Log24

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, November 12, 2017

Devil’s Music

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

From the groom in the previous post — Swedenborg Chapel

And from November 7 last year —

Monday, November 7, 2016

Subway Art for Times Square Church

Filed under: Uncategorized — m759 @ 9:11 PM 

Click images for related material.

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

"The Cardinal seemed a little preoccupied today."

Sunday, April 10, 2011

Bedeviled

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

From tonight's online New York Times

John McCracken, Sculptor of Geometric Forms, Dies at 76

McCracken died in Manhattan on Friday, April 8.

From Christopher Knight in tonight's online LA Times

the works embody perceptual and philosophical conundrums. The colored planks stand on the floor like sculptures….

McCracken was bedeviled by Stanley Kubrick's famously obscure science-fiction epic, "2001: A Space Odyssey," with its iconic image of an ancient monolith floating in outer space. The 1968 blockbuster was released two years after the artist made his first plank.

"At the time, some people thought I had designed the monolith or that it had been derived from my work," he told art critic Frances Colpitt of the coincidence in a 1998 interview.

Two photos of McCracken's 1967 Black Plank  seem relevant—

November 28, 2010 (Click to enlarge)

http://www.log24.com/log/pix11/110410-McCrackenPlank1967400w.jpg

December 28, 2010 (Click to enlarge)

http://www.log24.com/log/pix11/110410-McCracken-NatGallery-NothingToSeeHere-400w.jpg

Material that an artist might view as related, if only synchronistically—

Two posts in this journal on the dates the photos were taken—
The Embedding on November 28 and Dry Bones on December 28.

The photos are of an exhibition titled "There is nothing to see here" at the
National Gallery of Art, October 30, 2010-April 24, 2011 —

Click to enlarge.

http://www.log24.com/log/pix11/110410-NothingToSee-400w.jpg

For related nihilism from the National Gallery, see "Pictures of Nothing" in this journal.

Some less nihilistic illustrations—

The Meno  Embedding

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

A photo by one of the artists whose work is displayed above beside McCracken's—

http://www.log24.com/log/pix11/110410-Sugimoto-AndoChurch.jpg

"Accentuate the Positive."
 — Clint Eastwood

Saturday, March 18, 2017

News Search

Filed under: General — Tags: — m759 @ 12:30 PM

See also, in this journal, "Before the Devil Knows You're Dead."

Saturday, April 5, 2014

A Princeton Half-Hour

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

"May you be in heaven a full half-hour
before the devil knows you're dead ."

Related material:
Yesterday's posts of 12 PM and 8 PM,
and the life of Philip Seymour Hoffman.

Monday, June 1, 2020

The Gefter Boundary

Filed under: General — Tags: , — m759 @ 5:09 PM

The message was clear: having a finite frame of reference
creates the illusion of a world, but even the reference frame itself
is an illusion. Observers create reality, but observers aren’t real.
There is nothing ontologically distinct about an observer, because
you can always find a frame in which that observer disappears:
the frame of the frame itself, the boundary of the boundary.”

— Amanda Gefter in 2014, quoted here on Mayday 2020.

Image- Josefine Lyche work (with 1986 figures by Cullinane) in a 2009 exhibition in Oslo

See as well the previous post.

A Graveyard Smash: Galois Geometry Meets Nordic Aliens

Filed under: General — Tags: — m759 @ 4:02 PM

See also Vril Chick.

Thursday, May 28, 2020

Unity Game

Filed under: General — Tags: , , — m759 @ 11:59 PM

“Old men ought to be explorers.” — T. S. Eliot

“Everybody’s lost but me!” — Young Indiana Jones, quoted
in a book review (“Knox Peden on Martin Hägglund”) in
Sydney Review of Books  on May 26 . . .

” Here I am reminded of the words of
the young Indiana Jones alone in the desert,
decades before the Last Crusade:
‘Everybody’s lost but me.’ “

 Related remarks — Now You See It, Now You Don’t.

Finite Geometry at GitHub

Filed under: General — Tags: , , — m759 @ 5:04 PM

My website on finite geometry is now available
on GitHub at http://m759.github.io/ . The part
of greatest interest to coders is also at
https://repl.it/@m759/View-4x4x4#index.html .

Wednesday, May 20, 2020

Cue the Violins

Filed under: General — Tags: , — m759 @ 11:36 PM

In memory of a music editor.

Blangsted reportedly died on May 1.
See also that date in this journal, among
other posts tagged The Next Level.

Raiders of the Lost Unity

Filed under: General — Tags: — m759 @ 2:21 PM

Mathematics as a Black Art

Filed under: General — Tags: , , — m759 @ 11:34 AM

Sunday, May 17, 2020

Gran Torino

Filed under: General — Tags: — m759 @ 3:55 PM

Source citation for an article quoted here last night

Hegel’s Conceptual Group Action —

A check of that source yields the seal of the University of Torino —

Related material —

Saturday, May 16, 2020

Bullshit Studies

Filed under: General — Tags: — m759 @ 11:59 PM

From https://www.mathunion.org/outreach/logos/versions-all-logos

Click the logo for some IMU history.

Related bullshit —

Hegel’s Conceptual Group Action

Click the banner below for the background of the logo

Thursday, May 7, 2020

Now You See It, Now You Don’t

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

Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to explain
how art fits into our subject and what we mean by beauty.”

— Sir Michael Atiyah, quoted here on April 4, 2016

 

Illustrations, from the American Mathematical Society Spring
2020 book sale, of a book scheduled to be published May 28.

Kant as Diamond Cutter

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

“He wished Kant were alive. Kant would have appreciated it.
That master diamond cutter.”

— Robert M. Pirsig, Zen and the Art of Motorcycle Maintenance , Part III.

Kant’s  “category theory” —

“In the Transcendental Analytic, Kant deduces the table of twelve categories, or pure concepts of the understanding….

The categories must be ‘schematized’ because their non-empirical origin in pure understanding prevents their having the sort of sensible content that would connect them immediately to the objects of experience; transcendental schemata are mediating representations that are meant to establish the connection between pure concepts and appearances in a rule-governed way. Mathematical concepts are discussed in this context since they are unique in being pure but also sensible concepts: they are pure because they are strictly a priori  in origin, and yet they are sensible since they are constructed in concreto . ”

— Shabel, Lisa, “Kant’s Philosophy of Mathematics”, The Stanford Encyclopedia of Philosophy  (Spring 2016 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/spr2016/entries/kant-mathematics/>.

See also The Diamond Theorem and Octad.us.

Wednesday, May 6, 2020

The Ship of Theseus”…

Filed under: General — Tags: , — m759 @ 2:17 PM

is a philosophical conundrum  discussed this morning  in the weblog of
David Justice.

A related statement of this “problem of identity,” from posts
in this  weblog tagged “For Banff 2009” yesterday afternoon

Remarks related to the ship of Theseus

Tuesday, May 5, 2020

In Memoriam: Jan Saxl, Mathematician

Filed under: General — Tags: — m759 @ 1:03 PM

For the conference, see
https://www.birs.ca/events/2009/5-day-workshops/09w5030 .

For the Church of Synchronology:
Time for you to see the field.” (Posts now tagged “For Banff 2009”)

Notes for a Wake

Filed under: General — Tags: — m759 @ 11:52 AM

If Not Sublime

Filed under: General — Tags: — m759 @ 4:24 AM

This is perhaps the same Robert Mezey, poet at Pomona College,
who reportedly died on April 25.

See a Pomona link, the Fano Hallows, from this journal on that date.

Monday, May 4, 2020

Raiders of the Lost Sublime

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

The Man Behind the Counter

Filed under: General — Tags: — m759 @ 7:06 PM

The title is a phrase from the Suzanne Vega song in the previous post.

Always busy counting . . . .” — Tagline at Peter J. Cameron’s weblog.

“This morning brought the news that Jan Saxl died on Saturday.”

Peter J. Cameron today

A search for Saxl in this  weblog yields a post related to a topic in
Wolfram Neutsch’s book Coordinates.  See Saturday’s post
Turyn’s Octad Theorem: The Next Level.”

Related narrative from the Saturday post —

Related narrative from Sunday’s Westworld finale —

The Thing and I …

Filed under: General — Tags: — m759 @ 3:00 PM

Continues.

Update at 4 PM —

“It is always
Nice to see you”
Says the man
Behind the counter

— Suzanne Vega. “Tom’s Diner”

Tom Stall’s diner in  “A History of  Violence” (30 September 2005).

This  journal on 30 September 2005 —

“This place ain’t doing me any good.
I’m in the wrong town,
I should be in Hollywood.”

— Dylan, “Things Have Changed

Sunday, May 3, 2020

Crichton Time

Filed under: General — Tags: , — m759 @ 2:37 PM


Into the Westworld

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

He beomes aware of something else… some other presence.
“Anybody here?” he says.
I am here.
He almost jumps, it is so loud. Or it seems loud. Then he wonders if
he has heard anything at all.
“Did you speak?”
No.
How are we communicating? he wonders.
The way everything communicates with everything else.
Which way is that?
Why do you ask if you already know the answer?

— Sphere   by Michael Crichton, Harvard ’64

Saturday, May 2, 2020

Beat the Clock

Filed under: General — Tags: — m759 @ 12:27 PM

Turyn’s Octad Theorem: The Next Level*

Filed under: General — Tags: — m759 @ 11:24 AM

From  the obituary of a game inventor  who reportedly died
on Monday, February 25, 2013 —

” ‘He was hired because of the game,’ Richard Turyn,
a mathematician who worked at Sylvania, told
the Washington Post in a 2004 feature on Diplomacy.”

* For the theorem, see Wolfram Neutsch,  Coordinates .
(Published by de Gruyter, 1996. See pp. 761-766.)

Having defined (pp. 751-752) the Miracle Octad Generator (MOG)
as a 4×6 array to be used with Conway’s “hexacode,” Neutsch says . . .

“Apart from the three constructions of the Golay codes
discussed at length in this book (lexicographic and via
the MOG or the projective line), there are literally
dozens of alternatives. For lack of space, we have to
restrict our attention to a single example. It has been
discovered by Turyn and can be connected in a very
beautiful way with the Miracle Octad Generator….

To this end, we consider the natural splitting of the MOG into
three disjoint octads L, M, R (‘left’, ‘middle’, and ‘right’ octad)….”

— From page 761

The theorem of Turyn”  is on page 764

Friday, May 1, 2020

The H-State

Filed under: General — Tags: , , — m759 @ 9:27 PM

Related pure mathematics

The Escape from Plato’s Cave to . . .

See also Numberland and Walpurgisnacht Geometry.

Bullshit Studies

Filed under: General — Tags: , , — m759 @ 2:55 PM

The following passage is from Amanda Gefter’s  Trespassing
on Einstein’s Lawn  (Bantam Books, 2014).

“You know the story of Plato’s cave?” my father asked. “All the prisoners are chained up in the cave and they can’t see the real world outside, only the shadows on the wall? That’s supposed to be a negative thing, like they’ll never know reality. But the truth is, you have to be stuck inside a limited reference frame for there to be any reality at all! If you weren’t chained to your light cone, you’d see nothing. The H-state.”

I nodded. “You’d have no information. You need the broken symmetry, the shadow, to have information and information gives rise to the world. It from bit.”

I couldn’t help but grin with excitement. The message was clear: having a finite frame of reference creates the illusion of a world, but even the reference frame itself is an illusion. Observers create reality, but observers aren’t real. There is nothing ontologically distinct about an observer, because you can always find a frame in which that observer disappears: the frame of the frame itself, the boundary of the boundary.

“If physicists discover an invariant someday, the game will be up,” my father mused. “That would rule out the hypothesis that the universe is really nothing.”

That was true. But so far, at least, every last invariant had gone the way of space and time, rendered relative and observer-dependent. Spacetime, gravity, electromagnetism, the nuclear forces, mass, energy, momentum, angular momentum, charge, dimensions, particles, fields, the vacuum, strings, the universe, the multiverse, the speed of light— one by one they had been downgraded to illusion. As the surface appearance of reality fell away, only one thing remained. Nothing.

My path to Gefter’s father’s musing led from a quotation attributed,
probably falsely, to John Archibald Wheeler on page 52 of Octavio
Paz’s  Claude Lévi-Strauss: An Introduction  (Cornell, 1970)

There is a point at which

‘something is nothing and nothing is something.’

The quote may actually be by AP writer John Barbour reporting
on a 1967 American Physical Society talk by Wheeler, “The End
of Time.”

Gefter mentions Wheeler 369 times:

See as well Introduction to Quantum Woo.

Sunday, April 26, 2020

The Triangle of Art

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

This post was suggested by yesterday morning’s link to  The Fano Hallows.

Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to explain
how art fits into our subject and what we mean by beauty.”

— Sir Michael Atiyah, quoted here on April 4, 2016

A symbol related to  The Fano Hallows

Sunday, April 19, 2020

Easter Egg for Wittgenstein

Filed under: General — Tags: — m759 @ 1:24 PM

A language game on Orthodox Easter —

See also Geometric Theology and Trinity Staircase.

Friday, April 17, 2020

Under the April Snow…

Filed under: General — Tags: — m759 @ 3:26 PM

Nonsense for Streisand:

Thursday, April 16, 2020

Snow White’s Time in Space

Filed under: General — Tags: , — m759 @ 2:32 PM

Kristen Stewart — 'Freezin' my ass off'

See as well Pi Day 2020.

Tuesday, March 17, 2020

Geometric Theology

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

Before time began” — Optimus Prime

IMAGE- The Trinity of Max Black (a 3-set, with its eight subsets arranged in a Hasse diagram that is also a cube)

See also posts tagged Aitchison.

 

Thursday, March 12, 2020

Letter to the World

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

Pace  Emily Dickinson.

Play Date

Filed under: General — Tags: — m759 @ 7:15 PM

Today's 4:02 AM ET post, "Steinfeld as Rose the Hat,"
suggests a review —

A more impressive woman in white

Update of 8 PM ET —

Beckinsale gives Oct. 5, 2001, as the date of the New York
premiere of the film "Serendipity."  Synchronology check:

Beckinsale's premiere date — Oct. 5, 2001 — is incorrect.
The film was released  on that date, but its New York premiere
was actually on Oct. 3, 2001. See Getty Images.

Sunday, March 8, 2020

A Tool

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

Detail: 

Click on the above date for further details.

Sunday, March 1, 2020

Same Staircase, Different Day

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

Freeman Dyson on his staircase at Trinity College
(University of Cambridge) and on Ludwig Wittgenstein:

“I held him in the highest respect and was delighted
to find him living in a room above mine on the same
staircase. I frequently met him walking up or down
the stairs, but I was too shy to start a conversation.”

Frank Close on Ron Shaw:

“Shaw arrived there in 1949 and moved into room K9,
overlooking Jesus Lane. There is nothing particularly
special about this room other than the coincidence that
its previous occupant was Freeman Dyson.”

— Close, Frank. The Infinity Puzzle  (p. 78).
Basic Books. Kindle Edition.

See also other posts now tagged Trinity Staircase.

Illuminati enthusiasts  may enjoy the following image:

'Ex Fano Apollinis'- Fano plane, eightfold cube, and the two combined.

Sunday, February 23, 2020

The Representation of Reality

Filed under: General — Tags: — m759 @ 1:36 PM

"Although art is fundamentally everywhere and always the same,
nevertheless two main human inclinations, diametrically opposed
to each other, appear in its many and varied expressions. ….
The first aims at representing reality objectively, the second subjectively." 

Mondrian, 1936  [Links added.]

An image search today (click to enlarge) —

Image search for 'Eightfold Cube'

Friday, January 17, 2020

September Morn

Filed under: General — Tags: — m759 @ 10:17 PM

Epigraph from Ch. 4 of Design Theory , Vol. I:

"Es is eine alte Geschichte,
 doch bleibt sie immer neu 
"
 —Heine (Lyrisches Intermezzo  XXXIX)

This epigraph was quoted here earlier on
the morning of September 1, 2011.

Design Theory

Filed under: General — Tags: , — m759 @ 12:57 PM

On a recently deceased professor emeritus of architecture
at Princeton —

"… Maxwell  'established the school as a principal
center of design research, history and theory.' ”

"This is not the Maxwell you're looking for."

Friday, January 10, 2020

The Lotos Rose

Filed under: General — Tags: , , — m759 @ 12:01 AM

See the title in this journal.

Thursday, January 9, 2020

Universal History (Adapted)

Filed under: General — Tags: , , — m759 @ 11:05 PM

"One of the more fortuitous encounters of late-20th-century popular culture —
almost up there with Lennon meets McCartney and Taylor meets Burton —
took place on Labor Day 1965, at Jane Fonda’s Malibu beach house. The
actress was hosting a daylong bash at which her father, Henry’s,
generation mingled uneasily with her Hollywood hippie friends. The Byrds
played in the backyard. A young comedian-turned-film director named Mike
Nichols was approached by an improv comic-turned-itinerant writer named
Buck Henry, who asked how he was doing. Nichols dourly looked around
at all the proto-Summer of Love vibes and said, 'Here, under the shadow
of the great tree, I have found peace.'

Henry immediately recognized a sardonic East Coast kindred spirit trapped
in Lotusland . . . ."

—  Ty Burr,  Boston Globe  staff, January 9, 2020, 10:34 AM

Adapt or Die

Filed under: General — Tags: , , — m759 @ 11:53 AM

For Hollywood

Maybe.

For Emily Yahr (see second item above) —

Buck Henry reportedly died yesterday, January 8, 2020.
This journal on that date a year earlier —

Sunday, December 29, 2019

Springer Link

Filed under: General — Tags: , , — m759 @ 5:08 PM

Related reading

"I closed my eyes and saw the number 137—
so very close to the reciprocal of alpha—
on the chest of the runner in Van Cortlandt Park.
Should I start the story there? "

— Alpert, Mark.  Saint Joan of New York
(Science and Fiction) (p. 103).
Springer International Publishing. Kindle edition. 

Cover detail:

See as well St. Joan in this  journal.

Friday, December 27, 2019

The Secret Life of Mark Alpert

Filed under: General — Tags: — m759 @ 9:45 AM

Booklist on Final Theory :

"Alpert, an editor for Scientific American , laces his high-IQ
doomsday thriller with clearly explicated and hauntingly beautiful
scientific theories…."

Booklist on The Omega Theory

"Alpert’s follow-up to his acclaimed first novel, Final Theory  (2008),
continues the adventures of science historian David Swift."

See as well this  journal on June 1, 2008.

God in the Object…

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

. . . Pace  Wallace Stevens.

"The history of the universe can thus be seen as
an endless chain of changes, but Aquinas argued
that there must be some transcendent entity that
initiated the chain, something that is itself
unchanging and that already possesses all of the
properties that worldly objects can come to possess.
He also claimed that this entity must be eternal;
because it is the root of all causes, nothing else
could’ve caused it. And unlike all worldly objects,
the transcendent entity is necessary—it must  exist."

— Mark Alpert in Scientific American, 12/23/2019

Wednesday, December 11, 2019

Klein Quadric

Filed under: General — Tags: , — m759 @ 1:06 AM

The architecture of the recent post 
Geometry of 6 and 8 is in part
a reference to the Klein quadric.

Tuesday, December 10, 2019

Eternal Color

Filed under: General — Tags: , , — m759 @ 7:34 PM

For the above title, see posts tagged Eternal Color.

From this evening's online New York Times

Related imaterial —

A scene from the film of the above book —

Sunday, December 8, 2019

Geometry of 6 and 8

Filed under: General — Tags: , — m759 @ 4:03 AM

Just as
the finite space PG(3,2) is
the geometry of the 6-set, so is
the finite space PG(5,2)
the geometry of the 8-set.*

Selah.

* Consider, for the 6-set, the 32
(16, modulo complementation)
0-, 2-, 4-, and 6-subsets,
and, for the 8-set, the 128
(64, modulo complementation)
0-, 2-, 4-, 6-, and 8-subsets.

Update of 11:02 AM ET the same day:

See also Eightfold Geometry, a note from 2010.

Thursday, December 5, 2019

The Fontana Arches

Filed under: General — Tags: — m759 @ 4:17 PM

(For Harlan Kane)

See also earlier posts tagged The Fontana Arches.

Wednesday, December 4, 2019

Spielerei

Filed under: General — Tags: , , — m759 @ 7:18 PM

(See the title in this journal.)

Also on the above YouTube date — July 12, 2010 —

 

In Memoriam

Filed under: General — Tags: , , — m759 @ 12:35 PM

Also on the above date — October 28, 2013 —

Nicht Spielerei

Filed under: General — Tags: , — m759 @ 10:22 AM

(See the title in earlier posts.)

Monday, December 2, 2019

“Show me all  the blueprints.”

Filed under: General — Tags: , — m759 @ 11:32 PM

Aesthetics at Harvard

Filed under: General — Tags: , , , — m759 @ 4:05 PM

"What the piece of art is about is the gray space in the middle."

— David Bowie, as quoted in the above Crimson  piece.

Bowie's "gray space" is the space between the art and the beholder.

I prefer the gray space in the following figure —

Some small Galois spaces (the Cullinane models)

Context:  The Trinity Stone  (Log24, June 4, 2018).

Gray Space

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

See as well a search for Gray Space in this journal.

Related material:  The Schwartz Omega .

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

— Richard Evan Schwartz

D8: The Black Queen’s Square

Filed under: General — Tags: , — m759 @ 3:45 AM

The previous post quoted some dialogue from Victor Hugo's
novel about the French Revolution, Ninety-Three.

This suggests a look at the following non-fiction book:

Compare and contrast with the novel The Eight , by Katherine Neville,
about chess and the French Revolution.

Neville's birthday, April 4, plays a major role in her novel. The dies natalis 
(in the Roman Catholic sense) of the above Birth of the Chess Queen 
author, on the other hand, was reportedly November 20, 2019.

Following a link in this journal from November 20 leads to remarks 
that might interest the subjects of an upcoming film, "The Two Popes."

Sunday, December 1, 2019

Dialogue

Filed under: General — Tags: , — m759 @ 10:09 PM

Space Song

Filed under: General — Tags: , — m759 @ 1:50 PM

From this journal on the above date — April 13, 2014 (Gray Space) —

Review of Seeing Gray , a book by pastor Adam Hamilton
of the United Methodist Church of the Resurrection
in Leawood, Kansas, a suburb of Kansas City—

“Adam Hamilton invites us to soulful gray space
between polarities, glorious gray space that is holy,
mysterious, complex, and true. Let us find within
our spirits the courage and humility to live and learn
in this faithful space, to see gray, to discern a more
excellent way.”

—Review by United Methodist Bishop Hope Morgan Ward

The above flashback was suggested by CBS Sunday Morning today —

See also Romanesque in this journal.

Wednesday, October 30, 2019

For The October Country

Filed under: General — Tags: — m759 @ 2:01 AM

The above document was linked to here on Dec. 8, 2008

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 .

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 29, 2019

Stage Direction: “Comments Off.”

Filed under: General — Tags: , — m759 @ 11:29 AM

The previous post dealt with “magic” cubes, so called because of the
analogous “magic” squares. Douglas Hofstadter has written about a
different, physical , object, promoted as “the  Magic Cube,” that Hofstadter
felt embodied “a deep invariant”:

Friday, September 27, 2019

The Black List

Filed under: General — Tags: , — m759 @ 11:46 AM

"… Max Black, the Cornell philosopher, and others have pointed out
how 'perhaps every science must start with metaphor and end with
algebra, and perhaps without the metaphor there would never have
been any algebra' …."

— Max Black, Models and Metaphors, Cornell U. Press, 1962,
page 242, as quoted in Dramas, Fields, and Metaphors, by 
Victor Witter Turner, Cornell U. Press, paperback, 1975, page 25
 

Metaphor —

Algebra —

The 16 Dirac matrices form six anticommuting sets of five matrices each (Arfken 1985, p. 214):

1. alpha_1alpha_2alpha_3alpha_4alpha_5,

2. y_1y_2y_3y_4y_5,

3. delta_1delta_2delta_3rho_1rho_2,

4. alpha_1y_1delta_1sigma_2sigma_3,

5. alpha_2y_2delta_2sigma_1sigma_3,

6. alpha_3y_3delta_3sigma_1sigma_2.

SEE ALSO:  Pauli Matrices

REFERENCES:

Arfken, G. Mathematical Methods for Physicists, 3rd ed.  Orlando, FL: Academic Press, pp. 211-217, 1985.

Berestetskii, V. B.; Lifshitz, E. M.; and Pitaevskii, L. P. "Algebra of Dirac Matrices." §22 in Quantum Electrodynamics, 2nd ed.  Oxford, England: Pergamon Press, pp. 80-84, 1982.

Bethe, H. A. and Salpeter, E. Quantum Mechanics of One- and Two-Electron Atoms.  New York: Plenum, pp. 47-48, 1977.

Bjorken, J. D. and Drell, S. D. Relativistic Quantum Mechanics.  New York: McGraw-Hill, 1964.

Dirac, P. A. M. Principles of Quantum Mechanics, 4th ed.  Oxford, England: Oxford University Press, 1982.

Goldstein, H. Classical Mechanics, 2nd ed.  Reading, MA: Addison-Wesley, p. 580, 1980.

Good, R. H. Jr. "Properties of Dirac Matrices." Rev. Mod. Phys. 27, 187-211, 1955.

Referenced on Wolfram|Alpha:  Dirac Matrices

CITE THIS AS:

Weisstein, Eric W.  "Dirac Matrices."

From MathWorld— A Wolfram Web Resource. 
http://mathworld.wolfram.com/DiracMatrices.html

Desiring the exhilarations of changes:
The motive for metaphor, shrinking from
The weight of primary noon,
The A B C of being,

The ruddy temper, the hammer
Of red and blue, the hard sound—
Steel against intimation—the sharp flash,
The vital, arrogant, fatal, dominant X.

— Wallace Stevens, "The Motive for Metaphor"

Tuesday, September 24, 2019

American Mathematical Society News

Filed under: General — Tags: — m759 @ 10:15 AM

  "Specifically, on May 6 of this year …." …

Friday, August 30, 2019

The Coxeter Aleph

Filed under: General — Tags: — m759 @ 4:21 AM

(Continued)

The previous post displayed part of a page from
a newspaper published the day Olivia Newton-John
turned 21 — Friday, September 26, 1969.

A meditation, with apologies to Coleridge:

In Xanadu did Newton-John
A stately pleasure-square decree
Where Aleph the sacred symbol ran
Through subsquares measureless to man.

A related video —

Beware, beware, her flashing eyes, her floating hair:

Set design —

As opposed to block design

Tuesday, August 27, 2019

Tiger Leaps

Filed under: General — Tags: , — m759 @ 10:33 AM

Walter Benjamin on 'a tiger's leap into the past'

See also Tiger in this journal, esp.

The image “http://www.log24.com/log/pix05/050123-Tiger.JPG” cannot be displayed, because it contains errors.   and . . .

other "Death and the Spirit" posts.

Friday, August 16, 2019

Nocciolo

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

(Continued)

IMAGE- 'Nocciolo': A 'kernel' for Pascal's Hexagrammum Mysticum: The 15 2-subsets of a 6-set as points in a Galois geometry.

A revision of the above diagram showing
the Galois-addition-table structure —

Related tables from August 10

See "Schoolgirl Space Revisited."

Saturday, August 10, 2019

Schoolgirl Space* Revisited:

Filed under: General — Tags: , — m759 @ 10:51 PM

The Square "Inscape" Model of
the Generalized Quadrangle W(2)

Click image to enlarge.

* The title refers to the role of PG (3,2) in Kirkman's schoolgirl problem.
For some backstory, see my post Anticommuting Dirac Matrices as Skew Lines
and, more generally, posts tagged Dirac and Geometry.

Frases de Cómic

Filed under: General — Tags: — m759 @ 4:03 PM

Thursday, August 8, 2019

Release Date

Filed under: General — Tags: — m759 @ 4:17 AM

From this  journal on that date

Piano roll for "I am sixteen going on seventeen" —

Saturday, July 27, 2019

Make America Strange Again

Filed under: General — Tags: — m759 @ 10:59 PM

Suggested by the previous post, "The Swarm" —

“‘Oracle, why did you write
The Grasshopper Lies Heavy?
What are we supposed to learn?'”

— Philip K. Dick

“She began throwing the coins.“

I Ching Hexagram 61: Inner Truth

The Swarm

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

“Design Is How It Works.” — Steve Jobs

Filed under: General — Tags: — m759 @ 9:59 PM

The Stephen King Intelligence Test

Filed under: General — Tags: , — m759 @ 4:41 PM


Related images —

See also other posts tagged Arti Facts.

This  post was suggested by those posts and by the following
attempt at humor —

Tonality

Filed under: General — Tags: — m759 @ 12:19 PM
 

https://languagelog.ldc.upenn.edu/nll/?p=22045

. . . the ancient Chinese made music the pinnacle of wisdom.

There was a Classic of Music (Yuè jīng 樂經), but it was lost already by the Han Dynasty (206 BC-220 AD).  Yet we know the esteem in which music was held by the ancient Chinese from passages in the classics like the following:

The Master heard the Shao in Qi and for three months did not notice the taste of the meat he ate. He said, 'I never dreamt the joys of music could reach such heights.'(D.C. Lau)

Analects  7.14

Shao is the music of the mythical emperor, Shun 舜.

Qi one of the Warring States.

Extensive commentary here.

Another instance of the sublimity of music in ancient China is the description of the performance of the Yellow Emperor's "The Pond of Totality" in chapter 14 of the Zhuang Zi  (see Victor H. Mair, tr., Wandering on the Way  [Bantam, 1994; University of Hawaii Press, 1998], pp. 132-136, available here).

Conversely, language studies in traditional China were referred to as "minor learning" (xiǎoxué 小學).

November 5, 2015 @ 4:05 pm · Filed by Victor Mair under Language and musicLanguage and philosophy

See also Ervin Wilson in Wikipedia, and a Log24 post from
the date of his death — December 8, 2016.

Monday, July 22, 2019

The Four Diamonds Meet the Five Red Herrings

Filed under: General — Tags: — m759 @ 11:34 AM

Lord Peter Wimsey (Balliol 1912) on the Balliol-Trinity rivalry at Oxford:

See also Balliol College in the post subtitled Spidey Goes to Church.

Signed, Sealed, Delivered

Filed under: General — Tags: — m759 @ 5:58 AM

Visual Languages

Filed under: General — Tags: — m759 @ 4:48 AM

Good question. See also this  journal on the above date —

September 15, 2018.

Space, Time, Form

Filed under: General — Tags: — m759 @ 3:33 AM

Magic cube and corresponding hexagram, or Star of David, with faces mapped to lines and edges mapped to points

Click the image for some remarks on a related novel.

Saturday, July 20, 2019

Humor for Professor X

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

Thursday, July 18, 2019

Split

Filed under: General — Tags: — m759 @ 6:55 PM

"My words say split, but my words they lie" — Bruce Springsteen

See a search for "Split" in this journal.

Tuesday, July 16, 2019

Schoolgirl Space for Quantum Mystics

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

From a post on St. Andrew's Day, 2017

See also "E-Numbers" and "E-Girls."

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.

Saturday, July 13, 2019

Which Roof?

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

Related material — Tetrahedron vs. Square and Cézanne's Greetings

Compare and contrast:

A figure from St. Patrick's Day 2004 that might represent a domed  roof

Inscribed Carpenter's Square:

In Latin, NORMA

and a cinematic "Fire Temple" from 2019

Friday, July 12, 2019

“Hello, old friend.” — Anthony Hopkins, Westworld S2 E6

Filed under: General — Tags: , — m759 @ 9:11 PM

See as well the previous post.

Ex Fano* Continues.

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

Professor Corbató reportedly died today.

* See Ex Fano  in this journal.

Instrumental

Filed under: General — Tags: — m759 @ 6:00 AM

The title is from a New York Times  correction in the previous post.

Introduction

'The Eddington Song'

Main Body

Piano roll for "I am sixteen going on seventeen" —

Medal of 9/15/06

Backstory:  Posts tagged Root Circle.

Thursday, July 11, 2019

Small World Continues.

Filed under: General — Tags: — m759 @ 11:08 PM

The book Ex-Prodigy  quoted above is by Norbert Wiener.

The Fano Correction

Friday, July 5, 2019

The Motive for Metaphor

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

"János Bolyai was a nineteenth-century mathematician who
set the stage for the field  of non-Euclidean geometry."

Transylvania Now , October 26, 2018

 

From  Coxeter and the Relativity Problem

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

Desiring the exhilarations of changes:
The motive for metaphor, shrinking from
The weight of primary noon,
The A B C of being,

The ruddy temper, the hammer
Of red and blue, the hard sound—
Steel against intimation—the sharp flash,
The vital, arrogant, fatal, dominant X.

Wallace Stevens, "The Motive for Metaphor"

Wednesday, June 19, 2019

Diagram

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

This post was suggested by the July 2006 posts now tagged Crazy Diamond.

Tuesday, June 18, 2019

The 7/11 Manifesto

Filed under: General — Tags: — m759 @ 10:07 PM

See 7/11, 2006.

Related material — Dabblers in the Collective Unconscious.

The Crooning

Filed under: General — Tags: — m759 @ 12:55 PM

For fans of the story theory of truth  

 A "tale as old as time . . ."
 — Song lyric,  Beauty and the Beast

Nicholas Hoult as X-Men "Beast" Hank McCoy

See also the previous post, "Equals Tolkien?"

Related material The real  McCoy

Equals Tolkien?

Filed under: General — Tags: — m759 @ 12:34 PM

Paris Review

Filed under: General — Tags: — m759 @ 3:33 AM

"The loveliness of Paris seems somehow sadly gay." — Song lyric

Stewart also starred in "Equals" (2016). From a synopsis —

"Stewart plays Nia, a writer who works at a company that extols
the virtues of space exploration in a post-apocalyptic society.
She falls in love with the film's main character, Silas (Nicholas Hoult),
an illustrator . . . ."

Space art in The Paris Review

For a different sort of space exploration, see Eightfold 1984.

June Moon Croon

Filed under: General — Tags: — m759 @ 12:02 AM

Monday, June 17, 2019

The Callahan Turtle

Filed under: General — Tags: , — m759 @ 10:54 PM

By Stephen King

IMAGE- Herbert John Ryser, 'Combinatorial Mathematics' (1963), page 1

Salvation by Grace

Filed under: General — Tags: , — m759 @ 3:18 PM

High Society

Filed under: General — Tags: , — m759 @ 1:16 PM

From the date in the above Google search result —

 See as well Jeremy Gray in this  journal.

Thursday, June 13, 2019

The Reality Blocks

Filed under: General — Tags: — m759 @ 9:24 PM

The new Log24 tag "Eightfold Metaphysics" used in the previous post
suggests a review of posts that were tagged "The Reality Blocks" on May 24.

Then there is, of course, the May 24 death of Murray Gell-Mann, who
hijacked from Buddhism the phrase "eightfold way."

See Gell-Mann in this journal and May 24, 2003.

Tuesday, June 11, 2019

Relish

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

"Center loosens, forms again elsewhere." — Roger Zelazny

See also an image in memory of the Coppertone artist
from a post of May 18, 2006

The image “http://www.log24.com/log/pix06A/060517-StarAndDiamond.bmp” cannot be displayed, because it contains errors.

“My God, it’s
full of stars!”

Monday, June 10, 2019

Revelations of a Transcendental Subject

Filed under: General — Tags: — m759 @ 10:45 PM

(Namely, Plato's ghost)

Background: "Transcendental subject" is Kant's term for, more or less, the self.

Rosalind Krauss in 1978 —

"To get inside the systems of this work,
whether LeWitt's or Judd's or Morris's,
is precisely to enter
a world without a center,
a world of substitutions and transpositions 
nowhere legitimated by the revelations
of a transcendental subject. This is the strength
of this work, its seriousness, and its claim to modernity." 

More from Krauss —

A book by an author with somewhat wider "cultural experience" —

See also "Plato's Diamond" in this journal.

Saturday, June 8, 2019

Nowhere Legitimated

Filed under: General — Tags: — m759 @ 10:28 PM

Art Object, continued and continued

Filed under: General — Tags: , — m759 @ 1:21 PM

Notes on a remark by Chuanming Zong

See as well posts mentioning "An Object of Beauty."

Update of 12 AM June 11 — A screenshot of this post 
is now available at  http://dx.doi.org/10.17613/hqk7-nx97 .

Sunday, June 2, 2019

Interpenetration

Filed under: General — Tags: — m759 @ 10:32 PM

A remark on coordinatization linked to by John Baez today —

This suggests a more historical perspective:

See as well a search for Interpenetration in this  journal.

Tuesday, May 28, 2019

Hogwash for Brattlefilm

Filed under: General — Tags: — m759 @ 10:54 PM

Related material — Khurana in this journal and
in The Harvard Crimson .

“Simplification in Its Purest Form”

Filed under: General — Tags: — m759 @ 10:08 PM

For students of the Hogwash School of Witchcraft and Wizardry —

"Elementary particles are the most fundamental
building blocks of nature, and their study
would seem to be an expression of simplification
in its purest form." 

— Sean Carroll in The New York Times  today 
in an opinion piece titled "The Physicist Who
Made Sense of the Universe"

Related remarks:  See a Log24 search for "Simpli".

The Khurana Embarrassment

Filed under: General — Tags: — m759 @ 12:36 PM

(A title for Harlan Kane)

Friday, May 24, 2019

Stevens and the Hoary Sages

Filed under: General — Tags: — m759 @ 11:05 AM

See also other posts now tagged The Reality Blocks.

Tuesday, May 21, 2019

Inside the White Cube

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

(Continued)

Monday, May 20, 2019

The Bond with Reality

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


"The bond with reality is cut."

— Hans Freudenthal, 1962

Indeed it is.

Sunday, May 19, 2019

The Building Blocks of Geometry

Filed under: General — Tags: , — m759 @ 10:21 PM

From "On the life and scientific work of Gino Fano
by Alberto Collino, Alberto Conte, and Alessandro Verra,
ICCM Notices , July 2014, Vol. 2 No. 1, pp. 43-57 —

" Indeed, about the Italian debate on foundations of Geometry, it is not rare to read comments in the same spirit of the following one, due to Jeremy Gray13. He is essentially reporting Hans Freudenthal’s point of view:

' When the distinguished mathematician and historian of mathematics Hans Freudenthal analysed Hilbert’s  Grundlagen he argued that the link between reality and geometry appears to be severed for the first time in Hilbert’s work. However, he discovered that Hilbert had been preceded by the Italian mathematician Gino Fano in 1892. . . .' "

13 J. Gray, "The Foundations of Projective Geometry in Italy," Chapter 24 (pp. 269–279) in his book Worlds Out of Nothing , Springer (2010).


Restoring the severed link —

Structure of the eightfold cube

See also Espacement  and The Thing and I.
 

Related material —

 
 

Friday, May 10, 2019

I Ching g6

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

For fans of Resonance Science

"When the men on the chessboard
get up and tell you where to go …."

Thursday, May 9, 2019

Defense Against the Dark Arts

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

F. Lanier Graham chess set (king-queen arrangement by the Wachowskis)

Monday, May 6, 2019

Possibilities

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

Related materialThe last three posts —

The Crimson Abyss,
Transgressive Politics at Harvard, and
"Thousand" Rhetoric

— as well as Saturday's The Chinese Jars of Shing-Tung Yau.

Sunday, May 5, 2019

“Thousand” Rhetoric

Filed under: General — Tags: — m759 @ 1:28 PM

Compare and contrast —

"A Thousand Possibilities"
— Title of a Harvard Crimson  May 3 column

See also The Thousand  in this  journal.

Transgressive Politics at Harvard

Filed under: General — Tags: , — m759 @ 1:02 PM

Related material —

See also the previous post.

The Crimson Abyss

Filed under: General — Tags: — m759 @ 12:20 PM

Continues.

The following conference has just ended.

Yau's actual 70th birthday was April 4.

Saturday, May 4, 2019

The Long Strange Trip of Abstract Art

Filed under: General — Tags: , — m759 @ 9:23 PM

In memory of an art collector's April 24 death —

See also Log24 on April 24

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