Log24

Friday, September 9, 2016

There IS such a thing

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

http://gregegan.customer.netspace.net.au/APPLETS/29/NonSimple4E.gif

See also Dueling Formulas,  Sinner or Saint?,  and The Zero Obit.

Tuesday, October 15, 2019

Inside the Fire Temple

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

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

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

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

From posts tagged Tetrahedron vs. Square

This  journal on October 3

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

Illustration (central detail   from the above tetrahedral figure) —

Thursday, October 3, 2019

The Overbye Metaphors

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

(For Harlan Kane)

"Once Mr. Overbye identifies a story, he said, the work is
in putting it in terms people can understand. 'Metaphors
are very important to the way I write,' he said. The results
are vivid descriptions that surpass mere translation."

— Raillan Brooks in The New York Times  on a Times
science writer, October 17, 2017.  Also on that date —

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

See as well The Black List (Log24, September 27).

Wednesday, September 18, 2019

The Perpetual Identity Crisis

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

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

Midrash — An image posted here on August 6

Sunday, August 25, 2019

Design Theory

Filed under: General — Tags: , , — m759 @ 7:58 PM

"Mein Führer Steiner"

See Hitler Plans and Quadruple System.

"There is  such a thing as a quadruple system."

— Saying adapted from a 1962 young-adult novel

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, August 3, 2019

Crosswicks Revisited

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

"There is  such a thing as a four-set."

Tuesday, April 9, 2019

Zero Dark Nine:

Filed under: General — Tags: — m759 @ 12:09 AM

The Crosswicks Curse Continues . . .

"There is  such a thing as geometry."

— Saying adapted from a 1962 young-adult novel.

Friday, February 15, 2019

The Gifts Reserved for Age

Filed under: General — m759 @ 3:30 PM

"But, as the passage now presents no hindrance
     To the spirit unappeased and peregrine
     Between two worlds become much like each other…."

T. S. Eliot, Four Quartets

Related obituary:

https://www.nytimes.com/2019/02/15/obituaries/tom-cade-dead.html

Related date:

"as of Feb. 6, 2019" (from a post at 12 AM ET Feb. 7) —

"There is  such a thing as a four-dimensional finite affine space."
— Saying adapted from a 1962 young-adult novel by Madeleine L'Engle

Monday, August 7, 2017

Pathbreaking

Filed under: General — m759 @ 7:20 PM

From Blockbuster, a post of Friday, August 4, 2017 —

The article suggests a look at  a July 3 Times  review of the life
of Jan Fontein, a former Boston Museum of Fine Arts director —

"Mr. Fontein’s time as director coincided with
the nationwide rise of the blockbuster exhibition,
and he embraced the concept. 'There was such a thing
as a contemplative museum, but I don’t think that can
survive anymore,' he told Newsweek  in 1978."

From The New York Times  this evening —

"Mr. Roth made his mark at the Victoria and Albert
with record-breaking exhibitions focused on
David Bowie in 2013, Alexander McQueen in 2015
and The Beatles and the youth revolution of the 1960s
in 2016."

Related material —

Record-breaking in this journal and Sunday in the Park with Death.

Friday, August 4, 2017

Blockbuster

Filed under: General — m759 @ 12:48 PM

This post was suggested by a  New York Times  article online today
about an upcoming exhibition at Boston's Museum of Fine Arts —

"A version of this article appears in print on August 6, 2017,
on Page AR2 of the New York edition with the headline:

Art;  Woodblock Smackdown!."

The article suggests a look at  a July 3 Times  review of the life of
Jan Fontein, a former Boston Museum of Fine Arts director —

"Mr. Fontein’s time as director coincided with
the nationwide rise of the blockbuster exhibition,
and he embraced the concept. 'There was such a thing
as a contemplative museum, but I don’t think that can
survive anymore,' he told Newsweek  in 1978."

Fontein died at 89 on May 19, 2017. See Dharmadhatu — a Log24 post
of July 4, 2017 — and its link to posts tagged May 19 Gestalt.

Sunday, June 4, 2017

In Memory of the Time Cube Page*

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

From this journal on August 18, 2015, "A Wrinkle in Terms" —

For two misuses by John Baez of the phrase “permutation group”
at the n-Category Café, see “A Wrinkle in the Mathematical Universe
and “Re: A Wrinkle…” —

“There is  such a thing as a permutation group.”
— Adapted from A Wrinkle in Time , by Madeleine L’Engle

* See RIP, Time Cube at gizmodo.com (September 1, 2015).

Wednesday, October 19, 2016

The Crosswicks Curse Continues

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

"There is  such a thing as 1906 "

Monday, October 17, 2016

A Wrinkle in Space

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

"There is  such a thing as a counting-pattern."

— Saying adapted from a young-adult novel

See also the previous post and

Saturday, June 18, 2016

Midnight in Herald Square

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

In memory of New Yorker  artist Anatol Kovarsky,
who reportedly died at 97 on June 1.

Note the Santa, a figure associated with Macy's at Herald Square.

See also posts tagged Herald Square, as well as the following
figure from this journal on the day preceding Kovarsky's death.

A note related both to Galois space and to
the "Herald Square"-tagged posts —

"There is  such a thing as a length-16 sequence."
— Saying adapted from a young-adult novel.

Friday, April 8, 2016

Space Cross

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

For George Orwell

Illustration from a book on mathematics —

This illustrates the Galois space  AG(4,2).

For some related spaces, see a note from 1984.

"There is  such a thing as a space cross."
— Saying adapted from a young-adult novel

Thursday, March 24, 2016

The Nervous Set*

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

The previous post suggests a review of the saying
"There is  such a thing as a 4-set."

* Title of a 1959 musical

Friday, January 29, 2016

Excellent Adventure*

Filed under: General — Tags: — m759 @ 9:29 PM

(Continued from Dec. 9, 2013)

"…it would be quite a long walk
for him if he had to walk straight across."

The image “http://www.log24.com/log/pix07A/070831-Ant1.gif” cannot be displayed, because it contains errors.

Swiftly Mrs. Who brought her hands… together.

"Now, you see," Mrs. Whatsit said,
"he would be  there, without that long trip.
That is how we travel."

The image “http://www.log24.com/log/pix07A/070831-Ant2.gif” cannot be displayed, because it contains errors.

– A Wrinkle in Time 
Chapter 5, "The Tesseract"

From a media weblog yesterday, a quote from the video below —

"At 12:03 PM Eastern Standard Time, January 12th, 2016…."

This  weblog on the previous day (January 11th, 2016) —

"There is  such a thing as harmonic analysis of switching functions."

— Saying adapted from a young-adult novel

* For some backstory, see a Caltech page.

Monday, January 11, 2016

Space Oddity

Filed under: General,Geometry — Tags: — m759 @ 3:15 PM

It is an odd fact that the close relationship between some
small Galois spaces and small Boolean spaces has gone 
unremarked by mathematicians.  

A Google search today for "Galois spaces" + "Boolean spaces"
yielded, apart from merely terminological sources, only some
introductory material I have put on the Web myself. 

Some more sophisticated searches, however led to a few
documents from the years 1971 – 1981 …

"Harmonic Analysis of Switching Functions" ,
by Robert J. Lechner, Ch. 5 in A. Mukhopadhyay, editor,
Recent Developments in Switching Theory , Academic Press, 1971.

"Galois Switching Functions and Their Applications,"
by B. Benjauthrit and I. S. Reed,
JPL Deep Space Network Progress Report 42-27 , 1975

D.K. Pradhan, “A Theory of Galois Switching Functions,”
IEEE Trans. Computers , vol. 27, no. 3, pp. 239-249, Mar. 1978

"Switching functions constructed by Galois extension fields,"
by Iwaro Takahashi, Information and Control ,
Volume 48, Issue 2, pp. 95–108, February 1981

An illustration from the Lechner paper above —

"There is  such a thing as harmonic analysis of switching functions."

— Saying adapted from a young-adult novel

Tuesday, August 18, 2015

A Wrinkle in Terms

Filed under: General,Geometry — m759 @ 8:23 AM

The phrase “the permutation group Sn” refers to a
particular  group of permutations that act on an
-element set N— namely, all  of them. For a given n ,
there are, in general, many  permutation groups that
act on N.  All but one are smaller than S.

In other words, the phrase “the permutation group Sn
does not  imply that “Sn ” is a symbol for a structure
associated with n  called “the  permutation group.”
It is instead a symbol for “the symmetric  group,” the largest
of (in general) many permutation groups that act on N.

This point seems to have escaped John Baez.

For two misuses by Baez of the phrase “permutation group” at the
n-Category Café, see “A Wrinkle in the Mathematical Universe”
and “Re: A Wrinkle…” —

“There is  such a thing as a permutation group.”
— Adapted from A Wrinkle in Time , by Madeleine L’Engle

Saturday, June 27, 2015

A Single Finite Structure

Filed under: General,Geometry — m759 @ 12:15 PM

"It is as if one were to condense
all trends of present day mathematics
onto a single finite structure…."

— Gian-Carlo Rota, foreword to
A Source Book in Matroid Theory ,
Joseph P.S. Kung, Birkhäuser, 1986

"There is  such a thing as a matroid."

— Saying adapted from a novel by Madeleine L'Engle

Related remarks from Mathematics Magazine  in 2009 —

See also the eightfold cube —

The Eightfold Cube

 .

Friday, June 19, 2015

Footnote

Filed under: General,Geometry — m759 @ 9:29 PM

There is  such a thing as geometry.*

* Proposition adapted from A Wrinkle in Time , by Madeleine L'Engle.

Friday, January 9, 2015

Fourth Right

Filed under: General — Tags: — m759 @ 1:00 AM

In memory of Rod Taylor, who
reportedly died at 84 on Wednesday,
the seventh day of 2015 —

And there is  such a thing as a 4-set.

Thursday, December 4, 2014

The Pony Argument

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

The title was suggested by this morning's post "Follow This."

From a previous incarnation of my home website, m759.com —
The second of the site's three pages mentions authors
Alfred Bester and Zenna Henderson :

"Bester and Henderson are particularly good at
fictional accounts of telepathy. The noted Harvard
philosopher W. V. Quine doubts such a thing exists,
but I prefer the 'There must be a pony' argument."

Related material: The date Nov. 27, 2014, in a web search today 


 in The Washington Post 

and in this  journal

Friday, November 28, 2014

The Soul of Stanford

Filed under: General — m759 @ 6:00 PM

A search for background on the academic
author cited in the previous post yields…

"The debate is, in the words of one professor,
'a struggle for the intellectual soul of Stanford.'"

Some may doubt there is such a thing.
See Marissa Mayer in this journal…

and in Vogue  (a story dated August 16, 2013)—

IMAGE- Marissa Mayer on numbers in Vogue magazine

Friday, November 7, 2014

The Crosswicks Curse…

Filed under: General — m759 @ 7:00 AM

Continues.

There is  such a thing as an MBTI Tesseract.

See a thread at http://www.typologycentral.com/forums/
from August 17 and 18, 2010.

See also this journal on those dates: The Kermode Game.

Monday, September 1, 2014

Mathematics, Not Theology

Filed under: General — m759 @ 5:00 PM

(Continued)

“A set having three members is a single thing
wholly constituted by its members but distinct from them.
After this, the theological doctrine of the Trinity as
‘three in one’ should be child’s play.”

— Max Black, Caveats and Critiques: Philosophical Essays
in Language, Logic, and Art
 , Cornell U. Press, 1975

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

“There is  such a thing as a three-set.”
— Saying adapted from a novel by Madeleine L’Engle

Monday, August 4, 2014

A Wrinkle in Space

Filed under: General,Geometry — Tags: , — m759 @ 10:30 AM

"There is  such a thing as a tesseract." — Madeleine L'Engle

An approach via the Omega Matrix:

http://www.log24.com/log/pix10A/100619-TesseractAnd4x4.gif

See, too, Rosenhain and Göpel as The Shadow Guests .

Thursday, May 8, 2014

Wrinkles in Time

Filed under: General — Tags: — m759 @ 2:25 PM

Rivka Galchen, in a piece mentioned here in June 2010

On Borges:  Imagining the Unwritten Book 

"Think of it this way: there is a vast unwritten book that the heart reacts to, that it races and skips in response to, that it believes in. But it’s the heart’s belief in that vast unwritten book that brought the book into existence; what appears to be exclusively a response (the heart responding to the book) is, in fact, also a conjuring (the heart inventing the book to which it so desperately wishes to respond)."

Related fictions

Galchen's "The Region of Unlikeness" (New Yorker , March 24, 2008)

Ted Chiang's "Story of Your Life." A film adaptation is to star Amy Adams.

… and non-fiction

"There is  such a thing as a 4-set." — January 31, 2012

Friday, March 14, 2014

The Search for Charles Wallace

Filed under: General — Tags: , — m759 @ 2:19 PM

The search in the previous post for the source of a quotation from Poincaré yielded, as a serendipitous benefit, information on an interesting psychoanalyst named Wilfred Bion (see the Poincaré  quotation at a webpage on Bion). This in turn suggested a search for the source of the name of author Madeleine L'Engle's son Bion, who may have partly inspired L'Engle's fictional character Charles Wallace.  Cynthia Zarin wrote about Bion in The New Yorker  of April 12, 2004 that

"According to the family, he is the person for whom L’Engle’s insistence on blurring fiction and reality had the most disastrous consequences."

Also from that article, material related to the name Bion and to what this journal has called "the Crosswicks Curse"*—

"Madeleine L’Engle Camp was born in 1918 in New York City, the only child of Madeleine Hall Barnett, of Jacksonville, Florida, and Charles Wadsworth Camp, a Princeton man and First World War veteran, whose family had a big country place in New Jersey, called Crosswicks. In Jacksonville society, the Barnett family was legendary: Madeleine’s grandfather, Bion Barnett, the chairman of the board of Jacksonville’s Barnett Bank, had run off with a woman to the South of France, leaving behind a note on the mantel. Her grandmother, Caroline Hallows L’Engle, never recovered from the blow. ….

… The summer after Hugh and Madeleine were married, they bought a dilapidated farmhouse in Goshen, in northwest Connecticut. Josephine, born in 1947, was three years old when they moved permanently to the house, which they called Crosswicks. Bion was born just over a year later."

* "There is  such a thing as a tesseract."

Thursday, February 27, 2014

The Crosswicks Curse

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

(Continued)

"There is  such a thing as a tesseract."

— Saying from Crosswicks

IMAGE- From Dmitri Tymoczko's 'Geometry of Music,' Chopin and a tesseract

See also March 5, 2011.

Adapted from the above passage —

"So did L'Engle understand four-dimensional geometry?"

No and Yes.

Tuesday, July 16, 2013

Space Itself

Filed under: General,Geometry — Tags: — m759 @ 10:18 AM

"How do you get young people excited
about space? How do you get them interested
not just in watching movies about space,
or in playing video games set in space
but in space itself?"

Megan Garber in The AtlanticAug. 16, 2012

One approach:

"There is  such a thing as a tesseract" and
Diamond Theory in 1937.

See, too, Baez in this journal.

Tuesday, June 4, 2013

Cover Acts

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

The Daily Princetonian  today:

IMAGE- 'How Jay White, a Neil Diamond cover act, duped Princeton'

A different cover act, discussed here  Saturday:

IMAGE- The diamond theorem affine group of order 322,560, published without acknowledgment of its source by the Mathematical Association of America in 2011

See also, in this journal, the Galois tesseract and the Crosswicks Curse.

"There is  such a thing as a tesseract." — Crosswicks saying

Wednesday, May 1, 2013

The Crosswicks Curse

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

(Continued)

"There is  such a thing as a tesseract." —A novel from Crosswicks

Related material from a 1905 graduate of Princeton,
"The 3-Space PG(3,2) and Its Group," is now available
at Internet Archive (1 download thus far).

The 3-space paper is relevant because of the
connection of the group it describes to the
"super, overarching" group of the tesseract.

Sunday, April 21, 2013

Sermon

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

"There is  such a thing as a figure in four dimensions."

Adapted from a novel

Saturday, April 13, 2013

Princeton’s Christopher Robin

The title is that of a talk (see video) given by
George Dyson at a Princeton land preservation trust,
reportedly on March 21, 2013.  The talk's subtitle was
"Oswald Veblen and the Six-hundred-acre Woods."

Meanwhile

Thursday, March 21, 2013

Geometry of Göpel Tetrads (continued)

m759 @ 7:00 PM

An update to Rosenhain and Göpel Tetrads in PG(3,2)
supplies some background from
Notes on Groups and Geometry, 1978-1986,
and from a 2002 AMS Transactions  paper.

IMAGE- Göpel tetrads in an inscape, April 1986

Related material for those who prefer narrative
to mathematics:

Log24 on June 6, 2006:

 

The Omen:


Now we are 
 

6!

Related material for those who prefer mathematics
to narrative:

What the Omen narrative above and the mathematics of Veblen
have in common is the number 6. Veblen, who came to
Princeton in 1905 and later helped establish the Institute,
wrote extensively on projective geometry.  As the British
geometer H. F. Baker pointed out,  6 is a rather important number
in that discipline.  For the connection of 6 to the Göpel tetrads
figure above from March 21, see a note from May 1986.

See also last night's Veblen and Young in Light of Galois.

"There is  such a thing as a tesseract." — Madeleine L'Engle

Saturday, March 16, 2013

The Crosswicks Curse

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

Continues.

From the prologue to the new Joyce Carol Oates
novel Accursed

"This journey I undertake with such anticipation
is not one of geographical space but one of Time—
for it is the year 1905 that is my destination.

1905!—the very year of the Curse."

Today's previous post supplied a fanciful link
between the Crosswicks Curse of Oates and
the Crosswicks tesseract  of Madeleine L'Engle.

The Crosswicks Curse according to L'Engle
in her classic 1962 novel A Wrinkle in Time —

"There is  such a thing as a tesseract."

A tesseract is a 4-dimensional hypercube that
(as pointed out by Coxeter in 1950) may also 
be viewed as a 4×4 array (with opposite edges
identified).

Meanwhile, back in 1905

For more details, see how the Rosenhain and Göpel tetrads occur naturally
in the diamond theorem model of the 35 lines of the 15-point projective
Galois space PG(3,2).

See also Conwell in this journal and George Macfeely Conwell in the
honors list of the Princeton Class of 1905.

Puzzles

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

For readers of The Daily Princetonian :

IMAGE- 4x4 array in 'Ancient Jewels' puzzle

(From a site advertised in the
Princetonian  on March 11, 2013)

For readers of The Harvard Crimson :

IMAGE- Harvard Crimson ad, Easter Sunday, 2008: 'Finite projective geometry as a graphic grammar of abstract design'

For some background, see Crimson Easter Egg and the Diamond 16 Puzzle.

For some (very loosely) related narrative, see Crosswicks in this journal
and the Crosswicks Curse  in a new novel by Joyce Carol Oates.

"There is  such a thing as a tesseract."
— Crosswicks author Madeleine L'Engle

Thursday, February 21, 2013

Galois Space

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

(Continued)

The previous post suggests two sayings:

"There is  such a thing as a Galois space."

— Adapted from Madeleine L'Engle

"For every kind of vampire, there is a kind of cross."

Thomas Pynchon

Illustrations—

(Click to enlarge.)

Saturday, July 7, 2012

Étude

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

 IMAGE- Google Books ad for 'Geometric Etudes in Combinatorial Mathematics,' by Alexander Soifer

IMAGE- Triangle cut into four congruent subtriangles

For remarks related by logic, see the square-triangle theorem.

For remarks related by synchronicity, see Log24 on
the above publication date,  June 15, 2010.

According to Google (and Soifer's page xix), Soifer wants to captivate
young readers.

Whether young readers should  be captivated is open to question.

"There is  such a thing as a 4-set."

Update of 9:48 the same morning—

Amazon.com says Soifer's book was published not on June 15, but on
 June 29 , 2010
(St. Peter's Day).

Tuesday, February 28, 2012

Page

Filed under: General — m759 @ 1:06 PM

A followup to this morning's post Stolen Glory— 

Columbia's Butler Library "plays a role in
Paul Auster's 2009 novel Invisible ,
where the novel's main protagonist, Adam Walker,
takes a job as a 'page' in the library's stacks." —Wikipedia

Part I (from Feb. 24)—

IMAGE- 'Wild nights are my glory'- Peter Woit, Mrs. Whatsit, and the tesseract

Part II— (Click to enlarge)

IMAGE- Page quoting Mrs. Whatsit: 'There IS such a thing as a tesseract.'

For the page's source, see Butler Library.

Tuesday, January 31, 2012

Tesseract

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

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

Tesseract formed from a 4-set—

IMAGE- Tesseract.


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

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

Update of August 12, 2012:

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

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."

Thursday, November 3, 2011

Ockham’s Bubbles–

Filed under: General,Geometry — m759 @ 10:30 AM

Mathematics and Narrative, continued

"… a vision invisible, even ineffable, as ineffable as the Angels and the Universal Souls"

— Tom Wolfe, The Painted Word , 1975, quoted here on October 30th

"… our laughable abstractions, our wryly ironic po-mo angels dancing on the heads of so many mis-imagined quantum pins."

— Dan Conover on September 1st, 2011

"Recently I happened to be talking to a prominent California geologist, and she told me: 'When I first went into geology, we all thought that in science you create a solid layer of findings, through experiment and careful investigation, and then you add a second layer, like a second layer of bricks, all very carefully, and so on. Occasionally some adventurous scientist stacks the bricks up in towers, and these towers turn out to be insubstantial and they get torn down, and you proceed again with the careful layers. But we now realize that the very first layers aren't even resting on solid ground. They are balanced on bubbles, on concepts that are full of air, and those bubbles are being burst today, one after the other.'

I suddenly had a picture of the entire astonishing edifice collapsing and modern man plunging headlong back into the primordial ooze. He's floundering, sloshing about, gulping for air, frantically treading ooze, when he feels something huge and smooth swim beneath him and boost him up, like some almighty dolphin. He can't see it, but he's much impressed. He names it God."

— Tom Wolfe, "Sorry, but Your Soul Just Died," Forbes , 1996

"… Ockham's idea implies that we probably have the ability to do something now such that if we were to do it, then the past would have been different…"

Stanford Encyclopedia of Philosophy

"Today is February 28, 2008, and we are privileged to begin a conversation with Mr. Tom Wolfe."

— Interviewer for the National Association of Scholars

From that conversation—

Wolfe : "People in academia should start insisting on objective scholarship, insisting  on it, relentlessly, driving the point home, ramming it down the gullets of the politically correct, making noise! naming names! citing egregious examples! showing contempt to the brink of brutality!"

As for "mis-imagined quantum pins"…
This 
journal on the date of the above interview— February 28, 2008

http://www.log24.com/log/pix08/080228-Wooters2.jpg

Illustration from a Perimeter Institute talk given on July 20, 2005

The date of Conover's "quantum pins" remark above (together with Ockham's remark above and the above image) suggests a story by  Conover, "The Last Epiphany," and four posts from September 1st, 2011—

BoundaryHow It WorksFor Thor's Day,  and The Galois Tesseract.

Those four posts may be viewed as either an exploration or a parody of the boundary between mathematics and narrative.

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

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.

http://www.log24.com/log/pix11B/110911-NYTobitsSm.jpg

From this  journal on August thirty-first—

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

http://www.log24.com/log/pix11B/110831-JohannSchmidt.jpg

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

Thursday, September 1, 2011

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.

Sunday, February 20, 2011

Brightness at Noon (continued)

Filed under: General — m759 @ 12:00 PM

IMAGE- 'There IS such a thing as a tesseract.'

Monday, November 1, 2010

Seasons of…

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

A Year of Magical Thinking

In memory of Theodore Chaikin Sorensen, who died at noon in New York on Halloween —

Two posts from All Saints' Day, 2009 —

October Endgame and Indignation and Laughter in Toronto.

Related material: New York Lottery on All Hallows' Eve this  year —

Midday 896,  Evening 384.

"Man is a system that transforms itself." (Paul Valéry, Cahiers , Vol. 2, page 896)

"There is  such a thing as a tesseract." (Madeleine L'Engle. See 384 on Halloween 2006.)

Monday, June 21, 2010

Test

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

From a post by Ivars Peterson, Director
of Publications and Communications at
the Mathematical Association of America,
at 19:19 UTC on June 19, 2010—

Exterior panels and detail of panel,
Michener Gallery at Blanton Museum
in Austin, Texas—

http://www.log24.com/log/pix10A/100621-MichenerGalleryPanel.jpg

Peterson associates the four-diamond figure
with the Pythagorean theorem.

A more relevant association is the
four-diamond view of a tesseract shown here
on June 19 (the same date as Peterson's post)
in the "Imago Creationis" post—

Image-- The Four-Diamond Tesseract

This figure is relevant because of a
tesseract sculpture by Peter Forakis—

http://www.log24.com/log/pix09A/091220-ForakisHypercube.jpg

This sculpture was apparently shown in the above
building— the Blanton Museum's Michener gallery—
as part of the "Reimagining Space" exhibition,
September 28, 2008-January 18, 2009.

The exhibition was organized by
Linda Dalrymple Henderson, Centennial Professor
in Art History at the University of Texas at Austin
and author of The Fourth Dimension and
Non-Euclidean Geometry in Modern Art
(Princeton University Press, 1983;
new ed., MIT Press, 2009).

For the sculptor Forakis in this journal,
see "The Test" (December 20, 2009).

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

Thursday, May 6, 2010

Mathematics and Narrative, continued…

Filed under: General — m759 @ 5:01 AM

The Unfolding

A post for Florencio Campomanes,
former president of the World Chess Federation.

Campomanes died at 83 in the Philippines
at 1:30 PM local time (1:30 AM Manhattan time)
on Monday, May 3, 2010.

From this journal on the date of his death —

"There is such a thing as a tesseract."
Madeleine L'Engle

Image by Christopher Thomas at Wikipedia
Unfolding of a hypercube and of a cube —

http://www.log24.com/log/pix10A/100506-Hcube_fold.gif

Image--Chess game from 'The Seventh Seal'

Related material from a story of the Philippines —

Image-- Alex Garland on how a hypercube unfolds to what he calls a tesseract

Monday, May 3, 2010

An Ordinary Evening

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

“…geometrically organized, with the parts labeled”

— Ursula K. Le Guin on what she calls “the Euclidean utopia

“There is such a thing as a tesseract.”

Madeleine L’Engle

Related material– Diamond Theory, 1937

Sunday, December 20, 2009

The Test

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

Dies Natalis of
Emil Artin

From the September 1953 Bulletin of the American Mathematical Society

Emil Artin, in a review of Éléments de mathématique, by N. Bourbaki, Book II, Algebra, Chaps. I-VII–

"We all believe that mathematics is an art. The author of a book, the lecturer in a classroom tries to convey the structural beauty of mathematics to his readers, to his listeners. In this attempt he must always fail. Mathematics is logical to be sure; each conclusion is drawn from previously derived statements. Yet the whole of it, the real piece of art, is not linear; worse than that its perception should be instantaneous. We all have experienced on some rare occasions the feeling of elation in realizing that we have enabled our listeners to see at a moment's glance the whole architecture and all its ramifications. How can this be achieved? Clinging stubbornly to the logical sequence inhibits the visualization of the whole, and yet this logical structure must predominate or chaos would result."

Art Versus Chaos

http://www.log24.com/log/pix09A/091220-ForakisHypercube.jpg
From an exhibit,
"Reimagining Space
"

The above tesseract (4-D hypercube)
sculpted in 1967 by Peter Forakis
provides an example of what Artin
called "the visualization of the whole."

For related mathematical details see
Diamond Theory in 1937.

"'The test?' I faltered, staring at the thing.
'Yes, to determine whether you can live
in the fourth dimension or only die in it.'"
Fritz Leiber, 1959

See also the Log24 entry for
Nov. 26,  2009, the date that
Forakis died.

"There is such a thing
as a tesseract."
Madeleine L'Engle, 1962

Monday, January 26, 2009

Monday January 26, 2009

Filed under: General — m759 @ 6:00 AM
Episode One

For the Hole in the Wall Gang:

http://www.log24.com/log/pix09/090126-Map.jpg

Shopkeeper: Good morning, sir. And what can I do for you then?
Prisoner: I’d like a map of this area.
Shopkeeper: Map? Colour or black and white?
Prisoner: Just a map.
Shopkeeper: Map…

He pauses to remember where he keeps such a thing.

Shopkeeper: Ah. Black and white…

He produces a map from a cupboard.

Shopkeeper: There we are, sir. I think you’ll find that shows everything.

The map is labelled “map of your village.” The Prisoner opens it; it shows the village bordered by “the mountains”: there are no external geographical names.

Prisoner: I… I meant a larger map.
Shopkeeper: Only in colour, sir. Much more expensive.
Prisoner: That’s fine.

The shopkeeper fetches him a colour map as inadequate as the last. It folds out as a larger sheet of paper, but still mentions only “the mountains,” “the sea,” and “the beach,” together with the title “your village.”

Prisoner: Er, that’s not what I meant. I meant a… a larger area.
Shopkeeper: No, we only have local maps, sir. There’s no demand for any others. You’re new here, aren’t you?

— Comment at 
The Word magazine,
January 16, 2009

Comment by m759,
January 16, 2009:

“In the pictures of the old masters, Max Picard wrote in The World of Silence, people seem as though they had just come out of the opening in a wall… “

— Annie Dillard in
For the Time Being

“Shopkeeper:
Only in colour, sir.
Much more expensive.

Prisoner:
That’s fine.”

Thursday, July 31, 2008

Thursday July 31, 2008

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

“Put bluntly, who is kidding whom?”

Anthony Judge, draft of
“Potential Psychosocial Significance
of Monstrous Moonshine:
An Exceptional Form of Symmetry
as a Rosetta Stone for
Cognitive Frameworks,”
dated September 6, 2007.

Good question.

Also from
September 6, 2007 —
the date of
Madeleine L’Engle‘s death —

 
Pavarotti takes a bow
Related material:

1. The performance of a work by
Richard Strauss,
Death and Transfiguration,”
(Tod und Verklärung, Opus 24)
by the Chautauqua Symphony
at Chautauqua Institution on
July 24, 2008

2. Headline of a music review
in today’s New York Times:

Welcoming a Fresh Season of
Transformation and Death

3. The picture of the R. T. Curtis
Miracle Octad Generator
on the cover of the book
Twelve Sporadic Groups:

Cover of 'Twelve Sporadic Groups'

4. Freeman Dyson’s hope, quoted by
Gorenstein in 1986, Ronan in 2006,
and Judge in 2007, that the Monster
group is “built in some way into
the structure of the universe.”

5. Symmetry from Plato to
the Four-Color Conjecture

6. Geometry of the 4×4 Square

7. Yesterday’s entry,
Theories of Everything

Coda:

There is such a thing

Tesseract
     as a tesseract.

— Madeleine L’Engle

Cover of The New Yorker, April 12, 2004-- Roz Chast, Easter Eggs

For a profile of
L’Engle, click on
the Easter eggs.

Tuesday, May 20, 2008

Tuesday May 20, 2008

Filed under: General — m759 @ 1:06 PM
The Unembarrassed Peddler

(For readers of
the previous entry
who would like to
know more about
purchasing the
Brooklyn Bridge)


From yesterday’s New York Times, in an obituary of a teacher of reporters:

“He was a stickler for spelling, insisting that students accurately compose dictated sentences, like this one: ‘Outside a cemetery sat a harassed cobbler and an embarrassed peddler, gnawing on a desiccated potato and gazing on the symmetry of a lady’s ankle with unparalleled ecstasy.'”

Spelling Your Way
To Success

Chapter I:
“gnawing on a  
  desiccated potato”

From the website
Blue Star Traders:
How the ancient crystal skull Synergy came to the Western World…

This skull first came to light when it was acquired about two and a half decades ago by a European businessman and avid hiker, as he traveled around Central and South America.  He acquired the skull from a very old native man, in a tiny village in the Andes, near the borders of Peru, Bolivia and Chile. He was just passing through, and had come upon the small settlement while looking for a place to stay for the night.  He wandered into the village and was greeted with smiles and an invitation to share a meal.

This gentleman, George, speaks several languages, and he usually has at least a few words in common with most of the people he meets in his travels– enough to get by, anyway.  Although he didn’t speak the same language as most of the people in this isolated village, there was an instant connection between them, and they managed with the smattering of Spanish and Portuguese that a few of them knew. In need of shelter for the night, George was offered a spot for his sleeping bag, near the fire, in the dwelling of an elderly man.

After a peaceful evening in the old man’s company, George gratefully accepted a simple breakfast and got ready to take his leave.  As he thanked the man for his generous hospitality, the elder led George to an old chest. Opening the crumbling wooden lid, he took out the crystal skull, touched it reverently, and handed it to George.  Awed by an artifact of such obvious antiquity, beauty and value, yet uncertain what he was expected to do with it, George tried to hand it back.  But the old man urged it upon him, making it clear that he was to take it with him. 

Curious about the history of such a thing, George tried to find out what the villagers knew about it. One young fellow explained in halting Spanish that  the skull had come into the possession of a much loved Catholic nun, in Peru.  She was quite old when she died in the early 1800’s, and she had given it to the old man’s “Grandfather” when he was just a boy.  (Note: It’s hard to say if this was really the man’s grandfather, or just the honorary title that many natives use to designate an ancestor or revered relative.)  The nun told the boy and his father that the skull was “an inheritance from a lost civilization” and, like the Christian cross, it was a symbol of the transcendence of Soul over death.  She said that it carried the message of immortal life and the illumination that we may discover when we lose our fear of death.  She gave it to the boy and his father, asking them to safeguard it until the “right” person came to get it– and share its message with the world.  It had been brought to that land from “somewhere else” and needed to wait until the right person could help it to continue its journey. “Your heart will know the person,” she said. 

“What a strange story,” thought George.

From elespectador.com:

“… ‘Supercholita’  tiene sobre todo una clara vocación divulgadora de la cultura andina. No en vano Valdez recibió su primer premio por explicar mediante este personaje cómo se cocina el ‘chuño,’ una típica patata deshidratada muy consumida en el altiplano boliviano.”

Chapter II:
“gazing on the symmetry
 of a lady’s ankle”

From “Sinatra: A Man
and His Music, Part II”
(reshown. prior to
“It Happened in Brooklyn,”
by Turner Classic Movies
on Sunday, May 11, 2008):

“Luck, be a lady tonight.”

From wordinfo.info:

astragalo-, astragal-
(Greek: anklebone, talus ball of ankle joint; dice, die [the Greeks made these from ankle bones])

astragalomancy, astragyromancy
Divination with dice, knuckle bones, stones, small pieces of wood, or ankle bones which were marked with letters, symbols, or dots. Using dice for divination is a form of astragalomancy.

Chapter III:
“unparalleled ecstasy”


Bright Star —

Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte

— Rubén Darío  

Bright Star and Crystal Skull

Image adapted from
Blue Star Traders


Related material:

The New York Lottery
  mid-day number yesterday–
719– and 7/19.

Monday, May 5, 2008

Monday May 5, 2008

Filed under: General — Tags: , — m759 @ 11:07 AM
Lottery Sermon

"And take upon's
the mystery of things
 as if we were God's spies"
King Lear  

PA Lottery Sunday, May 4, 2008: mid-day 170, evening 144

From Log24 on Aug. 19, 2003
and on Ash Wednesday, 2004:
a reviewer on
An Instance of the Fingerpost::

"Perhaps we are meant to
see the story as a cubist
   retelling of the crucifixion."

From Log24 on
Michaelmas 2007:

Kate Beckinsale (in 'Pearl Harbor') pointing to an instance of the number 144

Google searches suggested by
Sunday's PA lottery numbers
(mid-day 170, evening 144)
and by the above
figure of Kate Beckinsale
pointing to an instance of
the number 144 —

Click to enlarge:

Search for the meaning of 170 and 144, the PA lottery numbers of Sunday, May 4, 2008

Related material:

Beckinsale in another film
(See At the Crossroads,
Log24, Dec. 8, 2006):

"For every kind of vampire,
there is a kind of cross."
Gravity's Rainbow  
 
Kate Beckinsale in Underworld: Evolution

 

Kate Beckinsale, adapted from
poster for Underworld: Evolution
(DVD release date 6/6/6)
 
There is such a thing
as a tesseract.

"It was only in retrospect
that the silliness
became profound."

— Review of  
Faust in Copenhagen

From the conclusion of
Joan Didion's 1970 novel
  Play It As It Lays

Cover of 'Play It As It Lays'

"I know what 'nothing' means,
and keep on playing."

From Play It As It Lays,
the paperback edition of 1990
  (Farrar, Straus and Giroux) —

Page 170:

"By the end of a week she was thinking constantly
about where her body stopped and the air began,
about the exact point in space and time that was the
difference between Maria and other. She had the sense
that if she could get that in her mind and hold it for

170  

even one micro-second she would have what she had
come to get."

"The page numbers
are generally reliable."

Michaelmas 2007   

Sunday, March 2, 2008

Sunday March 2, 2008

Filed under: General — m759 @ 10:31 PM
Practical Magic

Halloween 2005:

“They don’t understand
what it is to be awake,
To be living
on several planes at once
Though one cannot speak
with several voices at once.”

— T. S. Eliot,
The Family Reunion

Margaret Wertheim with fellow tesseract authors

Several voices:

Margaret Wertheim in today’s
Los Angeles Times and at
The Pearly Gates of Cyberspace
,

Linda Dalrymple Henderson, and

Madeleine L’Engle and husband.

From Wertheim’s Pearly Gates:

Wertheim's 'Pearly Gates of Cyberspace,' page 200
“There is such a thing
as a tesseract.”

Madeleine L’Engle   

Tuesday, October 16, 2007

Tuesday October 16, 2007

Filed under: General — m759 @ 10:00 AM
In memory of
Harish-Chandra,
who died at 60
on this date in 1983

  The image “http://www.log24.com/log/pix07A/071016-Harish-Chandra.jpg” cannot be displayed, because it contains errors.
Harish-Chandra in 1981
(Photo by Herman Landshof)

Recent Log24 entries have parodied the use of the phrase “deep beauty” as the title of the Oct. 3-4 physics symposium of that name, which was supported by a grant from the John Templeton Foundation and sponsored by the Department of Philosophy at Princeton University.
Such parody was in part suggested by the symposium’s sources of financial and academic support. This support had, in the view of some, the effect of linking the symposium’s topic, the mathematics of quantum theory, with both religion (the Templeton Foundation) and philosophy (a field sometimes associated in popular thought– though not at Princeton— with quantum mysticism.)

As a corrective to the previous parodies here, the following material on the mathematician Harish-Chandra may help to establish that there is, in fact, such a thing as “deep beauty”– if not in physics, religion, or philosophy, at least in pure mathematics.

MacTutor History of Mathematics:

“Harish-Chandra worked at the Institute of Advanced Study at Princeton from 1963. He was appointed IBM-von Neumann Professor in 1968.”

R. P. Langlands (pdf, undated, apparently from a 1983 memorial talk):

“Almost immediately upon his arrival in Princeton he began working at a ferocious pace, setting standards that the rest of us may emulate but never achieve. For us there is a welter of semi-simple groups: orthogonal groups, symplectic groups, unitary groups, exceptional groups; and in our frailty we are often forced to treat them separately. For him, or so it appeared because his methods were always completely general, there was a single group. This was one of the sources of beauty of the subject in his hands, and I once asked him how he achieved it. He replied, honestly I believe, that he could think no other way. It is certainly true that he was driven back upon the simplifying properties of special examples only in desperate need and always temporarily.”

“It is difficult to communicate the grandeur of Harish-Chandra’s achievements and I have not tried to do so. The theory he created still stands– if I may be excused a clumsy simile– like a Gothic cathedral, heavily buttressed below but, in spite of its great weight, light and soaring in its upper reaches, coming as close to heaven as mathematics can. Harish, who was of a spiritual, even religious, cast and who liked to express himself in metaphors, vivid and compelling, did see, I believe, mathematics as mediating between man and what one can only call God. Occasionally, on a stroll after a seminar, usually towards evening, he would express his feelings, his fine hands slightly upraised, his eyes intent on the distant sky; but he saw as his task not to bring men closer to God but God closer to men. For those who can understand his work and who accept that God has a mathematical side, he accomplished it.”

For deeper views of his work, see

  1. Rebecca A. Herb, “Harish-Chandra and His Work” (pdf), Bulletin of the American Mathematical Society, July 1991, and
  2. R. P. Langlands, “Harish-Chandra, 1923-1983” (pdf, 28 pp., Royal Society memoir, 1985)

Friday, September 21, 2007

Friday September 21, 2007

Filed under: General — Tags: , — m759 @ 8:28 AM
Word and Object

"We may recall the ideal of 'dryness' which we associate with the symbolist movement, with writers such as T. E. Hulme and T. S. Eliot, with Paul Valery, with Wittgenstein. This 'dryness' (smallness, clearness, self-containedness) is a nemesis of Romanticism…. The temptation of art… is to console. The modern writer… attempts to console us by myths or by stories."

— Iris Murdoch  

"The consolations of form,
the clean crystalline work"

— Iris Murdoch, 
"Against Dryness"

"As a teacher Quine
was carefully organized,
precise, and conscientious,
but somewhat dry
in his classroom style."

Harvard Gazette 

Word:

The image “http://www.log24.com/log/pix07A/070921-Connectives.gif” cannot be displayed, because it contains errors.

Object:

The image “http://www.log24.com/log/pix07A/070921-Lindenbaum-Tarski.gif” cannot be displayed, because it contains errors.

Myth and Story:

The five entries ending
on Jan. 27, 2007

"There is such a thing
as a tesseract."
Madeleine L'Engle  
 

Friday, September 7, 2007

Friday September 7, 2007

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

The New York Times online,
Friday, Sept. 7, 2007:

Madeleine L’Engle,
Children’s Writer,
Is Dead

"Madeleine L’Engle, who in writing more than 60 books, including childhood fables, religious meditations and science fiction, weaved emotional tapestries transcending genre and generation, died Thursday [Sept. 6, 2007] in Connecticut. She was 88.

Her death, of natural causes, was announced today by her publisher, Farrar, Straus and Giroux."

More >>

Related material:

Log24 entries of
August 31

"That is how we travel."

The image “http://www.log24.com/log/pix07A/070831-Ant2.gif” cannot be displayed, because it contains errors.

A Wrinkle in Time,
Chapter 5,
"The Tesseract"

— and of 
September 2
(with update of
 September 5)–

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

Sunday, September 2, 2007

Sunday September 2, 2007

Filed under: General,Geometry — Tags: — m759 @ 5:11 PM

Comment at the
n-Category Cafe

Re: This Week’s Finds in Mathematical Physics (Week 251)

On Spekkens’ toy system and finite geometry

Background–

  • In “Week 251” (May 5, 2007), John wrote:
    “Since Spekkens’ toy system resembles a qubit, he calls it a “toy bit”. He goes on to study systems of several toy bits – and the charming combinatorial geometry I just described gets even more interesting. Alas, I don’t really understand it well: I feel there must be some mathematically elegant way to describe it all, but I don’t know what it is…. All this is fascinating. It would be nice to find the mathematical structure that underlies this toy theory, much as the category of Hilbert spaces underlies honest quantum mechanics.”
  • In the n-Category Cafe ( May 12, 2007, 12:26 AM, ) Matt Leifer wrote:
    “It’s crucial to Spekkens’ constructions, and particularly to the analog of superposition, that the state-space is discrete. Finding a good mathematical formalism for his theory (I suspect finite fields may be the way to go) and placing it within a comprehensive framework for generalized theories would be very interesting.”
  • In the n-category Cafe ( May 12, 2007, 6:25 AM) John Baez wrote:
    “Spekkens and I spent an afternoon trying to think about his theory as quantum mechanics over some finite field, but failed — we almost came close to proving it couldnt’ work.”

On finite geometry:

The actions of permutations on a 4 × 4 square in Spekkens’ paper (quant-ph/0401052), and Leifer’s suggestion of the need for a “generalized framework,” suggest that finite geometry might supply such a framework. The geometry in the webpage John cited is that of the affine 4-space over the two-element field.

Related material:

Update of
Sept. 5, 2007

See also arXiv:0707.0074v1 [quant-ph], June 30, 2007:

A fully epistemic model for a local hidden variable emulation of quantum dynamics,

by Michael Skotiniotis, Aidan Roy, and Barry C. Sanders, Institute for Quantum Information Science, University of Calgary. Abstract: "In this article we consider an augmentation of Spekkens’ toy model for the epistemic view of quantum states [1]…."
 

Skotiniotis et al. note that the group actions on the 4×4 square described in Spekkens' paper [1] may be viewed (as in Geometry of the 4×4 Square and Geometry of Logic) in the context of a hypercube, or tesseract, a structure in which adjacency is isomorphic to adjacency in the 4 × 4 square (on a torus).

Hypercube from the Skotiniotis paper:

Hypercube

Reference:

[1] Robert W. Spekkens, Phys. Rev. A 75, 032110 (2007),

Evidence for the epistemic view of quantum states: A toy theory
,

Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5 (Received 11 October 2005; revised 2 November 2006; published 19 March 2007.)

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

Thursday, June 21, 2007

Thursday June 21, 2007

Filed under: General,Geometry — Tags: — m759 @ 12:07 PM
Let No Man
Write My Epigraph

(See entries of June 19th.)

"His graceful accounts of the Bach Suites for Unaccompanied Cello illuminated the works’ structural logic as well as their inner spirituality."

Allan Kozinn on Mstislav Rostropovich in The New York Times, quoted in Log24 on April 29, 2007

"At that instant he saw, in one blaze of light, an image of unutterable conviction…. the core of life, the essential pattern whence all other things proceed, the kernel of eternity."

— Thomas Wolfe, Of Time and the River, quoted in Log24 on June 9, 2005

"… the stabiliser of an octad preserves the affine space structure on its complement, and (from the construction) induces AGL(4,2) on it. (It induces A8 on the octad, the kernel of this action being the translation group of the affine space.)"

— Peter J. Cameron, "The Geometry of the Mathieu Groups" (pdf)

"… donc Dieu existe, réponse!"

— Attributed, some say falsely,
to Leonhard Euler
 
"Only gradually did I discover
what the mandala really is:
'Formation, Transformation,
Eternal Mind's eternal recreation'"

(Faust, Part Two, as
quoted by Jung in
Memories, Dreams, Reflections)

 

Wolfgang Pauli as Mephistopheles

"Pauli as Mephistopheles
in a 1932 parody of
Goethe's Faust at Niels Bohr's
institute in Copenhagen.
The drawing is one of
many by George Gamow
illustrating the script."
Physics Today

 

"Borja dropped the mutilated book on the floor with the others. He was looking at the nine engravings and at the circle, checking strange correspondences between them.

'To meet someone' was his enigmatic answer. 'To search for the stone that the Great Architect rejected, the philosopher's stone, the basis of the philosophical work. The stone of power. The devil likes metamorphoses, Corso.'"

The Club Dumas, basis for the Roman Polanski film "The Ninth Gate" (See 12/24/05.)

"Pauli linked this symbolism
with the concept of automorphism."

The Innermost Kernel
 (previous entry)

And from
"Symmetry in Mathematics
and Mathematics of Symmetry
"
(pdf), by Peter J. Cameron,
a paper presented at the
International Symmetry Conference,
Edinburgh, Jan. 14-17, 2007,
we have

The Epigraph–

Weyl on automorphisms
(Here "whatever" should
of course be "whenever.")

Also from the
Cameron paper:

Local or global?

Among other (mostly more vague) definitions of symmetry, the dictionary will typically list two, something like this:

• exact correspondence of parts;
• remaining unchanged by transformation.

Mathematicians typically consider the second, global, notion, but what about the first, local, notion, and what is the relationship between them?  A structure M is homogeneous if every isomorphism between finite substructures of M can be extended to an automorphism of M; in other words, "any local symmetry is global."

Some Log24 entries
related to the above politically
(women in mathematics)–

Global and Local:
One Small Step

and mathematically–

Structural Logic continued:
Structure and Logic
(4/30/07):

This entry cites
Alice Devillers of Brussels–

Alice Devillers

"The aim of this thesis
is to classify certain structures
which are, from a certain
point of view, as homogeneous
as possible, that is which have
  as many symmetries as possible."

"There is such a thing
as a tesseract."

Madeleine L'Engle 

Monday, May 21, 2007

Monday May 21, 2007

Filed under: General,Geometry — Tags: — m759 @ 4:00 PM
No Royal Roads
Illustration from a
1980 article at JSTOR:

Coxeter as King of Geometry

A more recent royal reference:

"'Yau wants to be the king of geometry,' Michael Anderson, a geometer at Stony Brook, said. 'He believes that everything should issue from him, that he should have oversight. He doesn't like people encroaching on his territory.'" –Sylvia Nasar and David Gruber in The New Yorker, issue dated Aug. 28, 2006

Wikipedia, Cultural references to the Royal Road:

"Euclid is said to have replied to King Ptolemy's request for an easier way of learning mathematics that 'there is no royal road to geometry.' Charles S. Peirce, in his 'How to Make Our Ideas Clear' (1878), says 'There is no royal road to logic, and really valuable ideas can only be had at the price of close attention.'"

Related material:

Day Without Logic
(March 8, 2007)

and
The Geometry of Logic
(March 10, 2007)
:

The image “http://www.log24.com/log/pix07/070521-Tesseract.gif” cannot be displayed, because it contains errors.

There may be
no royal roads to
geometry or logic,
but…

"There is such a thing
as a tesseract."
— Madeleine L'Engle, 
A Wrinkle in Time

Thursday, May 3, 2007

Thursday May 3, 2007

Filed under: General,Geometry — Tags: — m759 @ 3:00 PM
A Web
of Links

"Some postmodern theorists like to talk about the relationship between 'intertextuality' and 'hypertextuality'; intertextuality makes each text a 'mosaic of quotations' [Kristeva, Desire in Language, Columbia U. Pr., 1980, 66] and part of a larger mosaic of texts, just as each hypertext can be a web of links and part of the whole World-Wide Web." —Wikipedia
 

The image “http://www.log24.com/log/pix07/070503-Tiffany.jpg” cannot be displayed, because it contains errors.

Related material

Day Without Logic,
Introduction to Logic,
The Geometry of Logic,
Structure and Logic,
Spider-Man and Fan:

The image “http://www.log24.com/log/pix07/070503-Devillers.jpg” cannot be displayed, because it contains errors.

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

Friday, April 20, 2007

Friday April 20, 2007

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

In Grand Rapids today

"… Bush spoke and answered audience questions for nearly 90 minutes inside East Grand Rapids High School in suburban Grand Rapids….

After leaving the school, Bush's motorcade stopped at the Gerald R. Ford Presidential Museum in downtown Grand Rapids, where he stood silently for a few moments after placing a bouquet of white roses at Ford's burial site on the museum grounds. The 38th president, who grew up in Grand Rapids, died Dec. 26 at age 93."

Multispeech

Mich. Lottery Apr. 20, 2007: Day 019, Night 001

 

For the meaning of the lottery icons
above, see this morning's entry and
an entry that it links to —
Time's Labyrinth continued
of March 8, 2007.

For the meaning of multispeech,
see the entries of
All Hallows' Eve, 2005:

Tesseract on the cover of The Gameplayers of Zan
 
"There is such a thing
as a tesseract."
A Wrinkle in Time 
 

Saturday, March 10, 2007

Saturday March 10, 2007

Filed under: General,Geometry — m759 @ 2:00 AM
Tesseract

A new page at finitegeometry.org,
The Geometry of Logic,
includes the following figure:

The 16 binary connectives arranged in a tesseract

There is such a thing
as a tesseract.

— Madeleine L’Engle
 

Friday, December 8, 2006

Friday December 8, 2006

Filed under: General — Tags: — m759 @ 9:00 AM
An Instance
of the Fingerpost
The image “http://www.log24.com/log/pix06B/061208-Date.jpg” cannot be displayed, because it contains errors.
 
"CRUCIAL (from Lat. crux, a cross),
that which has the form of a cross…
 From Francis Bacon's expression
instantia crucis (taken, as he says, from
the finger-post or crux at cross-roads)"
 
Encyclopaedia Britannica,
the classic 11th edition (1911)
 
"For every kind of vampire,
there is a kind of cross."
Gravity's Rainbow  
 
The image “http://www.log24.com/log/pix06A/060614-EvolutionBegins2.jpg” cannot be displayed, because it contains errors.

Kate Beckinsale, adapted from
poster for Underworld: Evolution
(DVD release date 6/6/6)

 
There is such a thing
as a tesseract.
A Wrinkle in Time  
 
Related material:
 
The tesseract on the cover of
The Gameplayers of Zan
(All Hallows' Eve, 2005), and
 
A Last Stitch in Time…or
A Map of the Map
of Kierkegaard's World:

"Appropriating the Button-molder's
words to Peer Gynt, he would say,
'We'll meet at the next crossroads…
and then we'll see–
I won't say more.'"

Monday, November 13, 2006

Monday November 13, 2006

Filed under: General — m759 @ 8:23 PM
Cognitive Blend:

Casino Royale
and
Time in the Rock

PA lottery Nov. 13, 2006: Mid-day 726, Evening 329
 
In today’s cognitive blend,
the role of Casino Royale
is played by the
Pennsylvania Lottery,
which points to 7/26,
Venus at St. Anne’s
(title of the closing chapter
of That Hideous Strength).

The role of
Time in the Rock
is played by a
Log24 entry of 3/29,
Diamond Theory in 1937.

There is such a thing
as a tesseract.

Tuesday, October 31, 2006

Tuesday October 31, 2006

Filed under: General,Geometry — Tags: , , — m759 @ 11:00 PM
To Announce a Faith

From 7/07, an art review from The New York Times:

Endgame Art?
It's Borrow, Sample and Multiply
in an Exhibition at Bard College

"The show has an endgame, end-time mood….

I would call all these strategies fear of form…. the dismissal of originality is perhaps the oldest ploy in the postmodern playbook. To call yourself an artist at all is by definition to announce a faith, however unacknowledged, in some form of originality, first for yourself, second, perhaps, for the rest of us.

Fear of form above all means fear of compression– of an artistic focus that condenses experiences, ideas and feelings into something whole, committed and visually comprehensible."

— Roberta Smith

It is doubtful that Smith
 would consider the
following "found" art an
example of originality.

It nevertheless does
"announce a faith."


The image “http://www.log24.com/log/pix06A/061031-PAlottery2.jpg” cannot be displayed, because it contains errors.


"First for yourself"

Today's mid-day
Pennsylvania number:
707

See Log24 on 7/07
and the above review.
 

"Second, perhaps,
for the rest of us"

Today's evening
Pennsylvania number:
384

This number is an
example of what the
reviewer calls "compression"–

"an artistic focus that condenses
 experiences, ideas and feelings
into something
whole, committed
 and visually comprehensible."

"Experiences"

See (for instance)

Joan Didion's writings
(1160 pages, 2.35 pounds)
on "the shifting phantasmagoria
which is our actual experience."

"Ideas"

See Plato.

"Feelings"

See A Wrinkle in Time.

"Whole"

The automorphisms
of the tesseract
form a group
of order 384.

"Committed"

See the discussions of
groups of degree 16 in
R. D. Carmichael's classic
Introduction to the Theory
of Groups of Finite Order
.

"Visually comprehensible"

See "Diamond Theory in 1937,"
an excerpt from which
is shown below.

The image “http://www.log24.com/theory/images/Carmichael440abbrev.gif” cannot be displayed, because it contains errors.

The "faith" announced by
the above lottery numbers
on All Hallows' Eve is
perhaps that of the artist
Madeleine L'Engle:

"There is such a thing
as a tesseract.
"

Wednesday, March 29, 2006

Wednesday March 29, 2006

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

The image “http://www.log24.com/theory/images/Carmichael440.gif” cannot be displayed, because it contains errors.
Note: Carmichael's reference is to
A. Emch, "Triple and multiple systems, their geometric configurations and groups," Trans. Amer. Math. Soc. 31 (1929), 25–42.

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

Wednesday, February 1, 2006

Wednesday February 1, 2006

Filed under: General — m759 @ 9:49 AM

“Good Night and Good Luck”

— Morgan Freeman, closing remarks  
at the Screen Actors Guild Awards
on January 29, 2006

The image “http://www.log24.com/log/pix06/060201-SAG1.jpg” cannot be displayed, because it contains errors.
A History
of Death
The image “http://www.log24.com/log/pix06/060201-SAG2.jpg” cannot be displayed, because it contains errors.

Review by Stephen Hunter  
of “Good Night and Good Luck”–

“The film, therefore, is like
a child’s view of these events,
untroubled by complexity,
hungry for myth and simplicity.”

“A larger poem for a larger audience….
A mythological form, a festival sphere,
A great bosom, beard and being,
alive with age.”

— Wallace Stevens, quoted in 
Log24, January 29, 2006

The image “http://www.log24.com/log/pix06/060131-PolarSanta1.jpg” cannot be displayed, because it contains errors.

Stephen Hunter on Senator McCarthy:

He “forever tarnished
by association the reputations
of the security services
charged with keeping us safe
from the actual–
yes, Virginia, there was
such a thing– Red menace.”

Thursday, April 14, 2005

Thursday April 14, 2005

Filed under: General — m759 @ 8:00 AM
Final Arrangements,
continued
From Log24, April 5, 2005:

Father Richard John Neuhaus yesterday argued that John Paul II should be called “the Great.”
 
Neuhaus stated that “If any phrase encapsulates the message that John Paul declared to the world, it is probably ‘prophetic humanism.'”  If there is such a thing, it is probably best exemplified by the I Ching.  For further details, see Hitler’s Still Point.

See also last Saturday’s entry,
Prophetic Humanism.

The image “http://www.log24.com/log/pix05/050414-Huan.jpg” cannot be displayed, because it contains errors.

Tuesday, April 5, 2005

Tuesday April 5, 2005

Filed under: General — Tags: — m759 @ 1:28 AM
The Garden of Good and Evil
continued

"Just the facts, ma'am" — Joe Friday

See the entry Lucky (?) Numbers of Saturday, April 2, 2005, 11:07 AM ET, for links to a few facts about the historical role of the Number of the Beast in the Pennsylvania Lottery.

The Pennsylvania Lottery mid-day drawings take place at about 1:10 PM ET.

Pope John Paul II died on Saturday, April 2, at 2:37 PM ET. 

Thus the final PA drawing of his lifetime was on that Saturday afternoon.

The winning mid-day number that day was…

034.

In the I Ching, this is the number of
The Power of the Great.

Father Richard John Neuhaus yesterday argued that John Paul II should be called "the Great."

 

Neuhaus stated that "If any phrase encapsulates the message that John Paul declared to the world, it is probably 'prophetic humanism.'"  If there is such a thing, it is probably best exemplified by the I Ching.  For further details, see Hitler's Still Point.

Father Neuhaus's argument included the following mysterious phrase:

"God's unfolding covenant with Abraham, Isaac, Jacob and Jesus."

Compare the following two passages from Holy Scripture:

Genesis 22:13

"…behold behind him
a ram caught in a thicket by his horns"

I Ching Hexagram 34

"A goat butts against a hedge
And gets its horns entangled."

A topic for discussion by the foolish:

In the current historical situation,
who is Isaac and who is the goat?

From yet another Holy Scripture,
a topic for discussion by the wise: 

“Anyone can create a pretty little bamboo garden in the world. But I doubt that the gardener would succeed in incorporating the world in his bamboo grove.”

Wednesday, September 15, 2004

Wednesday September 15, 2004

Filed under: General — m759 @ 2:56 PM

On Translation

From Illuminations, by Walter Benjamin, translated by Harry Zohn:

“If there is such a thing as a language of truth, the tensionless and even silent depository of the ultimate truth which all thought strives for, then this language of truth is– the true language.  And this very language, whose divination and description is the only perfection a philosopher can hope for, is concealed in concentrated fashion in translations.  There is no muse of philosophy, nor is there one of translation.  But despite the claims of sentimental artists, these two are not banausic.  For there is a philosophical genius that is characterized by a yearning for that language which manifests itself in translations: ‘Les langues imparfaites en cela que plusieurs, manque la suprême: penser étant écrire sans accessoires, ni chuchotement mais tacite encore l’immortelle parole, la diversité, sur terre, des idiomes empêche personne de proférer les mots qui, sinon se trouveraient, par une frappe unique, elle-même matériellement la vérité.’*  If what Mallarmé evokes here is fully fathomable to a philosopher, translation, with its rudiments of such a language, is midway between poetry and doctrine.  Its products are less sharply defined, but it leaves no less of a mark on history.”

* “The imperfection of languages consists in their plurality, the supreme one is lacking: thinking is writing without accessories or even whispering, the immortal word still remains silent; the diversity of idioms on earth prevents everybody from uttering the words which otherwise, at one single stroke, would materialize as truth.’

Stéphane Mallarmé / Crise de vers

(The Benjamin is from a copy of Illuminations I purchased exactly 12 years ago, on Sept. 15, 1992.)

Friday, March 19, 2004

Friday March 19, 2004

Filed under: General,Geometry — m759 @ 2:35 PM

Geometry of the 4×4 Square:

http://log24.com/theory/geometry.html

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

Monday, May 12, 2003

Monday May 12, 2003

Filed under: General — m759 @ 6:10 PM

The Tony Nominations

Dannie Abse quoting Robert Penn Warren:

“The name of the story will be Time,
But you must not pronounce its name.
Tell me a story of deep delight.”

 Dannie Abse

Abse deserves a Tony Smith award¹ for his play Pythagoras.

Frank Rich on Bush’s Top Gun speech:

“Only hours before President Bush’s prime-time speech came news of what Variety headlined on Page 1 as ‘Regime Change’ in Hollywood — the departure of the [West Wing] creator, the writer Aaron Sorkin.”

 George W. Bush

President Bush deserves a Tony Smith award² for his performance aboard the aircraft carrier Abraham Lincoln.

 Madeleine L’Engle on the religion of Cubism:

“There is such a thing as a tesseract.”

 Madeleine L’Engle

L’Engle, former librarian at the Cathedral of St. John the Divine in New York City, deserves a Tony Smith award³ for insisting on the existence of the tesseract, or 4-dimensional cube, as an object of conceptual art.

L’Engle is perhaps the best defender of the religious, or “story,” theory of truth, as opposed to the “diamond” theory of truth. (See my earlier May 12 entry, “Death and Truth,” which deals with the bishop of L’Engle’s cathedral.)

¹ See Tony Smith on mathematics.

² See Tony Smith on foreign policy.

³ See Tony Smith on conceptual art.

Wednesday, March 19, 2003

Wednesday March 19, 2003

Filed under: General,Geometry — m759 @ 4:04 AM


Aptheker

  A Look at the Rat

In memory of Herbert Aptheker, theoretician of the American Communist Party, who died on St. Patrick’s Day, 2003 —

From The New Yorker, issue dated March 24, 2003, Louis Menand on Edmund Wilson’s To the Finland Station:

“Wilson did know what was going on in the Soviet Union in the nineteen-thirties, as his pages on Stalin in To the Finland Station make clear. The problem wasn’t with Stalin; the problem was with Lenin, the book’s ideal type of the intellectual as man of action. Wilson admitted that he had relied on publications controlled by the Party for his portrait of Lenin. (Critical accounts were available; for example, the English translation of the émigré Mark Landau-Aldanov’s Lenin was published, by Dutton, in 1922.) Lenin could create an impression of selfless humanitarianism; he was also a savage and ruthless politician—a ‘pail of milk of human kindness with a dead rat at the bottom,’ as Vladimir Nabokov put it to Wilson in 1940, after reading To the Finland Station.  In the introduction to the 1972 edition, Wilson provided a look at the rat. He did not go on to explain in that introduction that the most notorious features of Stalin’s regime—the use of terror, the show trials, and the concentration camps—had all been inaugurated by Lenin. To the Finland Station begins with Napoleon’s betrayal of the principles of the French Revolution; it should have ended with Lenin’s betrayal of European socialism.” 

From Herbert Aptheker, “More Comments on Howard Fast“:

“We observe that in the list of teachers whom Howard Fast names as most influential in his own life there occur the names of fourteen individuals from Jefferson to Bernard Shaw, Upton Sinclair to Marx, Douglass to Engels, but there is no room for Lenin.
   He is, I think, an important teacher, too; indeed, in my view, Lenin is the greatest figure in the whole galaxy of world revolutionary leaders. He is, certainly, the greatest analyzer of and fighter against imperialism.”

For more on Howard Fast, see my entry
“Death Knell” of March 13, 2003

For a look at the pail of milk, see
the New Yorker cover in Geometry for Jews.

For a more cheerful look at geometry
on this St. Joseph’s Day, see
Harry J. Smith’s

Tesseract Site.

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

Friday, November 29, 2002

Friday November 29, 2002

Filed under: General — m759 @ 7:00 AM

On Madeleine L’Engle’s birthday:

There is such a thing as a tesseract.

Wednesday, September 4, 2002

Wednesday September 4, 2002

Filed under: General — m759 @ 3:21 AM

Birthday of AI guru John McCarthy

If you enter the question “Is there such a thing as artificial intelligence?” as a Google search phrase, you will be referred to ALICE, a chat robot. ALICE’s possible answers include “Yes,” “No,” and “Maybe.” Another search strategy leads to the following Google Directory page:

Computers > Artificial Intelligence >
Neural Networks > Companies.

This page, unlike ALICE, suggests that the appropriate answer to our question is the punch line to an old computer joke:  “There is now.”

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