Friday, September 30, 2011


Filed under: General — m759 @ 3:48 PM

"A Phenomenological Perspective,"
Ch. 2 in The Star and the Whole:
Gian-Carlo Rota on Mathematics and Phenomenology 
by Fabrizio Palombi, A K Peters/CRC Press, 2011—

"Rota is convinced that one of the fundamental tasks of phenomenology is that of highlighting the primordiality of sense. In his words, if 'many disputes among philosophers are disputes about primordiality' then 'phenomenology is yet another dispute about what is most primordial' (Rota, 1991a,* p. 54). In this way he evidently does not intend to deny the existence of matter, of objects, or of that objective dimension proper to science, in favor of a spiritualist option, but rather to posit as primordial another dimension of the world connected with contexts and with roles, which is considered primordial because each one of us is confronted with it primordially."

* The End of Objectivity: The Legacy of Phenomenology ,
Lectures by Rota at MIT 1974-1991, 457 pages,
MIT Mathematics Department, Cambridge, MA

"The Ultimate, Apocalyptic Laptop"
by George Johnson
Published: September 5, 2000, by The New York Times

"In a paper in the current issue of Nature , Dr. Lloyd describes the ultimate laptop— a computer as powerful as the laws of physics will allow. So energetic is this imaginary machine that using it would be like harnessing a thermonuclear reaction. In the most extreme version of this computer supreme, so much computational circuitry would be packed into so small a space that the whole thing would collapse and form a tiny black hole, an object so dense that not even light can escape its gravity."

Related material: Rota and "Black Hole" in this journal, as well as the Sator Square.

Thursday, September 29, 2011

Michaelmas Moonshine

Filed under: General — m759 @ 11:48 PM

Lottery hermeneutics for Michaelmas—

New York Lottery the evening of Thursday, Sept. 29, 2011—

499 and 6985.

For 499, see post  499 in this journal ("Angel Night," Sept. 29, 2002).

For 6985, see a Sunday, 6/9/85, review of Amy's Eyes , a children's book by Richard Kennedy.

"Especially for the adult reader, the narrator's musings comprise many of the book's great pleasures. He discusses the seductiveness of numerology, the 'Wayward Daughter of Mathematics'….''

Edwin J. Kenney Jr.

Wednesday, September 28, 2011

Mathieu Moonshine

Filed under: General — m759 @ 2:56 PM

The properties of the Mathieu group M 24 have recently interested some physicists

(Click to enlarge.)

IMAGE- July 2011 conference on Mathieu moonshine

For some related papers, see Mendeley.com.

Tuesday, September 27, 2011

The Lost Plot

Filed under: General — m759 @ 8:00 PM

Daredevil work—


For Dan Brown, author of The Lost Symbol


Al Pacino and Keanu Reeves in Devil's Advocate
(Syfy channel, 9 PM tonight)

"Klaatu barada nikto."

Monday, September 26, 2011

Inner and Outer

Filed under: General,Geometry — m759 @ 1:00 PM

For T.S. Eliot's Birthday

Last night's post "Transformation" was suggested in part
by the title of a Sunday New York Times  article on
George Harrison, "Within Him, Without Him," and by
the song title "Within You Without You" in the post
Death and the Apple Tree.

Related material— "Hamlet's Transformation"—

Hamlet, 2.2:

Something have you heard
Of Hamlet’s transformation; so call it,
Sith nor the exterior nor the inward man
Resembles that it was….”

A transformation:

The image “http://www.log24.com/theory/images/DTinscapes4-Trans.gif” cannot be displayed, because it contains errors.

Click on picture for details.

See also, from this year's Feast of the Transfiguration,
Correspondences and Happy Web Day.

For those who prefer the paganism of Yeats to
the Christianity of Eliot, there is the sequel to
"Death and the Apple Tree," "Dancers and the Dance."

Sunday, September 25, 2011


Filed under: General — m759 @ 11:01 PM

Random thoughts from A Story (September 13th)—


April 24, 2003
A Terrible Beauty


August 26, 2002
Round Lights


April 24, 2005
Today's Sermon


March 24, 2008
Death and the Apple Tree

A September 24th story from The Washington Post

"James R. Adams was author or co-author of seven books,
beginning in 1971 with 'The Sting of Death.'"

Adams, an Episcopal priest, died on September 13th.

The September 13th Log24 post from which the above numbered links were
taken was in memory of film producer John Calley, who also died on that day.

A quote from Adams on the resurrected body of Christ:

"This new body had peculiar powers."

A quote from Calley:

"Let me put you in this unit."

Saturday, September 24, 2011

Live from New York, It’s…

Filed under: General — m759 @ 11:29 PM


For instance…

Click to enlarge


See also Plato's Ghost.

Kernels of Being

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

For the Pope in Germany

"We wish to see Jesus. For somehow we know, we suspect, we intuit, that if we see Jesus we will see what Meister Eckhart might call “The Divine Kernel of Being”— that Divine Spark of God’s essence, God’s imago Dei, the image in which we are created. We seem to know that in seeing Jesus we just might find something essential about ourselves."

—The Reverend Kirk Alan Kubicek, St. Peter’s at Ellicott Mills, Maryland, weblog post of Saturday, March 28, 2009, on a sermon for Sunday, March 29, 2009

See also this journal in March 2009.

Related non-theology—

Weyl on coordinate systems, Cassirer on the kernel of being, and A Study in Art Education.

Ariadne and the Exorcist

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

The title describes two philosophical events (one major, one minor) from the same day— Thursday, July 5, 2007. Some background from 2001:

"Are the finite simple groups, like the prime numbers, jewels strung on an as-yet invisible thread? And will this thread lead us out of the current labyrinthine proof to a radically new proof of the Classification Theorem?" (p. 345)

— Ronald Solomon,  "A Brief History of the Classification of Finite Simple Groups," Bulletin of the American Mathematical Society , Vol. 38 No. 3 (July 2001), pp. 315-352

The major event— On July 5, 2007, Cambridge University Press published Robert T. Curtis's Symmetric Generation of Groups.*

Curtis's book does not purport to lead us out of Solomon's labyrinth, but its publication date may furnish a Jungian synchronistic clue to help in exiting another  nightmare labyrinth— that of postmodernist nominalism.

The minor event— The posting of Their Name is Legion in this journal on July 5, 2007.

* This is the date given by Amazon.co.uk and by BookDepository.com. Other sources give a later July date, perhaps applicable to the book's publication in the U.S. rather than Britain.

Friday, September 23, 2011

Kali’s Thunder

Filed under: General — m759 @ 9:00 PM

NY Lottery this evening—


Continued from Themes of Sept. 17—

"And I heard, as it were, the noise of thunder."

Lottery hermeneutics for this evening—

747 — Source of the sound effect in a post of April 28, 2009:

"And Kali, ‘The Dark One,’ addressed him with the voice of a cloud of thunder…"
The King and the Corpse , by Heinrich Zimmer

3695 — Number of a post on Steps Toward Salvation  (Dec. 14, 2008)


Filed under: General — m759 @ 3:01 PM

NY Lottery this afternoon—


See also post 1115.


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

For those who prefer Nick Stahl (star of "The Speed of Thought"— see previous post)
to Keanu Reeves as a savior figure, here is a still from another film with Stahl as savior—

IMAGE- Schwarzenegger carries coffin in 'Terminator 3: Rise of the Machines'

Backstory —

IMAGE- NY Times obits for Julius Blank and A. Richard Turner

See also a Log24 post from the day of Blank's death, The Uploading.

Times Square Church

Filed under: General — Tags: , — m759 @ 2:56 AM

Continued from April 29, 2011


Related material— Inception and Feed Them on Your Dreams.

Thursday, September 22, 2011

Product Placement

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

And the Irving Thalberg Award goes to…


Robert De Niro as Irving Thalberg in "The Last Tycoon"

Text and Context—



Jerky Treats  for dogs



"Mad Dog and Glory" (March 5, 1993)


"Point of No Return" (March 19, 1993) —
Note Jerky Treats  in background.

A possible acceptance speech for the Thalberg Award—

"Let me put you in this unit." — John Calley, via Buck Henry

Sicilian Reflections

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


Related material—


See also yesterday's Symmetric Generation.

New Day Nina

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

In memoriam


Bridget Fonda in "Point of No Return."  This is a 1993 remake of 1990's "La Femme Nikita ,"
virtually the same scene-by-scene, but with two nice new touches: the young assassin's code name
is Nina, after Nina Simone, and she happily shops as Simone sings "new day" in the soundtrack.

Wednesday, September 21, 2011

The Forking

Filed under: General — m759 @ 10:16 PM

"When you come to a fork in the road, take it." — Yogi Berra


This evening's NY Lottery numbers were 375 and 3141.

Subjective interpretations—

There seems to be only one relevant result of a Google search for "375 Log24"—


There are, however, two  relevant interpretations of the number 3141—

1. The Saturday Evening Post  3/1/41 article by Jack Alexander on AA—

"The members of Alcoholics Anonymous do not pursue or coddle
a malingering prospect, and they know the strange tricks of the alcoholic
as a reformed swindler knows the art of bamboozling."

2. Post number 3141 in this  journal— Aesthetics for Jesuits.

The Gleaming (continued)

Filed under: General — m759 @ 3:33 PM

Happy Birthday to Bill Murray (61) and Stephen King (64).

Symmetric Generation

Suggested by yesterday's Relativity Problem Revisited and by Cassirer on Objectivity

From Symmetric Generation of Groups , by R.T. Curtis (Cambridge U. Press, 2007)—

"… we are saying much more than that G M 24 is generated by
some set of seven involutions, which would be a very weak
requirement. We are asserting that M 24 is generated by a set
of seven involutions which possesses all the symmetries of L3(2)
acting on the points of the 7-point projective plane…."
Symmetric Generation , p. 41

"It turns out that this approach is particularly revealing and that
many simple groups, both sporadic and classical, have surprisingly
simple definitions of this type."
Symmetric Generation , p. 42

See also (click to enlarge)—


Cassirer's remarks connect the concept of objectivity  with that of object .

The above quotations perhaps indicate how the Mathieu group M 24 may be viewed as an object.

"This is the moment which I call epiphany. First we recognise that the object is one  integral thing, then we recognise that it is an organised composite structure, a thing  in fact: finally, when the relation of the parts is exquisite, when the parts are adjusted to the special point, we recognise that it is that  thing which it is. Its soul, its whatness, leaps to us from the vestment of its appearance. The soul of the commonest object, the structure of which is so adjusted, seems to us radiant. The object achieves its epiphany."

— James Joyce, Stephen Hero

For a simpler object "which possesses all the symmetries of L3(2) acting on the points of the 7-point projective plane…." see The Eightfold Cube.

For symmetric generation of L3(2) on that cube, see A Simple Reflection Group of Order 168.

Tuesday, September 20, 2011

1949 Revisited

Filed under: General — m759 @ 6:18 PM

Hermann Weyl's 1949 remarks in this morning's post
were made at an event on March 19 in honor of Einstein's
70th birthday five days earlier.

Somehow the conclusion of Margaret Atwood's 1988 novel
Cat's Eye  seems appropriate:


Relativity Problem Revisited

Filed under: General,Geometry — Tags: , , , , — m759 @ 4:00 AM

A footnote was added to Finite Relativity


Weyl on what he calls the relativity problem

IMAGE- Weyl in 1949 on the relativity problem

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

– Hermann Weyl, 1949, “Relativity Theory as a Stimulus in Mathematical Research

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

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

…. A note of Feb. 20, 1986, supplied an example of such coordinatizations in finite geometry. In that note, the group of mediating transformations acted directly on  coordinates within a 4×4 array. When the 4×4 array is embedded in a 4×6 array, a larger and more interesting group, M 24 (containing the original group), acts on the larger array.  There is no obvious solution to Weyl’s relativity problem for M 24.  That is, there is no obvious way* to apply exactly 24 distinct transformable coordinate-sets (or symbol-strings ) to the 24 array elements in such a way that the natural group of mediating transformations of the 24 symbol-strings is M 24. ….

Footnote of Sept. 20, 2011:

* R.T. Curtis has, it seems, a non-obvious way that involves strings of seven symbols.  His abstract for a 1990 paper says that in his construction “The generators of M 24 are defined… as permutations of twenty-four 7-cycles in the action of PSL2(7) on seven letters….”

See “Geometric Interpretations of the ‘Natural’ Generators of the Mathieu groups,” by R.T. Curtis,  Mathematical Proceedings of the Cambridge Philosophical Society  (1990), Vol. 107, Issue 01, pp. 19-26. (Rec. Jan. 3, 1989, revised Feb. 3, 1989.) This paper was published online on Oct. 24, 2008.

Some related articles by Curtis:

R.T. Curtis, “Natural Constructions of the Mathieu groups,” Math. Proc. Cambridge Philos. Soc.  (1989), Vol. 106, pp. 423-429

R.T. Curtis. “Symmetric Presentations I: Introduction, with Particular Reference to the Mathieu groups M 12  and M 24” In Proceedings of 1990 LMS Durham Conference ‘Groups, Combinatorics and Geometry’  (eds. M. W. Liebeck and J. Saxl),  London Math. Soc. Lecture Note Series 165, Cambridge University Press, 1992, pp. 380–396

R.T. Curtis, “A Survey of Symmetric Generation of Sporadic Simple Groups,” in The Atlas of Finite Groups: Ten Years On , (eds. R.T. Curtis and R.A. Wilson), London Math. Soc. Lecture Note Series 249, Cambridge University Press, 1998, pp. 39–57

Monday, September 19, 2011

A Sunset Number

Filed under: General — m759 @ 3:33 PM


as indicated by Gravity's Rainbow— the end of page 742

by the New York Lottery on the evening of June 17, 2010—


and by this journal on the afternoon of that day

Image-- Detail of New Yorker cover 'Finish Line,' double fiction issue of June 14 & 21, 2010

Detail from New Yorker  cover (fiction issue of June 2010)

Sunday, September 18, 2011

What Rough Beast

Filed under: General — m759 @ 9:00 PM

Lurching Toward Decision


"Suskind… nails, I think, Obama's intellectual blind spot. Indeed, Obama himself nails it, telling Suskind that he was too inclined to search for 'the perfect technical answer' to the myriad of complex issues coming at him."

Frank Rich on Ron Suskind's new book about the White House, Confidence Men

Very distantly related material—

From "Confidence Game," an Oct. 12, 2008, post in this journal, a quasi-European perspective—

Juliette Binoche in 'Blue'  Animated 2x2 kaleidoscope figures from Diamond Theory

Kaleidoscope turning…
Shifting pattern
within unalterable structure…

– Roger Zelazny, Eye of Cat   

See also …

Gravity’s Rainbow , Penguin Classics, 1995, page 742:

"… knowing his Tarot, we would expect to look among the Humility, among the gray and preterite souls, to look for him adrift in the hostile light of the sky, the darkness of the sea….

Now there’s only a long cat’s-eye of bleak sunset left over the plain tonight, bright gray against a purple ceiling of clouds, with an iris of


Anatomy of a Cube

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

R.D. Carmichael’s seminal 1931 paper on tactical configurations suggests
a search for later material relating such configurations to block designs.
Such a search yields the following

“… it seems that the relationship between
BIB [balanced incomplete block ] designs
and tactical configurations, and in particular,
the Steiner system, has been overlooked.”
— D. A. Sprott, U. of Toronto, 1955


The figure by Cullinane included above shows a way to visualize Sprott’s remarks.

For the group actions described by Cullinane, see “The Eightfold Cube” and
A Simple Reflection Group of Order 168.”

Update of 7:42 PM Sept. 18, 2011—

From a Summer 2011 course on discrete structures at a Berlin website—

A different illustration of the eightfold cube as the Steiner system S(3, 4, 8)—


Note that only the static structure is described by Felsner, not the
168 group actions discussed (as above) by Cullinane. For remarks on
such group actions in the literature, see “Cube Space, 1984-2003.”

Alpha and Omega

Filed under: General,Geometry — Tags: — m759 @ 2:22 AM


A transcription—

"Now suppose that α  is an element of order 23 in M 24 ; we number the points of Ω
as the projective line , 0, 1, 2, … , 22 so that α : i i  + 1 (modulo 23) and fixes . In
fact there is a full L 2 (23) acting on this line and preserving the octads…."

— R. T. Curtis, "A New Combinatorial Approach to M 24 ,"
Mathematical Proceedings of the Cambridge Philosophical Society  (1976), 79: 25-42

Midnight in Dostoevsky

Filed under: General — m759 @ 12:00 AM

"But Thou knewest not, it seems, that no sooner would man reject
miracle than he would reject God likewise, for he seeketh less
God than 'a sign' from Him." —The Grand Inquisitor

Update of 1:44 AM—

Seek and Ye Shall Find …

(Click images for larger context)


  The Holy Office —


Saturday, September 17, 2011

As It Were

Filed under: General — m759 @ 5:45 PM

Historic plane crashes at West Virginia air show

"The T-28 aircraft crashed at about 2:40 p.m.
during an acrobatic demonstration at the 2011
Thunder Over the Blue Ridge Open House & Air Show…."

See also Themes

"And I heard, as it were, the noise of thunder."


Filed under: General — m759 @ 12:00 PM

The previous two posts, Baggage and The Uploading, suggest
a review of Wroclaw's native son Ernst Cassirer.

The Uploading

Filed under: General — m759 @ 11:32 AM

(Continued from March 9.)

A detail from "Feist Sings 1, 2, 3, 4"—

"Uploaded by SesameStreet on Jul 18, 2008"

Those who prefer, as Weyl put it,
the hard core of objectivity"
to, as Eddington put it,
"the colorful tale of the subjective storyteller mind"
may consult this journal on the same day… July 18, 2008.


Filed under: General — m759 @ 10:48 AM

Dateline WROCLAW


See also "Feist Sings 1, 2, 3, 4."


Filed under: General — m759 @ 9:48 AM

"It's A and Z's Kingdom Come" (after The Man in Black)—


See also midnight's Themes.


Filed under: General — m759 @ 12:00 AM

"'Ain’t No Grave' is Johnny’s final studio recording. The album and its title track
deal heavily with themes of mortality, resurrection, and everlasting life.
The Johnny Cash Project pays tribute to these themes."

And  sells products!

IMAGE- Banner ad with thejohnnycashproject.com run all day Sept. 16, 2011, at NY Times

Click image to enlarge.

I prefer "And I heard, as it were, the noise of thunder."

Friday, September 16, 2011

Lights, Camera…

Filed under: General — m759 @ 7:20 PM

IMAGE- 'Euro Ministers Are Stalled as Geithner Calls for Action'

IMAGE- Black Widow in 'Iron Man 2'

Click on Black Widow for a video.

"I want one."

Art, Jung, Toronto

Filed under: General,Geometry — Tags: — m759 @ 9:29 AM


Related art—

Midnight's Icons,


Faust in Toronto,



McLuhan 'tetrad' figure with four diamonds surrounding a fifth, the medium

McLuhan and the Four Diamonds

(Wikipedia figure)

"Visually, a tetrad can be depicted as
 four diamonds forming an X…."

" X never, ever, marks the spot."


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

Background: Jung's Aion in this journal discusses this
figure from finite geometry's diamond theorem


Fig. A

This resembles a figure that served Jung
as a schema of the Self


Fig. B

Fig. A, with color variations, serves as the core
of many automatically generated Identicons
a different sort of self-symbol.

Examples from Sept. 6 at MathOverflow

http://www.log24.com/log/pix11B/110915-ChuangGravatar.png     http://www.log24.com/log/pix11B/110915-JacobLurieGravatar.png

A user wanting to custom-tailor his self-symbol should consider
the following from the identicon service Gravatar

1. User Submissions.  " you hereby do and shall grant to Automattic a worldwide, perpetual, irrevocable, royalty-free and fully-paid, transferable (including rights to sublicense) right to perform the Services (e.g., to use, modify, reproduce, distribute, prepare derivative works of, display, perform, and otherwise fully exercise and exploit all intellectual property, publicity, and moral rights with respect to any User Submissions, and to allow others to do so)."

Sounds rather Faustian.

Thursday, September 15, 2011

Life Story

Filed under: General — m759 @ 12:00 PM

(Continued from Tuesday's "A Story")

Tuesday's story, a collection of four random posts, was
suggested by Tuesday's NY lottery numbers.

That story leads, by association, to Frame Tale in
a post of 2:02 AM on Sunday, May 23, 2010. For related
material, see Death Story, a post of 9:40 PM that same Sunday.

Wednesday's numbers—


—suggest a counter-story…

Escape to Pine Mountain

A website on films about Latter-Day Saints (i.e., Mormons) asks

Was "Escape to Witch Mountain"
based on Zenna Henderson's "People" stories?

The lottery numbers above suggest the names of three women—
none, as far as I know, with any Mormon background—
who might rightly be called, without capital letters,
"latter-day saints"…

  1. Madeleine L'Engle (see 529 as the date 5/29),
  2. June Christy (a singer first recorded on 5/4/45), and
  3. Mary Rockwell Hook, architect of Pine Mountain
    Settlement School, who died on 9/8/78.

These three lives, taken together, may serve as
an antidote to the Death Story mentioned above.

"We tell ourselves stories…" — Joan Didion
"Therefore choose life." —God

Tuesday, September 13, 2011

A Story

Filed under: General — m759 @ 10:00 PM

For film producer and studio head John Calley, who died today at 81—

"When Death tells a story, you really have to listen." —The Book Thief  (cover)


New York Lottery on Tuesday, September 13th, 2011

This suggests the following random thoughts—


April 24, 2003
A Terrible Beauty


August 26, 2002
Round Lights


April 24, 2005
Today's Sermon


March 24, 2008
Death and the Apple Tree

Day 256

Filed under: General,Geometry — m759 @ 2:56 PM

Today is day 256 of 2011, Programmers' Day.

Yesterday, Monday, R. W. Barraclough's website pictured the Octad of the Week—


" X never, ever, marks the spot."

See also The Galois Tesseract.

Monday, September 12, 2011

Key Paper Now Online

Filed under: General — m759 @ 11:07 PM

The 3-Space PG(3,2) and Its Group

by George M. Conwell, Annals of Mathematics ,
Second Series, Vol. 11, No. 2 (Jan., 1910), pp. 60-76

Article Stable URL: http://www.jstor.org/stable/1967582

At least for now, this paper may be downloaded without
signing in or making a payment. Click the "View PDF" link.

Update of Sept. 13— From Library Journal  on Sept. 7—

The JSTOR journal archive announced today that it is making nearly 500,000 public domain journal articles from more than 220 journals—or about six percent of JSTOR's total content—freely available for use by "anyone, without registration and regardless of institutional affiliation."

The material, entitled Early Journal Content, will be rolled out in batches starting today over the course of one week. It includes content published in the United States before 1923 and international content published before 1870, which ensures that all the content is firmly in the public domain. JSTOR, in an announcement, said that the move was "a first step in a larger effort to provide more access options" to JSTOR content for independent scholars and others unaffiliated with universities.

Spider Flagship

Filed under: General — m759 @ 9:48 PM

Microsoft plans Windows 8 compatibility with mobile devices

"This won't be just another upgrade. Windows 8 is nothing less than the linchpin to Microsoft's strategy for keeping Windows relevant— if not resurgent— as the shift to the post-PC computing era unfolds.

'The stakes are huge,' says Charles King, principal analyst at research firm Pund-IT. 'The company must play outside its comfort zone, but if Microsoft succeeds, the potential opportunities could be significant.'"

Byron Acohido in USA TODAY this evening

Yesterday's 7:20 AM Google News—


From Cliff Robertson's 1958 TV classic "Days of Wine and Roses"—


From Fritz Leiber's 1959 sci-fi classic "Damnation Morning" —

She drew from her handbag a pale grey gleaming implement
that looked by quick turns to me like a knife, a gun,
a slim sceptre, and a delicate branding iron— especially when
its tip sprouted an eight-limbed star of silver wire.

“The test?” I faltered, staring at the thing.

“Yes, to determine whether you can live in the fourth dimension or only die in it.”

Sunday, September 11, 2011

First Lady

Filed under: General — m759 @ 7:59 AM

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


From this  journal on August thirty-first—

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


Hugo Weaving also played Agent Smith
in The Matrix Trilogy .

For Cynthia Zarin, biographer of Madeleine L'Engle

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

Saturday, September 10, 2011

Cliff Robertson

Filed under: General — m759 @ 9:29 PM


Eight is a gate.


NY Lottery Saturday, Sept. 10, 2011

The sound of singing grows louder.

Friday, September 9, 2011

Galois vs. Rubik

(Continued from Abel Prize, August 26)

IMAGE- Elementary Galois Geometry over GF(3)

The situation is rather different when the
underlying Galois field has two rather than
three elements… See Galois Geometry.

Image-- Sugar cube in coffee, from 'Bleu'

The coffee scene from “Bleu”

Related material from this journal:

The Dream of
the Expanded Field

Image-- 4x4 square and 4x4x4 cube

A Beginning

Filed under: General — Tags: , — m759 @ 5:29 AM

From MIT Commencement Day, 2011—

A symbol related to Apollo, to nine, and to "nothing"

A minimalist favicon—

IMAGE- Generic 3x3 square as favicon

This miniature 3×3 square— http://log24.com/log/pix11A/110518-3x3favicon.ico — may, if one likes,
be viewed as the "nothing" present at the Creation. 
See Feb. 19, 2011, and Jim Holt on physics.

Thursday, September 8, 2011


Filed under: General — m759 @ 8:00 PM

IMAGE- American Mathematical Society obituaries for Patrick C. Fischer and Frank Birtel published Sept. 8, 2011

For Fischer's fellow database enthusiast Codd, see Cross-Referenced (April 24, 2003).

For Birtel's fellow pseudoscience enthusiast Frank Tipler, see the artist's comment linked to in Romancing the Omega (April 19, 2011)—

"Omega Point" is a term used by mathematical physicist Frank Tipler for what he maintains is the ultimate fate of the universe required by the laws of physics."

Josefine Lyche, 2009

Starring the Diamond

Filed under: General,Geometry — m759 @ 2:02 PM

"In any geometry satisfying Pappus's Theorem,
the four pairs of opposite points of 83
are joined by four concurrent lines.
— H. S. M. Coxeter (see below)

Continued from Tuesday, Sept. 6

The Diamond Star


The above is a version of a figure from Configurations and Squares.

Yesterday's post related the the Pappus configuration to this figure.

Coxeter, in "Self-Dual Configurations and Regular Graphs," also relates Pappus to the figure.

Some excerpts from Coxeter—


The relabeling uses the 8 superscripts
from the first picture above (plus 0).
The order of the superscripts is from
an 8-cycle in the Galois field GF(9).

The relabeled configuration is used in a discussion of Pappus—


(Update of Sept. 10, 2011—
Coxeter here has a note referring to page 335 of
G. A. Miller, H. F. Blichfeldt, and L. E. Dickson,
Theory and Applications of Finite Groups , New York, 1916.)

Coxeter later uses the the 3×3 array (with center omitted) again to illustrate the Desargues  configuration—


The Desargues configuration is discussed by Gian-Carlo Rota on pp. 145-146 of Indiscrete Thoughts

"The value  of Desargues' theorem and the reason  why the statement of this theorem has survived through the centuries, while other equally striking geometrical theorems have been forgotten, is in the realization that Desargues' theorem opened a horizon of possibilities  that relate geometry and algebra in unexpected ways."

Wednesday, September 7, 2011

The Most Important Configuration

Filed under: General,Geometry — m759 @ 6:30 PM

A search for some background on Gian-Carlo Rota's remarks
in Indiscrete Thoughts * on a geometric configuration
leads to the following passages in Hilbert and Cohn-Vossen's
classic Geometry and the Imagination


These authors describe the Brianchon-Pascal configuration
of 9 points and 9 lines, with 3 points on each line
and 3 lines through each point, as being
"the most important configuration of all geometry."

Thus it seems worthwhile to relate it to the web page
on square configurations referenced here Tuesday.

The Encyclopaedia of Mathematics , ed. by Michiel Hazewinkel,
supplies a summary of the configuration apparently
derived from Hilbert and Cohn-Vossen


My own annotation at right above shows one way to picture the
Brianchon-Pascal points and lines— regarded as those of a finite,
purely combinatorial , configuration— as subsets of the nine-point
square array discussed in Configurations and Squares. The
rearrangement of points in the square yields lines that are in
accord with those in the usual square picture of the 9-point
affine plane.

A more explicit picture—


The Brianchon-Pascal configuration is better known as Pappus's  configuration,
and a search under that name will give an idea of its importance in geometry.

* Birkhäuser Boston, 1998 2nd printing, p. 145


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

An 1978 counterexample to Richard J. Trudeau's
1987 "Story Theory of truth" is now online.

Tuesday, September 6, 2011

Midnight in the Garden (continued)

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

The Diamond Star


See Configurations and Squares.

Monday, September 5, 2011


Filed under: General — m759 @ 11:30 AM

At Heaven’s Gate

Sunday, Sept. 4, 2011, RSS at 23:59 EDT:
Peter Woit's weblog Not Even Wrong

"Lisa Randall’s new book is about to come out, it’s entitled
Knocking on Heaven’s Door: How Physics and Scientific Thinking
Illuminate the Universe and the Modern World

Angels & Demons  (the film)—

As she enters the lab she reacts in horror
as she sees an eyeball lying on the floor…

Click images for some backstory



Woit on Randall

"She has taken on the role of a public face of physics,
 and has written a book which is in part a very general defense
 of science and the materialist, rationalist world-view
 that modern science is based on."

See also yesterday's "The Stone" column in The New York Times

"What Is Naturalism?"

I prefer philosophy enacted by Reba.

* A reference to Dan Brown, not Marianne Williamson

Sunday, September 4, 2011

The Found Numbers

Filed under: General — m759 @ 10:01 PM

(A sequel to The Lost Word)

IMAGE-NY Lottery Evening numbers, Sunday, Sept. 4, 2011: 038 and 1919

NY Lottery, Sunday, Sept. 4, 2011

The number "38" might refer, notably, to

  • The old postal code  of Harvard University,
  • the caliber  of one type of handgun…

… and, less notably, to a page number
     that appears in most books.

T.S. Eliot's essay "Tradition and the Individual Talent"
was first published in 1919.*

For some background on that year, see the Harvard Class  of 1919.

For a notable instance of the page number  38, see
Poetry and Belief in the Work of T.S. Eliot ,
by Kristian Smidt (first published in 1949 by
the Norwegian Academy of Science and Letters).

* "Tradition and the Individual Talent" in The Egoist
   at the Modernist Journals Project:
   Part I in Vol. 6, No. 4 (Sept. 1919),
   Parts II-III in Vol. 6, No. 5 (Dec. 1919)

The Lost Word

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

In memory of Bible translator Eugene Nida, who died on August 25

An excerpt from Log24 on that date

IMAGE- Aschbacher on the 2-local geometry of M24

Related biblical material from DailyGrail.com

"Dan Brown certainly packed a lot into the 500-plus pages of The Lost Symbol . But perhaps the key element to the story is the search for the ‘Lost Word,’ and— in the final pages— Robert Langdon’s discovery as to what that actually means. In the early chapters, Langdon explains to Sato that the Lost Word was 'one of Freemasonry’s most enduring symbols'…

…a single word, written in an arcane language that man could no longer decipher. The Word, like the Mysteries themselves, promised to unveil its hidden power only to those enlightened enough to decrypt it. “It is said,” Langdon concluded, “that if you can possess and understand  the Lost Word . . . then the Ancient Mysteries will become clear to you.”

and from Amazon.com.

Saturday, September 3, 2011

The Galois Tesseract (continued)

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

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

Here is some supporting material—


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

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

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

The affine structure appears in the 1979 abstract mentioned above—

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

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

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

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

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

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

(Received July 20 1987)

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

* For instance:

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

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

Friday, September 2, 2011


Filed under: General — m759 @ 11:07 PM

This afternoon's post

"… If the cliffhanger is done well, it will not
simply introduce a wholly unprepared turn
into the narrative (a random death, a new character,
an entirely unanticipated obstacle) but rather
tighten the configuration of known elements…."

— Michel Chaouli in "How Interactive Can Fiction Be?"

This evening's Los Angeles Times obituary for Gary Hammer—

"He had been called the Indiana Jones of horticulture,
 breaking his leg while hanging from a cliff in Mexico…."

Click to enlarge


For commentary on the first obituary above,
see Odin's Day (Wednesday, August 31).

"An unlikely story? Perhaps."
— Hans Furth, quoted here on
the date of Hammer's death (Sunday, August 7)


Filed under: General,Geometry — m759 @ 1:44 PM

Sarah Tomlin in a Nature  article on the July 12-15 2005 Mykonos meeting on Mathematics and Narrative—

"Today, Mazur says he has woken up to the power of narrative, and in Mykonos gave an example of a 20-year unsolved puzzle in number theory which he described as a cliff-hanger. 'I don’t think I personally understood the problem until I expressed it in narrative terms,' Mazur told the meeting. He argues that similar narrative devices may be especially helpful to young mathematicians…."

Michel Chaouli in "How Interactive Can Fiction Be?" (Critical Inquiry  31, Spring 2005), pages 613-614—

"…a simple thought experiment….*

… If the cliffhanger is done well, it will not simply introduce a wholly unprepared turn into the narrative (a random death, a new character, an entirely unanticipated obstacle) but rather tighten the configuration of known elements to such a degree that the next step appears both inevitable and impossible. We feel the pressure rising to a breaking point, but we simply cannot foresee where the complex narrative structure will give way. This interplay of necessity and contingency produces our anxious— and highly pleasurable— speculation about the future path of the story. But if we could determine that path even slightly, we would narrow the range of possible outcomes and thus the uncertainty in the play of necessity and contingency. The world of the fiction would feel, not open, but rigged."

* The idea of the thought experiment emerged in a conversation with Barry Mazur.

Barry Mazur in the preface to his 2003 book Imagining Numbers

"But the telltale adjective real  suggests two things: that these numbers are somehow real to us and that, in contrast, there are unreal  numbers in the offing. These are the imaginary numbers

The imaginary  numbers are well named, for there is some imaginative work to do to make them as much a part of us as the real numbers we use all the time to measure for bookshelves. 

This book began as a letter to my friend Michel Chaouli. The two of us had been musing about whether or not one could 'feel' the workings of the imagination in its various labors. Michel had also mentioned that he wanted to 'imagine imaginary numbers.' That very (rainy) evening, I tried to work up an explanation of the idea of these numbers, still in the mood of our conversation."

See also The Galois Quaternion and 2/19.

IMAGE- NY Lottery evening numbers Thursday, Sept. 1, 2011 were 144 and 0219

New York Lottery last evening

Mathematics and Narrative (continued)

Filed under: General — m759 @ 11:00 AM

IMAGE- Death on 10/10/10

See also this journal on October 10, 2010.

Thursday, September 1, 2011

The Galois Tesseract

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

Click to enlarge

IMAGE- The Galois Tesseract, 1979-1999

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

For Thor’s Day

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

If I Had a Hammer
(Eine Alte Geschichte )

How It Works

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

“Design is how it works.” — Steven Jobs (See Symmetry and Design.)

“By far the most important structure in design theory is the Steiner system S(5, 8, 24).”
— “Block Designs,” by Andries E. Brouwer

IMAGE- Harvard senior thesis on Mathieu groups, 2010, and supporting material from book 'Design Theory'

The name Carmichael is not to be found in Booher’s thesis.  A book he does  cite for the history of S(5,8,24) gives the date of Carmichael’s construction of this design as 1937.  It should  be dated 1931, as the following quotation shows—

From Log24 on Feb. 20, 2010

“The linear fractional group modulo 23 of order 24•23•11 is often represented as a doubly transitive group of degree 24 on the symbols ∞, 0, 1, 2,…, 22. This transitive group contains a subgroup of order 8 each element of which transforms into itself the set ∞, 0, 1, 3, 12, 15, 21, 22 of eight elements, while the whole group transforms this set into 3•23•11 sets of eight each. This configuration of octuples has the remarkable property that any given set of five of the 24 symbols occurs in one and just one of these octuples. The largest permutation group Γ on the 24 symbols, each element of which leaves this configuration invariant, is a five-fold transitive group of degree 24 and order 24•23•22•21•20•48. This is the Mathieu group of degree 24.”

– R. D. Carmichael, “Tactical Configurations of Rank Two,” in American Journal of Mathematics, Vol. 53, No. 1 (Jan., 1931), pp. 217-240

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

Es is eine alte Geschichte,
doch bleibt sie immer neu

—Heine (Lyrisches Intermezzo  XXXIX)

See also “Do you like apples?

Boundary (continued*)

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

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

From a weblog post on Captain America and Thor

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

That's Entertainment  weblog, August 14, 2011

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

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

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

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

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

* Continued from August 30.

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