Thursday, September 19, 2019

Marvel Gospel

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

Sound familiar?

Saturday, June 9, 2018

Through Marvel Comics, Darkly

Filed under: General — m759 @ 12:00 PM

Click the above for "Cloak and Dagger" in this  journal.

Monday, January 18, 2021


Filed under: General — Tags: — m759 @ 6:46 PM

“Take out the papers and the trash”

—The first line of the song Yakety-Yak (1958).

Related cultural observation —

The above passage is from “The Matrix,” a post of Nov. 23, 2017 —

Thursday, January 7, 2021

The Primes of Miss Jean Valentine

Filed under: General — m759 @ 5:24 PM

Valentine reportedly died on December 29, 2020.

Related dialogue from “The Mirror Has Two Faces” (1996) —

– It’s interesting how coupling appears in nature and mathematics.

– You were talking about pairs …

– Oh, the twin-prime conjecture. It explores pairs of prime numbers.
Those only divisible by themselves. Three-five. Five-seven.
Not seven-nine …

– Nine can be divided by three.

– That’s right.  And … and so on.  It was discovered that pairs were
often separated by …

– One number in between.

– Exactly. Did you read my book?

– No, I’m sorry.

– That’s okay. This is marvellous.

– A first date like a game show.

– I didn’t mean to lecture.

– I’m sorry, I didn’t mean to call it a date.

Twin-Prime Dates —

December 31  and December 29, 2020.

“Though the Witch knew the Deep Magic, there is a magic deeper still
which she did not know.  Her knowledge goes back only to the dawn
of time.  But if she could have looked a little further back, into the
stillness and the darkness before Time dawned…she would have known
that when a willing victim who had committed no treachery was killed
in a traitor’s stead, the Table would crack and Death itself would start
working backwards.”

– C. S. Lewis, The Lion, the Witch and the Wardrobe , as quoted at
https://apologyanalogy.com/death-working-backwards/ .

Monday, January 4, 2021


Filed under: General — Tags: — m759 @ 10:05 PM

“Mr. Breuer’s audiences had to be willing to embrace,
or at least shrug off, some quantity of abstruseness
in his productions. Yet there was often a rapturous,
cacophonous beauty to them. At their best … they
worked on spectators like enchantments.

You can sense that effect in Margo Jefferson’s
New York Times review of “Red Beads” (2005) …
that she called ‘theater as sorcery; it is a crossroads
where artistic traditions meet to invent a marvelous
common language.’ “

Laura Collins-Hughes, Jan. 4, 2021

I prefer . . .

Wednesday, June 3, 2020


Filed under: General — Tags: , — m759 @ 5:35 PM

And the sailor said Brandy, you’re a fine girl . . . .

Monday, March 30, 2020

Annals of Ugly Design

Filed under: General — m759 @ 5:33 PM

The Boston Globe  Saturday on Friday’s death of one of
the two architects of Boston City Hall

A gifted storyteller, Mr. McKinnell liked to recount
the response of renowned architect Philip Johnson to
City Hall. “ ‘Absolutely marvelous. … I think it’s wonderful.
… And it’s so ugly!’ ” Mr. McKinnell told Pasnik, adding:
“We thought that was the greatest praise we could get.”

See more ugliness from this  journal on Friday

See also this journal on the death of the other  City Hall architect.

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 —

Friday, September 27, 2019

Algebra for Schoolgirls

Filed under: General — Tags: , — m759 @ 8:37 AM

The 15 points of the finite projective 3-space PG(3,2)
arranged in tetrahedral form:

The letter labels, but not the tetrahedral form,
are from The Axioms of Projective Geometry , by
Alfred North Whitehead (Cambridge U. Press, 1906).

The above space PG(3,2), because of its close association with
Kirkman's schoolgirl problem, might be called "schoolgirl space."

Screen Rant  on July 31, 2019:

A Google Search sidebar this morning:

Apocalypse Soon!

Wednesday, September 18, 2019

Powers of X

Filed under: General — m759 @ 11:45 AM

Screen Rant  on July 31, 2019 —

The above space PG(3,2), because of its close association with
Kirkman's schoolgirl problem, might be called "schoolgirl space."

See as well a Log24 post from the above Screen Rant  date —

The Epstein Chronicles, or:  Z  is for Zorro .

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

Saturday, September 7, 2019

Fury Road: Jukebox Hero*

Filed under: General — m759 @ 10:11 PM

"Fury: OK. Your turn. Prove you're not a Skrull.

[Carol calmly blasts energy out of her fists towards a jukebox
on the opposite side of the room before resting her head
back on her fist. Fury continues to look towards the jukebox,
both concerned and confused]

Fury: And how is that supposed to prove to me that you're not a Skrull?

Carol: It's a photon blast.

Fury: And…?

Carol: A Skrull can't do that. So a full-bird colonel turned spy turned
SHIELD agent must have pretty high security clearance. Where's Pegasus?

[The scene changes to a black car driving through an empty highway
next to a mountain, before changing to the inside of the car showing Fury
driving and Carol in the passenger seat]

Fury: So the Skrulls are alien races which infiltrate and overtake alien planets.
And you're a Kree, a race of noble warriors.

Carol: Heroes. Noble warrior heroes."

* See also this morning's post in memory of guitarist Jimmy Johnson
   as well as . . .

Wednesday, August 28, 2019


Filed under: General — m759 @ 10:56 PM

Influenz , by Paul Klee —

"One of the most influential of modern thinkers . . . ."

As is Stan Lee . . .

"Watch the trailer." — This journal on Eliza Doolittle Day, 2012 .

Carol and Thor... 'easily the best part' in 'Avengers: Endgame' trailer

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.

Saturday, May 18, 2019

Plan 9

Filed under: General — m759 @ 11:36 AM



. . . .

Still, Wouk, a month away from his 95th birthday, knows he cannot write forever. He has described The Language God Talks  as a "summing up," even if he is toying with the idea of writing a sequel. Earnestly written and very brief, it is an unusual work—partly a quick trip through developments in cosmology, partly an episodic memoir, partly an essay on faith and science. At the end, it portrays an imagined conversation between Wouk and the scientist Richard Feynman: historical fiction about the drama of the believer and the skeptic. In real life, Wouk met Feynman while researching the atom bomb for War and Remembrance . Feynman wasn't interested in fiction; he called calculus "the language God talks." But during a summer at the Aspen Institute, the two men spent hours talking, and Wouk has been thinking about his exchanges with Feynman and other scientists ever since. He even tried to learn calculus.

Feynman was a secular Jew, and yet something about the way he saw the world resonated with the observantly religious novelist. One day Wouk came across an interview in which Feynman said, "It doesn't seem to me that this fantastically marvelous universe … can merely be a stage so that God can watch human beings struggle for good and evil—which is the view that religion has. The stage is too big for the drama." The huge stage and the human drama: "This is the subject I've been thinking about my whole life," Wouk says.

. . . .


Related remarks on language —

Wednesday, May 1, 2019

For the First of May

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

"The purpose of mathematics cannot be derived from an activity 
inferior to it but from a higher sphere of human activity, namely,

 Igor Shafarevitch in 1973

"The hint half guessed, the gift half understood, is Incarnation."

— T. S. Eliot in Four Quartets

See also Ultron Cube.

Sunday, March 17, 2019


Filed under: General — m759 @ 4:02 PM

"Watch the trailer." — This journal on Eliza Doolittle Day, 2012.

Carol and Thor... 'easily the best part' in 'Avengers: Endgame' trailer

Midrash — March 14 remarks on geometry from Christchurch, New Zealand

Monday, March 11, 2019

Overarching Metanarratives

Filed under: General — 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)

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 "

Saturday, December 22, 2018

“It’s a war every day.”

Filed under: General — m759 @ 2:41 PM

The title is from a New York Times  story  online this afternoon.

A recent pop-culture use of the word "war" —

The six "infinity stones" sought in the above war
suggest a review of the "six points of general position
in real projective 4-space" mentioned in today's earlier
post "Cremona-Richmond."  See as well Ron Shaw
in that post and in the infinity-related book below —

Tuesday, August 21, 2018

Available Light

Filed under: General — Tags: — m759 @ 12:16 PM


Click the image above for an interview dated Nov. 9, 2015.
Also on that date —


Thursday, August 9, 2018

“Just” a Metaphor?

Filed under: General — m759 @ 11:34 AM

    Marvel's Jessica Jones, S1 E13


These are questions to which we might hope
the New York Times 's film critics know the answers.

Wednesday, August 8, 2018

Annals of Phenomenology

Filed under: General — Tags: — m759 @ 4:44 PM

The Return of Purple Man
Or:  This Just In

Detail from a post of yesterday morning taken from
the laptop of private investigator Jessica Jones —


The above image, together with yesterday's date, suggested, rather
fancifully, yesterday morning's later post on just intonation, "Seventh."

The nemesis of Jessica Jones, the Purple Man


A New York Times  piece today* is related to both just intonation and
the color purple —


* Published at 1:50 PM ET —
itemprop="datePublished" content="2018-08-08T17:50:30.000Z"

Monday, July 9, 2018

Annals of Ontology

Filed under: General — m759 @ 12:22 PM

The Thing and I  continues.

"… the Quantum Realm wouldn’t really become a 'thing' 
in Marvel’s comic book mythology until the end of that
decade [the 1970s], and the arrival of a toy license at
the publisher."

— Graeme McMillan in The Hollywood Reporter  Saturday

Sunday, July 8, 2018


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

Eric Temple Bell, 'The Development of Mathematics'

See also Solomon's  cube.

* Title suggested by a 2011 dystopian novel.

Sunday, April 29, 2018


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

From the online New York Times  this afternoon:

Disney now holds nine of the top 10
domestic openings of all time —
six of which are part of the Marvel
Cinematic Universe. “The result is
a reflection of 10 years of work:
of developing this universe, creating
stakes as big as they were, characters
that matter and stories and worlds that
people have come to love,” Dave Hollis,
Disney’s president of distribution, said
in a phone interview.

From this  journal this morning:

"But she felt there must be more to this
than just the sensation of folding space
over on itself. Surely the Centaurs hadn't
spent ten years telling humanity how to 
make a fancy amusement-park ride
There had to be more—"

Factoring Humanity , by Robert J. Sawyer,
Tom Doherty Associates, 2004 Orb edition,
page 168

"The sensation of folding space . . . ."

Or unfolding:

Click the above unfolded space for some background.

Saturday, March 3, 2018


Filed under: General — m759 @ 1:00 PM

For the Marvel Comics surgeon  Dr. Stephen Strange

For a real-life surgeon who reportedly died on Feb. 24,
a quotation from this journal on that date —

"What of the night
That lights and dims the stars?
Do you know, Hans Christian,
Now that you see the night?"

— The concluding lines of "Sonatina to Hans Christian,"
by Wallace Stevens (in Harmonium  (second edition, 1931))

Related material —

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

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.

Friday, July 29, 2016

The Cumberbatch Question

Filed under: General — Tags: — m759 @ 2:29 PM

"How do I get from here to there?"

See also Compass in this journal.

Saturday, March 5, 2016

The Looming

Filed under: General — m759 @ 3:14 PM

Material related to the title:

  • From the post Edifice (March 1, 2016) —

"Euclid's edifice loomed in my consciousness
as a marvel among sciences, unique in its clarity
and unquestionable validity."
—Richard J. Trudeau in 
   The Non-Euclidean Revolution  (1986)

Tuesday, March 1, 2016


Filed under: General — m759 @ 12:00 PM

"Euclid's edifice loomed in my consciousness as a marvel among
sciences, unique in its clarity and unquestionable validity."
—Richard J. Trudeau in The Non-Euclidean Revolution  (1986)

On 'The Public Square,' from 'Edgar Allan Poe, Wallace Stevens, and the Poetics of American Privacy'

See also Edifice in this journal and last night's architectural post.

Sunday, January 31, 2016

Forgotten Lore

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

Continued from Sunday, January 24, 2016

Wikipedia on Io in Greek mythology
(a precursor to Marvel Comics) —

"Walter Burkert [18] notes that the story of Io was told
in the ancient epic tradition at least four times….

    18. Burkert, Homo Necans  (1974) 1983:
          164 note 14, giving bibliography."

An "io" story I prefer — m24.io.

Saturday, January 30, 2016

Cartoon Theology*

Filed under: General — m759 @ 10:00 AM

For Evangeline

(Some background — See Limerick in this journal.
See also "He's a mad scientist and I'm his beautiful daughter.")

"There was a young lady named Bright…."

"You read too slow, Daddy," she complained. She was childishly irritable about it. "You say a word. Then I think a long time. Then you say another word."

I knew what she meant. I remember, when I was a child, my thoughts used to dart in and out among the slowly droning words of any adult. Whole patterns of universes would appear and disappear in those brief moments.

"So?" I asked.

"So," she mocked me impishly. "You teach me to read. Then I can think quick as I want."

"Quickly," I corrected in a weak voice. "The word is 'quickly,' an adverb."

She looked at me impatiently, as if she saw through this allegedly adult device to show up a youngster's ignorance. I felt like the dope!

— From  "Star, Bright" by Mark Clifton 

Related material — The Quick and the Dead

* For example, from the Marvel Comics realm

Monday, November 2, 2015

The Devil’s Offer

Filed under: General,Geometry — Tags: , — m759 @ 11:09 AM

This is a sequel to the previous post and to the Oct. 24 post
Two Views of Finite Space.  From the latter —

” ‘All you need to do is give me your soul:
give up geometry and you will have this
marvellous machine.’ (Nowadays you
can think of it as a computer!) “

George Boole in image posted on All Souls' Day 2015

Saturday, October 24, 2015

Two Views of Finite Space

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

The following slides are from lectures on “Advanced Boolean Algebra” —

The small Boolean  spaces above correspond exactly to some small
Galois  spaces. These two names indicate approaches to the spaces
via Boolean algebra  and via Galois geometry .

A reading from Atiyah that seems relevant to this sort of algebra
and this sort of geometry —

” ‘All you need to do is give me your soul:  give up geometry
and you will have this marvellous machine.’ (Nowadays you
can think of it as a computer!) “

Related material — The article “Diamond Theory” in the journal
Computer Graphics and Art , Vol. 2 No. 1, February 1977.  That
article, despite the word “computer” in the journal’s title, was
much less about Boolean algebra  than about Galois geometry .

For later remarks on diamond theory, see finitegeometry.org/sc.

Friday, September 11, 2015

Omega Wrinkle:

Filed under: General — m759 @ 9:42 PM

A Phrase That Haunts

From this journal on August 23, 2013

Illustration from New York Times  review 
of the novel Point Omega —

IMAGE- NY Times headline 'A Wrinkle in Time' with 24 Hour Psycho and Point Omega scene

From the print version of The New York Times Sunday Book Review
dated Sept. 13, 2015 —

The online version, dated Sept. 11, 2015 —

From the conclusion of the online version —

On the above print  headline, "Wrinkles in Time,"
that vanished in the online version —

"Now you see it, now you don't"
is not a motto one likes to see demonstrated
by a reputable news firm.

Related material:  Jews Telling Stories.

Wednesday, May 6, 2015

Narrative Metaphysics

Filed under: General — m759 @ 11:01 PM

(Continued from Dec. 13, 2014.)

David Lavery's enthusiasm today for the Marvel Comics
"Infinity Stones" suggests a review of The Foundation Stone
mentioned in the post Narrative Metaphysics of 12/13/2014.

See as well "Many Dimensions" in this journal.

Thursday, April 9, 2015


Filed under: General — m759 @ 6:27 PM

From an informative April 7 essay in The Nation —

In his marvelous book Is That a Fish in Your Ear?: Translation and the Meaning of Everything , David Bellos demonstrates many of the ways that translation is not only possible but ubiquitous, so thoroughly woven into the fabric of our daily lives—from classrooms to international financial markets, from instruction manuals to poems—that if translation were somehow to become impossible, the world would descend into the zombie apocalypse faster than you can say “je ne sais quoi ."

— "Forensic Translation," by Benjamin Paloff

See also searches in this  journal for Core and for Kernel.
See as well Fabric Design and Symplectic.

Tuesday, January 27, 2015

Drama Club

Filed under: General — m759 @ 3:27 AM

Julianne Moore at the Screen Actors Guild awards
on Sunday evening:

"When I was 17, I decided I wanted to be
an actor. It didn't seem possible because
I'd never met a real actor," Moore said.
"So I want to say to all the kids in the
drama club, you guys are the real actors."

On the main character of the new film "Birdman"

"Thomson is clearly talented, yet unable to get out of
the shadow of his superhero role. He is filled with
a simmering rage as Robert Downey Jr. appears
on the TV, arguably the highest profile actor alive
courtesy of a role in the Marvel films."

— Grant Pearsall at The Snapper

A midrash for Robert:

See The Stars My Destination and Cube of Ultron.

Thursday, January 1, 2015

New Year’s Greeting from Franz Kafka

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

An image that led off the year-end review yesterday in
the weblog of British combinatorialist Peter J. Cameron:

See also this  weblog's post final post of 2014,
with a rectangular array illustrating the six faces
of a die, and Cameron's reference yesterday to
a die-related post

"The things on my blog that seem to be
of continuing value are the expository
series like the one on the symmetric group
(the third post in this series was reblogged
by Gil Kalai last month, which gave it a new
lease of life)…."

A tale from an author of Prague:

The Emperor—so they say—has sent a message, directly from his death bed, to you alone, his pathetic subject, a tiny shadow which has taken refuge at the furthest distance from the imperial sun. He ordered the herald to kneel down beside his bed and whispered the message into his ear. He thought it was so important that he had the herald repeat it back to him. He confirmed the accuracy of the verbal message by nodding his head. And in front of the entire crowd of those who’ve come to witness his death—all the obstructing walls have been broken down and all the great ones of his empire are standing in a circle on the broad and high soaring flights of stairs—in front of all of them he dispatched his herald. The messenger started off at once, a powerful, tireless man. Sticking one arm out and then another, he makes his way through the crowd. If he runs into resistance, he points to his breast where there is a sign of the sun. So he moves forward easily, unlike anyone else. But the crowd is so huge; its dwelling places are infinite. If there were an open field, how he would fly along, and soon you would hear the marvelous pounding of his fist on your door. But instead of that, how futile are all his efforts. He is still forcing his way through the private rooms of the innermost palace. He will never he win his way through. And if he did manage that, nothing would have been achieved. He would have to fight his way down the steps, and, if he managed to do that, nothing would have been achieved. He would have to stride through the courtyards, and after the courtyards the second palace encircling the first, and, then again, stairs and courtyards, and then, once again, a palace, and so on for thousands of years. And if he finally did burst through the outermost door—but that can never, never happen—the royal capital city, the centre of the world, is still there in front of him, piled high and full of sediment. No one pushes his way through here, certainly not with a message from a dead man. But you sit at your window and dream of that message when evening comes.

See also a passage quoted in this  weblog on the original
date of Cameron's Prague image, July 26, 2014 —

"The philosopher Graham Harman is invested in
re-thinking the autonomy of objects and is part 
of a movement called Object-Oriented-Philosophy
(OOP)." — From “The Action of Things,” a 2011
M.A. thesis at the Center for Curatorial Studies,
Bard College, by Manuela Moscoso 

— in the context of a search here for the phrase
     "structure of the object." An image from that search:

Sunday, November 30, 2014

View from the Bottom

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

Reality's Mirror: Exploring the Mathematics of Symmetry —

"Here is a book that explains in laymen language
what symmetry is all about, from the lowliest snowflake
and flounder to the lofty group structures whose
astonishing applications to the Old One are winning
Nobel prizes. Bunch's book is a marvel of clear, witty
science writing, as delightful to read as it is informative
and up-to-date. The author is to be congratulated on
a job well done." — Martin Gardner

"But, sweet Satan, I beg of you, a less blazing eye!"

— Rimbaud,  A Season in Hell

"… the lowliest snowflake and flounder…." 
      — Martin Gardner

Thomas Mann on the deathly precision of snowflakes

Britannica article, 'Flounder'

Thursday, August 7, 2014

The Omega Mystery

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

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

Related material:

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

A scene from the film of the above book —

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

— Richard Evan Schwartz

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

Friday, July 11, 2014

Back to 1955

Filed under: General — m759 @ 10:00 AM

Nick Fury takes the Tesseract…

… which travels back to 1955
(see The Call Girls, Nov. 3, 2013)…

IMAGE- Cover design by Robert Flynn of 'The Armed Vision,' a 1955 Vintage paperback by Stanley Edgar Hyman

Above: A 1955 cover design by Robert Flynn.

Images from December 1955…

… and a fictional image imagined in an earlier year:

Thursday, March 13, 2014


Filed under: General — m759 @ 12:00 PM

A Presbyterian meditation —

A scene from the film of the above book —

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

Richard Evan Schwartz

For more of the story, see Golay in this journal.

Tuesday, December 24, 2013

Through a Mirror, Darkly

Filed under: General — Tags: — m759 @ 12:26 PM

Review of a book first published in 1989—

Reality's Mirror: Exploring the Mathematics of Symmetry —

"Here is a book that explains in laymen language
what symmetry is all about, from the lowliest snowflake
and flounder to the lofty group structures whose
astonishing applications to the Old One are winning
Nobel prizes. Bunch's book is a marvel of clear, witty
science writing, as delightful to read as it is informative
and up-to-date. The author is to be congratulated on
a job well done." — Martin Gardner

A completely different person whose name
mirrors that of the Mathematics of Symmetry  author —

IMAGE- Daily Princetonian, Dec. 23, 2013

See also this  journal on the date mentioned in the Princetonian .

"Always with a little humor." — Yen Lo

Monday, October 21, 2013

Edifice Complex

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

New! Improved!

"Euclid's edifice loomed in my consciousness 
as a marvel among sciences, unique in its
clarity and unquestionable validity." 
—Richard J. Trudeau in
   The Non-Euclidean Revolution  (First published in 1986)

Readers of this journal will be aware that Springer's new page
advertising Trudeau's book, pictured above, is a bait-and-switch
operation. In the chapter advertised, Trudeau promotes what he
calls "the Diamond Theory of Truth" as a setup for his real goal,
which he calls "the Story Theory of Truth."

For an earlier use of the phrase "Diamond Theory" in
connection with geometry, see a publication from 1977.

Saturday, April 13, 2013


Filed under: General — m759 @ 6:16 PM

Two friends from Brooklyn —

"… both marveled at early Ingmar Bergman movies."

One of the friends' "humor was inspired by
surrealist painters and Franz Kafka."


"Most of Marvel's fictional characters operate in
a single reality known as the Marvel Universe…."

This journal yesterday


Related material:  The Cosmic Cube.

Wednesday, September 19, 2012

Art Wars (continued)

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

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

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

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

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

Some related art

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

and, more recently

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

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

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

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


For Odin’s Day*

Filed under: General — m759 @ 12:00 PM

(Mathematics and Narrative, continued)

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


Sabol died yesterday.


An art gallery in Oslo is exhibiting a tesseract.


The Jewel of Odin's Treasure Room

(Click to enlarge.)

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

Thursday, September 13, 2012


Filed under: General — m759 @ 2:00 AM

Yesterday's online Los Angeles Times  
on a film that inspired recent protests in Cairo—

The film… was shown on June 23
to an audience of less than 10
at a theater on Hollywood Boulevard,
a source familiar with the screening said….
The screening was at The Vine Theater,
which rents itself out for private screenings,
said one person involved in the theater.

An image from this journal on that same day, June 23

IMAGE- Rudolf Koch's version of the 'double cross' symbol

    Source: Rudolf KochThe Book of Signs

For some background on the symbol, see Damnation Morning.

See also Don Henley's Hollywood hymn "Garden of Allah."

Update of 8 PM Sept. 13, 2012—

Other sources give the film's screening date not as June 23,
2012, but rather as June 30, 2012. (BBC News, LAWEEKLY blogs)

The following post from this journal on that  date may or
may not have some religious relevance.

Saturday, June 30, 2012


Filed under: Uncategorized — m759 @ 7:20 PM

"… to snare the spirits of mankind in nets of magic"

— The aim of the artist, according to Thomas Wolfe

Related entertainment—

High-minded— Many Dimensions .

Not so high-minded— The Cosmic Cube .

Saturday, June 30, 2012


Filed under: General — m759 @ 7:20 PM

"… to snare the spirits of mankind in nets of magic"

— The aim of the artist, according to Thomas Wolfe 

Related entertainment—

High-minded— Many Dimensions .

Not so high-minded— The Cosmic Cube .

Friday, January 27, 2012

Mathematics and Narrative (continued)

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

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

Circles Disturbed: The Interplay of Mathematics and Narrative

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

Timeline of the Marvel Cinematic Universe — Norway, March 1942

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

Tesseracts Disturbed — (Click to enlarge)

Detail of Tesseracts Disturbed —

Narrative of the detail—

See Tesseract in this journal and Norway, May 2010

The Oslo Version and Annals of Conceptual Art.

"Oh, what a tangled web we weave…"

Sunday, January 22, 2012

The Presbyterian Exorcist

Filed under: General — Tags: — m759 @ 9:26 AM

(Backstory— Presbyterian in this journal)

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

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

Exercise— Discuss the above paragraph's vulgarity.

Discuss also the more robust vulgarity of Marvel Entertainment

Context— "Marvel" in this journal, and The Cosmic Cube.

Sunday, August 28, 2011

The Cosmic Part

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

Yesterday’s midday post, borrowing a phrase from the theology of Marvel Comics,
offered Rubik’s mechanical contrivance as a rather absurd “Cosmic Cube.”

A simpler candidate for the “Cube” part of that phrase:


The Eightfold Cube

As noted elsewhere, a simple reflection group* of order 168 acts naturally on this structure.

“Because of their truly fundamental role in mathematics,
even the simplest diagrams concerning finite reflection groups
(or finite mirror systems, or root systems—
the languages are equivalent) have interpretations
of cosmological proportions.”

Alexandre V. Borovik in “Coxeter Theory: The Cognitive Aspects

Borovik has a such a diagram—


The planes in Borovik’s figure are those separating the parts of the eightfold cube above.

In Coxeter theory, these are Euclidean hyperplanes. In the eightfold cube, they represent three of seven projective points that are permuted by the above group of order 168.

In light of Borovik’s remarks, the eightfold cube might serve to illustrate the “Cosmic” part of the Marvel Comics phrase.

For some related theological remarks, see Cube Trinity in this journal.

Happy St. Augustine’s Day.

* I.e., one generated by reflections : group actions that fix a hyperplane pointwise. In the eightfold cube, viewed as a vector space of 3 dimensions over the 2-element Galois field, these hyperplanes are certain sets of four subcubes.

Wednesday, August 17, 2011

Marginal Remarks

Filed under: General — m759 @ 11:17 AM

Today's Google Doodle is in honor of Fermat's birthday—


"I have discovered a truly marvelous proof of this theorem,
 which this doodle is too small to contain."
— Google's caption

Another marginal remark, from a link target in last night's "Ein Kampf"—

"We are talking about the spatial and temporal phenomenon of language,
not about some non-spatial, non-temporal chimera [Note in margin:
Only it is possible to be interested in a phenomenon in a variety of ways]."

— Ludwig Wittgenstein, Philosophical Investigations  (1953),  Section 108

Related material on spatial and temporal phenomena—

A Dec. 29, 2010, comment to a Dec. 26 weblog post on
"Unexpected Connections in Mathematics"—


Connoisseurs of synchronicities  in the phenomena of language may note that
these December dates mark the feasts of St. Stephen and St. Thomas Becket.

From the feast of the latter, two Log24 posts: Toy Stories and True Grid.

Those less enchanted by pop math than Google may prefer to observe
two other birthdays today— those of Robert De Niro and of Sean Penn:


Tuesday, May 10, 2011

Groups Acting

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

The LA Times  on last weekend's film "Thor"—

"… the film… attempts to bridge director Kenneth Branagh's high-minded Shakespearean intentions with Marvel Entertainment's bottom-line-oriented need to crank out entertainment product."

Those averse to Nordic religion may contemplate a different approach to entertainment (such as Taymor's recent approach to Spider-Man).

A high-minded— if not Shakespearean— non-Nordic approach to groups acting—

"What was wrong? I had taken almost four semesters of algebra in college. I had read every page of Herstein, tried every exercise. Somehow, a message had been lost on me. Groups act . The elements of a group do not have to just sit there, abstract and implacable; they can do  things, they can 'produce changes.' In particular, groups arise naturally as the symmetries of a set with structure. And if a group is given abstractly, such as the fundamental group of a simplical complex or a presentation in terms of generators and relators, then it might be a good idea to find something for the group to act on, such as the universal covering space or a graph."

— Thomas W. Tucker, review of Lyndon's Groups and Geometry  in The American Mathematical Monthly , Vol. 94, No. 4 (April 1987), pp. 392-394

"Groups act "… For some examples, see

Related entertainment—

High-minded— Many Dimensions

Not so high-minded— The Cosmic Cube


One way of blending high and low—

The high-minded Charles Williams tells a story
in his novel Many Dimensions about a cosmically
significant cube inscribed with the Tetragrammaton—
the name, in Hebrew, of God.

The following figure can be interpreted as
the Hebrew letter Aleph inscribed in a 3×3 square—


The above illustration is from undated software by Ed Pegg Jr.

For mathematical background, see a 1985 note, "Visualizing GL(2,p)."

For entertainment purposes, that note can be generalized from square to cube
(as Pegg does with his "GL(3,3)" software button).

For the Nordic-averse, some background on the Hebrew connection—

Saturday, May 7, 2011


Filed under: General — m759 @ 8:48 AM

24 Frames column in the LA Times

Critical Mass: 'Thor' swings his hammer and the critics scream

May 6, 2011 |  2:55 pm

Bridges are the key theme of this weekend's "Thor," a film that bridges us from the doldrums of spring releases to the flashier, if not better, world of summer blockbusters. It also serves as another step in the bridge from the first "Iron Man" in 2008 to next summer's superhero all-star jam, "The Avengers." And within the film itself, a superhero actioner about the Norse god of thunder and his adventures in his home of Asgard and on Earth, a rainbow bridge connects the well-regarded Asgard sections of the film with the less successful Earth sections, set in Puente Antiguo, N.M. (which means "Old Bridge").

According to Times critic Kenneth Turan, the film also attempts to bridge director Kenneth Branagh's high-minded Shakespearean intentions with Marvel Entertainment's bottom-line-oriented need to crank out entertainment product.

— Patrick Kevin Day

Related material: Kate and Thor in The Turning  and A Bridge Too Far.

Saturday, December 4, 2010

Damnation on 42nd Street

Filed under: General — m759 @ 5:24 AM

Yesterday's New York Lottery— Midday 042, Evening 919.

Here 042 may be seen as referring to New York's 42nd Street…

Below, West 42nd St., facing north, from yesterday's New York Times


Related material —

That story is part of the Change War  saga by Fritz Leiber, notably represented by Leiber's 1957 novel The Big Time.

See also Comic Book Resources on the new comic-book  series Spider-Man: Big Time

CBR: “Big Time” is this title of this new era of “Amazing Spider-Man.” Why choose that title? What exactly is it referring to?

DAN SLOTT: “Big Time” refers to more than “Amazing Spider-Man,” it also refers to other Spider-Projects: “Astonishing Spider-Man/Iron Man,” the new Norman Osborn mini, and the all-new “Spider-Girl!” With “Amazing,” “Big Time” takes on a lot of meanings. In this book, everything is bigger: bigger stakes for Peter Parker, bigger threats for Spider-Man, and a much bigger comic. We are expanding to 30 pages of material, twice a month!

As for yesterday's evening NY lottery number 919, see 9/19.

Friday, August 7, 2009

Friday August 7, 2009

Filed under: General — m759 @ 9:29 PM

Angel and Beast

Screenwriter Frank Pierson spoke at
Chautauqua Institution this morning.

The gist of his remarks may be found
in an undated graduation speech
at WarnerSisters.com.

His suggested motto for filmmakers:
   "To reach and touch
     the angel in the beast."

The Chautauquan Daily
Friday, August 7, 2009
by Sara Toth, staff writer —

"Pierson listed his favorite movies as
the Italian and French films that, after
World War II, captivated him and his
'Those movies were overwhelmingly
fascinating to us, and changed the way
in which we saw movies, and the way we
saw our lives and what we wanted to do
with ourselves,' he said. 'There were so
many that were absolutely marvelous.'
Such movies are not made any
more, Pierson said, and the quality
of the movies now pale in comparison
to those of the 1970s and 1980s.
About once a year, the Coen brothers
release a movie, and Woody Allen
'occasionally' makes a good film,
Pierson said. But the mainstream movies
that are shown in the multiplexes now
are geared toward only one audience:
young men with disposable incomes.
'That's really catering so extensively
to a rather limited audience-- a mentally
retarded and emotionally stunted
audience at that-- that there's not a lot left
over for the rest of us,' Pierson said."

Tuesday, August 5, 2008

Tuesday August 5, 2008

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

Cover of  'The Last Theorem,' a novel by Arthur C. Clarke and Frederik Pohl

The Last Theorem
, a novel by
Arthur C. Clarke and Frederik Pohl

From the publisher's description:

"The Last Theorem is a story of one man’s mathematical obsession, and a celebration of the human spirit and the scientific method. It is also a gripping intellectual thriller….

In 1637, the French mathematician Pierre de Fermat scrawled a note in the margin of a book about an enigmatic theorem: 'I have discovered a truly marvelous proof of this proposition which this margin is too narrow to contain.' He also neglected to record his proof elsewhere. Thus began a search for the Holy Grail of mathematics– a search that didn’t end until 1994, when Andrew Wiles published a 150-page proof. But the proof was burdensome, overlong, and utilized mathematical techniques undreamed of in Fermat’s time, and so it left many critics unsatisfied– including young Ranjit Subramanian, a Sri Lankan with a special gift for mathematics and a passion for the famous 'Last Theorem.'

When Ranjit writes a three-page proof of the theorem that relies exclusively on knowledge available to Fermat, his achievement is hailed as a work of genius, bringing him fame and fortune…."

For a similar third-world fantasy about another famous theorem, see the oeuvre of Ashay Dharwadker.

Note the amazing conclusion of Dharwadker's saga (thus far)–

Dharwadker devises a proof of the four-color theorem that leads to…

Grand Unification
of the Standard Model
with Quantum Gravity!

For further background, see

Ashay Dharwadker
  and Usenet Postings.

Clarke lived in Sri Lanka (formerly Ceylon) from 1956 until his death last March.

For another connection with Sri Lanka, see

Location, Location, Location
(July 13, 2005) and
Bagombo Snuff Box
(May 7, 2006).

Saturday, July 14, 2007

Saturday July 14, 2007

Filed under: General — m759 @ 4:07 AM
A Note from the
Catholic University
of America

The August 2007 issue of Notices of the American Mathematical Society contains tributes to the admirable personal qualities and mathematical work of the late Harvard professor George Mackey.  For my own tributes, see Log24 on March 17, 2006April 29, 2006, and March 10, 2007.  For an entry critical of Mackey’s reductionism– a philosophical, not mathematical, error– see Log24 on May 23, 2007 (“Devil in the Details”).

Here is another attack on reductionism, from a discussion of the work of another first-rate mathematician, the late Gian-Carlo Rota of MIT:

“Another theme developed by Rota is that of ‘Fundierung.’ He shows that throughout our experience we encounter things that exist only as founded upon other things: a checkmate is founded upon moving certain pieces of chess, which in turn are founded upon certain pieces of wood or plastic. An insult is founded upon certain words being spoken, an act of generosity is founded upon something’s being handed over. In perception, for example, the evidence that occurs to us goes beyond the physical impact on our sensory organs even though it is founded upon it; what we see is far more than meets the eye. Rota gives striking examples to bring out this relationship of founding, which he takes as a logical relationship, containing all the force of logical necessity. His point is strongly antireductionist. Reductionism is the inclination to see as ‘real’ only the foundation, the substrate of things (the piece of wood in chess, the physical exchange in a social phenomenon, and especially the brain as founding the mind) and to deny the true existence of that which is founded. Rota’s arguments against reductionism, along with his colorful examples, are a marvelous philosophical therapy for the debilitating illness of reductionism that so pervades our culture and our educational systems, leading us to deny things we all know to be true, such as the reality of choice, of intelligence, of emotive insight, and spiritual understanding. He shows that ontological reductionism and the prejudice for axiomatic systems are both escapes from reality, attempts to substitute something automatic, manageable, and packaged, something coercive, in place of the human situation, which we all acknowledge by the way we live, even as we deny it in our theories.”

Robert Sokolowski, foreword to Rota’s Indiscrete Thoughts

Father Robert Sokolowski

Father Robert Sokolowski

Fr. Robert Sokolowski, Ph.D., is Professor of Philosophy at The Catholic University of America in Washington, D.C. Ordained a Roman Catholic priest in 1962, he is internationally recognized and honored for his work in philosophy, particularly phenomenology. In 1994, Catholic University sponsored a conference on his work and published several papers and other essays under the title, The Truthful and the Good, Essays In Honor of Robert Sokolowski.

Thomas Aquinas College newsletter

The tributes to Mackey are contained in the first of two feature articles in the August 2007 AMS Notices.  The second feature article is a review of a new book by Douglas Hofstadter.  For some remarks related to that article, see Thursday’s Log24 entry “Not Mathematics but Theology.”

Monday, June 19, 2006

Monday June 19, 2006

Filed under: General — m759 @ 4:00 PM

A Reply to John Updike

See Updike on digitized snippets.

The following four snippets were pirated from the end of MathPages Quotations, compiled by Kevin Brown.

They are of synchronistic interest in view of the previous two Log24 entries, which referred (implicitly) to a Poe story and (explicitly) to Pascal.

"That is another of your odd notions,"
said the Prefect, who had the fashion
of calling everything 'odd' that was
beyond his comprehension, and thus
lived amid an absolute legion of 'oddities.'
Edgar Allan Poe

I knew when seven justices could not
take up a quarrel, but when the parties
were met themselves, one of them
thought but of an If, as, 'If you said so,
then I said so'; and they shook hands
and swore brothers. Your If is the only
peacemaker; much virtue in If.

I have made this letter longer than usual
because I lack the time to make it shorter.
Blaise Pascal

S'io credessi che mia risposta fosse
a persona che mai tornasse al mondo,
questa fiamma staria senza piu scosse.
Ma per cio che giammai di questo fondo
non torno vivo alcun, s'i'odo il vero,
senza tema d'infamia ti rispondo.
Dante, 1302

For translations of the Dante (including one by Dorothy Sayers), see everything2.com.

An anonymous author there notes that Dante describes a flame in which is encased a damned soul. The flame vibrates as the soul speaks:

If I thought that I were making
Answer to one that might return to view
The world, this flame should evermore
cease shaking.

But since from this abyss, if I hear true,
None ever came alive, I have no fear
Of infamy, but give thee answer due.

-- Dante, Inferno, Canto 27, lines 61-66,
translated by Dorothy Sayers

Updike says,

“Yes, there is a ton of information on the web but much of it is grievously inaccurate, unedited, unattributed and juvenile. The electronic marvels that abound around us serve, I have the impression, to inflame what is most informally and non-critically human about us. Our computer screens stare back at us with a kind of giant, instant aw-shucks, disarming in its modesty.”

Note Updike’s use of “inflame.”

For an aw-shucks version of “what is most informally and non-critically human about us,” as well as a theological flame, see both the previous entry and the above report from Hell.

Note that the web serves also to correct material that is inaccurate, unedited, unattributed, and juvenile. For examples, see Mathematics and Narrative. The combination of today’s entry for Pascal’s birthday with that web page serves both to light one candle and to curse the darkness.

Thursday, January 26, 2006

Thursday January 26, 2006

Filed under: General,Geometry — m759 @ 9:00 AM
In honor of Paul Newman’s age today, 81:

On Beauty

Elaine Scarry, On Beauty (pdf), page 21:

“Something beautiful fills the mind yet invites the search for something beyond itself, something larger or something of the same scale with which it needs to be brought into relation. Beauty, according to its critics, causes us to gape and suspend all thought. This complaint is manifestly true: Odysseus does stand marveling before the palm; Odysseus is similarly incapacitated in front of Nausicaa; and Odysseus will soon, in Book 7, stand ‘gazing,’ in much the same way, at the season-immune orchards of King Alcinous, the pears, apples, and figs that bud on one branch while ripening on another, so that never during the cycling year do they cease to be in flower and in fruit. But simultaneously what is beautiful prompts the mind to move chronologically back in the search for precedents and parallels, to move forward into new acts of creation, to move conceptually over, to bring things into relation, and does all this with a kind of urgency as though one’s life depended on it.”

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

The above symbol of Apollo suggests, in accordance with Scarry’s remarks, larger structures.   Two obvious structures are the affine 4-space over GF(3), with 81 points, and the affine plane over GF(32), also with 81 points.  Less obvious are some related projective structures.  Joseph Malkevitch has discussed the standard method of constructing GF(32) and the affine plane over that field, with 81 points, then constructing the related Desarguesian projective plane of order 9, with 92 + 9 + 1 = 91 points and 91 lines.  There are other, non-Desarguesian, projective planes of order 9.  See Visualizing GL(2,p), which discusses a spreadset construction of the non-Desarguesian translation plane of order 9.  This plane may be viewed as illustrating deeper properties of the 3×3 array shown above. To view the plane in a wider context, see The Non-Desarguesian Translation Plane of Order 9 and a paper on Affine and Projective Planes (pdf). (Click to enlarge the excerpt beow).

The image “http://www.log24.com/theory/images/060126-planes2.jpg” cannot be displayed, because it contains errors.

See also Miniquaternion Geometry: The Four Projective Planes of Order 9 (pdf), by Katie Gorder (Dec. 5, 2003), and a book she cites:

Miniquaternion geometry: An introduction to the study of projective planes, by T. G. Room and P. B. Kirkpatrick. Cambridge Tracts in Mathematics and Mathematical Physics, No. 60. Cambridge University Press, London, 1971. viii+176 pp.

For “miniquaternions” of a different sort, see my entry on Visible Mathematics for Hamilton’s birthday last year:

The image “http://www.log24.com/theory/images/Quaternions2.jpg” cannot be displayed, because it contains errors.


Wednesday, November 16, 2005

Wednesday November 16, 2005

Filed under: General — m759 @ 4:04 PM

Adam Gopnik on C. S. Lewis in this week’s New Yorker:

“Lewis began with a number of haunted images….”

“The best of the books are the ones… where the allegory is at a minimum and the images just flow.”

“‘Everything began with images,’ Lewis wrote….”

“We go to the writing of the marvellous, and to children’s books, for stories, certainly, and for the epic possibilities of good and evil in confrontation, not yet so mixed as they are in life. But we go, above all, for imagery: it is the force of imagery that carries us forward. We have a longing for inexplicable sublime imagery….”

“The religious believer finds consolation, and relief, too, in the world of magic exactly because it is at odds with the necessarily straitened and punitive morality of organized worship, even if the believer is, like Lewis, reluctant to admit it. The irrational images– the street lamp in the snow and the silver chair and the speaking horse– are as much an escape for the Christian imagination as for the rationalist, and we sense a deeper joy in Lewis’s prose as it escapes from the demands of Christian belief into the darker realm of magic. As for faith, well, a handful of images is as good as an armful of arguments, as the old apostles always knew.”

Related material:

The image “http://www.log24.com/log/pix05B/051116-Time.jpg” cannot be displayed, because it contains errors.

Click on pictures for details.

See also Windmills and
Verbum sat sapienti?
as well as

an essay

 at Calvin College
on Simone Weil,
Charles Williams,
Dante, and
the way of images.”

Sunday, April 11, 2004

Sunday April 11, 2004

Filed under: General,Geometry — m759 @ 3:28 PM

Good Friday and
Descartes’s Easter Egg

“The use of z, y, x . . . to represent unknowns is due to René Descartes, in his La géometrie (1637)…. In a paper on Cartesian ovals, prepared before 1629, x alone occurs as unknown…. This is the earliest place in which Descartes used one of the last letters of the alphabet to represent an unknown.”

— Florian Cajori, A History of Mathematical Notations. 2 volumes. Lasalle, Illinois: The Open Court Publishing Co., 1928-1929. (Vol. 1, page 381)

This is from


Descartes’s Easter Egg is found at

EggMath: The Shape of an Egg —
Cartesian Ovals

An Easter Meditation
on Humpty Dumpty

The following is excerpted from a web page headed “Catholic Way.”  It is one of a series of vicious and stupid Roman Catholic attacks on Descartes.  Such attacks have been encouraged by the present Pope, who today said “may the culture of life and love render vain the logic of death.”

The culture of life and love is that of the geometry (if not the philosophy) of Descartes.  The logic of death is that of Karol Wojtyla, as was made very clear in the past century by the National Socialist Party, which had its roots in Roman Catholicism.

Humpty Dumpty sat on a wall.
Humpty Dumpty had a great fall

“In the century just completed, the human race found itself in a position not unlike the scrambled mess at the base of an imaginary English wall….

… we are heirs to a humanity that is broken, fractured, confused, unsure of what to make of itself….

 … ‘postmodernism’ is merely the articulation of the fractured, dissipated state of the human being…. 

Without relating a history of modern philosophy, our unfortunate human shell has suffered a continual fragmentation for a period of roughly 500 years. (You philosophers out there will recognize immediately that I am referring to the legacy of René Descartes.) And this fragmentation has been a one-way street: one assault after another on the integrity and dignity of the human person until you have, well, the 20th Century.

But now it’s the 21st Century.

The beauty … the marvel … the miracle of our time is the possibility that gravity will reverse itself: Humpty Dumpty may be able, once again, to assume his perch.”

—  Ted Papa,
Raising Humpty Dumpty


The upper part
of the above icon
is from EggMath.
For the lower part,
see Good Friday.

Sunday, March 14, 2004

Sunday March 14, 2004

Filed under: General,Geometry — m759 @ 3:28 PM

Clarity and Certainty

“At the age of 12 I experienced a second wonder of a totally different nature: in a little book* dealing with Euclidean plane geometry, which came into my hands at the beginning of a schoolyear. Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which — though by no means evident — could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty [Klarheit und Sicherheit] made an indescribable impression upon me….  For example I remember that an uncle told me the Pythagorean theorem before the holy geometry booklet* had come into my hands. After much effort I succeeded in ‘proving’ this theorem on the basis of the similarity of triangles … for anyone who experiences [these feelings] for the first time, it is marvellous enough that man is capable at all to reach such a degree of certainty and purity [Sicherheit und Reinheit] in pure thinking as the Greeks showed us for the first time to be possible in geometry.”

— from “Autobiographical Notes” in Albert Einstein: Philosopher-Scientist, edited by Paul Arthur Schilpp

“Although our intellect always longs for clarity and certainty, our nature often finds uncertainty fascinating.”

— Carl von Clausewitz at Quotes by Clausewitz

For clarity and certainty, consult All About Altitudes (and be sure to click the “pop it up” button).

For murkiness and uncertainty, consult The Fog of War.

Happy birthday, Albert.

* Einstein’s “holy geometry booklet” was, according to Banesh Hoffman, Lehrbuch der Geometrie zum Gebrauch an höheren Lehranstalten, by Eduard Heis (Catholic astronomer and textbook writer) and Thomas Joseph Eschweiler.

Thursday, January 29, 2004

Thursday January 29, 2004

Filed under: General,Geometry — Tags: , — m759 @ 2:30 PM

in the Theory of Design

"Whether or not we can follow the theorist in his demonstrations, there is one misunderstanding we must avoid at all cost.  We must not confuse the analyses of geometrical symmetries with the mathematics of combination and permutation….

The earliest (and perhaps the rarest) treatise on the theory of design drives home this insight with marvellous precision."

— E. H. Gombrich, 1979, in
   The Sense of Order

This is perhaps the stupidest remark I have ever read.  The "treatise on the theory of design" that Gombrich refers to is

  • Dominique Douat, Méthode Pour Faire une Infinité de Desseins Differents…. Paris, 1722.

For some background, see

Truchet Tiling,  

Truchet & Types:
Tiling Systems and Ornaments
, and

Douat Designs

Certain of the Truchet/Douat patterns have rather intriguing mathematical properties, sketched in my website Diamond Theory.  These properties become clear if and only if we we do what Gombrich declares that we must not do:  "confuse the analyses of geometrical symmetries with the mathematics of combination and permutation."  (The verb "confuse" should, of course, be replaced by the verb "combine.")

Saturday, August 16, 2003

Saturday August 16, 2003

Filed under: General,Geometry — Tags: — m759 @ 2:16 AM

My Personal Thorny Crown

Kirk Varnedoe, 57, art historian and former curator of the Museum of Modern Art, died Thursday, August 14, 2003.

From his New York Times obituary:

" 'He loved life in its most tangible forms, and so for him art was as physical and pleasurable as being knocked down by a wave,' said Adam Gopnik, the writer and a former student of his who collaborated on Mr. Varnedoe's first big show at the Modern, 'High & Low.' 'Art was always material first — it was never, ever bound by a thorny crown of ideas.' "

For some background on the phrase "thorny crown of ideas," see the web page


The phrase "thorny crown of ideas" is also of interest in the light of recent controversy over Mel Gibson's new film, "The Passion."

For details of the controversy, see Christopher Orlet's Aug. 14 essay at Salon.com,

Mel Gibson vs. "The Jews"

For a real "thorny crown of ideas," consider the following remarks by another art historian:

"Whether or not we can follow the theorist in his demonstrations, there is one misunderstanding we must avoid at all cost.  We must not confuse the analyses of geometrical symmetries with the mathematics of combination and permutation….

The earliest (and perhaps the rarest) treatise on the theory of design drives home this insight with marvellous precision."

— E. H. Gombrich, 1979, in
   The Sense of Order

This is perhaps the most stupid remark I have ever read.  The "treatise on the theory of design" that Gombrich refers to is

  • Dominique Douat, Methode pour faire une infinité de desseins differents avec des carreaux mipartis de deux couleurs par une ligne diagonale : ou observations du Pere Dominique Douat Religieux Carmes de la Province de Toulouse sur un memoire inséré dans l'Histoire de l'Académie Royale des Sciences de Paris l'année 1704, présenté par le Reverend Sebastien Truchet religieux du même ordre, Academicien honoraire, imprimé chez Jacques Quillau, Imprimeur Juré de l'Université, Paris 1722.

This is the title given at the web page

Truchet & Types:
Tiling Systems and Ornaments

which gives some background. 

Certain of the Truchet/Douat patterns have rather intriguing mathematical properties, sketched in my website Diamond Theory.  These properties become clear if and only we we do what Gombrich moronically declares that we must not do:  "confuse the analyses of geometrical symmetries with the mathematics of combination and permutation."  (The verb "confuse" should, of course, be replaced by the verb "combine.") 

What does all this have to do with

Mel Gibson vs. "The Jews" ?

As jesting Pilate seems to have realized, whenever Jews (or, for that matter, Christians) tell stories, issues of truth may arise.  Such issues, as shown by current events in that damned Semitic Hell-on-Earth that used to be referred to as "the Holy Land," can be of life-and-death importance.

Scene from
The Passion

The Roman soldiers may have fashioned a physical crown of thorns, but the Jews are quite capable of fashioning a very uncomfortable crown of, as Gopnik says, "ideas."

Here is an example.

"Ernst Hans Josef Gombrich, who as an author went by the name E. H. Gombrich, was born in Vienna in 1909….

The Gombrich family was Jewish, but his parents felt this had no particular relevance. In later years Mr. Gombrich said that whether someone was Jewish or not was a preoccupation for the Gestapo."

— Michael Kimmelman's obituary for Gombrich in the New York Times. Kimmelman is chief art critic for the New York Times and author of the Times's Aug. 15 Varnedoe obituary.

The web page Understanding cited above contains a link to

Pilate, Truth, and Friday the Thirteenth,

a page combining some religious remarks with a quotation of an extremely patronizing and superficial reference to my own work (and, in passing, to Truchet/Douat patterns).

This reference, and the above-quoted remark by Gombrich, constitute my own modest claim to what the Jew Gopnik jokingly calls a "thorny crown of ideas."

To me it is no joke.

This partly accounts for the rather strained quality of the attempt at humor in a web page I put together yesterday in response to Varnedoe's obituary:

Fahne Hoch, Macbeth!

Another reason for the strained quality is my being struck by the synchronicity of reading Varnedoe's obituary shortly after I had done a journal entry related to the death in July of an earlier Museum of Modern Art curator.  Like Robert A. Heinlein, I think the God of the Jews is a lousy deity and an even worse father figure.  I do, however, believe in synchronicity.

Saturday, December 7, 2002

Saturday December 7, 2002

Filed under: General — m759 @ 9:30 PM


Shall we read?

From Contact, by Carl Sagan:

  “You mean you could decode a picture hiding in pi
and it would be a mess of Hebrew letters?”
  “Sure.  Big black letters, carved in stone.”
  He looked at her quizzically.
  “Forgive me, Eleanor, but don’t you think
you’re being a mite too… indirect? 
You don’t belong to a silent order of Buddhist nuns. 
Why don’t you just tell your

From The Nation – Thailand
Sat Dec 7 19:36:00 EST 2002:

New Jataka books
blend ethics and art

Published on Dec 8, 2002

“The Ten Jataka, or 10 incarnations of the Lord Buddha before his enlightenment, are among the most fascinating religious stories….

His Majesty the King wrote a marvellous book on the second incarnation of the Lord Buddha…. It has become a classic, with the underlying aim of encouraging Thais to pursue the virtue of perseverance.

For her master’s degree at Chulalongkorn University’s Faculty of Arts, Her Royal Highness Princess Maha Chakri Sirindhorn wrote a dissertation related to the Ten Jataka of the Buddha. Now with the 4th Cycle Birthday of Princess Sirindhorn approaching on April 2, 2003, a group of artists, led by prominent painter Theeraphan Lorpaiboon, has produced a 10-volume set, the “Ten Jataka of Virtues”, as a gift to the Princess.

Once launched on December 25, the “Ten Jataka of Virtues” will rival any masterpiece produced in book form….”

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

Monday, August 5, 2002

Monday August 5, 2002

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

History, Stephen said….

The Modern Word

— To really know a subject you've got to learn a bit of its history….

John Baez, August 4, 2002

We both know what memories can bring;
They bring diamonds and rust.

—  Joan Baez, April 1975 

All sorts of structures that can be defined for finite sets have analogues for the projective geometry of finite fields….

Clearly this pattern is trying to tell us something; the question is what. As always, it pays to focus on the simplest case, since that's where everything starts.

John Baez, August 4, 2002

In the beginning was the word….

The Gospel according to Saint John

The anonymous author of John makes liberal use of allegory and double-entendre to illustrate this theme.

The Gospel of John

Born yesterday: Logician John Venn

Venn considered three discs R, S, and T as typical subsets of a set U. The intersections of these discs and their complements divide U into 8 nonoverlapping regions….

History of Mathematics at St. Andrews

Who would not be rapt by the thought of such marvels?….

Saint Bonaventure on the Trinity

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