Tuesday, January 31, 2012


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

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

Tesseract formed from a 4-set—

IMAGE- Tesseract.

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

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

Update of August 12, 2012:

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

Monday, January 30, 2012

Enda’s Game*

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

The following passage by Tolkien was suggested by a copy of next Sunday's New York Times Book Review  that arrived in the mail today. (See Orson Scott Card's remarks on page 26— "Uncle Orson"— and the Review 's concluding essay "Grand Allusion.")

"Lastly, tengwesta  [system or code of signs] has also become an impediment. It is in Incarnates clearer and more precise than their direct reception of thought. By it also they can communicate easily with others, when no strength is added to their thought: as, for example, when strangers first meet. And, as we have seen, the use of 'language' soon becomes habitual, so that the practice of ósanwe  (interchange of thought) is neglected and becomes more difficult. Thus we see that the Incarnate tend more and more to use or to endeavour to use ósanwe  only in great need and urgency, and especially when lambe  is unavailing. As when the voice cannot be heard, which comes most often because of distance. For distance in itself offers no impediment whatever to ósanwe . But those who by affinity might well use ósanwe  will use lambe  when in proximity, by habit or preference. Yet we may mark also how the 'affine' may more quickly understand the lambe  that they use between them, and indeed all that they would say is not put into words. With fewer words they come swifter to a better understanding. There can be no doubt that here ósanwe  is also often taking place; for the will to converse in lambe  is a will to communicate thought, and lays the minds open. It may be, of course, that the two that converse know already part of the matter and the thought of the other upon it, so that only allusions dark to the stranger need be made; but this is not always so. The affine** will reach an understanding more swiftly than strangers upon matters that neither have before discussed, and they will more quickly perceive the import of words that, however numerous, well-chosen, and precise, must remain inadequate."

* "If a poem catches a student's interest at all, he or she should damned well be able to look up an unfamiliar word in the dictionary…."

   — Elizabeth Bishop, quoted in the essay "Grand Allusion" mentioned above. For a brief dictionary of most of the unfamiliar words in this post's title and in the above passage, see Vinyar Tengwar  39 (July 1998). This is copyrighted but freely available on the Web.

** The word "affine" has connotations not intended by Tolkien. See that word in this journal. See also page 5 of next Sunday's Times Book Review , which contains a full-page ad for the 50th anniversary edition of A Wrinkle in Time . "There is  such a thing as a tesseract."

Sunday, January 29, 2012


Filed under: General,Geometry — m759 @ 6:00 PM

Weblog posts of two prominent mathematicians today discussed
what appears to be a revolution inspired by the business practices
of some commercial publishers of mathematics.

See Gowers and Cameron.

My own concern is more with the so-called "Non-Euclidean Revolution"
described by Richard Trudeau in a book of that title (Birkhäuser, 1987).

A 1976 document relevant to the concerns in the Trudeau book—

Though not as well known as another document discussing
"self-evident" truths, Cameron's remarks are also of some
philosophical interest.

They apply to finite  geometry, a topic unknown to Euclid,
but nevertheless of considerable significance for the foundations  
of mathematics.

"The hand of the creative artist, laid upon the major premise,
 rocks the foundations of the world." — Dorothy Sayers

Sunday School

Filed under: General,Geometry — m759 @ 9:00 AM

Beach with Palms

The following images were suggested by the
"other things" link in yesterday's "287501346" post.

From June 10, 2008

Two frames from the Jodie Foster film "Contact"—

See Storyline and Time Fold.

Saturday, January 28, 2012

The Sweet Smell of Avon

Filed under: General,Geometry — m759 @ 9:48 AM

IMAGE- NY Times on 'Narrowing the Definition of Autism'

The twin topics of autism and of narrowing definitions
suggested the following remarks.

The mystical number "318" in the pilot episode
of Kiefer Sutherland's new series about autism, "Touch,"
is so small that it can easily apply (as the pilot
illustrated) to many different things: a date, a
time, a bus number, an address, etc.

The last 3/18 Log24 post— Defining Configurations
led, after a false start and some further research,
to the writing of the webpage Configurations and Squares.

An image from that page—

IMAGE- Coxeter 3x3 array with rows labeled 287/501/346.

Interpreting this, in an autistic manner, as the number
287501346 lets us search for more specific items
than those labeled simply 318.

The search yields, among other things, an offer of
Night Magic Cologne  (unsold)—

IMAGE- Online offer of Avon Night Magic Cologne- 'The mystery and magic of the night is yours.'

For further mystery and magic, see, from the date
the Night Magic offer closed— May 8, 2010— "A Better Story."
See also the next day's followup, "The Ninth Gate."

Friday, January 27, 2012


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

See also the links at the end of this morning's post.


Filed under: General — Tags: — m759 @ 12:21 PM


Notices of the American Mathematical Society

See also Rosetta Stone in this  journal.

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…"

Thursday, January 26, 2012

Change Illustrated

Filed under: General — m759 @ 7:59 PM

"Something is happening to our town."

— Mayor of Pleasantville

Related material— Alicia Keys's birthday,
the ending of Midnight in Paris , and Rocket Billie.

See also a Sinatra song uploaded on this date
(Paul Newman's birthday) last year.

Found in Space

Filed under: General — m759 @ 6:00 PM

"You have a clear and lively voice."
           — Midnight in Paris

Related material— Today's previous post.


Filed under: General — m759 @ 9:57 AM

NYT > Obituaries Dick Tufeld, Robot Voice in TV’s ‘Lost in Space,’ Dies at 85
    Wed Jan 25, 2012 23:42 from NYT Obituaries  By Bruce Weber

   "Mr. Tufeld possessed one of Hollywood’s most often-heard
   disembodied voices, especially from the 1950s through the 1970s."

In memoriam— A link from the date of Tufeld's death

After Midnight

Filed under: General — m759 @ 12:16 AM

IMAGE- NYT obits shortly after midnight on Jan. 26, 2012

Wednesday, January 25, 2012

A Larger City

Filed under: General — m759 @ 10:09 AM

By Penelope Lively
229 pages. Viking. $26.95.

Review by Michiko Kakutani
in The New York Times ,
online Jan. 23, 2012

As a historian, Henry acknowledges that he has “a soft spot for what is known as the Cleopatra’s nose theory of history— the proposal that had the nose of Cleopatra been an inch longer, the fortunes of Rome would have been different.” It’s a bit of a reductio ad absurdum, he admits, but nonetheless “a reference to random causality that makes a lot of sense when we think about the erratic sequence of events that we call history.”

What Ms. Lively has done in this captivating volume is to use all her copious storytelling gifts to show how a similar kind of random causality rules individual lives, how one unlucky event can set off unexpected chain reactions, how the so-called butterfly effect— whereby the flapping of a tiny butterfly’s wings can supposedly lead to a huge storm elsewhere in the world— ripples through the ebb and flow of daily life.

Rhetorical question—

"Why walk when you can fly?"
— Mary Chapin Carpenter

Rhetorical answer—

Two excerpts from a webpage on random walks

A drunk man will find his way home,
but a drunk bird may get lost forever.

— Shizuo Kakutani

Now we move to a larger city

IMAGE- 'Now we move to a larger city...' illustrated by 4x4 grid with dots signifying extension of the grid

State of a Nation

Filed under: General — m759 @ 2:19 AM

Happy birthday to…

Alicia Keys in 'Superwoman' video

Tuesday, January 24, 2012

The Infinity Point

Filed under: General,Geometry — m759 @ 2:20 PM

From Labyrinth of the Line (March 2, 2011)—

"… construct the Golay code by taking the 24 points
to be the points of the projective line F23 ∪ {}…."

— Robert A. Wilson

A simpler projective line— a Galois geometry
model of the line F2 ∪ {}—

Image- The Three-Point Line: A Finite Projective Geometry

Here we may consider  to be modeled*
by the third square above— the Galois window .

* Update of about 1 AM Jan. 25, 2012—
  This infinity-modeling is of course a poetic conceit,
  not to be taken too seriously. For a serious 
  discussion of points at infinity and finite fields,
  see (for instance) Daniel Bump's "The Group GL(2)."

The Screwing

Filed under: General,Geometry — Tags: , — m759 @ 7:59 AM

"Debates about canonicity have been raging in my field
(literary studies) for as long as the field has been
around. Who's in? Who's out? How do we decide?"

— Stephen Ramsay, "The Hermeneutics of Screwing Around"

An example of canonicity in geometry—

"There are eight heptads of 7 mutually azygetic screws, each consisting of the screws having a fixed subscript (from 0 to 7) in common. The transformations of LF(4,2) correspond in a one-to-one manner with the even permutations on these heptads, and this establishes the isomorphism of LF(4,2) and A8. The 35 lines in S3 correspond uniquely to the separations of the eight heptads into two complementary sets of 4…."

 — J.S. Frame, 1955 review of a 1954 paper by W.L. Edge,
"The Geometry of the Linear Fractional Group LF(4,2)"

Thanks for the Ramsay link are due to Stanley Fish
(last evening's online New York Times ).

For further details, see The Galois Tesseract.

Monday, January 23, 2012

How It Works

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


J. H. Conway in 1971 discussed the role of an elementary abelian group
of order 16 in the Mathieu group M24. His approach at that time was
purely algebraic, not geometric—

IMAGE- J. H. Conway in 1971 discussed the role of the elementary abelian group of order 16 in the Mathieu group M24. His approach then was purely algebraic, not geometric.

For earlier (and later) discussions of the geometry  (not the algebra )
of that order-16 group (i.e., the group of translations of the affine space
of 4 dimensions over the 2-element field), see The Galois Tesseract.

How Stuff Works

Filed under: General — m759 @ 3:48 PM

"Design is how it works." —Steve Jobs

Website logo—

IMAGE- Website logo- 'How Stuff Works: We figure it out so you don't have to'

Screenshot from How Stuff Works—

IMAGE- Christ in the Last Judgment, from 'How Stuff Works'

IMAGE- 'Apple's Mind-Bogglingly Greedy and Evil License Agreement'

(Click image for details.)

From "A Device Worthy of a Gothic Novel,"
Chapter XVI of The Club Dumas,
by Arturo Perez-Reverte (1993),
Vintage International, April 1998….
the basis of the 1999 Roman Polanski film
The Ninth Gate

Aren't you going to give me a document to sign?"
"A document?"
"Yes. It used to be called a pact. Now it would be a contract
with lots of small print, wouldn't it? 'In the event of litigation,
the parties are to submit to the jurisdiction of the courts of…'
That's a funny thing. I wonder which court covers this."


Filed under: General — m759 @ 5:01 AM

"Yo sé de un laberinto griego que es una línea única, recta."
 —Borges, "La Muerte y la Brújula"

"I know of one Greek labyrinth which is a single straight line."
—Borges, "Death and the Compass"

Sunday, January 22, 2012


Filed under: General,Geometry — Tags: , — m759 @ 8:09 PM

From life's box of chocolates

Happy birthday to Piper Laurie.

* Those who prefer their
souvenirs without sentiment
may consult the quaternions.

Year of the Dragon

Filed under: General — m759 @ 3:33 PM

In China, the Year of the Dragon
has now begun. See Holy Field
in this journal.

Pound Sign

Filed under: General — m759 @ 1:00 PM

See Metamagical Themas


— as well as the pound sign, alias hash .


Filed under: General — m759 @ 12:00 PM



"The Cardinal seemed a little preoccupied today."

The New Yorker , May 13, 2002

Card Trick

Filed under: General — m759 @ 11:30 AM

An image suggested by a book at
Princeton University Press—


Click image for details.

See also a somewhat deeper book from Princeton.


Filed under: General — m759 @ 10:00 AM

Kindergarten Theology

(Log24, St. Bridget's Day, 2008)

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.

Saturday, January 21, 2012

The Oxford Murders

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


Blame It on Trajan

Wikipedia on the 2008 film The Oxford Murders

IMAGE- Tall column of images from Log24, headed by permutahedron pictures

Christmas Eve image search
suggested by Stevens's phrase
"diamond globe."

(Larger version: 2 MB)

Friday, January 20, 2012

Cock Tale

Filed under: General — m759 @ 10:00 PM

(Continued from November 12, 2005 and June 7, 2011)

IMAGE- 1970 photo used in New Yorker of Jan. 23, 2012, with added photo credit information from a Philippe Sollers website

Related material— "Labyrinth," a fiction
by the late Roberto Bolaño
in the current New Yorker 

"There's no photo credit."

The Nothing That Is

Filed under: General — m759 @ 9:00 PM

"The 'one' with whom the reader has identified himself
has now become 'the listener, who listens in the snow';
he has become the snow man, and he knows winter
with a mind of winter, knows it in its strictest reality,
stripped of all imagination and human feeling.
But at that point when he sees the winter scene
reduced to absolute fact, as the object not of the mind,
but of the perfect perceptual eye that sees
'nothing that is not there,' then the scene,
devoid of  its imaginative correspondences,
has become 'the nothing that is.'"

Robert Pack, Wallace Stevens:
An Approach to His Poetry and Thought
New Brunswick: Rutgers UP, 1958.


IMAGE- The Ninefold Square at Ninevine.net


Filed under: General — m759 @ 8:00 PM
Saturday, November 12, 2005

— m759 @ 8:00 PM


A Singer 7-Cycle

“… problems are the poetry of chess.
   They demand from the composer
   the same virtues that characterize
   all worthwhile art:
   originality, invention, 
   harmony, conciseness,
   complexity, and
   splendid insincerity.”

   —Vladimir Nabokov

Soul Souvenirs

Filed under: General — m759 @ 7:00 PM

"If the fault is with the soul, the sovereigns
Of the soul must likewise be at fault, and first.
If the fault is with the souvenirs, yet these
Are the soul itself. And the whole of the soul, Swenson,
As every man in Sweden will concede,
Still hankers after lions, or, to shift,
Still hankers after sovereign images."

— Wallace Stevens

"Friday night and the lights are low"

See also American Music Award.

Brightness at Noon

Filed under: General — m759 @ 12:00 PM


See "harmonic analysis" in Mathematical Imagery and elsewhere in this journal.

Poetry and Thought*

Filed under: General — m759 @ 10:30 AM

* Title courtesy of George Steiner.
   For the "thought" part, see Plato's diamond
   in last night's Mathematical Imagery.

Thursday, January 19, 2012

Mathematical Imagery

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

Bourgain and Tao

From the Crafoord Prize website

Related meta -mathematical image from Diamond Theory

Mathematical  image related to combinatorics—

See also permutahedron in this journal.


Filed under: General — m759 @ 8:08 PM

Those impressed by George Steiner's remark on Hegel in the previous post may consult…

(Click to enlarge.)

(The Christian Examiner.  Volume LXXX. New Series, Volume I.  January, March, May, 1866.
New York: James Miller, Publisher, 522, Broadway.  Boston: Walker, Fuller, & Co.

By C. C. Everett, pp. 196-207.

A review of…

The Secret of Hegel, being the Hegelian System in Origin, Principle, Form, and Matter.
By James Hutchinson Sterling. In two volumes.
London: Longman, Green, Longman, Roberts, & Green. 1865. 8vo, 2 vols.)

On Hegel, from the review—

"He starts not from the beginning, but from the heart, of the world.
There never was a time when this pure Being— which, in its
undivided absoluteness, is indistinguishable from nothing;
as pure, unbroken light is indistinguishable from darkness—

was by itself alone; but this absolute Being is yet the foundation
and the groundwork of whatever is."

For more on Hegel's logic, see Marxists.org.

See also Steiner on chess and Lenin in The New Yorker
(September 7, 1968, page 133).


Filed under: General — m759 @ 2:02 PM

Perception vs. Inception


"Where philosophy and literature mesh, where they are litigious toward one another in form or matter, these echoes of origin can be heard. The poetic genius of abstract thought is lit, is made audible. Argument, even analytic, has its drumbeat. It is made ode. What voices the closing movements of Hegel’s Phenomenology  better than Edith Piaf’s non de non , a twofold negation which Hegel would have prized?

This essay is an attempt to listen more closely."

— George Steiner, The Poetry of Thought 


Edith Piaf's rien de rien . See also Is Nothing Sacred?

Square Triangles

Filed under: General,Geometry — Tags: — m759 @ 1:26 PM

MathWorld.Wolfram.com has an article titled "Square-Triangle Theorem."

An article of my own, whose HTML title was previously "Triangles are Square," has been retitled accordingly.

What Rough Beast

Filed under: General — m759 @ 8:28 AM


"On the Internet, nobody knows…"

Related material— Story Theory and the Number of the Beast

Wednesday, January 18, 2012

For Esther

Filed under: General — m759 @ 7:23 AM



Tuesday, January 17, 2012


Filed under: General,Geometry — m759 @ 8:48 PM

In memory of Bach interpreter
Gustav Leonhardt



Khora as Synchronicity

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

A search for khora  + tao  yields a paper on Derrida—



A check of the above date— Nov. 18, 2010— yields…

Thursday, November 18, 2010

Frontiers of Speculation

 m759 @ 8:02 AM

Peter Woit has a post on Scientific American 's new Garrett Lisi article, "A Geometric Theory of Everything."

The Scientific American  subtitle is "Deep down, the particles and forces of the universe are a manifestation of exquisite geometry."

See also Rhetoric (Nov. 4, 2010) and Exquisite Geometries (May 19, 2009).

Related material on the temptation of physics
for a pure mathematician—

This morning's post on khora  and Cardinal Manning, and,
from Hawking's birthday this year, Big Apple.

Within this  post, by leading us to the apple,
Derrida as usual plays the role of Serpent.

Manning and Khora

Filed under: General — m759 @ 1:26 AM

A weblog post from Saturday, Jan. 14, 2012—

"Today is the 120th anniversary of Cardinal Henry Edward Manning's death."

A Reluctant Sinner  (Thanks to Andrew Cusack for the link.)

If Manning is a saint, then Saturday was his feast day.

Some background— Manning in this journal.

See also Saturday's Derrida at Villanova. The link there to
previous posts on that topic leads to a post on Derrida's promotion
of his neologism différance as a version of Plato's khôra.

I prefer Manning's discussion of a closely related concept,
the scholastic philosophers' materia prima .

See Hugh R. King's 1956 paper sneering at the scholastics'
concept, and Heisenberg's much better-informed remarks
on the related concept of potentia

IMAGE- Excerpt from 'The Metaphysical Thought of Thomas Aquinas' by John F. Wippel

For a related fictional account of a religious quest for "possibilities"
and "excluded middles" between "zeroes and ones," see
Ingraffia on The Crying of Lot 49 .

Monday, January 16, 2012

Mapping Problem

Filed under: General,Geometry — Tags: , — m759 @ 5:10 PM

Thursday's post Triangles Are Square posed the problem of
finding "natural" maps from the 16 subsquares of a 4×4 square
to the 16 equilateral subtriangles of an edge-4 equilateral triangle.


Here is a trial solution of the inverse problem—


(Click for larger version.)

Exercise— Devise a test for "naturality" of
such mappings and apply it to the above.

Sunday, January 15, 2012


Filed under: General — m759 @ 11:00 AM

IMAGE- Book cover symbolizing postmodern theory with a minus sign and biblical theology with a plus sign

"Accentuate the Positive."

— Clint Eastwood, soundtrack album for
"Midnight in the Garden of Good and Evil."

Requiem Maas

Filed under: General — m759 @ 10:30 AM

Headline from yesterday evening's New York Times  obituaries—

Frederica Sagor Maas, Silent-Era Scriptwriter, Dies at 111.

For Maas… Past Tense (Jan. 7, 2012) and its link to Dogma.

Related material—

Last night's SNL, this morning's Entertainment Break,
and — in the context of DogmaCatholics Believe.

From an LA Times  story about Maas on January 7

"Many of the screenplays she and her husband wrote between 1938 and 1950 were never produced. Hopeless, humiliated and having little money, the couple drove to a hilltop overlooking Hollywood with the intention of committing suicide in their Plymouth. Clutching each other, they started sobbing and realized that 'none of these things mattered. We had each other,' wrote Maas…."

Entertainment Break

Filed under: General — m759 @ 10:00 AM


Related material– Saturday night's Derrida at Villanova and Villanueva.

Sunday School

Filed under: General — m759 @ 9:00 AM

Identity Crisis— Or, Tilting at Whirligigs 

Saturday, January 14, 2012

Derrida at Villanova

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

"As Derrida said at Villanova,
"We wait for something we would not like to wait for.
That is another name for death."

— Brian D. Ingraffia, "Is the Postmodern Post-Secular?,"
p. 50 in Postmodern Philosophy and Christian Thought ,
ed. by Merold Westphal, Indiana University Press, 1999, pp. 44-68

See also Derrida at Villanova in this journal.

The link to Ingraffia's remarks was suggested by
this evening's New York Times  obituaries—


Defining Form (continued)

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

Detail of Sylvie Donmoyer picture discussed
here on January 10


The "13" tile may refer to the 13 symmetry axes
in the 3x3x3 Galois cube, or the corresponding
13 planes through the center in that cube. (See
this morning's post and Cubist Geometries.)

Damnation Morning*

Filed under: General,Geometry — Tags: , — m759 @ 5:24 AM


The following is adapted from a 2011 post

IMAGE- Galois vs. Rubik

* The title, that of a Fritz Leiber story, is suggested by
   the above picture of the symmetry axes of the square.
   Click "Continued" above for further details. See also
   last Wednesday's Cuber.

Friday, January 13, 2012

Mysteries of Faith

Filed under: General — m759 @ 6:00 PM


Wiener on Paley

… he was already recognised as the ablest of the group of young English mathematicians who have been inspired by the genius of G H Hardy and J E Littlewood. In a group notable for its brilliant technique, no one had developed this technique to a higher degree than Paley. Nevertheless he should not be though of primarily as a technician, for with this ability he combined creative power of the first order. As he himself was wont to say, technique without 'rugger tactics' will not get one far, and these rugger tactics he practised to a degree that was characteristic of his forthright and vigorous nature.

The Telegraph  today on British mystery author Reginald Hill—

"After National Service between 1955 and 1957,
he went up on a scholarship to St Catherine’s College, Oxford,
where he played rugby…."

Further details—


"Unsheathe your dagger definitions." — James Joyce

Some context— St. Catherine in this journal and her feast day last year.


Filed under: General — Tags: — m759 @ 8:28 AM

Click logos for related persons.

IMAGE- Logo of St. Peter's College, Oxford

IMAGE- Logo of St. John's College, Oxford

Some related news.

Background from this journal—

Collegiality, That Hideous Strength , and The Oxford Murders .

See also…

"The heart of the book is the conveying of a meaningful understanding
of where mathematical results originated…."

Thursday, January 12, 2012

Collegiality in Action

Filed under: General — m759 @ 10:30 PM


"The transcripts of the 2006 meetings, released after a standard five-year delay,
clearly show some of the nation’s pre-eminent economic minds did not fully understand
the basic mechanics of the economy that they were charged with supervising."

— Binyamin Appelbaum in the Jan. 12 online New York Times

Academics may recall other examples of comfortably ignorant collegiality.

Triangles Are Square

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

Coming across John H. Conway's 1991*
pinwheel  triangle decomposition this morning—


— suggested a review of a triangle decomposition result from 1984:

IMAGE- Triangle and square, each with 16 parts

Figure A

(Click the below image to enlarge.)

IMAGE- 'Triangles Are Square,' by Steven H. Cullinane (American Mathematical Monthly, 1985)

The above 1985 note immediately suggests a problem—

What mappings of a square  with c 2 congruent parts
to a triangle  with c 2 congruent parts are "natural"?**

(In Figure A above, whether the 322,560 natural transformations
of the 16-part square map in any natural way to transformations
of the 16-part triangle is not immediately apparent.)

* Communicated to Charles Radin in January 1991. The Conway
  decomposition may, of course, have been discovered much earlier.

** Update of Jan. 18, 2012— For a trial solution to the inverse
    problem, see the "Triangles are Square" page at finitegeometry.org.

Wednesday, January 11, 2012


“Examples galore of this feeling must have arisen in the minds of the people who extended the Magic Cube concept to other polyhedra, other dimensions, other ways of slicing.  And once you have made or acquired a new ‘cube’… you will want to know how to export a known algorithm , broken up into its fundamental operators , from a familiar cube.  What is the essence of each operator?  One senses a deep invariant lying somehow ‘down underneath’ it all, something that one can’t quite verbalize but that one recognizes so clearly and unmistakably in each new example, even though that example might violate some feature one had thought necessary up to that very moment.  In fact, sometimes that violation is what makes you sure you’re seeing the same thing , because it reveals slippabilities you hadn’t sensed up till that time….

… example: There is clearly only one sensible 4 × 4 × 4 Magic Cube.  It is the  answer; it simply has the right spirit .”

— Douglas R. Hofstadter, 1985, Metamagical Themas: Questing for the Essence of Mind and Pattern  (Kindle edition, locations 11557-11572)

See also Many Dimensions in this journal and Solomon’s Cube.

Language Game

Filed under: General,Geometry — Tags: , — m759 @ 8:08 AM

Tension in the Common Room

IMAGE- 'Launched from Cuber' scene in 'X-Men: First Class'

In memory of population geneticist James F. Crow,
who died at 95 on January 4th.

Tuesday, January 10, 2012


Filed under: General — m759 @ 6:29 PM

"'Interpenetration'" — Stanley Fish in yesterday evening's online New York Times

"You want Frye's with that?" — A recent humanities graduate



Defining Form

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

(Continued from Epiphany and from yesterday.)

Detail from the current American Mathematical Society homepage


Further detail, with a comparison to Dürer’s magic square—

http://www.log24.com/log/pix12/120110-Donmoyer-Still-Life-Detail.jpg http://www.log24.com/log/pix12/120110-DurerSquare.jpg

The three interpenetrating planes in the foreground of Donmoyer‘s picture
provide a clue to the structure of the the magic square array behind them.

Group the 16 elements of Donmoyer’s array into four 4-sets corresponding to the
four rows of Dürer’s square, and apply the 4-color decomposition theorem.
Note the symmetry of the set of 3 line diagrams that result.

Now consider the 4-sets 1-4, 5-8, 9-12, and 13-16, and note that these
occupy the same positions in the Donmoyer square that 4-sets of
like elements occupy in the diamond-puzzle figure below—


Thus the Donmoyer array also enjoys the structural  symmetry,
invariant under 322,560 transformations, of the diamond-puzzle figure.

Just as the decomposition theorem’s interpenetrating lines  explain the structure
of a 4×4 square , the foreground’s interpenetrating planes  explain the structure
of a 2x2x2 cube .

For an application to theology, recall that interpenetration  is a technical term
in that field, and see the following post from last year—

Saturday, June 25, 2011 

Theology for Antichristmas

— m759 @ 12:00 PM

Hypostasis (philosophy)

“… the formula ‘Three Hypostases  in one Ousia
came to be everywhere accepted as an epitome
of the orthodox doctrine of the Holy Trinity.
This consensus, however, was not achieved
without some confusion….” —Wikipedia



Click for further details:



Monday, January 9, 2012

M Theory

Filed under: General — Tags: — m759 @ 7:59 AM

Yesterday's All About Eve post featured Pope John Paul II
with his close friend and confidant Jerzy Kluger.
Their counterparts Xavier and Magneto in the recent film
"X-Men: First Class," together with Catholic doctrine on telepathy,
suggest  the following meditations.

Douglas Hofstadter on interpenetration

IMAGE- 'Interpenetration' in Douglas Hofstadter's 'I Am a Strange Loop'

— as well as Trinity in this journal.

First the punchline—

Script M (interpreted by some scanners as '771.')

Then the joke.

Sunday, January 8, 2012

Big Apple

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


    “…the nonlinear characterization of Billy Pilgrim
    emphasizes that he is not simply an established
    identity who undergoes a series of changes but
    all the different things he is at different times.”

A 2x4 array of squares

This suggests that the above structure
be viewed as illustrating not eight  parts
but rather 8! = 40,320 parts.


"The Cardinal seemed a little preoccupied today."

The New Yorker , May 13, 2002

See also a note of May 14 , 2002.

All About Eve (continued)

Filed under: General — m759 @ 4:23 AM

Literary symbolism, offered without comment—

http://www.log24.com/log/pix12/120108-NYTfront305AM.jpg http://www.log24.com/log/pix12/120108-NYTobits.jpg


Log24 post of January 14, 2011


Saturday, January 7, 2012


Filed under: General,Geometry — Tags: , — m759 @ 7:14 PM

Today's Google Doodle for the 100th birthday of Charles Addams—


A doodle from this year's Feast of the Epiphany


A doodle based on today's previous post and on a post for Twelfth Night, 2003

IMAGE- Quilt blocks- Devil's Claws and Yankee Puzzle

IMAGE- 'I'm a Yankee Doodle Dandy' musical notes

Past Tense

Filed under: General — Tags: — m759 @ 4:09 PM

From a post that was written for Twelfth Night

Bernhard Weiss on the philosophy of Michael Dummett—

" … debates about realism, that is, those debates that ask
whether or not one or another aspect of the world is independent
of the way we represent that aspect to ourselves. For example,
is there a realm of mathematical entities that exists fully formed
independently of our mathematical activity? Are there facts about
the past that our use of the past tense aims to capture?"

Yes and Yes.

See also The Whirligig of Time in this journal.

Number and Form

Filed under: General — Tags: — m759 @ 10:09 AM

A link for Josefine Lyche in memory of Anne Tyng.

Fearful Cold Intelligence

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

"Dreams are sleep's watchful brother, of death's fraternity,
heralds, watchmen of that coming night, and our attitude
toward them may be modeled upon Hades, receiving, hospitable,
yet relentlessly deepening, attuned to the nocturne, dusky, and
with a fearful cold intelligence that gives permanent shelter
in his house to the incurable conditions of human being."

— James Hillman, conclusion of
The Dream and the Underworld  (Harper & Row, 1979)

In memory of Raymond Edward Alan Christopher Paley

IMAGE- 'Note on the Mathieu Group M12' by Marshall Hall, Jr.

Related material— Mathieu Symmetry.

Little Morning

Filed under: General — Tags: — m759 @ 5:01 AM

IMAGE- Conclusion of Hillman's 'The Dream and the Underworld'

… que cantaba el rey David.

Friday, January 6, 2012


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

An example for the Feast of the Epiphany*
IMAGE- Cathy Hull, detail from cover of Hillman's 'The Dream and the Underworld'

For one approach to defining this form, see Diamond Star.

* And for Pomona College

Defining Form

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

IMAGE- MLA session, 'Defining Form,' chaired by Colleen Rosenfeld of Pomona College

Some related resources from Malcolm Lowry

"…his eyes ranged the Consul's books disposed quite neatly… on high shelves around the walls: Dogme et Ritual de la Haute Magie , Serpent and Siva Worship in Central America , there were two long shelves of this, together with the rusty leather bindings and frayed edges of the numerous cabbalistic and alchemical books, though some of them looked fairly new, like the Goetia of the Lemegaton of Solomon the King , probably they were treasures, but the rest were a heterogeneous collection…."

Under the Volcano , Chapter VI

— and from Matilde Marcolli

Seven books on analytical psychology

See also Marcolli in this morning's previous post, The Garden Path.

For the relevance of alchemy to form, see Alchemy in this journal.

The Garden Path

Filed under: General — Tags: — m759 @ 4:01 AM

"Not all those who have sought to decode the symbolism of the Tarot pack
have been occultists; some have been serious scholars…."

— Michael Dummett, The Game of Tarot , Ch. 20

“Eliot by his own admission took the ‘still point of the turning world’
in Burnt Norton  from the Fool in Williams’s The Greater Trumps .”

— Humphrey Carpenter, The Inklings , Ballantine Books, 1981, p. 106

From a talk on April 16, 2010, in Cuernavaca


Related material—

IMAGE- 'A Walk in the Noncommutative Garden'

See also The Martial Art of Giving Talks.

(Thanks to Lieven Le Bruyn for his Twelfth Night post on this topic.)

Thursday, January 5, 2012

Crown Archetype

Filed under: General — m759 @ 3:09 PM

"Publisher: Crown Archetype (November 1, 2011)"

— Amazon.com on Piper Laurie's new self-portrait

See also last year's For All Hallows Day and today's previous post.

Some context:  "God's Girlfriend" in this journal
and "Shouts & Murmurs" in The New Yorker
of January 9th, 2012—


   Life is  like a box of chocolates.

ILLUSTRATION: Maximilian Bode


Filed under: General,Geometry — m759 @ 6:00 AM

From a review of Truth and Other Enigmas , a book by the late Michael Dummett—

"… two issues stand out as central, recurring as they do in many of the
essays. One issue is the set of debates about realism, that is, those debates that ask
whether or not one or another aspect of the world is independent of the way we
represent that aspect to ourselves. For example, is there a realm of mathematical
entities that exists fully formed independently of our mathematical activity? Are
there facts about the past that our use of the past tense aims to capture? The other
issue is the view
which Dummett learns primarily from the later Wittgenstein
that the meaning of an expression is fully determined by its use, by the way it
is employed by speakers. Much of his work consists in attempts to argue for this
thesis, to clarify its content and to work out its consequences. For Dummett one
of the most important consequences of the thesis concerns the realism debate and
for many other philosophers the prime importance of his work precisely consists
in this perception of a link between these two issues."

Bernhard Weiss, pp. 104-125 in Central Works of Philosophy , Vol. 5,
ed. by John Shand,
McGill-Queen's University Press, June 12, 2006

The above publication date (June 12, 2006) suggests a review of other
philosophical remarks related to that date. See …


For some more-personal remarks on Dummett, see yesterday afternoon's
"The Stone" weblog in The New York Times.

I caught the sudden look of some dead master….

Four Quartets

Wednesday, January 4, 2012


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

I revised the cubes image and added a new link to
an explanatory image in posts of Dec. 30 and Jan. 3
(and at finitegeometry.org). (The cubes now have
quaternion "i , j , k " labels and the cubes now
labeled "k " and "-k " were switched.)

I found some relevant remarks here and here.

Tuesday, January 3, 2012

Dark Fields

Filed under: General — m759 @ 10:10 PM

"He had come a long way to this blue lawn,
and his dream must have seemed so close
that he could hardly fail to grasp it.
He did not know that it was already behind him,
somewhere back in that vast obscurity beyond the city,
where the dark fields of the republic rolled on under the night."

The Great Gatsby


See also St. Andrew's Day, 2011, in this journal.


Filed under: General,Geometry — Tags: , , — m759 @ 7:48 AM

In memory of artist Ronald Searle

IMAGE- Ronald Searle, 'Pythagoras puzzled by one of my theorums,' from 'Down with Skool'

Searle reportedly died at 91 on December 30th.

From Log24 on that date

IMAGE- Quaternion group acting on an eightfold cube

Click the above image for some context.

Update of 9:29 PM EST Jan. 3, 2012



From RationalWiki

Theorum (rhymes with decorum, apparently) is a neologism proposed by Richard Dawkins in The Greatest Show on Earth  to distinguish the scientific meaning of theory from the colloquial meaning. In most of the opening introduction to the show, he substitutes "theorum" for "theory" when referring to the major scientific theories such as evolution.

Problems with "theory"

Dawkins notes two general meanings for theory; the scientific one and the general sense that means a wild conjecture made up by someone as an explanation. The point of Dawkins inventing a new word is to get around the fact that the lay audience may not thoroughly understand what scientists mean when they say "theory of evolution". As many people see the phrase "I have a theory" as practically synonymous with "I have a wild guess I pulled out of my backside", there is often confusion about how thoroughly understood certain scientific ideas are. Hence the well known creationist argument that evolution is "just  a theory" – and the often cited response of "but gravity is also just  a theory".

To convey the special sense of thoroughness implied by the word theory in science, Dawkins borrowed the mathematical word "theorem". This is used to describe a well understood mathematical concept, for instance Pythagoras' Theorem regarding right angled triangles. However, Dawkins also wanted to avoid the absolute meaning of proof associated with that word, as used and understood by mathematicians. So he came up with something that looks like a spelling error. This would remove any person's emotional attachment or preconceptions of what the word "theory" means if it cropped up in the text of The Greatest Show on Earth , and so people would (in "theory ") have no other choice but to associate it with only the definition Dawkins gives.

This phrase has completely failed to catch on, that is, if Dawkins intended it to catch on rather than just be a device for use in The Greatest Show on Earth . When googled, Google will automatically correct the spelling to theorem instead, depriving this very page its rightful spot at the top of the results.

See also


Some backgound— In this journal, "Diamond Theory of Truth."

Monday, January 2, 2012

The Lovely Bones*

Filed under: General — Tags: , — m759 @ 8:48 AM

An adaptation for the late Barbara Lea

Man's spirit will be flesh-bound, when found at best,
But úncúmberèd: meadow-dówn is nót distréssed
For a ráinbow fóoting it nor shé for her bónes rísen.

— After Gerard Manley Hopkins, Society of Jesus

* "And I began to see things in a way that let me hold the world without me in it."

    — Alice Sebold

Sunday, January 1, 2012

Sunday Shul

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

"… myths are stories, and like all narratives
they unravel through time, whereas grids
are not only spatial to start with,
they are visual structures that explicitly reject
a narrative or sequential reading of any kind."

— Rosalind Krauss in "Grids,"
October  (Summer 1979), 9: 50-64.


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

The Ninefold Square

See Coxeter and the Aleph and Ayn Sof

Mathematics and Narrative, Illustrated


Times Square Church

Filed under: General — m759 @ 3:33 AM


IMAGE- NY Times online front page, New Year's Day 2012

Both Ears…


And the Tale


For another version of the tale, click
the "Continued" link from last year's final post.

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