Log24

Tuesday, July 31, 2012

Happy Birthday, J. K. Rowling

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

IMAGE- Google Book Search for 'Malcom Lowry' + 'thinking of treadmills'

See also a July 19 death, this journal on that date, and
Down the Up Staircase.

Logo

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

IMAGE- 'Yankee Puzzle' quilt block pattern on cover of Northrop Frye's 'Anatomy of Criticism'

On Universals and
A Passage to India
 :
 
"The universe, then, is less intimation
than cipher: a mask rather than a revelation
in the romantic sense. Does love meet with love?
Do we receive but what we give? The answer is
surely a paradox, the paradox that there are
Platonic universals beyond, but that the glass
is too dark to see them. Is there a light beyond
the glass, or is it a mirror only to the self?
The Platonic cave is even darker than Plato
made it, for it introduces the echo, and so
leaves us back in the world of men, which does
not carry total meaning, is just a story of events."
 
– Betty Jay, reader's guide to A Passage to India

http://www.log24.com/log/pix08/080413-Marabar.jpg

Judy Davis in the Marabar Caves

The above image is from this journal on Sunday, April 13, 2008.

The preceding cover of a book by Northrop Frye was suggested
by material in this journal from February 2003.

See also Yankee Puzzle and Doodle Dandy.

Monday, July 30, 2012

Logos

Filed under: Uncategorized — m759 @ 10:48 PM

Related material:

Frank J. Prial on the late singer Tony Martin

— and, on Jan. 1, 2005, on beverage marketing:

Every picture tells a story.

Happy birthday to Hilary Swank.

Geometry and Death

Filed under: Uncategorized — m759 @ 2:23 PM

(Continued)

A Necessary Truth—

James Singer, "A Theorem in Finite Projective Geometry
and Some Applications to Number Theory," Transactions
of the American Mathematical Society  
43 (1938), 377-385.

A Contingent Truth—

Singer Tony Martin reportedly died Friday evening, July 27, 2012.

In his memory, some references to a "Singer 7-Cycle."

See also this journal 7 years prior to Martin's death.

Something to Read

Filed under: Uncategorized — m759 @ 1:00 PM

(Continued)

Eric M. Friedlander, President of the
American Mathematical Society (AMS),
in the March 2011 AMS Notices 

"I think the best thing the AMS does by far is the Notices .
It could easily be in all doctors’ and dentists’ offices."

Notices : "Really?"

Friedlander: "It could be."

Related material from this journal:

Olivier as Dr. Christian Szell

The icosahedron (a source of duads and synthemes)

Is it safe?"

 Annals of Art Education: 
     Geometry and Death

Unnecessary* Truth?

Filed under: Uncategorized — m759 @ 12:26 PM

"There is no question about what arithmetic is for
or why it is supported. Society cannot proceed
without it. Addition, subtraction, multiplication,
division, percentages: though not all citizens can
deal fluently with all of them, we make the
assumption that they can when necessary.
Those who cannot are sometimes at a disadvantage.

Algebra, though, is another matter."

— Underwood Dudley in the Notices of the
American Mathematical Society
, May 2010
:
"What Is Mathematics For?" 

A less nuanced remark from the American
Mathematical Society (AMS) today—

"The answer to the recent Op-Ed piece
in The New York Times  entitled
'Is Algebra Necessary?'
is resoundingly YES!"

— Eric Friedlander, AMS president

* A review of philosophical terminology—

"The distinction between necessary truth
and contingent truth is a version of Leibniz 's 
distinction between truths of reason and truths
of fact. A necessary truth must be true and
could not be false, whatever way the world is. 
It is true in itself. A contingent truth, on the other 
hand, depends upon the empirical world and might 
have been false had the world been different." 

— The Blackwell Dictionary of Western Philosophy

Sunday, July 29, 2012

The Galois Tesseract

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

(Continued)

The three parts of the figure in today's earlier post "Defining Form"—

IMAGE- Hyperplanes (square and triangular) in PG(3,2), and coordinates for AG(4,2)

— share the same vector-space structure:

   0     c     d   c + d
   a   a + c   a + d a + c + d
   b   b + c   b + d b + c + d
a + b a + b + c a + b + d   a + b + 
  c + d

   (This vector-space a b c d  diagram is from  Chapter 11 of 
    Sphere Packings, Lattices and Groups , by John Horton
    Conway and N. J. A. Sloane, first published by Springer
    in 1988.)

The fact that any  4×4 array embodies such a structure was implicit in
the diamond theorem (February 1979). Any 4×4 array, regarded as
a model of the finite geometry AG(4, 2), may be called a Galois tesseract.
(So called because of the Galois geometry involved, and because the
16 cells of a 4×4 array with opposite edges identified have the same
adjacency pattern as the 16 vertices of a tesseract (see, for instance,
Coxeter's 1950 "Self-Dual Configurations and Regular Graphs," figures
5 and 6).)

A 1982 discussion of a more abstract form of AG(4, 2):

Source:

The above 1982 remarks by Brouwer may or may not have influenced
the drawing of the above 1988 Conway-Sloane diagram.

Defining Form

Filed under: Uncategorized — Tags: — m759 @ 2:00 PM

IMAGE- Hyperplanes (square and triangular) in PG(3,2), and coordinates for AG(4,2)

Background: Square-Triangle Theorem.

For a more literary approach, see "Defining Form" in this journal
and a bibliography from the University of Zaragoza.

Saturday, July 28, 2012

Claremont Review

Filed under: Uncategorized — m759 @ 3:00 PM

On painter Karl Benjamin of Claremont, California,
who reportedly died on Thursday

He played them music
and everything was concentrated and timeless
and all were artists 'til the bell rang.

Another remark from Claremont—

"'Once upon a time' used to be a gateway to
a land that was inviting precisely because
it was timeless, like the stories it introduced
and their ageless lessons about the human condition."

– Dorothea Israel Wolfson, 
   Claremont Review of Books, Summer 2006

Benjamin was a professor emeritus at Pomona College.

Hey

Filed under: Uncategorized — m759 @ 12:00 PM

The end of the beginning of the London Games 
suggests other games —

Shadows (July 14) —

IMAGE- G. H. Hardy around 1900, strongly resembling Paul McCartney

A Game of Shadows — "You know my methods."

Related religious material —

The Feast of Saint  Jude,  2011.

Friday, July 27, 2012

Olympics Special

Filed under: Uncategorized — Tags: — m759 @ 7:20 PM

Quoted in some remarks yesterday on geometry—

IMAGE- Eric Temple Bell on 'Solomon's Seal' as a 'highly special topic'

From posts linked to this morning—

IMAGE- 'Jewel in the Crown'- MAA version of the Crown of Geometry

The Source— 

IMAGE- Coxeter as King of Geometry

Raiders of the Lost Ring

Filed under: Uncategorized — Tags: , — m759 @ 11:30 AM

Wikipedia on a magical ring

IMAGE- Wikipedia article, 'Seal of Solomon'

Background—  The Ring and the Stone, a story linked to here Wednesday.

"By then he was familiar with the work of the Vienna Actionists….
He once said that he had his first taste of the movement
when he heard the screams of his mother’s dental patients
from her office next door to the family’s apartment."

Obituary of a Viennese artist who reportedly died Wednesday

"Is it safe?"

Thursday, July 26, 2012

Solomon’s Seal

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

(Mathematics and Narrative, continued)

Narrative—

The Ring and The Stone from yesterday's post, and…

"In Medieval Jewish, Christian and Islamic legends,
the Seal of Solomon was a magical signet ring
said to have been possessed by King Solomon…."

— Wikipedia article, Seal of Solomon

Mathematics—

IMAGE- Eric Temple Bell on the mathematics of 'Solomon's Seal' (in his 'Development of Mathematics')

A fact related to the mathematical
"Solomon's seal" described above by Bell:

IMAGE- J.W.P. Hirschfeld on the mathematics of 'Solomon's Seal', with reference to Edge on the same topic

The reference to Edge is as follows—

[3] Edge, W. L., Quadrics over GF(2) and
their relevance for the cubic surface group
,
Canadian J. Maths. 11 (1959) ….

(This reference relates Hirschfeld's remarks
quoted above to the 64-point affine space
illustrated below (via the associated
63-point projective  space PG (5, 2)).

As for the narrative's Stone… 

See Solomon's Cube.

IMAGE- 'Solomon's Cube'

Wednesday, July 25, 2012

Tale

Filed under: Uncategorized — m759 @ 11:00 AM

The Ring and the Stone

Tuesday, July 24, 2012

Biograph

Filed under: Uncategorized — m759 @ 11:30 AM

(Continued from July 22)

IMAGE- 'Follow the Ring' note from the film 'Inside Man'

Manhattan, July 22, 2012 — "Once upon a time, in a quiet corner
of the Middle East, there lived a shepherd named Gyges.  Despite
the hardships in his life Gyges was relatively satisfied with his meager
existence. Then, one day, he found a ring buried in a nearby cave."
Read more…

Joker

Filed under: Uncategorized — m759 @ 9:48 AM

"I want you on the Swansea lathe today."
— Boss of the Christ figure in "The Machinist" (2004)

Related material in this journal—

Dylan Thomas and Modern Times

Time and Date

Filed under: Uncategorized — m759 @ 1:30 AM

IMDb trivia page on "The Machinist" (2004)—

"The time of 1:30 AM is significant throughout the movie.
Trevor often notices something out of the ordinary at this time.
During the 1 hour 30 minute mark in the movie,
the major plot twist is revealed."

As for the date  1/30… See Tolkien on telepathy.

(Backstory: The Gospel According to Father Hardon )

A Reappearing Number

Filed under: Uncategorized — m759 @ 12:31 AM

(Continued)

Part I:

Christian Bale as a Spanish Christ-parody in "The Machinist" (2004)

Part II:

"The expression 'the devil is in the details'
is turned on its head in the exhibit 'The Sacred Made Real'
at Washington's National Gallery of Art…." 
Catholic News Service, 2010

Part III:

Tonight's New York Times  obituaries

Part IV:

"My goodness, there must be a hole in this glass."
— Maria in "The Machinist" (2004)

Monday, July 23, 2012

In Memoriam

Filed under: Uncategorized — m759 @ 9:00 PM

Wilson Pickett and the Holy…

IMAGE- Triangular logo of the NASA Gravity Recovery and Interior Laboratory

L'Chaim

23 Puzzle

Filed under: Uncategorized — m759 @ 12:48 PM

For those who like puzzles—

What film's page at IMDb recommends
the following "also liked" choices?

The Interpreter (2005), Identity (2003), The Game (1997),
The Ghost Writer (2010), The Machinist (2004), The Number 23 (2007)

A similar puzzle: Related Books

Manchurian Symphony

Filed under: Uncategorized — m759 @ 12:00 PM

"Always with a little humor"
— Manchurian Candidate

"Sound of a Theorem"

In Memoriam

Ding

Sunday, July 22, 2012

Art Wars

Filed under: Uncategorized — m759 @ 9:16 PM

For art collector Herbert Vogel,
who reportedly died today

IMAGE- Herbert Vogel with abstract half-circle art in 1978

Philip Kennicott in The Washington Post , July 3, 2009— 

"The Vogels help allay deep cultural fears
within the art world— fears that art is elitist,
or some kind of confidence game,
or not a serious endeavor (a fear that has
dogged art since at least the time of Plato)."

Some related material from finitegeometry.org,
offered without comment—

IMAGE- 1975 half-circle art by Cullinane based on work in 1960s by Swiss artist Paul Talman

Biograph

Filed under: Uncategorized — m759 @ 10:01 AM

"One ring to bring them all…"
— J. R. R. Tolkien, Catholic author

Today in History, July 22, by The Associated Press—

"In 1934, bank robber John Dillinger was shot to death
by federal agents outside Chicago's Biograph Theater,
where he had just seen the Clark Gable movie
'Manhattan Melodrama.'"

From a  Manhattan Melodrama

"Follow the Ring" 

Piatigorsky died on Sunday, July 15. Notes in this  journal from that date—

Backstory—

Saturday, July 21, 2012

Republican

Filed under: Uncategorized — m759 @ 8:25 PM

Colin Moynihan on the late Alexander Cockburn

"His attachment to left-wing journalism— and controversy—
was forged very early on. His father, Claud Cockburn, while
covering the Spanish Civil War for The Daily Worker , joined
the Republican forces fighting the rebellion of Francisco
Franco. (Claud Cockburn, under a pseudonym, also wrote
novels, including Beat the Devil , which was made into a
film with Humphrey Bogart and which his son used as the
title of his column in The Nation .)"

A video linked to in this evening's earlier post suggests
an antidote to the film oeuvre  of Guillermo del Toro—
in particular, to Pan's Labyrinth.

IMAGE- Google sidebar on 'Pan's Labyrinth' director

Ticket

Filed under: Uncategorized — m759 @ 7:00 PM

For a leftist who wrote well

Here's your ticket, pack your bags,
            time for jumpin' overboard 
Transportation is here 

Talking Heads lyric

Photo Opportunity

Filed under: Uncategorized — m759 @ 10:21 AM

The New York Times  online front page
update of 10:11 this morning provides
a sort of antidote to Saturday morning
cartoons.

The update's time suggests a check
of this  journal's most recent post
with the date  10/11. It turns out
to be a meditation on art and
the speed of perception.

Related material:

A linked-to post, Twenty-Four.

Saturday Morning Cartoon

Filed under: Uncategorized — m759 @ 9:00 AM

"A Saturday morning cartoon is the colloquial term
for the animated television programming that has
typically been scheduled on Saturday mornings
on the major American television networks from
the 1960s to the present…." —Wikipedia

Martin Gardner in the Notices of the
American Mathematical Society 
,
June/July 2005:

“I did a column in Scientific American 
on minimal art, and I reproduced one of
Ed Rinehart’s [sic ] black paintings. 
Of course, it was just a solid square of
pure black.”

Black square 256x256

Click on picture for details.

For a cartoon graveyard

IMAGE- LA Times obits for two Saturday Night Live writers

Friday, July 20, 2012

Speaking the Unspeakable

Filed under: Uncategorized — m759 @ 4:28 PM
 
http://www.latimes.com/news/politics/
la-pn-romney-colorado-shooting-is-
unspeakable-tragedy-20120720,0,1226295.story 

Romney: Colorado shooting is 'unspeakable tragedy'

By Seema Mehta 
July 20, 2012, 11:15 a.m.


BOW, N.H.—Hours after a shooter killed a dozen people
in a Colorado cinema, Mitt Romney scrubbed a scheduled
campaign rally Friday and instead offered his somber
condolences and prayers to the victims and their 
families.

“Our hearts break with the sadness of this unspeakable
tragedy. Ann and I join the president and first lady
and all Americans in offering our deepest condolences
to those whose lives were shattered in a few moments, 
a few moments of evil in Colorado,” Romney told a few
hundred people gathered at a lumber yard here.

“I stand before you today not as a man running for 
office but as a father and grandfather, a husband, an
American. This is a time for each of us to look into 
our hearts and remember how much we love one another 
and how much we love and how much we care for our great
country. There’s so much love and goodness in the heart
of America.”

Romney, wearing a navy blue suit and blue tie, spoke
for four minutes. The trademark campaign banners with
slogans such as “Believe in America” or “Obama’s Upside-
Down Economy” were gone, leaving a handful of American
flags as the backdrop. Before Romney spoke, Father 
Christian Tutor, an Anglican Catholic priest, led a 
prayer. ...


Caped Crusader

I Mean, Seriously…

Filed under: Uncategorized — m759 @ 8:00 AM

See also related juvenile humor, as well as Aurora in this  journal.

Thursday, July 19, 2012

But Seriously…

Filed under: Uncategorized — m759 @ 9:00 PM

For those who prefer fiction:

"Many Dimensions  (1931) — An evil antiquarian illegally purchases
the fabled Stone of Suleiman (Williams uses this Muslim form
rather than the more familiar King Solomon) from its Islamic guardian
in Baghdad and returns to England to discover not only that the Stone
can multiply itself infinitely without diminishing the original, but that it
also allows its possessor to transcend the barriers of space and time."

Wikipedia article on the author Charles Williams

Midsummer Nightmare

Filed under: Uncategorized — m759 @ 7:11 PM

A Passage to India… With Slides and Chanting

"Why art thou here, 
Come from the farthest Steppe of India?"

Midsummer Night's Dream 

"After graduating, Mr. Franken headed for Harvard,
while Mr. Davis chose the University of the Pacific
in Stockton, Calif., because, he said, he had heard
that it had a foreign study program in India, where
he hoped to smoke opium. (They did, and he did.)"

— Obituary of Saturday Night Live  writer Tom Davis
by Douglas Martin in this evening's online New York Times

"Frances Alenikoff, a dancer, choreographer and visual artist
whose performances often interwove movement with slides,
film, speaking, tape recordings and chant, died on June 23
in Southampton, N.Y. She was 91."

— Margalit Fox, online NY Times of July 8, 2012

Click for up-to-date context from the Times.

Un Coup de Dés

Filed under: Uncategorized — m759 @ 8:28 AM

IMAGE- NY lottery July 18, 2012: midday 926, 1261; evening 600, 0131.

Connoisseurs of hermeneutics will find that interpretations
of three of the above numbers are not hard to come by.

Consider, say, dates, post numbers in Log24, times, and so forth.

For the fourth number, 0131, try the following search:

http://www.m759.net/wordpress/?s=0131.

Wednesday, July 18, 2012

Joyce’s Nightmare

Filed under: Uncategorized — m759 @ 6:00 PM

"History, Stephen said…."

For a black widow —

See history in today's Boston.com
and Waldorf in this journal.

Incommensurables

Filed under: Uncategorized — m759 @ 9:48 AM

(Continued from Midsummer Eve)

"At times, bullshit can only be countered with superior bullshit."

— Norman Mailer, March 3, 1992, PBS transcript

"Just because it is a transition between incommensurables, the transition between competing paradigms cannot be made a step at a time, forced by logic and neutral experience. Like the gestalt switch, it must occur all at once (though not necessarily in an instant) or not at all."

Thomas Kuhn, The Structure of Scientific Revolutions , 1962, as quoted in The Enneagram of Paradigm Shifting

"In the spiritual traditions from which Jung borrowed the term, it is not the SYMMETRY of mandalas that is all-important, as Jung later led us to believe. It is their capacity to reveal the asymmetry that resides at the very heart of symmetry." 

The Enneagram as Mandala

I have little respect for Enneagram enthusiasts, but they do at times illustrate Mailer's maxim.

My own interests are in the purely mathematical properties of the number nine, as well as those of the next square, sixteen.

Those who prefer bullshit may investigate non-mathematical properties of sixteen by doing a Google image search on MBTI.

For bullshit involving nine, see (for instance) Einsatz  in this journal.

For non-bullshit involving nine, sixteen, and "asymmetry that resides at the very heart of symmetry," see Monday's Mapping Problem continued. (The nine occurs there as the symmetric  figures in the lower right nine-sixteenths of the triangular analogs  diagram.)

For non-bullshit involving psychological and philosophical terminology, see James Hillman's Re-Visioning Psychology .

In particular, see Hillman's "An Excursion on Differences Between Soul and Spirit."

Tuesday, July 17, 2012

For Black Widow

Filed under: Uncategorized — m759 @ 2:56 PM

I need a photo opportunity
I want a shot at redemption
Don't want to end up a cartoon
In a cartoon graveyard

— Paul Simon

See also the name Romanova
and the name Anastasia.

Putting the X in…

Filed under: Uncategorized — m759 @ 12:00 PM
 

Éxito

Or: The Death of Ivan Illich

See For David Lavery and the prequel Art Isn't Easy

"Time for you to see the field."

Hub

Filed under: Uncategorized — m759 @ 11:30 AM

GitHub

Got

Monday, July 16, 2012

Finishing Up at Noon

Filed under: Uncategorized — m759 @ 12:00 PM

(Continued.)

Jaws for Frank

Part I: October 8, 2010

http://www.log24.com/log/pix10B/101008-StartingOut.jpg

Above: Frank Langella in
"Starting Out in the Evening"

Right: Johnny Depp in
"The Ninth Gate"

http://www.log24.com/log/pix10B/101008-NinthGate.jpg

Part II: Noon Today

IMAGE- Moby Dick bites boat 'in twain' 

"The rest is the madness of art."

See also Patterns in the Carpets
and Saturday's Shadows.

Merit vs. Meritocracy

Filed under: Uncategorized — m759 @ 11:02 AM

The New York Times  online opinion today

"Merit has been traditionally equated with intelligence, industriousness, educational attainment, creativity and competency. In a meritocracy, formal qualifications provide opportunity, position is no longer ascribed by birth, and rewards flow to those who excel.

The rise of meritocratic competition as the preeminent means of social stratification in America has been hailed as a welcome advance because it replaced a society dominated by an upper class dependent on inherited wealth and status. The transition to meritocracy has, however, had unintended consequences. In the business sector, particularly, other less benign qualities emerge as essential to meritocratic success: aggressiveness, ruthlessness, dominance-seeking, victimizing behavior, acquisitiveness and the disciplined pursuit of self-interest." 

Journalism professor Thomas B. Edsall discussing remarks last December by Mitt Romney

Note the subtle shift here from "merit" to "meritocracy." Romney used the former word, not the latter.

Note also this sentence, aimed particularly at meritocratic New York Times  readers—

"In a meritocracy, formal qualifications provide opportunity… and rewards flow to those who excel."

Edsall lies. In a meritocracy, rewards flow to those who rubber-stamp "formal qualifications." See particularly Walter Kirn on meritocracy.

Edsall is pandering to Times  readers. Romney was pandering to a different group—

IMAGE- Mitt Romney Delivers Remarks to Republican Jewish Coalition

Mapping Problem continued

Filed under: Uncategorized — m759 @ 2:56 AM

Another approach to the square-to-triangle
mapping problem (see also previous post)—

IMAGE- Triangular analogs of the hyperplanes in the square model of PG(3,2)

For the square model referred to in the above picture, see (for instance)

Coordinates for the 16 points in the triangular arrays 
of the corresponding affine space may be deduced
from the patterns in the projective-hyperplanes array above.

This should solve the inverse problem of mapping,
in a natural way, the triangular array of 16 points 
to the square array of 16 points.

Update of 9:35 AM ET July 16, 2012:

Note that the square model's 15 hyperplanes S 
and the triangular model's 15 hyperplanes T —

— share the following vector-space structure —

   0     c     d   c + d
   a   a + c   a + d a + c + d
   b   b + c   b + d b + c + d
a + b a + b + c a + b + d   a + b + 
  c + d

   (This vector-space a b c d  diagram is from
   Chapter 11 of   Sphere Packings, Lattices
   and Groups
, by   John Horton Conway and
   N. J. A. Sloane, first published by Springer
   in 1988.)

Sunday, July 15, 2012

Mapping Problem

Filed under: Uncategorized — m759 @ 4:00 PM

A trial solution to the
square-to-triangle mapping problem

IMAGE- Mapping of square array to triangular array based on gnomons

Problem: Is there any good definition of "natural"
square-to-triangle mappings according to which
the above mapping is natural (or, for that matter,
un-natural)?

Squares Are Triangular

Filed under: Uncategorized — m759 @ 2:00 PM

"A figurate number… is a number
that can be represented by
a regular geometrical arrangement
of equally spaced points."

Eric W. Weisstein at Wolfram MathWorld

For example—

IMAGE- 16 points in a square array and in a triangular array

Call a convex polytope P  an n-replica  if  P  consists of
mutually congruent polytopes similar to P  packed together.

The square-triangle theorem (or lemma) says that

"Every triangle is an n-replica"
is true if and only if n  is a square.

Equivalently,

The positive integer n  is a square
if and only if every triangle is an n-replica.

(I.e., squares are triangular.)

This supplies the converse to the saying that

Triangles Are Square.

Saturday, July 14, 2012

Lemma

Filed under: Uncategorized — m759 @ 2:25 PM

IMAGE- 'Lemma (mathematics)' in Wikipedia

For example—

A letter to the editor of the American Mathematical Monthly
from the June-July 1985 issue has—

… a "square-triangle" lemma:

   ( t ∈ T , t  is an  -replica )
    if and only if  
n  is a square.

  [I.e., "Every triangle is an -replica"
   is true if and only if n  is a square.]

For definitions, see the 1985 letter in Triangles Are Square.

(The 1984 lemma discussed there has now, in response to an article
in Wolfram MathWorld, been renamed the square-triangle theorem .)

A search today for related material yielded the following—

"Suppose that one side of a triangle
has length . Then it can be cut
into n  2 congruent triangles which
are similar to the original one and
whose corresponding sides to the
side of length  have lengths 1."

This was supplied, without attribution, as part of the official solution
to Problem 3 in the 17th Asian Pacific Mathematics Olympiad
from March 2005. Apparently it seemed obvious to the composer
of the problem. As the 1985 letter notes, it may be not quite  obvious.

At any rate, it served in Problem 3 as a lemma , in the sense
described above by Wikipedia. See related remarks by Doron Zeilberger.

Shadows

Filed under: Uncategorized — m759 @ 11:01 AM

A tribute to Richard D. Zanuck in the style of Tim Burton—

Part I

IMAGE- G. H. Hardy around 1900, strongly resembling Paul McCartney

Part II

IMAGE- Beatles lyric 'There's a shadow hanging over me'

Part III

IMAGE- Richard D. Zanuck interview on 'Dark Shadows' film of director Tim Burton

The above Zanuck interview on Dark Shadows
was published on Midsummer Eve, June 23, 2012.

Also from June 23, 2012

IMAGE- 'Double Cross' symbol by Rudolf Koch from Midsummer Eve, 2012

Related material— Russell on Hardy.

Friday, July 13, 2012

Through

Filed under: Uncategorized — Tags: — m759 @ 5:48 PM

Oscar-winning producer
Richard Zanuck dies at 77

See also today's previous post,
Vanity Fair 's 1996 "Remains of the Dia,"
and english-for-students.com/dia.html.

Click to enlarge

Suddenly

Filed under: Uncategorized — Tags: — m759 @ 10:25 AM

1940 —

2003 —

"… Conceptualism — suddenly art
could be nothing more than an idea,
a thought on a piece of paper
that played in your head."

— Michael Kimmelman,
    "The Dia Generation,"
    The New York Times Magazine ,
    Sunday, April 6, 2003


Suddenly?

Thursday, July 12, 2012

Galois Space

Filed under: Uncategorized — Tags: — m759 @ 6:01 PM

An example of lines in a Galois space * —

The 35 lines in the 3-dimensional Galois projective space PG(3,2)—

(Click to enlarge.)

There are 15 different individual linear diagrams in the figure above.
These are the points of the Galois space PG(3,2).  Each 3-set of linear diagrams
represents the structure of one of the 35  4×4 arrays and also represents a line
of the projective space.

The symmetry of the linear diagrams accounts for the symmetry of the
840 possible images in the kaleidoscope puzzle.

* For further details on the phrase "Galois space," see
Beniamino Segre's "On Galois Geometries," Proceedings of the
International Congress of Mathematicians, 1958  
[Edinburgh].
(Cambridge U. Press, 1960, 488-499.)

(Update of Jan. 5, 2013— This post has been added to finitegeometry.org.)

Wednesday, July 11, 2012

Cuber

Filed under: Uncategorized — m759 @ 11:00 AM

(Continued)

For Pete Rustan, space recon expert, who died on June 28—

(Click to enlarge.)

See also Galois vs. Rubik and Group Theory Template.

Tuesday, July 10, 2012

Space Cadets

Filed under: Uncategorized — m759 @ 12:00 PM

From this journal on June 19, 2012

Walter Gropius on space—

"Was ist Raum, wie können wir ihn
 erfassen und gestalten?"

Walter Gropius,

The Theory and
Organization of the
Bauhaus
  (1923)

A book published on the same date—
June 19, 2012:

IMAGE- 'The Cryptos Conundrum,' by Chase Brandon

"… what Chalmers called the convergence of coincidence
a force majeure of unrelated events that shaped one's life,
that perhaps defined the concept of life itself.
He believed in the power of that force."

The Cryptos Conundrum , by Chase Brandon

See also Chase Brandon in Sunday's Huffington Post .

"I wrote another book."
— Robert De Niro as Harlan Kane

Euclid vs. Galois

Filed under: Uncategorized — m759 @ 11:01 AM

(Continued)

Euclidean square and triangle

Galois square and triangle

Background—

This journal on the date of Hilton Kramer's death,
The Galois Tesseract, and The Purloined Diamond.

Monday, July 9, 2012

Taps for Fatso

Filed under: Uncategorized — m759 @ 7:00 AM
 
IMAGE- Donna Reed and Montgomery Clift in 'From Here to Eternity'

A Story That Works

“There is the dark, eternally silent, unknown universe;
there are the friend-enemy minds shouting and whispering
their tales and always seeking the three miracles —

  • that minds should really touch, or
  • that the silent universe should speak, tell minds a story,
  • or (perhaps the same thing) that there should be a story
    that works, that is all hard facts, all reality, with
    no illusions and no fantasy;

and lastly, there is lonely, story-telling, wonder-questing,
mortal me.”

Dichtung ist Stiftung.

Sunday, July 8, 2012

Not Quite Obvious

Filed under: Uncategorized — m759 @ 11:00 AM

"That n 2 points fall naturally into a triangular array
is a not-quite-obvious fact which may have applications…
and seems worth stating more formally."

— Steven H. Cullinane, letter in the
American Mathematical Monthly 
1985 June-July issue

If the ancient Greeks had not been distracted by
investigations of triangular  (as opposed to square )
numbers, they might have done something with this fact.

A search for occurrences of the phrase

"n2 [i.e., n 2 ] congruent triangles" 

indicates only fairly recent (i.e., later than 1984) results.*

Some related material, updated this morning—

This suggests a problem
 

What mappings of a square  array of n 2 points to
a triangular  array of n 2 points are "natural"?

http://www.log24.com/log/pix12B/120708-SquareAndTriangle.jpg

In the figure above, whether
the 322,560 natural permutations
of the square's 16 points
map in any natural way to
  permutations of the triangle's 16 points
is not immediately apparent.

 

* Update of July 15, 2012 (11:07 PM ET)—

Theorem on " rep-" (Golomb's terminology)
triangles from a 1982 book—

IMAGE- Theorem (12.3) on Golomb and 'rep-k^2' triangles in book published in 1982-- 'Transformation Geometry,' by George Edward Martin

Saturday, July 7, 2012

Quartet

Filed under: Uncategorized — m759 @ 1:23 PM

"Euclid (Ancient Greek: Εὐκλείδης Eukleidēs), fl. 300 BC, 
also known as Euclid of Alexandria, was a Greek
mathematician, often referred to as the 'Father of Geometry.'"

— Wikipedia

A Euclidean quartet (see today's previous post)—

IMAGE- Triangle cut into four congruent subtriangles
Image by Alexander Soifer

See also a link from June 28, 2012, to a University Diaries  post
discussing "a perfection of thought."

Perfect means, among other things, completed .

See, for instance, the life of another Alexandrian who reportedly
died on the above date—

"Gabriel Georges Nahas was born in Alexandria, Egypt, on
 March 4, 1920…."

 — This afternoon's online New York Times

Étude

Filed under: Uncategorized — 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).

Thursday, July 5, 2012

Shard

Filed under: Uncategorized — Tags: — m759 @ 11:32 AM

A post suggested by an article on The Shard of London
in this morning's Wall Street Journal—

IMAGE- Video, 'Unknown. Liam Neeson Tribute,' 3:27 of 3:54, Hand with Shard

As for the "Personal Jesus" song that accompanies the above video tribute,
listen to Liam Neeson as the voice of Aslan in recent Narnia  films
and consider the saying of C. S. Lewis that Aslan is not a tame  lion.

Here Lewis may, if one likes, be regarded as the "inkling" of Heidegger
in last night's post—

Claves Regni Caelorum

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

Or: Night of Lunacy

From 9 PM Monday

IMAGE- Page 304 of Heidegger's 'Existence and Being' - Heidegger's essay 'Hölderlin and the Essence of Poetry,' tr. by Douglas Scott, publ. by Henry Regnery Company, Chicago, in 1949
Note that the last line, together with the page number, forms
a sort of key

The rest  of the story—

IMAGE- Heidegger quote continued, ending with reference to Hölderlin's 'night of lunacy'

For one reinterpretation of the page number 304, see a link— 
Sermon— from Tuesday's post Diamond Speech.

The linked-to sermon itself has a link, based on a rereading
of 304 as 3/04, to a post of March 4, 2004, with…

WW and ZZ

as rendered by figures from the Kaleidoscope Puzzle

     .

Yesterday morning the same letter-combinations occurred
in a presentation at CERN of a newly discovered particle—

IMAGE- 'High mass: WW, ZZ'

(Click for context.)

Since the particle under discussion may turn out to be the
God  particle, it seems fitting to interpret WW and ZZ as part
of an imagined requiem  High Mass.

Ron Howard, director of a film about CERN and the God particle,
may regard this imaginary Mass as performed for the late
Andy Griffith, who played Howard's father in a television series.

Others may prefer to regard the imaginary Mass as performed 
for the late John E. Brooks, S. J., who served as president of
The College of the Holy Cross, Worcester, Mass., for 24 years.

Griffith died Tuesday. Brooks died Monday.

For some background on the Holy Cross, see posts of
Sept. 14 (Holy Cross Day) and Sept. 15, 2010—

  1. Language Game,
  2. Wittgenstein, 1935, and
  3. Holy Cross Day Revisited.

For more lunacy, see…

Continue a search for thirty-three and three
— Katherine Neville, The Eight

Wednesday, July 4, 2012

Physical Poetry

Filed under: Uncategorized — m759 @ 9:14 PM

For a White House daughter…

See Sweater and Light.

From this evening's Capitol Fourth

She goes off with a great big bang
Boys I tell ya it's a beautiful thing

Related material— Big Bang  in this journal—

http://www.log24.com/log/pix11C/111005-PlancksCafe-CruzEnters.jpg

Detective Cruz enters Planck's Constant Café in "The Big Bang."

Poetic Physics

Filed under: Uncategorized — m759 @ 8:00 AM

IMAGE- July 4th Google search for Heuer + 'The Missing Cornerstone'

    See also Symbology, The Lost Cornerstone, and Now, Here's My Plan.

Western Union

Filed under: Uncategorized — m759 @ 2:02 AM

A story in two parts —

Part I (April 1986) and Part II (June 2012).

Tuesday, July 3, 2012

Calling

Filed under: Uncategorized — m759 @ 3:28 PM

IMAGE- Andy Griffith dead at 86 (Google News top story)

IMAGE- 'If you've got a message, call Western Union' --Clifford Odets, according to Harvard Crimson

IMAGE- Video: '1957 Face in the Crowd /message in movie'

—Rhodes, I want you to get to know people like that.
 I'd like to sort of take you under my wing and educate you.

—Shucks, General, I'm just a country boy.

Diamond Speech

Filed under: Uncategorized — m759 @ 6:23 AM

IMAGE- Resignation of Robert Diamond as Barclays CEO

"And when I think about the values
that are important to me today,
I think first about meritocracy."

Robert Diamond, Colby College '73, now
Chair of the Colby College Board of Trustees, in a
commencement address on Sunday, May 25, 2008

Other remarks on that Sunday —

Related material from Colby—

IMAGE- Colby College page on mathematician Fernando Q. Gouvea

See also an MAA report on Gouvea from June 6, 2012.

Monday, July 2, 2012

Stiftung

Filed under: Uncategorized — m759 @ 9:00 PM

Heidegger, "Hölderlin and the Essence of Poetry,"
translated by Douglas Scott, in Existence and Being  ,
Regnery, 1949, pp. 291-316—

IMAGE- Page 304 of Heidegger's 'Existence and Being' - Heidegger's essay 'Hölderlin and the Essence of Poetry,' tr. by Douglas Scott, publ. by Henry Regnery Company, Chicago, in 1949

See also Hexagram 36.

Hexagram 44 Revisited

Filed under: Uncategorized — m759 @ 12:00 PM

Images from a Google search suggested by
last night's post Coming to Meet, by the recent
film "Archie's Final Project," and by a Thursday,
June 28, 2012, Times Higher Education  piece, 
"Raiders of the Lost Archives"—

IMAGE- Diana Rigg in 'The Hospital' and in 'The Avengers'

    Log24, December 8, 2008 —

David Carradine displays a yellow book-- the Princeton I Ching.

"Let the fingers do the research."
          — Archive Raiders

Sunday, July 1, 2012

Shining Night*

Filed under: Uncategorized — m759 @ 11:59 PM
 

This evening's online New York Times

Doris Sams, Pro Baseball Star, Dies at 85

By RICHARD GOLDSTEIN

Published: July 1, 2012

Doris Sams, who pitched a perfect game and set a single-season home run record in the women’s professional baseball world of the 1940s and 50s that inspired the movie “A League of Their Own,” died Thursday in Knoxville, Tenn. She was 85. more>>

"High summer holds the earth." 

James Agee, quoted in a post by University Diaries
linked to here on Thursday.

See also For Taylor and Country Strong.

* A phrase from the quoted Agee poem

Coming to Meet

Filed under: Uncategorized — m759 @ 10:30 PM

Hexagram 44:
Coming to Meet

"This hexagram indicates a situation in which
the principle of darkness, after having been eliminated,
furtively and unexpectedly obtrudes again from within
and below. Of its own accord the female principle
comes to meet the male. It is an unfavourable and
dangerous situation, and we must understand and
promptly prevent the possible consequences.

The hexagram is linked with the fifth month
[June-July], because at the summer solstice
the principle of darkness gradually becomes
ascendant again."

— Richard Wilhelm 

To counteract the principle of darkness—

The Uploading (Friday— St. Peter's Day, 2012),

Thor's Light Bulb Joke, and …

 .

Parables

Filed under: Uncategorized — Tags: — m759 @ 2:48 AM

Death and the Compass  and The Library of Babel .

Related material in this journal—

Wag the Dogma and Widener.

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