instance —

Exercise: Describe, without seeing the rest of the article,
the rule used for connecting the balls above.

Wikipedia offers a much clearer picture of a (non-adinkra) tesseract —

      And then, more simply, there is the Galois tesseract

For parts of my own  world in June 2010, see this journal for that month.

The above Galois tesseract appears there as follows:

Image-- The Dream of the Expanded Field

See also the Klein correspondence in a paper from 1968
in yesterday's 2:54 PM ET post

Tuesday, August 23, 2022

Chair

Filed under: General — m759 @ 2:40 am

"Where fashion sits" — Song lyric

Wednesday, August 3, 2022

From the “Shifting Phantasmagoria” of Joan Didion

Filed under: General — Tags: , — m759 @ 12:45 pm

Also on July 30 . . .

Saturday, July 30, 2022

Literary Figures

Filed under: General — m759 @ 7:48 pm

"… the tesseract, identified with a figure too inclusive,
contradictory, and all-pervasive to be seen as a character,
connects multiple dimensions in a manner counter to
ordinary thought…."

— Catherine Flynn, "From Dowel to Tesseract" (2016)

As does the I Ching .

Summer Camp

Filed under: General — Tags: — m759 @ 1:07 am

Or: The Sontag Puzzlement

Wikipedia on "Heavenly Creatures"

"Juliet introduces Pauline to the idea of 'the Fourth World',
a Heaven without Christians where music and art are
celebrated. Juliet believes she will go there when she dies.
Certain actors and musicians have the status of saints in
this afterlife, such as singer Mario Lanza, with whom
both girls are obsessed."

   Related material — Sontag + Camp .

Saturday, March 26, 2022

Box Geometry: Space, Group, Art  (Work in Progress)

Filed under: General — Tags: , — m759 @ 2:06 am

Many structures of finite geometry can be modeled by
rectangular or cubical arrays ("boxes") —
of subsquares or subcubes (also "boxes").

Here is a draft for a table of related material, arranged
as internet URL labels.

Finite Geometry Notes — Summary Chart
 

Name Tag .Space .Group .Art
Box4

2×2 square representing the four-point finite affine geometry AG(2,2).

(Box4.space)

S4 = AGL(2,2)

(Box4.group)

 

(Box4.art)

Box6 3×2 (3-row, 2-column) rectangular array
representing the elements of an arbitrary 6-set.
S6  
Box8 2x2x2 cube or  4×2 (4-row, 2-column) array. S8 or Aor  AGL(3,2) of order 1344, or  GL(3,2) of order 168  
Box9 The 3×3 square. AGL(2,3) or  GL(2,3)  
Box12 The 12 edges of a cube, or  a 4×3  array for picturing the actions of the Mathieu group M12. Symmetries of the cube or  elements of the group M12  
Box13 The 13 symmetry axes of the cube. Symmetries of the cube.  
Box15 The 15 points of PG(3,2), the projective geometry
of 3 dimensions over the 2-element Galois field.
Collineations of PG(3,2)  
Box16 The 16 points of AG(4,2), the affine geometry
of 4 dimensions over the 2-element Galois field.

AGL(4,2), the affine group of 
322,560 permutations of the parts
of a 4×4 array (a Galois tesseract)

 
Box20 The configuration representing Desargues's theorem.    
Box21 The 21 points and 21 lines of PG(2,4).    
Box24 The 24 points of the Steiner system S(5, 8, 24).    
Box25 A 5×5 array representing PG(2,5).    
Box27 The 3-dimensional Galois affine space over the
3-element Galois field GF(3).
   
Box28 The 28 bitangents of a plane quartic curve.    
Box32 Pair of 4×4 arrays representing orthogonal 
Latin squares.
Used to represent
elements of AGL(4,2)
 
Box35 A 5-row-by-7-column array representing the 35
lines in the finite projective space PG(3,2)
PGL(3,2), order 20,160  
Box36 Eurler's 36-officer problem.    
Box45 The 45 Pascal points of the Pascal configuration.    
Box48 The 48 elements of the group  AGL(2,3). AGL(2,3).  
Box56

The 56 three-sets within an 8-set or
56 triangles in a model of Klein's quartic surface or
the 56 spreads in PG(3,2).

   
Box60 The Klein configuration.    
Box64 Solomon's cube.    

— Steven H. Cullinane, March 26-27, 2022

Tuesday, March 15, 2022

Midnight Wrinkle

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

See as well the life of a real  astrophysicist.

Update of 12:26 PM ET March 15:
Vide  other posts now tagged The Rosenhain Symmetry.

Monday, February 21, 2022

Stimulus

Filed under: General — Tags: — m759 @ 11:28 am

IMAGE- Weyl in 1949 on the relativity problem

"The class is objectively characterized, but not
the individual coordinate assignment."

Tell it to Watchduck, Hermann.

See a related remark by Quack5quack in Raiders of the Lost Coordinates.

Sunday, February 20, 2022

4×4 Nomenclature

Filed under: General — Tags: — m759 @ 2:35 am

The geometry of the 4×4 square may be associated with the name
Galois, as in "the Galois tesseract," or similarly with the name Kummer. 
Here is a Google image search using the latter name —

(Click to enlarge.)

 

Thursday, February 17, 2022

Four Dots, Six Lines

Filed under: General — Tags: — m759 @ 1:46 am

"There is  such a thing  as  a tesseract." 

— Mrs. Whatsit in  A Wrinkle in Time  (1962)

"Simplify, simplify." — Henry David Thoreau in Walden  (1854)

Von Franz representation of the I Ching's Hexagram 2, The Receptive
 

A Jungian on this six-line figure:

“They are the same six lines that exist in the I Ching…. Now observe the square more closely: four of the lines are of equal length, the other two are longer…. For this reason symmetry cannot be statically produced and a dance results.”
 
— Marie-Louise von Franz,
   Number and Time  (1970)

Tuesday, December 21, 2021

Grid Language, Grid Logic

Filed under: General — m759 @ 12:00 am

Grid Language —

Anna Schechtman on the grid and crossword puzzles

Grid Logic —

Tesseract  in this journal and "The Unmagicking."

Rosalind Krauss and Grid

Friday, December 10, 2021

Unhinged Melody

Filed under: General — Tags: — m759 @ 12:43 pm

The time of the previous post was 4:46 AM ET today.

Fourteen minutes later —

"I'm a groupie, really." — Murray Bartlett in today's online NY Times

The previous post discussed group actions on a 3×3 square array. A tune
about related group actions on a 4×4  square array (a Galois tesseract. . .

'The Eddington Song'

Sunday, December 5, 2021

The Venn Lotus

Filed under: General — m759 @ 11:20 pm

Venn Lotus and Galois Tesseract, by Steven H. Cullinane on 5 December 2021.

Saturday, December 4, 2021

Delphic Code

Filed under: General — Tags: — m759 @ 8:46 pm

Log24 on this date — December 4th — in 2008:

"I named this script ocode and chmod 755'd it to make it executable…"

— Software forum post on the OCR program Tesseract

The New York Times  at 7:55 PM ET today:

Sunday, April 11, 2021

Space Itself

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

See the title in this journal. This review was suggested by
a phrase of Catherine Flynn:

Saturday, April 10, 2021

Gap Dance in Utrecht

Filed under: General — Tags: , — m759 @ 2:05 pm

(See also Gap Dance elsewhere in this journal.)

"… the Wake  seemed to be everywhere
at the Utrecht Joyce Symposium."

"What I saw at the Symposium at Utrecht
were scholars working to close the gap
between the multifaceted complexity
of the text and the vastly greater complexity
of the readers experiencing it."

— "Along the Krommerun: The Twenty-Fourth International
James Joyce Symposium, Utrecht, The Netherlands,
15-20 June 2014," by Andrew Ferguson, University of Virginia.

"Central to these structural and aesthetic innovations,
however, is a mundane element: the wooden dowel.
The dowel is a small peg of variable length;
its ends lack distinct heads, allowing it work
in any direction. The dowels  remain hidden
in the Red Blue Chair as they connect rail to rail
and rail to  plank, invisible yet essential to the chair's
appearance and its defiance of convention and gravity.
Critics have noted the chair's flouting of the rules of
modern architectural semantics: Yves-Alain Bois writes
of the elements that function simultaneously in two ways,
as both supporting prop and supported  cantilever, as
subverting "the functionalist ethic of modernist
architecture — the dictum that would have one meaning
per sign". It is the dowel that allows the elements of
the chair to attain so subtly this semantic complexity.
The  chair's innovations are not technological,
but rather concern the arrangement  and deployment
of existing materials and elements. The dowel is
a modest but highly adaptable means of joining:
while the dovetail joint requires two equally sized
components, the mortise and tenon involves a male
and a female element, and the housed joint requires
an extended zone of contact, the dowel  neutrally
connects all kinds of elements to one another,
its single point allowing maximum freedom in
the orientation of the connected elements."

— Page 25 of "From Dowel to Tesseract,"
by Catherine Flynn. Source: European Joyce
Studies
, 2016, Vol. 24, A LONG THE
KROMMERUN: Selected Papers from the 
Utrecht James Joyce Symposium, pp. 20-45.

In Like Flynn . . .

Filed under: General — Tags: , — m759 @ 3:14 am

Continues.

Wednesday, February 17, 2021

Raiders of the Lost Coordinates . . .

Continues.

From other posts tagged Tetrahedron vs. Square —

"There is  such a thing as a 4-set."
— Saying adapted from a 1962 young-adult novel.

Illustration (central detail  a  from the above tetrahedral figure) —

A Harvard Variation
from Timothy Leary —

The topics of Harvard and Leary suggest some other cultural
history, from The Coasters"Poison Ivy" and "Yakety Yak."

Wednesday, November 18, 2020

La Chanson Fatale

Filed under: General — Tags: — m759 @ 11:31 pm

Harold Edwards, a founding co-editor of The Mathematical Intelligencer ,
reportedly  died at 84 on Tuesday, Nov. 10, 2020.

Images from this  journal on that date

"Surprise Party" revisited —

A Philippine meditation by Alex Garland quoted here on May 6, 2010

Monday, July 13, 2020

Unpoetic License

Filed under: General — Tags: , , — m759 @ 6:21 am

The above novel uses extensively the term “inscape.”
The term’s originator, a 19th-century Jesuit poet,
is credited . . . sort of.  For other uses of the term,
search for Inscape in this journal. From that search —

A quote from a 1962 novel

“There’s something phoney
in the whole setup, Meg thought.
There is definitely something rotten
in the state of Camazotz.”

Addendum for the Church of Synchronology

The Joe Hill novel above was published (in hardcover)
on Walpurgisnacht —April 30, 2013.  See also this journal
on that date.

Tuesday, April 21, 2020

Living Water

Filed under: General — m759 @ 6:47 pm

The Tesseract Timeline:
Where The Cube Has Been In The …

www.cinemablend.com › news › the-tesseract-timeline-…
Mar 13, 2019 – With HYDRA. In 1942, Johann Schmidt, a.k.a.
the Red Skull, arrived in Tønsberg to procure the Tesseract
from an ancient church. While he …

Related material from posts tagged Aqua
(suggested by a name in the previous post) —

Lene from Tønsberg

 

Saturday, March 7, 2020

The “Octad Group” as Symmetries of the 4×4 Square

From "Mathieu Moonshine and Symmetry Surfing" —

(Submitted on 29 Sep 2016, last revised 22 Jan 2018)
by Matthias R. Gaberdiel (1), Christoph A. Keller (2),
and Hynek Paul (1)

(1)  Institute for Theoretical Physics, ETH Zurich
(2)  Department of Mathematics, ETH Zurich

https://arxiv.org/abs/1609.09302v2 —

"This presentation of the symmetry groups Gi  is
particularly well-adapted for the symmetry surfing
philosophy. In particular it is straightforward to
combine them into an overarching symmetry group G
by combining all the generators. The resulting group is
the so-called octad group

G = (Z2)4  A8 .

It can be described as a maximal subgroup of M24
obtained by the setwise stabilizer of a particular
'reference octad' in the Golay code, which we take
to be O= {3,5,6,9,15,19,23,24} ∈ 𝒢24. The octad
subgroup is of order 322560, and its index in M24
is 759, which is precisely the number of
different reference octads one can choose."

This "octad group" is in fact the symmetry group of the affine 4-space over GF(2),
so described in 1979 in connection not with the Golay code but with the geometry
of the 4×4 square.* Its nature as an affine group acting on the Golay code was
known long before 1979, but its description as an affine group acting on
the 4×4 square may first have been published in connection with the
Cullinane diamond theorem and Abstract 79T-A37, "Symmetry invariance in a
diamond ring
," by Steven H. Cullinane in Notices of the American Mathematical
Society
, February 1979, pages A-193, 194.

* The Galois tesseract .

Update of March 15, 2020 —

Conway and Sloane on the "octad group" in 1993 —

Monday, February 24, 2020

Hidden Figure

Filed under: General — m759 @ 2:55 pm

“There is  such a thing as  ▦  .”

— Saying adapted from a 1962 young-adult novel.

Tuesday, January 28, 2020

Very Stable Kool-Aid

Filed under: General — Tags: , , — m759 @ 2:16 pm

Two of the thumbnail previews
from yesterday's 1 AM  post

"Hum a few bars"

"For 6 Prescott Street"

Further down in the "6 Prescott St." post, the link 5 Divinity Avenue
leads to

A Letter from Timothy Leary, Ph.D., July 17, 1961

Harvard University
Department of Social Relations
Center for Research in Personality
Morton Prince House
5 Divinity Avenue
Cambridge 38, Massachusetts

July 17, 1961

Dr. Thomas S. Szasz
c/o Upstate Medical School
Irving Avenue
Syracuse 10, New York

Dear Dr. Szasz:

Your book arrived several days ago. I've spent eight hours on it and realize the task (and joy) of reading it has just begun.

The Myth of Mental Illness is the most important book in the history of psychiatry.

I know it is rash and premature to make this earlier judgment. I reserve the right later to revise and perhaps suggest it is the most important book published in the twentieth century.

It is great in so many ways–scholarship, clinical insight, political savvy, common sense, historical sweep, human concern– and most of all for its compassionate, shattering honesty.

. . . .

The small Morton Prince House in the above letter might, according to
the above-quoted remarks by Corinna S. Rohse, be called a "jewel box."
Harvard moved it in 1978 from Divinity Avenue to its current location at
6 Prescott Street.

Related "jewel box" material for those who
prefer narrative to mathematics —

"In The Electric Kool-Aid Acid Test , Tom Wolfe writes about encountering 
'a young psychologist,' 'Clifton Fadiman’s nephew, it turned out,' in the
waiting room of the San Mateo County jail. Fadiman and his wife were
'happily stuffing three I-Ching coins into some interminable dense volume*
of Oriental mysticism' that they planned to give Ken Kesey, the Prankster-
in-Chief whom the FBI had just nabbed after eight months on the lam.
Wolfe had been granted an interview with Kesey, and they wanted him to
tell their friend about the hidden coins. During this difficult time, they
explained, Kesey needed oracular advice."

— Tim Doody in The Morning News  web 'zine on July 26, 2012**

Oracular advice related to yesterday evening's
"jewel box" post …

A 4-dimensional hypercube H (a tesseract ) has 24 square
2-dimensional faces
.  In its incarnation as a Galois  tesseract
(a 4×4 square array of points for which the appropriate transformations
are those of the affine 4-space over the finite (i.e., Galois) two-element
field GF(2)), the 24 faces transform into 140 4-point "facets." The Galois 
version of H has a group of 322,560 automorphisms. Therefore, by the
orbit-stabilizer theorem, each of the 140 facets of the Galois version has
a stabilizer group of  2,304 affine transformations.

Similar remarks apply to the I Ching  In its incarnation as  
a Galois hexaract , for which the symmetry group — the group of
affine transformations of the 6-dimensional affine space over GF(2) —
has not 322,560 elements, but rather 1,290,157,424,640.

* The volume Wolfe mentions was, according to Fadiman, the I Ching.

** See also this  journal on that date — July 26, 2012.

Monday, January 27, 2020

Jewel Box

Filed under: General — Tags: — m759 @ 9:02 pm

The phrase "jewel box" in a New York Times  obituary online this afternoon
suggests a review. See "And He Built a Crooked House" and Galois Tesseract.

Monday, October 21, 2019

Algebra and Space… Illustrated.

Filed under: General — Tags: , — m759 @ 4:26 pm

Related entertainment —

Detail:

   George Steiner

"Perhaps an insane conceit."

 

Perhaps.

 

See Quantum Tesseract Theorem .

 

Perhaps Not.

 

 See Dirac and Geometry .

Tuesday, October 15, 2019

Inside the Fire Temple

Filed under: General — Tags: , — m759 @ 8:00 pm

(The title refers to Log24 posts now tagged Fire Temple.)

In memory of a  New Yorker  cartoonist who
reportedly died at 97 on October 3, 2019  …

"Read something that means something." 
New Yorker  advertising slogan

From posts tagged Tetrahedron vs. Square

This  journal on October 3

"There is  such a thing as a 4-set."
— Saying adapted from a 1962 young-adult novel.

Illustration (central detail   from the above tetrahedral figure) —

Wednesday, October 9, 2019

The Joy of Six

Note that in the pictures below of the 15 two-subsets of a six-set,
the symbols 1 through 6 in Hudson's square array of 1905 occupy the
same positions as the anticommuting Dirac matrices in Arfken's 1985
square array. Similarly occupying these positions are the skew lines
within a generalized quadrangle (a line complex) inside PG(3,2).

Anticommuting Dirac matrices as spreads of projective lines

Related narrative The "Quantum Tesseract Theorem."

Sunday, September 22, 2019

Simplex Sigillum Veri

Filed under: General — m759 @ 5:21 am

(Continued from November 27, 2010.)

The previous post suggests a review of a Tilman Piesk
illustration, with the general form of a 4-simplex, from
the Wikipedia article titled Simplex .  As the article 
notes, the lines shown connecting points are those of a
tesseract.

Tetrahedral tesseract adapted from Piesk illustration in Wikipedia 'Simplex' article

Wednesday, September 18, 2019

The Perpetual Identity Crisis

Filed under: General — Tags: — m759 @ 8:56 am

"There is  such a thing as a 4-set." — Saying adapted
from a 1962 young-adult novel.

Midrash — An image posted here on August 6

Tuesday, August 6, 2019

Mathematics and Narrative:  The Crosswicks Curse Continues.

Filed under: General — m759 @ 7:03 pm

"There is  such a thing as a desktop."

— Saying adapted from a 1962 young-adult novel.

Sunday, July 14, 2019

Old Pathways in Science:

Filed under: General — Tags: , — m759 @ 12:37 pm

The Quantum Tesseract Theorem Revisited

From page 274 — 

"The secret  is that the super-mathematician expresses by the anticommutation
of  his operators the property which the geometer conceives as  perpendicularity
of displacements.  That is why on p. 269 we singled out a pentad of anticommuting
operators, foreseeing that they would have an immediate application in describing
the property of perpendicular directions without using the traditional picture of space.
They express the property of perpendicularity without the picture of perpendicularity.

Thus far we have touched only the fringe of the structure of our set of sixteen E-operators.
Only by entering deeply into the theory of electrons could I show the whole structure
coming into evidence."

A related illustration, from posts tagged Dirac and Geometry —

Anticommuting Dirac matrices as spreads of projective lines

Compare and contrast Eddington's use of the word "perpendicular"
with a later use of the word by Saniga and Planat.

Tuesday, July 9, 2019

Perception of Space

Filed under: General — Tags: , , , — m759 @ 10:45 am

(Continued)

The three previous posts have now been tagged . . .

Tetrahedron vs. Square  and  Triangle vs. Cube.

Related material —

Tetrahedron vs. Square:

Labeling the Tetrahedral Model  (Click to enlarge) —

Triangle vs. Cube:

and, from the date of the above John Baez remark —

Monday, July 8, 2019

Exploring Schoolgirl Space

See also "Quantum Tesseract Theorem" and "The Crosswicks Curse."

Thursday, July 4, 2019

From Devil’s Night 2014

Filed under: General — m759 @ 7:59 am

And now, General, time presses; and America is in a hurry.
Have you realized that though you may occupy towns and win battles,
you cannot conquer a nation? — The Devil's Disciple

A figure related to Dürer's "magic" square posted during Devil's Night —

Tuesday, April 9, 2019

Zero Dark Nine:

Filed under: General — Tags: , — m759 @ 12:09 am

The Crosswicks Curse Continues . . .

"There is  such a thing as geometry."

— Saying adapted from a 1962 young-adult novel.

Monday, March 11, 2019

Ant-Man Meets Doctor Strange

Filed under: General — m759 @ 1:22 pm

IMAGE- Concepts of Space

The 4×4 square may also be called the Galois Tesseract .
By analogy, the 4x4x4 cube may be called the Galois Hexeract .

"Think outside the tesseract.

Overarching Metanarratives

Filed under: General — Tags: — m759 @ 4:15 am

See also "Overarching + Tesseract" in this  journal. From the results
of that search, some context for the "inscape" of the previous post —

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)

Wednesday, March 6, 2019

The Relativity Problem and Burkard Polster

Filed under: General,Geometry — Tags: — m759 @ 11:28 am
 

From some 1949 remarks of Weyl—

"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."

— Hermann Weyl, "Relativity Theory as a Stimulus in Mathematical Research," Proceedings of the American Philosophical Society , Vol. 93, No. 7, Theory of Relativity in Contemporary Science: Papers Read at the Celebration of the Seventieth Birthday of Professor Albert Einstein in Princeton, March 19, 1949  (Dec. 30, 1949), pp. 535-541

Weyl in 1946—:

"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."

— Hermann Weyl, The Classical Groups , Princeton University Press, 1946, p. 16

For some context, see Relativity Problem  in this journal.

In the case of PG(3,2), there is a choice of geometric models 
to be coordinatized: two such models are the traditional
tetrahedral model long promoted by Burkard Polster, and
the square model of Steven H. Cullinane.

The above Wikipedia section tacitly (and unfairly) assumes that
the model being coordinatized is the tetrahedral model. For
coordinatization of the square model, see (for instance) the webpage
Finite Relativity.

For comparison of the two models, see a figure posted here on
May 21, 2014 —

Labeling the Tetrahedral Model  (Click to enlarge) —

"Citation needed" —

The anonymous characters who often update the PG(3,2) Wikipedia article
probably would not consider my post of 2014, titled "The Tetrahedral
Model of PG(3,2)
," a "reliable source."

Thursday, February 28, 2019

Wikipedia Scholarship

Filed under: General — Tags: , , — m759 @ 12:31 pm

Cullinane's Square Model of PG(3,2)

Besides omitting the name Cullinane, the anonymous Wikipedia author
also omitted the step of representing the hypercube by a 4×4 array —
an array called in this  journal a Galois  tesseract.

Sunday, February 17, 2019

See Also …

Filed under: General — Tags: — m759 @ 12:31 pm

"And the Führer digs for trinkets in the desert."

"See also Acht "
— Cambridge German-English Dictionary, article on "Elf "

Wednesday, December 12, 2018

An Inscape for Douthat

Some images, and a definition, suggested by my remarks here last night
on Apollo and Ross Douthat's remarks today on "The Return of Paganism" —

Detail of Feb. 20, 1986, note by Steven H. Cullinane on Weyl's 'relativity problem'

Kibler's 2008 'Variations on a theme' illustrated.

In finite geometry and combinatorics,
an inscape  is a 4×4 array of square figures,
each figure picturing a subset of the overall 4×4 array:


 

Related material — the phrase
"Quantum Tesseract Theorem" and  

A.  An image from the recent
      film "A Wrinkle in Time" — 

B.  A quote from the 1962 book —

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

Tuesday, November 13, 2018

Blackboard Jungle Continues.

Filed under: G-Notes,General,Geometry — Tags: , , — m759 @ 6:19 pm

From the 1955 film "Blackboard Jungle" —

From a trailer for the recent film version of A Wrinkle in Time

Detail of the phrase "quantum tesseract theorem":

From the 1962 book —

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

Related mathematics from Koen Thas that some might call a
"quantum tesseract theorem" —

Some background —

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

See also posts tagged Dirac and Geometry. For more
background on finite  geometry, see a web page
at Thas's institution, Ghent University.

Monday, October 15, 2018

History at Bellevue

Filed under: General,Geometry — Tags: , — m759 @ 9:38 pm

The previous post, "Tesserae for a Tesseract," contains the following
passage from a 1987 review of a book about Finnegans Wake

"Basically, Mr. Bishop sees the text from above
and as a whole — less as a sequential story than
as a box of pied type or tesserae for a mosaic,
materials for a pattern to be made."

A set of 16 of the Wechsler cubes below are tesserae that 
may be used to make patterns in the Galois tesseract.

Another Bellevue story —

“History, Stephen said, is a nightmare
from which I am trying to awake.”

— James Joyce, Ulysses

Tuesday, September 4, 2018

MBTI at the Church of St. Frank*

Filed under: General — m759 @ 10:45 am

(Suggested by the previous post)

http://www.log24.com/log/pix18/180904-MBTI-Tesseract-post-141107.jpg

* For the church, see Kermode + Garber.

Wednesday, June 27, 2018

Taken In

Filed under: General,Geometry — Tags: , , , — m759 @ 9:36 am

A passage that may or may not have influenced Madeleine L'Engle's
writings about the tesseract :

From Mere Christianity , by C. S. Lewis (1952) —

"Book IV – Beyond Personality:
or First Steps in the Doctrine of the Trinity"
. . . .

I warned you that Theology is practical. The whole purpose for which we exist is to be thus taken into the life of God. Wrong ideas about what that life is, will make it harder. And now, for a few minutes, I must ask you to follow rather carefully.

You know that in space you can move in three ways—to left or right, backwards or forwards, up or down. Every direction is either one of these three or a compromise between them. They are called the three Dimensions. Now notice this. If you are using only one dimension, you could draw only a straight line. If you are using two, you could draw a figure: say, a square. And a square is made up of four straight lines. Now a step further. If you have three dimensions, you can then build what we call a solid body, say, a cube—a thing like a dice or a lump of sugar. And a cube is made up of six squares.

Do you see the point? A world of one dimension would be a straight line. In a two-dimensional world, you still get straight lines, but many lines make one figure. In a three-dimensional world, you still get figures but many figures make one solid body. In other words, as you advance to more real and more complicated levels, you do not leave behind you the things you found on the simpler levels: you still have them, but combined in new ways—in ways you could not imagine if you knew only the simpler levels.

Now the Christian account of God involves just the same principle. The human level is a simple and rather empty level. On the human level one person is one being, and any two persons are two separate beings—just as, in two dimensions (say on a flat sheet of paper) one square is one figure, and any two squares are two separate figures. On the Divine level you still find personalities; but up there you find them combined in new ways which we, who do not live on that level, cannot imagine.

In God's dimension, so to speak, you find a being who is three Persons while remaining one Being, just as a cube is six squares while remaining one cube. Of course we cannot fully conceive a Being like that: just as, if we were so made that we perceived only two dimensions in space we could never properly imagine a cube. But we can get a sort of faint notion of it. And when we do, we are then, for the first time in our lives, getting some positive idea, however faint, of something super-personal—something more than a person. It is something we could never have guessed, and yet, once we have been told, one almost feels one ought to have been able to guess it because it fits in so well with all the things we know already.

You may ask, "If we cannot imagine a three-personal Being, what is the good of talking about Him?" Well, there isn't any good talking about Him. The thing that matters is being actually drawn into that three-personal life, and that may begin any time —tonight, if you like.

. . . .

But beware of being drawn into the personal life of the Happy Family .

https://www.jstor.org/stable/24966339

"The colorful story of this undertaking begins with a bang."

And ends with

Martin Gardner on Galois

"Galois was a thoroughly obnoxious nerd,
suffering from what today would be called
a 'personality disorder.'  His anger was
paranoid and unremitting."

Thursday, June 21, 2018

Models of Being

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

A Buddhist view —

“Just fancy a scale model of Being
made out of string and cardboard.”

— Nanavira Thera, 1 October 1957,
on a model of Kummer’s Quartic Surface
mentioned by Eddington

A Christian view —

A formal view —

From a Log24 search for High Concept:

See also Galois Tesseract.

Monday, June 11, 2018

Arty Fact

Filed under: General,Geometry — Tags: , , , , — m759 @ 10:35 pm

The title was suggested by the name "ARTI" of an artificial
intelligence in the new film 2036: Origin Unknown.

The Eye of ARTI —

See also a post of May 19, "Uh-Oh" —

— and a post of June 6, "Geometry for Goyim" — 

Mystery box  merchandise from the 2011  J. J. Abrams film  Super 8 

An arty fact I prefer, suggested by the triangular computer-eye forms above —

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

This is from the July 29, 2012, post The Galois Tesseract.

See as well . . .

Sunday, June 10, 2018

Pieces of April

Filed under: General — Tags: , , — m759 @ 12:25 am

This journal on April 16, 2018 —

Happy birthday to Pope Emeritus Benedict XVI.

Related material from another weblog in a post also dated April 16, 2018 —

"As I write this, it’s April 5, midway through the eight-day
festival of Passover. During this holiday, we Jews air our
grievances against the ancient Pharaoh who enslaved
and oppressed us, and celebrate the feats of strength
with which the Almighty delivered us from bondage —
wait a minute, I think I’m mixing up Passover with Festivus."
. . . .

"Next month: Time and Tesseracts."

From that next post, dated May 16, 2018 —

"The tesseract entered popular culture through
Madeleine L’Engle’s 'A Wrinkle in Time' . . . ."

The post's author, James Propp, notes that

" L’Engle caused some of her readers confusion
when one of the characters … the prodigy
Charles Wallace Murray [sic ] , declared 'Well, the fifth
dimension’s a tesseract.' "

Propp is not unfamiliar with prodigies:

"When I was a kid living in the Long Island suburbs,
I sometimes got called a math genius. I didn’t think
the label was apt, but I didn’t mind it; being put in
the genius box came with some pretty good perks."

— "The Genius Box," a post dated March 16, 2018

To me, Propp seems less like Charles Wallace
and more like the Prime Coordinator —

For further details, see the following synchronicity checks:

Propp March 16     Log24 March 16

Propp April 16        Log24 April 16

Propp May 16        Log24 May 16 .

Tuesday, May 1, 2018

Wake

Filed under: General — Tags: , — m759 @ 2:29 pm

Remarks on space from 1998 by sci-fi author Robert J. Sawyer quoted
here on Sunday (see the tag "Sawyer's Space") suggest a review of
rather similar remarks on space from 1977 by sci-fi author M. A. Foster
(see the tag "Foster's Space"):

Quoted here on September 26, 2012

"All she had to do was kick off and flow."

— The Gameplayers of Zan

"I'se so silly to be flowing but I no canna stay."

— Finnegans Wake

Another work by Sawyer —

Sunday, April 29, 2018

Sermon

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

'Imprisoned in a tesseract' in a 1998 science fiction novel

Wednesday, April 25, 2018

A Deathly Triangle

Filed under: General,Geometry — Tags: , , , — m759 @ 11:27 pm

'Imprisoned in a Tesseract,' a study of novelist James Blish

Thursday, March 29, 2018

“Before Creation Itself . . .”

Filed under: General,Geometry — Tags: , , , , — m759 @ 10:13 am

From the Diamond Theorem Facebook page —

A question three hours ago at that page

"Is this Time Cube?"

Notes toward an answer —

And from Six-Set Geometry in this journal . . .

Sunday, March 11, 2018

Blackboard Jungle Continues . . .

Filed under: General — m759 @ 10:28 am

. . . With intolerable disrespect for the word …
In particular, the word "theorem."
 

See also "Quantum Tesseract Theorem" in this  journal.

Thursday, March 8, 2018

Thanking the Academy…

Filed under: General,Geometry — m759 @ 9:11 pm

Continues.

Thursday, January 25, 2018

Beware of Analogical Extension

Filed under: General,Geometry — Tags: — m759 @ 11:29 am

"By an archetype  I mean a systematic repertoire
of ideas by means of which a given thinker describes,
by analogical extension , some domain to which
those ideas do not immediately and literally apply."

— Max Black in Models and Metaphors 
    (Cornell, 1962, p. 241)

"Others … spoke of 'ultimate frames of reference' …."
Ibid.

A "frame of reference" for the concept  four quartets

A less reputable analogical extension  of the same
frame of reference

Madeleine L'Engle in A Swiftly Tilting Planet :

"… deep in concentration, bent over the model
they were building of a tesseract:
the square squared, and squared again…."

See also the phrase Galois tesseract .

Wednesday, January 24, 2018

The Pentagram Papers

Filed under: General,Geometry — Tags: — m759 @ 12:40 pm

(Continued)

From a Log24 post of March 4, 2008 —

SINGER, ISAAC:
"Are Children the Ultimate Literary Critics?"
— Top of the News 29 (Nov. 1972): 32-36.

"Sets forth his own aims in writing for children and laments
'slice of life' and chaos in children's literature. Maintains that
children like good plots, logic, and clarity, and that they
have a concern for 'so-called eternal questions.'"

— An Annotated Listing of Criticism
by Linnea Hendrickson

"She returned the smile, then looked across the room to
her youngest brother, Charles Wallace, and to their father,
who were deep in concentration, bent over the model
they were building of a tesseract: the square squared,
and squared again: a construction of the dimension of time."

— A Swiftly Tilting Planet,
by Madeleine L'Engle

Cover of 'A Swiftly Tilting Planet' and picture of tesseract

For "the dimension of time," see A Fold in TimeTime Fold,
and Diamond Theory in 1937

A Swiftly Tilting Planet  is a fantasy for children 
set partly in Vespugia, a fictional country bordered by
Chile and Argentina.

Ibid.

The pen's point:

Wm. F. Buckley as Archimedes, moving the world with a giant pen as lever. The pen's point is applied to southern South America.
John Trever, Albuquerque Journal, 2/29/08

Note the figure on the cover of National Review  above —

A related figure from Pentagram Design

See, more generally,  Isaac Singer  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

Thursday, December 28, 2017

Rocky Start

Filed under: General — m759 @ 9:11 pm

The above prose suggests a musical alternative to the Dec. 21
Camazotz song in the posts tagged Quantum Tesseract Theorem . . .

 

Saturday, December 23, 2017

Search Result

Filed under: General — m759 @ 9:29 pm

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 ."

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!"

Thursday, October 19, 2017

Design Grammar***

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 10:22 pm

The elementary shapes at the top of the figure below mirror
the looking-glass property  of the classical Lo Shu square.

The nine shapes at top left* and their looking-glass reflection
illustrate the looking-glass reflection relating two orthogonal
Latin squares over the three digits of modulo-three arithmetic.

Combining these two orthogonal Latin squares,** we have a
representation in base three of the numbers from 0 to 8.

Adding 1 to each of these numbers yields the Lo Shu square.

Mirror symmetry of the ninefold Lo Shu magic square

* The array at top left is from the cover of
Wonder Years:
Werkplaats Typografie 1998-2008
.

** A well-known construction.

*** For other instances of what might be
called "design grammar" in combinatorics,
see a slide presentation by Robin Wilson.
No reference to the work of Chomsky is
intended.

Saturday, September 23, 2017

The Turn of the Frame

"With respect to the story's content, the frame thus acts
both as an inclusion of the exterior and as an exclusion
of the interior: it is a perturbation of the outside at the
very core of the story's inside, and as such, it is a blurring
of the very difference between inside and outside."

— Shoshana Felman on a Henry James story, p. 123 in
"Turning the Screw of Interpretation,"
Yale French Studies  No. 55/56 (1977), pp. 94-207.
Published by Yale University Press.

See also the previous post and The Galois Tesseract.

Sunday, August 27, 2017

Black Well

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

The “Black” of the title refers to the previous post.
For the “Well,” see Hexagram 48.

Related material —

The Galois Tesseract and, more generally, Binary Coordinate Systems.

Saturday, August 26, 2017

Aesthetic Distance

Filed under: General,Geometry — m759 @ 7:59 pm

Naive readers may suppose that this sort of thing is 
related to what has been dubbed "geometric group theory."

It is not. See posts now tagged Aesthetic Distance.

Sunday, July 23, 2017

The Partitioned Self

Filed under: General — m759 @ 6:00 am

Jung's self-symbol

http://www.log24.com/log/pix10A/100615-JungImago.gif

A meditation on Jung's self-symbol

Tuesday, July 11, 2017

Dialogue from Plato’s Cave

Filed under: General — Tags: , — m759 @ 10:15 am

At  scifi.stackexchange.com

Why was the Cosmic Cube named the Tesseract 
in the Marvel movie series? Is there any specific reason 
for the name change? According to me, Cosmic Cube
seems a nice and cooler name.

— Asked March 14, 2013, by Dhwaneet Bhatt
    
At least it wasn't called 'The AllSpark.' 
It's not out of the realm of possibility. 

— Solemnity, March 14, 2013

Saturday, June 3, 2017

Expanding the Spielraum (Continued*)

Filed under: General,Geometry — Tags: — m759 @ 1:13 pm

Or:  The Square

"What we do may be small, but it has
 a certain character of permanence."
— G. H. Hardy

* See Expanding the Spielraum in this journal.

Tuesday, May 23, 2017

Pursued by a Biplane

Filed under: General,Geometry — Tags: — m759 @ 9:41 pm

The Galois Tesseract as a biplane —

Cary Grant in 'North by Northwest'

Saturday, May 20, 2017

The Ludicrous Extreme

Filed under: General,Geometry — Tags: — m759 @ 1:04 am

From a review of the 2016 film "Arrival"

"A seemingly off-hand reference to Abbott and Costello
is our gateway. In a movie as generally humorless as Arrival,
the jokes mean something. Ironically, it is Donnelly, not Banks,
who initiates the joke, naming the verbally inexpressive
Heptapod aliens after the loquacious Classical Hollywood
comedians. The squid-like aliens communicate via those beautiful,
cryptic images. Those signs, when thoroughly comprehended,
open the perceiver to a nonlinear conception of time; this is
Sapir-Whorf taken to the ludicrous extreme."

Jordan Brower in the Los Angeles Review of Books

Further on in the review —

"Banks doesn’t fully understand the alien language, but she
knows it well enough to get by. This realization emerges
most evidently when Banks enters the alien ship and, floating
alongside Costello, converses with it in their picture-language.
She asks where Abbott is, and it responds — as presented
in subtitling — that Abbott 'is death process.'
'Death process' — dying — is not idiomatic English, and what
we see, written for us, is not a perfect translation but a
rendering of Banks’s understanding. This, it seems to me, is a
crucial moment marking the hard limit of a human mind,
working within the confines of human language to understand
an ultimately intractable xenolinguistic system."

For what may seem like an intractable xenolinguistic system to
those whose experience of mathematics is limited to portrayals
by Hollywood, see the previous post —

van Lint and Wilson Meet the Galois Tesseract.

The death process of van Lint occurred on Sept. 28, 2004.

See this journal on that date

Tuesday, May 2, 2017

Image Albums

Filed under: General,Geometry — Tags: , , , , , — m759 @ 1:05 pm

Pinterest boards uploaded to the new m759.net/piwigo

Diamond Theorem 

Diamond Theorem Correlation

Miracle Octad Generator

The Eightfold Cube

Six-Set Geometry

Diamond Theory Cover

Update of May 2 —

Four-Color Decomposition

Binary Galois Spaces

The Galois Tesseract

Update of May 3 —

Desargues via Galois

The Tetrahedral Model

Solomon's Cube

Update of May 8 —

Art Space board created at Pinterest

Saturday, March 25, 2017

Twin Pillars of Symmetry

Filed under: General,Geometry — Tags: — m759 @ 1:00 pm

The phrase "twin pillars" in a New York Times  Fashion & Style
article today suggests a look at another pair of pillars —

This pair, from the realm of memory, history, and geometry disparaged
by the late painter Mark Rothko, might be viewed by Rothko
as  "parodies of ideas (which are ghosts)." (See the previous post.)

For a relationship between a 3-dimensional simplex and the {4, 3, 3},
see my note from May 21, 2014, on the tetrahedron and the tesseract.

Sunday, March 19, 2017

Norwegian Sermon

Filed under: General — m759 @ 11:30 am

"And the Führer digs for trinkets in the desert."

See also the previous post.

Saturday, December 10, 2016

Folk Etymology

Images from Burkard Polster's Geometrical Picture Book

See as well in this journal the large  Desargues configuration, with
15 points and 20 lines instead of 10 points and 10 lines as above.

Exercise:  Can the large Desargues configuration be formed
by adding 5 points and 10 lines to the above Polster model
of the small configuration in such a way as to preserve
the small-configuration model's striking symmetry?  
(Note: The related figure below from May 21, 2014, is not
necessarily very helpful. Try the Wolfram Demonstrations
model
, which requires a free player download.)

Labeling the Tetrahedral Model (Click to enlarge) —

Related folk etymology (see point a  above) —

Related literature —

The concept  of "fire in the center" at The New Yorker , 
issue dated December 12, 2016, on pages 38-39 in the
poem by Marsha de la O titled "A Natural History of Light."

Cézanne's Greetings.

Friday, December 9, 2016

Snow Dance

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

See Ballet Blanc  in this journal.

For a darker perspective, click on the image below.

IMAGE- Detail of large 'Search for the Lost Tesseract' image with Amy Adams, Richard Zanuck, 'snowflake' structure, and white gloves

See also Cartier in The Hexagon of Opposition.

Happy birthday to Kirk Douglas.

Kirk Douglas in 'Diamonds'

Wednesday, October 5, 2016

Sources

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

From a Google image search yesterday

Sources (left to right, top to bottom) —

Math Guy (July 16, 2014)
The Galois Tesseract (Sept. 1, 2011)
The Full Force of Roman Law (April 21, 2014)
A Great Moonshine (Sept. 25, 2015)
A Point of Identity (August 8, 2016)
Pascal via Curtis (April 6, 2013)
Correspondences (August 6, 2011)
Symmetric Generation (Sept. 21, 2011)

Tuesday, August 16, 2016

Midnight Narrative

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

The images in the previous post do not lend themselves
to any straightforward narrative. Two portions of the
large image search are, however, suggestive —


Boulez and Boole      and

Cross and Boolean lattice.

The improvised cross in the second pair of images
is perhaps being wielded to counteract the
Boole of the first pair of images. See the heading
of the webpage that is the source of the lattice
diagram toward which the cross is directed —

Update of 10 am on August 16, 2016 —

See also Atiyah on the theology of
(Boolean) algebra vs. (Galois) geometry:

Friday, July 15, 2016

Autistic Enchantment*

Filed under: General,Geometry — Tags: , — m759 @ 10:29 am

Robert Nye, author of the novel Falstaffreportedly died
at 77 on July 2, 2016.

Harvey D. Heinz, expert on magic squares, cubes,
tesseracts, etc., reportedly died at 82 on July 6, 2013.

In memoriam —

From the date of Nye's death:

From Nye's book:

From the date of Heinz's death:

* See also a search for the title in this journal.

Wednesday, May 25, 2016

Framework

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

"Studies of spin-½ theories in the framework of projective geometry
have been undertaken before." — Y. Jack Ng  and H. van Dam
February 20, 2009

For one such framework,* see posts from that same date 
four years earlier — February 20, 2005.

* A 4×4 array. See the 19771978, and 1986 versions by 
Steven H. Cullinane,   the 1987 version by R. T. Curtis, and
the 1988 Conway-Sloane version illustrated below —

Cullinane, 1977

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

Cullinane, 1978

Cullinane, 1986

Curtis, 1987

Update of 10:42 PM ET on Sunday, June 19, 2016 —

The above images are precursors to

Conway and Sloane, 1988

Update of 10 AM ET Sept. 16, 2016 — The excerpt from the
1977 "Diamond Theory" article was added above.

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.

Tuesday, May 10, 2016

Game Theory

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

The following passage appeared in this journal
on the night of May 23-24, 2015.

IMAGE- A fictional vision of resurrection within a tesseract

The afternoon  of May 23, 2015, was significant
for devotees of mathematics and narrative.

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