Log24

Sunday, June 16, 2019

Master Plan from Outer Space

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

IMAGE- The large Desargues configuration and Desargues's theorem in light of Galois geometry

Sunday, February 17, 2019

For the Schoolgirls of 1959

Filed under: General — Tags: — m759 @ 1:35 PM

The dies natalis  of St. Buddy Holly was Feb. 3, 1959.

This  year on Feb. 3, a geometric illustration of the well-known
schoolgirl problem was added to a brand-new Wikipedia article
on the finite geometry PG(3,2).

Wednesday, September 13, 2017

Summer of 1984

Filed under: General,Geometry — Tags: , — m759 @ 9:11 AM

The previous two posts dealt, rather indirectly, with
the notion of "cube bricks" (Cullinane, 1984) —

Group actions on partitions —

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168

Another mathematical remark from 1984 —

For further details, see Triangles Are Square.

Tuesday, April 4, 2017

Plan 9 Continues

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

"Plan 9 deals with the resurrection of the dead."

— Bill Murray in "Ed Wood"
 

For The Church of Plan 9

(The plan , as well as the elevation ,
of the above structure is a 3×3 grid.)

Wednesday, June 24, 2015

Scoring Plan 9

Filed under: General — Tags: , — m759 @ 6:33 PM

In keeping with the resurrection themes of the
previous post and of "Plan 9 from Outer Space,"
here is a link to the soundtrack of "Field of Dreams."

Related material:

A post of March 11, 2014, on
truth, cornfields, and Rebecca Goldstein —
Dark Fields of the Republic.

R.I.P., James Horner.

Sunday, November 30, 2014

Two Physical Models of the Fano Plane

Filed under: General,Geometry — Tags: , — m759 @ 1:23 AM

The Regular Tetrahedron

The seven symmetry axes of the regular tetrahedron
are of two types: vertex-to-face and edge-to-edge.
Take these axes as the "points" of a Fano plane.
Each of the tetrahedron's six reflection planes contains 
two vertex-to-face axes and one edge-to-edge axis.
Take these six planes as six of the "lines" of a Fano
plane. Then the seventh line is the set of three 
edge-to-edge axes.

(The Fano tetrahedron is not original with me.
See Polster's 1998 A Geometrical Picture Book pp. 16-17.)

The Cube

There are three reflection planes parallel to faces
of the cube. Take the seven nonempty subsets of
the set of these three planes as the "points" of a
Fano plane. Define the Fano "lines" as those triples
of these seven subsets in which each member of
the triple is the symmetric-difference sum of the 
other two members.

(This is the eightfold cube  discussed at finitegeometry.org.)

Wednesday, November 26, 2014

A Tetrahedral Fano-Plane Model

Filed under: General,Geometry — Tags: — m759 @ 5:30 PM

Update of Nov. 30, 2014 —

It turns out that the following construction appears on
pages 16-17 of A Geometrical Picture Book , by 
Burkard Polster (Springer, 1998).

"Experienced mathematicians know that often the hardest
part of researching a problem is understanding precisely
what that problem says. They often follow Polya's wise
advice: 'If you can't solve a problem, then there is an
easier problem you can't solve: find it.'"

—John H. Conway, foreword to the 2004 Princeton
Science Library edition of How to Solve It , by G. Polya

For a similar but more difficult problem involving the
31-point projective plane, see yesterday's post
"Euclidean-Galois Interplay."

The above new [see update above] Fano-plane model was
suggested by some 1998 remarks of the late Stephen Eberhart.
See this morning's followup to "Euclidean-Galois Interplay" 
quoting Eberhart on the topic of how some of the smallest finite
projective planes relate to the symmetries of the five Platonic solids.

Update of Nov. 27, 2014: The seventh "line" of the tetrahedral
Fano model was redefined for greater symmetry.

Tuesday, September 16, 2014

Plan 9

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

(Continued from St. Augustine's Day, 2012)

"Plan 9 deals with the resurrection of the dead."

Epigraph to "No Great Magic," a story by Fritz Leiber:

 To bring the dead to life
Is no great magic.
Few are wholly dead:
Blow on a dead man’s embers
And a live flame will start.


—Graves

Friday, September 11, 2009

Friday September 11, 2009

Filed under: General — Tags: — m759 @ 1:00 PM
For 9/11

Cover of 'Underworld,' by Don DeLillo, First Edition, Advance Reader's Copy, 1997

 

Cover of Underworld,
 by Don DeLillo, First Edition,
 Advance Reader's Copy, 1997

"Time and chance
happeneth to them all."
Ecclesiastes 9:11  

Related material:

1. The previous entry, on
  Copenhagen physicist
Aage Bohr, and      
2. Notes from this journal
 from Bohr's birthday,
 June 19th, through  
        Midsummer Night, 2007…
 including notes on   
  Faust in Copenhagen
   3. Walpurgisnacht 2008 and
 Walpurgisnacht 2009

Friday September 11, 2009

Filed under: General,Geometry — Tags: — m759 @ 2:56 AM
Theology
 in memory of
physicist Aage Bohr,
who died at 87 on
Tuesday, Sept. 8, 2009

Swarthmore physics honors thesis, 136 pp., 2007–

Abstract:

"Quantum mechanics, which has no completely accepted interpretation but many seemingly strange physical results, has been interpreted in a number of bizarre and fascinating ways over the years. The two interpretations examined in this paper, [Aage] Bohr and [Ole] Ulfbeck's 'Genuine Fortuitousness' and Stuckey, Silberstein, and Cifone's 'Relational Blockworld,' seem to be two such strange interpretations; Genuine Fortuitousness posits that causality is not fundamental to the universe, and Relational Blockworld suggests that time does not act as we perceive it to act. In this paper, I analyze these two interpretations…."

Footnote 55, page 114:
 
"Thus far, I have been speaking in fairly abstract terms, which can sometimes be unhelpful on the issue of explaining anything about the structure of space-time. I want to pause for a moment to suggest a new potential view of the blockworld within a 'genuinely fortuitous' universe in more visual terms. Instead of the 'static spacetime jewel' of blockworld that is often invoked by eternalists to help their readers conceptualize of what a blockworld would 'look like' from the outside, now imagine that a picture on a slide is being projected onto the surface of this space-time jewel."

Interpolated figure
from Log24:

 

Juliette Binoche in 'Blue'  The 24 2x2 Cullinane Kaleidoscope animated images

Cf. August 5, 2009.


From the perspective of one inside the jewel, one might ask 'Why is this section blue while this section is black?,' and from within the jewel, one could not formulate an answer since one could not see the entire picture projected on the jewel; however, from outside the jewel, an observer (some analogue of Newton's God, perhaps, looking down on his 'sensorium' from the 5th dimension) could easily see the pattern and understand that all of the 'genuinely fortuitous' events inside the space-time jewel are, in fact, completely determined by the pattern in the projector."

— "Genuine Fortuitousness, Relational Blockworld, Realism, and Time" (pdf), by Daniel J. Peterson, Honors Thesis, Swarthmore College, December 13, 2007

Monday, September 7, 2009

Monday September 7, 2009

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

Magic Boxes

"Somehow it seems to fill my head with ideas– only I don't exactly know what they are!…. Let's have a look at the garden first!"

— A passage from Lewis Carroll's Through the Looking-Glass. The "garden" part– but not the "ideas" part– was quoted by Jacques Derrida in Dissemination in the epigraph to Chapter 7, "The Time before First."

Commentary
 on the passage:

Part I    "The Magic Box,"  shown on Turner Classic Movies earlier tonight

Part II: "Mimsy Were the Borogoves," a classic science fiction story:

"… he lifted a square, transparent crystal block, small enough to cup in his palm– much too small to contain the maze of apparatus within it. In a moment Scott had solved that problem. The crystal was a sort of magnifying glass, vastly enlarging the things inside the block. Strange things they were, too. Miniature people, for example– They moved. Like clockwork automatons, though much more smoothly. It was rather like watching a play."

Part III:  A Crystal Block

Cube, 4x4x4

Four coloring pencils, of four different colors

Image of pencils is by
Diane Robertson Design.

Related material:
"A Four-Color Theorem."

Part IV:

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

"Click on the Yellow Book."

Saturday, September 5, 2009

Saturday September 5, 2009

Filed under: General,Geometry — Tags: — m759 @ 10:31 PM
For the
Burning Man

'The Stars My Destination,' current edition (with cover slightly changed)

(Cover slightly changed.)

 
Background —

 
SAT
 
Part I:

Sophists (August 20th)

Part II:

VERBUM
SAT
SAPIENTI

Escher's 'Verbum'

Escher's Verbum


Solomon's Cube



Part III:

From August 25th

Equilateral triangle on a cube, each side's length equal to the square root of two

"Boo, boo, boo,
  square root of two.
"

Wednesday, August 19, 2009

Wednesday August 19, 2009

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

Group Actions, 1984-2009

From a 1984 book review:

"After three decades of intensive research by hundreds of group theorists, the century old problem of the classification of the finite simple groups has been solved and the whole field has been drastically changed. A few years ago the one focus of attention was the program for the classification; now there are many active areas including the study of the connections between groups and geometries, sporadic groups and, especially, the representation theory. A spate of books on finite groups, of different breadths and on a variety of topics, has appeared, and it is a good time for this to happen. Moreover, the classification means that the view of the subject is quite different; even the most elementary treatment of groups should be modified, as we now know that all finite groups are made up of groups which, for the most part, are imitations of Lie groups using finite fields instead of the reals and complexes. The typical example of a finite group is GL(n, q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled."

— Jonathan L. Alperin,
   review of books on group theory,
   Bulletin (New Series) of the American
   Mathematical Society
10 (1984) 121, doi:
   10.1090/S0273-0979-1984-15210-8
 

A more specific example:


Actions of GL(2,3) on a 3x3 coordinate-array

The same example
at Wolfram.com:

Ed Pegg Jr.'s program at Wolfram.com to display a large number of actions of small linear groups over finite fields

Caption from Wolfram.com:
 
"The two-dimensional space Z3×Z3 contains nine points: (0,0), (0,1), (0,2), (1,0), (1,1), (1,2), (2,0), (2,1), and (2,2). The 48 invertible 2×2 matrices over Z3 form the general linear group known as GL(2, 3). They act on Z3×Z3 by matrix multiplication modulo 3, permuting the nine points. More generally, GL(n, p) is the set of invertible n×n matrices over the field Zp, where p is prime. With (0, 0) shifted to the center, the matrix actions on the nine points make symmetrical patterns."

Citation data from Wolfram.com:

"GL(2,p) and GL(3,3) Acting on Points"
 from The Wolfram Demonstrations Project,
 http://demonstrations.wolfram.com/GL2PAndGL33ActingOnPoints/,
 Contributed by: Ed Pegg Jr"

As well as displaying Cullinane's 48 pictures of group actions from 1985, the Pegg program displays many, many more actions of small finite general linear groups over finite fields. It illustrates Cullinane's 1985 statement:

"Actions of GL(2,p) on a p×p coordinate-array have the same sorts of symmetries, where p is any odd prime."

Pegg's program also illustrates actions on a cubical array– a 3×3×3 array acted on by GL(3,3). For some other actions on cubical arrays, see Cullinane's Finite Geometry of the Square and Cube.
 

Monday, August 10, 2009

Monday August 10, 2009

Filed under: General — Tags: — m759 @ 9:29 PM
For Maine Preacher
Stephen King

Union colonel Joshua Chamberlain, on the way to the battle at Gettysburg, remembers his boyhood.

"Maine… is silent and cold.

Maine in the winter: air is darker, the sky is a deeper dark. A darkness comes with winter that these Southern people don't know. Snow falls so much earlier and in the winter you can walk in a snowfield among bushes, and visitors don't know that the bushes are the tops of tall pines, and you're standing in thirty feet of snow. Visitors. Once long ago visitors in the dead of winter: a preacher preaching hell-fire. Scared the fool out of me. And I resented it and Pa said I was right.

Pa.

When he thought of the old man he could see him suddenly in a field in the spring, trying to move a gray boulder. He always knew instinctively the ones you could move, even though the greater part was buried in the earth, and he expected you to move the rock and not discuss it. A hard and silent man, an honest man, a noble man. Little humor but sometimes the door opened and you saw the warmth within a long way off, a certain sadness, a slow, remote, unfathomable quality as if the man wanted to be closer to the world but did not know how. Once Chamberlain had a speech memorized from Shakespeare and gave it proudly, the old man listening but not looking, and Chamberlain remembered it still: 'What a piece of work is man… in action how like an angel!' And the old man, grinning, had scratched his head and then said stiffly, 'Well, boy, if he's an angel, he's sure a murderin' angel.' And Chamberlain had gone on to school to make an oration on the subject: Man, the Killer Angel. And when the old man heard about it he was very proud, and Chamberlain felt very good remembering it."

— Michael Shaara, The Killer Angels: A Novel of the Civil War

 

Monday August 10, 2009

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

Pictures Within Pictures

"The Chinese language is written in ideograms, pictures. Think of a DO NOT ENTER pictogram, a circle with a diagonal slash, a type of ideogram. It tells you what to do or not do, but not why. The why is part of a larger context, a bigger picture. Such is the nature of the Chinese language. Simple yet complex. Pictures within pictures."

Customer review at Amazon.com

See also the pictures in this journal on today's date five years ago.

Sunday, August 9, 2009

Sunday August 9, 2009

Filed under: General — Tags: — m759 @ 4:00 PM
 
WIVB TV Buffalo, 2:56 PM ET Aug. 9, 2009, Severe Weather Statement accompanied by Instant Action 'Ace of Aces' aerial dogfight game

 

See also "Chautauqua"
at Stormfront.org and
 the five entries ending with
"Unfriendly Persuasion"
this morning.

Background:
"Today's Sinner"

Sunday August 9, 2009

Filed under: General — Tags: — m759 @ 9:11 AM

Unfriendly Persuasion

"What disturbs Americans of all ideological persuasions is the fear that almost everything, not just government, is fixed or manipulated by some powerful hidden hand…."

Frank Rich in today's New York Times

Franz Liebkind (played by Kenneth Mars) in 'The Producers,' with helmet and pigeon

Author! Author!

Sunday August 9, 2009

Filed under: General — Tags: — m759 @ 2:45 AM

On the Waterfront

"On the Hoboken waterfront, people scattered as pieces of debris fell from the sky. A wheel from one of the aircraft lay on Hoboken's Sinatra Drive." —Associated Press, August 8, 2009

So set 'em up, Joe…

Thursday, August 6, 2009

Thursday August 6, 2009

Filed under: General,Geometry — Tags: — m759 @ 1:44 PM
A Fisher of Men
 
 
Cover, Schulberg's novelization of 'Waterfront,' Bantam paperback
Update: The above image was added
at about 11 AM ET Aug. 8, 2009.

 
Dove logo, First United Methodist Church of Bloomington, Indiana

From a webpage of the First United Methodist Church of Bloomington, Indiana–

 

Dr. Joe Emerson, April 24, 2005–

"The Ultimate Test"

— Text: I Peter 2:1-9

Dr. Emerson falsely claims that the film "On the Waterfront" was based on a book by the late Budd Schulberg (who died yesterday). (Instead, the film's screenplay, written by Schulberg– similar to an earlier screenplay by Arthur Miller, "The Hook"–  was based on a series of newspaper articles by Malcolm Johnson.)

"The movie 'On the Waterfront' is once more in rerun. (That’s when Marlon Brando looked like Marlon Brando.  That’s the scary part of growing old when you see what he looked like then and when he grew old.)  It is based on a book by Budd Schulberg."

 

Emerson goes on to discuss the book, Waterfront, that Schulberg wrote based on his screenplay–

"In it, you may remember a scene where Runty Nolan, a little guy, runs afoul of the mob and is brutally killed and tossed into the North River.  A priest is called to give last rites after they drag him out."

 

Hook on cover of Budd Schulberg's novel 'Waterfront' (NY Times obituary, detail)

New York Times today

Dr. Emerson flunks the test.

 

Dr. Emerson's sermon is, as noted above (Text: I Peter 2:1-9), not mainly about waterfronts, but rather about the "living stones" metaphor of the Big Fisherman.

My own remarks on the date of Dr. Emerson's sermon

The 4x6 array used in the Miracle Octad Generator of R. T. Curtis

Those who like to mix mathematics with religion may regard the above 4×6 array as a context for the "living stones" metaphor. See, too, the five entries in this journal ending at 12:25 AM ET on November 12 (Grace Kelly's birthday), 2006, and today's previous entry.

Thursday August 6, 2009

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

The Running

"Budd Schulberg, who wrote the award-winning screenplay for 'On the Waterfront' and created a classic American archetype of naked ambition, Sammy Glick, in his novel What Makes Sammy Run?, died on Wednesday. He was 95…."

Running man with blue background on the cover of 'Eye of Cat,' by Roger Zelazny

Log24, Dec. 16, 2003:

See, too, Blue Matrices, and
a link for Beethoven's birthday:

Juliette Binoche with musical score from Kieslowski's 'Blue'

Song for the
Unification of Europe

Wednesday, July 15, 2009

Wednesday July 15, 2009

Filed under: General — Tags: — m759 @ 11:09 AM
The Plot Thickens

Thanks to David Lavery
see previous entry— the
word for today is…
 
Cover of 'Zaddik,' a novel by David Rosenbaum

"As the story develops, an
 element of magical realism
 enters the picture."
Amazon review   

Related material:

For background on magical
realism, see the update to
today's previous entry.

See also
A Year of Magical Thinking
(June 6, 2009) and
the entries of May 19-22,
featuring Judy Davis in…

Poster for 'Diamonds' miniseries on ABC starting May 24, 2009

(Cf. St. Bridget's Day, 2003)

Wednesday July 15, 2009

Filed under: General — Tags: — m759 @ 4:01 AM
DETAIL
of obituaries page,
New York Times,
Monday morning:

Detail, obits page, NY Times Monday morning, July 13, 2009

Detail of arts page,
New York Times, Wednesday morning:

(Click ad for more on the Monday night death of Dash Snow.)

Arts page detail, morning of Wednesday, July 15, 2009

Headlines collage by Dash Snow

Hurt yet?
_________________

Update of 5:01 AM:

Lavery Hits
Literary Jackpot

From the top right of
this morning's online
New York Times front page:

Christoph Niemann on witchcraft and snow

Click on voodoo doll
for further details.

See also…

1. Monday's link to a
Wallace Stevens poem,
"Snow and Stars"

2. The conclusion of this
morning's Times obituary
for artist Dash Snow, which
gives his daughter's name…
"Secret."

3. David Lavery's excellent
analysis
of the classic
 Conrad Aiken story
"Silent Snow, Secret Snow."
 

Wednesday, June 17, 2009

Wednesday June 17, 2009

Filed under: General — Tags: — m759 @ 4:30 AM

Back to the Real

Colum McCann on yesterday’s history:

“Fiction gives us access to a very real history.”

The Associated Press thought for today:

“Journalism allows its readers to witness history; fiction gives its readers an opportunity to live it.”

— John Hersey, American author (born on this date in 1914, died 1993).

From John Hersey’s The Child Buyer (1960):

“I was wondering about that this morning… About forgetting. I’ve always had an idea that each memory was a kind of picture, an insubstantial picture. I’ve thought of it as suddenly coming into your mind when you need it, something you’ve seen, something you’ve heard, then it may stay awhile, or else it flies out, then maybe it comes back another time…. If all the pictures went out, if I forgot everything, where would they go? Just out into the air? Into the sky? Back home around my bed, where my dreams stay?”

“We keep coming back and coming back
To the real: to the hotel instead of the hymns….”

— Wallace Stevens

Hotel Bella Vista, Cuernavaca, Morelos, Mexico

Postcard from eBay
From Under the Volcano, by Malcolm Lowry, 1947, Chapter I: 

Faustus is gone: regard his hellish fall —
Shaken, M. Laruelle replaced the book on the table… he reached to the floor for a folded sheet of paper that had fluttered out of it. He picked the paper up between two fingers and unfolded it, turning it over. Hotel Bella Vista, he read. There were really two sheets of uncommonly thin hotel notepaper….

I sit now in a little room off the bar at four-thirty in the morning drinking ochas and then mescal and writing this on some Bella Vista notepaper I filched the other night…. But this is worst of all, to feel your soul dying. I wonder if it is because to-night my soul has really died that I feel at the moment something like peace. Or is it because right through hell there is a path, as Blake well knew, and though I may not take it, sometimes lately in dreams I have been able to see it? …And this is how I sometimes think of myself, as a great explorer who has discovered some extraordinary land from which he can never return to give his knowledge to the world: but the name of this land is hell. It is not Mexico of course but in the heart.

Thursday, June 4, 2009

Thursday June 4, 2009

Filed under: General — Tags: , — m759 @ 1:24 PM

The Grasshopper
Lies Heavy

David Carradine dies at 72

“‘Oracle, why did you write
The Grasshopper Lies Heavy?
What are we supposed to learn?'”

— Philip K. Dick

She began throwing the coins.

I Ching Hexagram 61: Inner Truth

Click on image
for further details.

Thursday June 4, 2009

Filed under: General — Tags: — m759 @ 10:30 AM
A Passage to Egypt
 

The First Draft: Reviews flood in
after Obama's Cairo speech

The conclusion:
a tribute to E. M. Forster

Test-- 'To prove you're a person and not a script'-- type the word 'connect.'

(Image only; not for use.)

Those who prefer spam scripts
to persons may consult
the entry from midnight.
 

Thursday June 4, 2009

Filed under: General,Geometry — Tags: — m759 @ 12:00 AM
Steps
continued from
October 16, 2008
 

New collection release:
Pattern in Islamic Art
from David Wade

October 16, 2008

David Wade has partnered with ARTstor to distribute approximately 1,500 images of Islamic art, now available in the Digital Library. These images illustrate patterns and designs found throughout the Islamic world, from the Middle East and Europe to Central and South Asia. They depict works Wade photographed during his travels, as well as drawings and diagrams produced for publication. Reflective of Wade's particular interest in symmetry and geometry, these images analyze and break down common patterns into their basic elements, thereby revealing the underlying principles of order and balance in Islamic art. Islamic artists and craftsmen employed these intricate patterns to adorn all types of surfaces, such as stone, brick, plaster, ceramic, glass, metal, wood, and textiles. The collection contains examples of ornamentation from monumental architecture to the decorative arts.

To view the David Wade: Pattern in Islamic Art collection: go to the ARTstor Digital Library, browse by collection, and click "David Wade: Pattern in Islamic Art;" or enter the Keyword Search: patterninislamicart.

For more detailed information about this collection, visit the David Wade: Pattern in Islamic Art collection page.

 
The above prose illustrates
the institutional mind at work.

Those who actually try to view
the Wade collection will
encounter the following warning:

To access the images in the ARTstor Digital Library you need to be affiliated with a participating institution (university, college, museum, public library or K-12 school).
You say
"go to the ARTstor Digital Library,"
I say
"theatlantic.com/doc/200305/lewis."
 

Tuesday, May 19, 2009

Tuesday May 19, 2009

Filed under: General,Geometry — Tags: , — m759 @ 7:20 PM
Exquisite Geometries

"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."

"Block Designs," 1995, by Andries E. Brouwer

"The Steiner system S(5, 8, 24) is a set S of 759 eight-element subsets ('octads') of a twenty-four-element set T such that any five-element subset of T is contained in exactly one of the 759 octads. Its automorphism group is the large Mathieu group M24."

The Miracle Octad Generator (MOG) of R.T. Curtis (webpage)

"… in 1861 Mathieu… discovered five multiply transitive permutation groups…. In a little-known 1931 paper of Carmichael… they were first observed to be automorphism groups of exquisite finite geometries."

William M. Kantor, 1981

The 1931 paper of Carmichael is now available online from the publisher for $10.
 

Saturday, April 25, 2009

Saturday April 25, 2009

Filed under: General,Geometry — Tags: — m759 @ 9:22 PM
State of Play

Russell Crowe in 'State of Play'

The Russell Crowe
Hotel Puzzle

by John Tierney

"Russell Crowe arrives at the Hotel Infinity looking tired and ornery. He demands a room. The clerk informs him that there are no vacancies…."

Footprints from California today
(all by a person or persons using Firefox browsers):

7:10 AM
http://m759.xanga.com/679142359/concepts-of-space/?
Concepts of Space: Euclid vs. Galois

8:51 AM
http://m759.xanga.com/689601851/art-wars-continued/?
Art Wars continued: Behind the Picture

1:33 PM
http://m759.xanga.com/678995132/a-riff-for-dave/?
A Riff for Dave: Me and My Shadow

2:11 PM
http://m759.xanga.com/638308002/a-death-of-kings/?
A Death of Kings: In Memory of Bobby Fischer

2:48 PM
http://m759.xanga.com/691644175/art-wars-in-review–/?
Art Wars in review– Through the Looking Glass: A Sort of Eternity

3:28 PM and
http://m759.xanga.com/684680406/annals-of-philosophy/?
Annals of Philosophy: The Dormouse of Perception

4:28 PM
http://m759.xanga.com/641536988/epiphany-for-roy-part-i/?
Epiphany for Roy, Part I

6:03 PM
http://m759.xanga.com/641949564/art-wars-continued/?
At the Still Point: All That Jazz

6:22 PM
http://m759.xanga.com/644330798/where-entertainment-is-not-god/?
Where Entertainment is Not God: The Just Word

7:14 PM
http://m759.xanga.com/643490468/happy-new-yorker-day/?
Happy New Yorker Day– Class Galore

7:16 PM
http://m759.xanga.com/643812753/the-politics-of-change/?
The Politics of Change: Jumpers
 

"Relax," said the night man.
"We are programmed to receive."
— Hotel California
 

Saturday April 25, 2009

Filed under: General — Tags: — m759 @ 11:09 AM
April is Awareness Month

for both

Mathematics and Autism.

Welcome to the
Black Hole Café

"Our lifelong friendship made me not only an admirer of the depth, scholarship, and sheer energy of his mathematical work (and of his ceaseless activities as an editorial entrepreneur on behalf of mathematics) but one in awe of his status as the ultimate relaxed sophisticate."
 

The late Jacob T. Schwartz 
  on Gian-Carlo Rota

Psychoshop

by Alfred Bester
and Roger Zelazny:

His manner was all charm and grace; pure café society….

He purred a chuckle. "My place. If you want to come, I'll show you."

"Love to. The Luogo Nero? The Black Place?"

"That's what the locals call it. It's really Buoco Nero, the Black Hole."

"Like the Black Hole of Calcutta?"

"No. Black Hole as in astronomy. Corpse of a dead star, but also channel between this universe and its next-door neighbor."


"After Davis and Hersh,
it will be hard to uphold
the Glasperlenspiel
view of mathematics."
— Gian-Carlo Rota  

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

http://www.log24.com/log/pix09/090425-AutismPuzzlePiece.jpg

AutismGear.com
 

Thursday, April 23, 2009

Thursday April 23, 2009

Filed under: General — Tags: — m759 @ 5:24 PM
Star Quality

Eight-pointed star, background image for the E! Online logo

This deliberately cryptic entry is to thank an anonymous reader in Sweden for the following footprint:

Sweden
Speedy
…&uid=37798719 4/23/2009
4:33 PM

“Speedy” is the browser name supplied to the server. The link is to a Columbus Day, 2003, entry with the song phrase “spinnin’ wheel, spinnin’ true.” The time is Eastern Daylight.

Related material:

Vide today’s midday PA lottery number, 177, the 1919 edition of The Oxford Book of English Verse, and the time (interpreted, in a Joycean manner, as a date) of this morning’s first entry.

Happy birthday to Judy Davis
and happy Day of the Book.

Oxford Book of English Verse 1250-1900

Thursday April 23, 2009

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

 

The Geometry
of Language

(continued from April 16)

Background:

Professor Arielle Saiber with chess set

Click on the image for an
interview with the author of
Giordano Bruno and
the Geometry of Language
.

Related material:

Joyce on language —

The sigla of 'Finnegans Wake'

Bruno, Joyce, and coincidentia oppositorum

Cullinane on geometry —

Geometry of the I Ching (for comparison to Joyce's 'sigla')

Click on images for details.
 

Thursday April 23, 2009

Filed under: General — Tags: — m759 @ 7:22 AM
Theology for Holst

“Timothy J. Holst, who joined the Ringling Brothers and Barnum & Bailey Circus as a lowly Keystone Kops clown, rose to the role of singing ringmaster, and ultimately became the show’s talent czar, died April 16 in São Paulo, Brazil, during a visit to sign up circus acts. He was 61.”

Tiene angel.

Timothy J. Holst, who died April 16, 2009
But seriously…. 

Wednesday, April 8, 2009

Wednesday April 8, 2009

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

Good's Singularity

Irving John "I.J." Good died Sunday, April 5, 2009.

The date of his death was also Palm Sunday and the day of the Academy of Country Music Awards.

Information from Wikipedia:

Good, 92, was a cryptanalyst at Bletchley Park during World War II.

"He was born as Isidore Jacob Gudak to a Jewish family in London. In his publications he was called I. J. Good. He studied mathematics at Jesus College, Cambridge, graduating in 1938. He did research work under G.H. Hardy and Besicovitch before moving to Bletchley Park in 1941 on completing his doctorate.

At Bletchley Park, he was initially in Hut 8 under the supervision of Alan Turing…"

[Related material: the death of Turing (a major fan of the Evil Queen in Snow White) and yesterday's entry]

Wikipedia states that "I. J. Good's vanity car license plate, hinting at his spylike wartime work, was '007 IJG'…. He played chess to county standard, and helped to popularise Go, an Asian boardgame, through a 1965 article in New Scientist (he had learned the rules from Turing). In 1965, he described a concept similar to today's meaning of technological singularity, in that it included in it the advent of superhuman intelligence:

Let an ultraintelligent machine be defined as a machine that can far surpass all the intellectual activities of any man however clever. Since the design of machines is one of these intellectual activities, an ultraintelligent machine could design even better machines; there would then unquestionably be an 'intelligence explosion,' and the intelligence of man would be left far behind. Thus the first ultraintelligent machine is the last invention that man need ever make….
— Good, I. J. (1965). 'Speculations Concerning the First Ultraintelligent Machine', Advances in Computers, Vol. 6."
"Some say the symbol
of Apple Computers,
the apple with a bite out of it,
is a nod to Alan Turing."

 

— from "Alan Turing and
the Apple
" at Flickr, uploaded
on Epiphany 2006 by guano

Alan Turing and the Apple

 

http://www.log24.com/log/pix09/090408-TuringApples.jpg

Above: Composite by "guano" at Flickr

Will: Do you like apples?     
Clark: Yeah.                       
Will: Well, I got her number.
 How do you like them apples?

— "Good Will Hunting

Happy Spy Wednesday.

Sunday, April 5, 2009

Sunday April 5, 2009

Filed under: General — Tags: — m759 @ 7:35 AM
Reba's Widget

Click to enlarge.

Reba McEntire to host Academy of Country Music Awards Sunday, April 5, 2009

Sunday April 5, 2009

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

About the People:
Race to Witch Mountain

"As Robert Kennedy once told a crowd of students in South Africa, it is a revolutionary world that we live in and, thus, it is young people who must take the lead– [applause]– because young people are unburdened by the biases or prejudices of the past."

President Obama in Strasbourg on Friday, April 3, 2009

"George Bernard Shaw once wrote, 'Some people see things as they are and say why? I dream things that never were and say, why not?'"

— Robert Kennedy, University of Kansas, March 18, 1968

George Bernard Shaw:

THE SNAKE. I can talk of many things. I am very wise. It was I who whispered the word to you that you did not know. Dead. Death. Die.

EVE [shuddering] Why do you remind me of it? I forgot it when I saw your beautiful hood. You must not remind me of unhappy things.

THE SERPENT. Death is not an unhappy thing when you have learnt how to conquer it.

EVE. How can I conquer it?

THE SERPENT. By another thing, called birth.

EVE. What? [Trying to pronounce it] B-birth?

THE SERPENT. Yes, birth.

EVE. What is birth?

THE SERPENT. The serpent never dies. Some day you shall see me come out of this beautiful skin, a new snake with a new and lovelier skin. That is birth.

EVE. I have seen that. It is wonderful.

THE SERPENT. If I can do that, what can I not do? I tell you I am very subtle. When you and Adam talk, I hear you say 'Why?' Always 'Why?' You see things; and you say 'Why?' But I dream things that never were; and I say 'Why not?' I made the word dead to describe my old skin that I cast when I am renewed. I call that renewal being born.

EVE. Born is a beautiful word.

THE SERPENT. Why not be born again and again as I am, new and beautiful every time?

EVE. I! It does not happen: that is why.

THE SERPENT. That is how; but it is not why. Why not?

EVE. But I should not like it. It would be nice to be new again; but my old skin would lie on the ground looking just like me; and Adam would see it shrivel up and–

THE SERPENT. No. He need not. There is a second birth.

EVE. A second birth?

THE SERPENT. Listen. I will tell you a great secret….

"Listen, I tell you a mystery…."
Saul of Tarsus   

About the People
(with apologies to
Zenna Henderson):

'Spaceships, Toddlers, Model T Cars, and Jars of Beer'

"We've got to stop meeting like this."

Saturday, March 21, 2009

Saturday March 21, 2009

Filed under: General — Tags: — m759 @ 11:30 PM
Interpreter's Booth

Tonight's online New York Times:

NY Times  online March 21, 2009: Pope in Angola tells clergy to work against belief in witchcraft

Click to enlarge.


Mary Karr,
"Facing Altars:
    Poetry and Prayer"–

"There is a body
on the cross
  in my church."


Sean Penn gives Nicole Kidman his card in 'The Interpreter'

Sean Penn and Nicole Kidman
in "The Interpreter."

Click to enlarge.

"My card."

"Is Heart of Darkness the story of Kurtz or the story of Marlow’s experience of Kurtz?  Was Marlow invented as a rhetorical device for heightening the meaning of Kurtz’s moral collapse, or was Kurtz invented in order to provide Marlow with the centre of his experience in the Congo?  Again a seamless web, and we tell ourselves that the old-fashioned question 'Who is the protagonist?' is a meaningless one."


Wayne C. Booth, p. 346 in
The Rhetoric of Fiction
(1961),
as quoted by Paul Wake in
"The Storyteller in Chance"

Saturday March 21, 2009

Filed under: General,Geometry — Tags: — m759 @ 12:25 AM

Counters in Rows

"Music, mathematics, and chess are in vital respects dynamic acts of location. Symbolic counters are arranged in significant rows. Solutions, be they of a discord, of an algebraic equation, or of a positional impasse, are achieved by a regrouping, by a sequential reordering of individual units and unit-clusters (notes, integers, rooks or pawns)."

— George Steiner
   (See March 10, "Language Game.")
 



For example:

Model of the 21-point projective plane consisting of the 1- and 2- subsets of a 6-set

Click to enlarge.

Context:

Notes on Finite Geometry
(Section on 6-set structures)
 

Friday, February 20, 2009

Friday February 20, 2009

Filed under: General,Geometry — Tags: , , — m759 @ 11:01 PM

The Cross
of Constantine

mentioned in
this afternoon's entry
"Emblematizing the Modern"
was the object of a recent
cinematic chase sequence
(successful and inspiring)
starring Mira Sorvino
at the Metropolitan
Museum of Art.

In memory of
Dr. Hunter S. Thompson,
dead by his own hand
on this date
four years ago

Rolling Stone memorial to Dr. Hunter S. Thompson

Click for details.

There is
another sort of object
we may associate with a
different museum and with
a modern Constantine
See "Art Wars for MoMA"
(Dec. 14, 2008).

This object, modern
rather than medieval,
is the ninefold square:

The ninefold square

It may suit those who,
like Rosalind Krauss
(see "Emblematizing"),
admire the grids of modern art
but view any sort of Christian
cross with fear and loathing.

For some background that
Dr. Thompson might appreciate,
see notes on Geometry and Death
in this journal, June 1-15, 2007,
and the five Log24 entries
 ending at 9 AM Dec. 10. 2006,
which include this astute
observation by J. G. Ballard:

"Modernism's attempt to build a better world with the aid of science and technology now seems almost heroic. Bertolt Brecht, no fan of modernism, remarked that the mud, blood and carnage of the first world war trenches left its survivors longing for a future that resembled a white-tiled bathroom."

Selah.

Friday February 20, 2009

Filed under: General — Tags: , — m759 @ 6:00 PM
A Kind of Cross

Descartes portrait

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

— Thomas Pynchon in  
Gravity's Rainbow

Descartes's Cross

Click for source.

Related material:

A memorial service
held at 2 PM today at the
U.S. Space & Rocket Center
in Huntsville, Alabama, and
 today's previous entry.

Friday February 20, 2009

Filed under: General,Geometry — Tags: , — m759 @ 2:01 PM
Emblematizing
 the Modern
 

The following meditation was
inspired by the recent fictional
recovery, by Mira Sorvino
in "The Last Templar,"

of a Greek Cross —
"the Cross of Constantine"–
and by the discovery, by
art historian Rosalind Krauss,
of a Greek Cross in the
art of Ad Reinhardt.

http://www.log24.com/log/pix09/090220-CrossOfDescartes.jpg

The Cross of Descartes  

Note that in applications, the vertical axis
of the Cross of Descartes often symbolizes
the timeless (money, temperature, etc.)
while the horizontal axis often symbolizes time.


T.S. Eliot:

"Men’s curiosity searches past and future
And clings to that dimension. But to apprehend
The point of intersection of the timeless
With time, is an occupation for the saint…."


There is a reason, apart from her ethnic origins, that Rosalind Krauss (cf. 9/13/06) rejects, with a shudder, the cross as a key to "the Pandora's box of spiritual reference that is opened once one uses it." The rejection occurs in the context of her attempt to establish not the cross, but the grid, as a religious symbol:
 

"In suggesting that the success [1] of the grid
is somehow connected to its structure as myth,
I may of course be accused of stretching a point
beyond the limits of common sense, since myths
are stories, and like all narratives they unravel
through time, whereas grids are not only spatial
to start with, they are visual structures
that explicitly reject a narrative
or sequential reading of any kind.

[1] Success here refers to
three things at once:
a sheerly quantitative success,
involving the number of artists
in this century who have used grids;
a qualitative success through which
the grid has become the medium
for some of the greatest works
of modernism; and an ideological
success, in that the grid is able–
in a work of whatever quality–
to emblematize the Modern."

— Rosalind Krauss, "Grids" (1979)

Related material:

Time Fold and Weyl on
objectivity and frames of reference.

See also Stambaugh on
The Formless Self
as well as
A Study in Art Education
and
Jung and the Imago Dei.

Tuesday, February 17, 2009

Tuesday February 17, 2009

Filed under: General,Geometry — Tags: , , — m759 @ 1:06 PM

Diamond-Faceted:
Transformations
of the Rock

A discussion of Stevens's late poem "The Rock" (1954) in Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, p. 120:

For Stevens, the poem "makes meanings of the rock." In the mind, "its barrenness becomes a thousand things/And so exists no more." In fact, in a peculiar irony that only a poet with Stevens's particular notion of the imagination's function could develop, the rock becomes the mind itself, shattered into such diamond-faceted brilliance that it encompasses all possibilities for human thought:

The rock is the gray particular of man's life,
The stone from which he rises, up—and—ho,
The step to the bleaker depths of his descents ...

The rock is the stern particular of the air,
The mirror of the planets, one by one,
But through man's eye, their silent rhapsodist,

Turquoise the rock, at odious evening bright
With redness that sticks fast to evil dreams;
The difficult rightness of half-risen day.

The rock is the habitation of the whole,
Its strength and measure, that which is near,
     point A
In a perspective that begins again

At B: the origin of the mango's rind.

                    (Collected Poems, 528)

A mathematical version of
this poetic concept appears
in a rather cryptic note
from 1981 written with
Stevens's poem in mind:

http://www.log24.com/log/pix09/090217-SolidSymmetry.jpg

For some explanation of the
groups of 8 and 24
motions referred to in the note,
see an earlier note from 1981.

For the Perlis "diamond facets,"
see the Diamond 16 Puzzle.

For a much larger group
of motions, see
Solomon's Cube.

As for "the mind itself"
and "possibilities for
human thought," see
Geometry of the I Ching.

Saturday, February 14, 2009

Saturday February 14, 2009

Filed under: General — Tags: — m759 @ 9:29 PM
The Devil
in the Details

 

Here are clearer pictures of
the Einstein-Gutkind letter
discussed here February 7.

The pictures are from
the Bloomsbury Auctions site.

http://www.log24.com/log/pix09/Einstein-Gutkind1954-1.jpg

http://www.log24.com/log/pix09/Einstein-Gutkind1954-2.jpg

The Bloomsbury Auctions caption for these images is as follows:

303. Einstein (Albert, theoretical physicist, 1879-1955) Autograph Letter signed to Eric B. Gutkind, in German, 1½pp. & envelope, 4to, Princeton, 3rd January 1954, thanking him for a copy of his book and expressing his view of God and Judaism, [The word God is for me nothing more than the expression and product of human weaknesses, the Bible a collection of honorable, but still primitive legends which are nevertheless pretty childish… . For me the Jewish religion like all other religions is an incarnation of the most childish superstitions. And the Jewish people to whom I gladly belong and with whose mentality I have a deep affinity have no different quality for me than all other people…], folds, slightly browned ; and a photograph of Gutkind, v.s., v.d.

est. £6000 – £8000

Einstein’s view of God and Judaism.
Eric B. Gutkind (1877-1965), philosopher; author of Choose Life: The Biblical Call to Revolt, 1952.
Albert Einstein – see also lot 497

Sold for £170000
Sale 649, 15th May 2008

Here is a close reading of the part of the letter itself that Bloomsbury gives in English, transcribed from the above images.

Line-by-line transcription of paragraph 2, starting at line 4 of that paragraph:                        

                   ... Das Wort Gott ist für mich nichts als Ausdruck
und Produkt menschlicher Schwächen, die Bibel eine Sammlung
ehrwürdiger, aber doch reichlich primitiver Legenden. Keine noch
so feinsinnige Auslegung kann (für mich) etwas daran ändern.
Diese verfeinerten Auslegungen sind naturgemäß höchst mannigfaltig
und haben so gut wie nichts mit dem Urtext zu schaffen. Für
mich ist die unverfälschte jüdische Religion, wie alle anderen
Religionen, eine Inkarnation des primitiven Aberglaubens. Und das
jüdische Volk, zu dem ich gern gehöre und mit dessen Mentalität ich
tief verwachsen bin, hat für mich doch keine andersartige
Qualität als alle anderen Völker. So weit meine Erfahrung reicht,
ist es auch um nichts besser als andere menschliche Gruppierungen,
wenn es auch durch Mangel an Macht gegen die schlimmsten
Auswüchse gesichert ist. Ansonsten kann ich nichts "Auserwähltes"
an ihm wahrnehmen.

The Guardian of May 13, 2008 stated that the following was "translated from German by Joan Stambaugh"–

... The word God is for me nothing more than the expression
and product of human weaknesses, the Bible a collection
of honourable, but still primitive legends which are nevertheless pretty childish. No
interpretation no matter how subtle can (for me) change this.
These subtilised interpretations are highly manifold
according to their nature and have almost nothing to do with the original text. For
me the Jewish religion like all other
religions is an incarnation of the most childish [German: primitiven] superstitions. And the
Jewish people to whom I gladly belong and with whose mentality I
have a deep affinity have no different
quality for me than all other people. As far as my experience goes,
they are also no better than other human groups,
although they are protected from the worst
cancers by a lack of power. Otherwise I cannot see anything 'chosen'
about them.

Phrases by Stambaugh that do not appear in the German text are highlighted.

Stambaugh, a philosophy professor, is the author of a work on Buddhism, The Formless Self. For some related material on young men who "go crying 'The world is myself, life is myself'" in May, see Wallace Stevens's "The Pediment of Appearance."
 

Wednesday, November 19, 2008

Wednesday November 19, 2008

Filed under: General,Geometry — Tags: , — m759 @ 5:01 PM

"Through the unknown,
remembered gate…."

Four Quartets

(Epigraph to the introduction,
Parallelisms of Complete Designs
by Peter J. Cameron,
Merton College, Oxford)

"It's still the same old story…."
— Song lyric

The Great Gatsby
Chapter 6:

"An instinct toward his future glory had led him, some months before, to the small Lutheran college of St. Olaf in southern Minnesota. He stayed there two weeks, dismayed at its ferocious indifference to the drums of his destiny, to destiny itself, and despising the janitor’s work with which he was to pay his way through."

There is a link to an article on St. Olaf College in Arts & Letters Daily  today:

"John Milton, boring? Paradise Lost  has a little bit of something for everybody. Hot sex! Hellfire! Some damned good poetry, too…" more»

The "more" link is to The Chronicle of Higher Education.

For related material on Paradise Lost  and higher education, see Mathematics and Narrative.

Wednesday November 19, 2008

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

Sympathy for Baird Bryant

"Pleased to meet you
Hope you guess my name
But what's puzzling you
Is the nature of my game"

The Rolling Stones

"'Don't you want to
hear him call your name
when you're standing
at the pearly gates?'
I told the Preacher 'Yes, I do,
but I hope he don't call today.'"

— Kenny Chesney, song at the CMA Awards on Wednesday, November 12, quoted here at 9:00 AM on Thursday, Novermber 13

Related material:

LA Times obituary for the experienced bohemian writer and filmmaker Baird Bryant, who died at 80 on Thursday, November 13. Bryant filmed parts of "Easy Rider" in 1968 and of the Altamont concert in 1969. He was apparently a member of the Harvard College Class of 1950.

A more complete account of Bryant's life

Thirty references to the Devil in a book by Bryant

Solace With Interruptions

(Log24 entries for November 12, 13, and 14 — the day before Bryant's death, the day of his death, and the day after)
 

Tuesday, August 19, 2008

Tuesday August 19, 2008

Filed under: General,Geometry — Tags: — m759 @ 8:30 AM
Three Times

"Credences of Summer," VII,

by Wallace Stevens, from
Transport to Summer (1947)

"Three times the concentred
     self takes hold, three times
The thrice concentred self,
     having possessed
The object, grips it
     in savage scrutiny,
Once to make captive,
     once to subjugate
Or yield to subjugation,
     once to proclaim
The meaning of the capture,
     this hard prize,
Fully made, fully apparent,
     fully found."

Stevens does not say what object he is discussing.

One possibility —

Bertram Kostant, Professor Emeritus of Mathematics at MIT, on an object discussed in a recent New Yorker:

"A word about E(8). In my opinion, and shared by others, E(8) is the most magnificent 'object' in all of mathematics. It is like a diamond with thousands of facets. Each facet offering a different view of its unbelievable intricate internal structure."

Another possibility —
 

The 4x4 square

  A more modest object —
the 4×4 square.

Update of Aug. 20-21 —

Symmetries and Facets

Kostant's poetic comparison might be applied also to this object.

The natural rearrangements (symmetries) of the 4×4 array might also be described poetically as "thousands of facets, each facet offering a different view of… internal structure."

More precisely, there are 322,560 natural rearrangements– which a poet might call facets*— of the array, each offering a different view of the array's internal structure– encoded as a unique ordered pair of symmetric graphic designs. The symmetry of the array's internal structure is reflected in the symmetry of the graphic designs. For examples, see the Diamond 16 Puzzle.

For an instance of Stevens's "three times" process, see the three parts of the 2004 web page Ideas and Art.

* For the metaphor of rearrangements as facets, note that each symmetry (rearrangement) of a Platonic solid corresponds to a rotated facet: the number of symmetries equals the number of facets times the number of rotations (edges) of each facet–

Platonic solids' symmetry groups

The metaphor of rearrangements as facets breaks down, however, when we try to use it to compute, as above with the Platonic solids, the number of natural rearrangements, or symmetries, of the 4×4 array. Actually, the true analogy is between the 16 unit squares of the 4×4 array, regarded as the 16 points of a finite 4-space (which has finitely many symmetries), and the infinitely many points of Euclidean 4-space (which has infinitely many symmetries).

If Greek geometers had started with a finite space (as in The Eightfold Cube), the history of mathematics might have dramatically illustrated Halmos's saying (Aug. 16) that

"The problem is– the genius is– given an infinite question, to think of the right finite question to ask. Once you thought of the finite answer, then you would know the right answer to the infinite question."

The Greeks, of course, answered the infinite questions first– at least for Euclidean space. Halmos was concerned with more general modern infinite spaces (such as Hilbert space) where the intuition to be gained from finite questions is still of value.
 

Saturday, May 19, 2007

Saturday May 19, 2007

Filed under: General — Tags: , , — m759 @ 9:29 AM
Point of View

"In a sense, too, Wallace Stevens has spent a lifetime writing a single poem. What gives his best work its astonishing power and vitality is the way in which a fixed point of view, maturing naturally, eventually takes in more than a constantly shifting point of view could get at.

The point of view is romantic, 'almost the color of comedy'; but 'the strength at the center is serious.'  Behind Wallace Stevens stand Wordsworth and Coleridge as well as Rimbaud and Mallarmé, and, surprisingly enough, La Fontaine and Pope. This poetic lineage is important only in so far as it proves that a master can claim the world as ancestor. Knowing where he stands, the poet can move as a free man in the company of free men."

Samuel French Morse, review 
of The Collected Poems
of Wallace Stevens, in
The New York Times
(October 3, 1954)
 
Related material

The point of view
expressed in Log24 on
  today's date in 2004:

For a related gloss on Stevens's remark
"the strength at the center is serious,"
see "Serious" (also on an October 3).

Saturday, August 19, 2006

Saturday August 19, 2006

Filed under: General — Tags: — m759 @ 4:28 PM
Metaphysical
Wonderlands

"With no means to verify its truth, superstring theory, in the words of Burton Richter, director emeritus of the Stanford Linear Accelerator Center, may turn out to be 'a kind of metaphysical wonderland.' Yet it is being pursued as vigorously as ever, its critics complain, treated as the only game in town."

— "The Inelegant Universe," by George Johnson, in the Sept. 2006 Scientific American

Some may prefer metaphysics of a different sort:

"To enter Cervantes’s world, we cross a threshold that is Shakespearean and quixotic into a metaphysical wonderland where time expands to become space and vast vaulted distances bend back on themselves, where the threads of fiction and the strands of history shuttle back and forth in the great loom of the artist’s imagination."

As wonderlands go, I personally prefer Clive Barker's Weaveworld.
 

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 March 29, 2006

Filed under: General — Tags: — m759 @ 12:00 PM
Darkness at Noon,
continued

It turns out that Medawar (see previous entry) also wrote a deeply hostile review of Koestler’s The Act of Creation.  (See Pluto’s Republic.)

There are plenty more like Medawar, so it may be that a further effort at documentation of Diamond Theory is needed.  See this evening’s entry, to follow.

Thursday, February 9, 2006

Thursday February 9, 2006

Filed under: General — Tags: — m759 @ 9:00 PM
The Vanishing (?) President
The image “http://www.log24.com/log/pix06/060209-Summers.jpg” cannot be displayed, because it contains errors.

Karen E. Fields, translator’s introduction to Elementary Forms of the Religious Life, by Emile Durkheim:

“Durkheim breathed the air of turn-of-the-century Paris, a place that fizzed with experiments in artistic representation, and a time when philosophy, science, and art existed in nothing like today’s isolation from one another.24

 

24  Judith Ryan provides an illuminating account of the links joining physics, psychology, philosophy, painting, and literature in The Vanishing Subject: Early Psychology and Literary Modernism, Chicago, University of Chicago Press, 1991.”

And today’s Crimson provides an illuminating account of Judith Ryan and (implicitly) forms of the religious life at Harvard.

Related material:
The Crucifixion of John O’Hara,
The Crimson Passion,
Supper at Eight,
Riddle.

Thursday February 9, 2006

Filed under: General — Tags: — m759 @ 12:24 PM

Space, Time, and Scarlett

From last night’s Grammy awards, lyrics performed by Christina Aguilera and Herbie Hancock:

“a place where there’s no space or time”
Leon Russell

Not bad, but as Kat358 noted on May 4, 2005,

The image “http://www.log24.com/log/pix06/060209-Blondes.jpg” cannot be displayed, because it contains errors.

“Scarlett Johansson does this ‘old Hollywood glam’ look much better.”

For a reference to the place described in Russell’s lyrics, see the riff on the number “265”
linked to in last night’s “Midnight in the Garden of the Soul.”

Related material– Jazz Improvisation:

“Once an appropriate group of people has been assembled, you must decide what to play.”

Thursday February 9, 2006

Filed under: General — Tags: — m759 @ 12:00 AM
Midnight in the Garden
of the Soul

(continued)This time slot, reserved at midnight,
seems to belong to Frank Goodman,
who, according to this morning’s
(3 AM) New York Times, was
“one of the last of the old-time
Broadway press agents.”

Related material:

Yesterday afternoon’s entry
on a fictional “press agent
for ‘The Garden of the Soul,'”
and the entry on death
and gardens from Friday,
Feb. 3, 2006, the day
Frank Goodman died.

Friday, August 19, 2005

Friday August 19, 2005

Filed under: General,Geometry — Tags: — m759 @ 2:00 PM

Mathematics and Narrative
continued

"There is a pleasantly discursive treatment of Pontius Pilate's unanswered question 'What is truth?'"

H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau's remarks on the "Story Theory" of truth as opposed to  the "Diamond Theory" of truth " in The Non-Euclidean Revolution

"I had an epiphany: I thought 'Oh my God, this is it! People are talking about elliptic curves and of course they think they are talking mathematics. But are they really? Or are they talking about stories?'"

An organizer of last month's "Mathematics and Narrative" conference

"A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the 'Story Theory' of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called 'true.' The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes*…."

Richard J. Trudeau in The Non-Euclidean Revolution

"'Deniers' of truth… insist that each of us is trapped in his own point of view; we make up stories about the world and, in an exercise of power, try to impose them on others."

— Jim Holt in this week's New Yorker magazine.  Click on the box below.

The image “http://www.log24.com/log/pix05B/050819-Critic4.jpg” cannot be displayed, because it contains errors.

* Many stripes

   "What disciplines were represented at the meeting?"
   "Apart from historians, you mean? Oh, many: writers, artists, philosophers, semioticians, cognitive psychologists – you name it."

 

An organizer of last month's "Mathematics and Narrative" conference

Thursday, August 19, 2004

Thursday August 19, 2004

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

The Tiffany Code

5:01:58 AM ET:

A link for Jill St. John's birthday —

The Geometrics of Brilliance

Twinkle, twinkle…

Beach reading for
 brilliant redheads…

and for everyone else:

Click on pictures for details.
 

Thursday August 19, 2004

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

Instantia Crucis

"Francis Bacon used the phrase instantia crucis, 'crucial instance,' to refer to something in an experiment that proves one of two hypotheses and disproves the other. Bacon's phrase was based on a sense of the Latin word crux, 'cross,' which had come to mean 'a guidepost that gives directions at a place where one road becomes two,' and hence was suitable for Bacon's metaphor."

The American Heritage® Dictionary of the English Language, Fourth Edition

The high notes hit by Harriet Wheeler, Jen Slocumb, and Alanis Morissette can, I am sorry to say, be excruciating. (See previous entry.) I greatly prefer the mellow tones of Mary Chapin Carpenter:

"I guess you're never really all alone,
        or too far from the pull of home,
An' the stars upon that painted dome
        still shine."

MCC, Grand Central Station

From an entry of 12/22/02:

 









White
Horse
Wings
As if

A white horse comes as if on wings.

— I Ching, Hexagram 22: Grace

See also

Plato, Pegasus, and the Evening Star,

Shining Forth, and

Music for Pegasus.

Carpenter's song quoted above
is from the album
Between Here and Gone,
released April 27, 2004.
 

Wednesday, May 19, 2004

Wednesday May 19, 2004

Filed under: General,Geometry — Tags: — m759 @ 2:00 PM

Style

In memory of Lynn H. Loomis:

The above diagram is from a
(paper) journal note of October 21, 1999.

It pictures the relationship of my own discovery, diamond theory (at center), to the field, harmonic analysis, of Professor Loomis, a writer whose style I have long admired.

A quotation from the 1999 note:

"…it is not impossible to draw a fairly sharp dividing line between our mental disposition in the case of esthetic response and that of the responses of ordinary life.  A far more difficult question arises if we try to distinguish it from the responses made by us to certain abstract mental constructions such as those of pure mathematics…. Perhaps the distinction lies in this, that in the case of works of art the whole end and purpose is found in the exact quality of the emotional state, whereas in the case of mathematics the purpose is the constatation of the universal validity of the relations without regard to the quality of the emotion accompanying apprehension.  Still, it would be impossible to deny the close similarity of the orientation of faculties and attention in the two cases."
— Roger Fry, Transformations (1926), Doubleday Anchor paperback, 1956, p. 8

In other words, appreciating mathematics is much like appreciating art.

(Digitized diagram courtesy of Violet.)

Tuesday, August 19, 2003

Tuesday August 19, 2003

Filed under: General — Tags: — m759 @ 10:23 PM

O'Hara's Fingerpost

In The New York Times Book Review of next Sunday (August 24, 2003), Book Review editor Charles McGrath writes that author John O'Hara

"… discovered a kind of story… in which a line of dialogue or even a single observed detail indicates that something crucial has changed."

From the Online Etymology Dictionary:

crucial – 1706, from Fr. crucial… from L. crux (gen. crucis) "cross." The meaning "decisive, critical" is extended from a logical term, Instantias Crucis, adopted by Francis Bacon (1620); the notion is of cross fingerboard signposts at forking roads, thus a requirement to choose.

The remainder of this note deals with the "single observed detail" 162.

 

162

Instantias Crucis

Francis Bacon says

"Among Prerogative Instances I will put in the fourteenth place Instances of the Fingerpost, borrowing the term from the fingerposts which are set up where roads part, to indicate the several directions. These I also call Decisive and Judicial, and in some cases, Oracular and Commanding Instances. I explain them thus. When in the investigation of any nature the understanding is so balanced as to be uncertain to which of two or more natures the cause of the nature in question should be assigned on account of the frequent and ordinary concurrence of many natures, instances of the fingerpost show the union of one of the natures with the nature in question to be sure and indissoluble, of the other to be varied and separable; and thus the question is decided, and the former nature is admitted as the cause, while the latter is dismissed and rejected. Such instances afford very great light and are of high authority, the course of interpretation sometimes ending in them and being completed. Sometimes these instances of the fingerpost meet us accidentally among those already noticed, but for the most part they are new, and are expressly and designedly sought for and applied, and discovered only by earnest and active diligence."

The original:

Inter praerogativas instantiarum, ponemus loco decimo quarto Instantias Crucis; translato vocabulo a Crucibus, quae erectae in biviis indicant et signant viarum separationes. Has etiam Instantias Decisorias et Judiciales, et in casibus nonnullis Instantias Oraculi et Mandati, appellare consuevimus. Earum ratio talis est. Cum in inquisitione naturae alicujus intellectus ponitur tanquam in aequilibrio, ut incertus sit utri naturarum e duabus, vel quandoque pluribus, causa naturae inquisitae attribui aut assignari debeat, propter complurium naturarum concursum frequentem et ordinarium, instantiae crucis ostendunt consortium unius ex naturis (quoad naturam inquisitam) fidum et indissolubile, alterius autem varium et separabile ; unde terminatur quaestio, et recipitur natura illa prior pro causa, missa altera et repudiata. Itaque hujusmodi instantiae sunt maximae lucis, et quasi magnae authoritatis; ita ut curriculum interpretationis quandoque in illas desinat, et per illas perficiatur. Interdum autem Instantiae Crucis illae occurrunt et inveniuntur inter jampridem notatas; at ut plurimum novae sunt, et de industria atque ex composito quaesitae et applicatae, et diligentia sedula et acri tandem erutae.

— Francis Bacon, Novum Organum, Book Two, "Aphorisms," Section XXXVI

A Cubist Crucifixion

An alternate translation:

"When in a Search of any Nature the Understanding stands suspended, the Instances of the Fingerpost shew the true and inviolable Way in which the Question is to be decided. These Instances afford great Light…"

From a review by Adam White Scoville of Iain Pears's novel titled An Instance of the Fingerpost:

"The picture, viewed as a whole, is a cubist description, where each portrait looks strikingly different; the failings of each character's vision are obvious. However, in a cubist painting the viewer often can envision the subject in reality. Here, even after turning the last page, we still have a fuzzy view of what actually transpired. Perhaps we are meant to see the story as a cubist retelling of the crucifixion, as Pilate, Barabbas, Caiaphas, and Mary Magdalene might have told it. If so, it is sublimely done so that the realization gradually and unexpectedly dawns upon the reader. The title, taken from Sir Francis Bacon, suggests that at certain times, 'understanding stands suspended' and in that moment of clarity (somewhat like Wordsworth's 'spots of time,' I think), the answer will become apparent as if a fingerpost were pointing at the way. The final narrative is also titled An Instance of the Fingerpost, perhaps implying that we are to see truth and clarity in this version. But the biggest mystery of this book is that we have actually have no reason to credit the final narrative more than the previous three and so the story remains an enigma, its truth still uncertain."

For the "162" enigma, see

Dogma,

The Matthias Defense, and

The Still Point and the Wheel.

See also the December 2001 Esquire and

the conclusion of my previous entry.
 

Tuesday August 19, 2003

Filed under: General,Geometry — Tags: — m759 @ 5:23 PM

Intelligence Test

From my August 31, 2002, entry quoting Dr. Maria Montessori on conciseness, simplicity, and objectivity:

Above: Dr. Harrison Pope, Harvard professor of psychiatry, demonstrates the use of the Wechsler Adult Intelligence Scale "block design" subtest.

Another Harvard psychiatrist, Armand Nicholi, is in the news lately with his book The Question of God: C.S. Lewis and Sigmund Freud Debate God, Love, Sex, and the Meaning of Life

 

Pope

Nicholi

Old
Testament
Logos

New
Testament
Logos

For the meaning of the Old-Testament logos above, see the remarks of Plato on the immortality of the soul at

Cut-the-Knot.org.

For the meaning of the New-Testament logos above, see the remarks of R. P. Langlands at

The Institute for Advanced Study.

For the meaning of life, see

The Gospel According to Jill St. John,

whose birthday is today.

"Some sources credit her with an I.Q. of 162."
 

Thursday, January 9, 2003

Thursday January 9, 2003

Filed under: General — Tags: , — m759 @ 4:48 PM

Balanchine's Birthday

Today seems an appropriate day to celebrate Apollo and the nine Muses.

From a website on Balanchine's and Stravinsky's ballet, "Apollon Musagete":

In his Poetics of Music (1942) Stravinsky says: "Summing up: What is important for the lucid ordering of the work– for its crystallization– is that all the Dionysian elements which set the imagination of the artist in motion and make the life-sap rise must be properly subjugated before they intoxicate us, and must finally be made to submit to the law: Apollo demands it."  Stravinsky conceived Apollo as a ballet blanc– a "white ballet" with classical choreography and monochromatic attire. Envisioning the work in his mind's eye, he found that "the absence of many-colored hues and of all superfluities produced a wonderful freshness." Upon first hearing Apollo, Diaghilev found it "music somehow not of this world, but from somewhere else above." The ballet closes with an Apotheosis in which Apollo leads the Muses towards Parnassus. Here, the gravely beautiful music with which the work began is truly recapitulated "on high"– ceaselessly recycled, frozen in time.

— Joseph Horowitz

 

 

Another website invoking Apollo:

The icon that I use… is the nine-fold square…. The nine-fold square has centre, periphery, axes and diagonals.  But all are present only in their bare essentials.  It is also a sequence of eight triads.  Four pass through the centre and four do not.  This is the garden of Apollo, the field of Reason…. 

In accordance with these remarks, here is the underlying structure for a ballet blanc:

A version of 'grid3x3.gif.'

This structure may seem too simple to support movements of interest, but consider the following (click to enlarge):

As Sir Arthur Quiller-Couch, paraphrasing Horace, remarks in his Whitsun, 1939, preface to the new edition of the Oxford Book of English Verse, "tamen usque recurret Apollo."

The alert reader will note that in the above diagrams, only eight of the positions move.

Which muse remains at the center?

Consider the remark of T. S. Eliot, "At the still point, there the dance is," and the fact that on the day Eliot turned 60, Olivia Newton-John was born.  How, indeed, in the words of another "sixty-year-old smiling public man," can we know the dancer from the dance?
 

Sunday, April 5, 2015

Night at the Museum and…

Filed under: General — m759 @ 11:52 PM

Plan 9 Resurrection

in memory of a labor leader who reportedly
died Sunday evening —

Saturday, June 29, 2019

That’s “Merry” … And Quite Contrary

Filed under: General — Tags: — m759 @ 10:59 PM

"John Horton Conway is a cross between
Archimedes, Mick Jagger and Salvador Dalí." 

The Guardian  paraphrasing Siobhan Roberts, 
                    

John Horton Conway and his Leech lattice doodle
in The Guardian . Photo: Hollandse Hoogte/Eyevine.

. . . .

"In junior school, one of Conway’s teachers had nicknamed him 'Mary'.
He was a delicate, effeminate creature. Being Mary made his life
absolute hell until he moved on to secondary school, at Liverpool’s
Holt High School for Boys. Soon after term began, the headmaster
called each boy into his office and asked what he planned to do with
his life. John said he wanted to read mathematics at Cambridge.
Instead of 'Mary' he became known as 'The Prof'. These nicknames
confirmed Conway as a terribly introverted adolescent, painfully aware
of his own suffering."  — Siobhan Roberts, loc. cit.

From the previous post

See as well this  journal on the above Guardian  date —

 

The Benson Epiphany

Filed under: General — Tags: — m759 @ 9:14 AM

Monday, June 17, 2019

The Callahan Turtle

Filed under: General — Tags: , — m759 @ 10:54 PM

By Stephen King

IMAGE- Herbert John Ryser, 'Combinatorial Mathematics' (1963), page 1

Salvation by Grace

Filed under: General — Tags: , — m759 @ 3:18 PM

High Society

Filed under: General — Tags: , — m759 @ 1:16 PM

From the date in the above Google search result —

 See as well Jeremy Gray in this  journal.

Sunday, June 16, 2019

Eliot’s Perpetual Motion Structure*

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

From a date described by Peter Woit in his post
"Not So Spooky Action at a Distance" (June 11) —

See also The Lost Well.

 * "As a Chinese jar…." — Four Quartets

Tuesday, May 14, 2019

Change

Filed under: General — Tags: — m759 @ 10:02 PM

"Is there no change of death in paradise?" — Wallace Stevens 

From a New Yorker  book review dated May 13 —

"In 'Field Flowers,' Glück’s flower scoffs that 'absence of change' 
is humanity’s 'poor idea of heaven.' But the religious believer
might object that Hägglund’s idea of eternity is equally poor." 

Here James Wood is reviewing Martin Hägglund’s
This Life: Secular Faith and Spiritual Freedom 
(Pantheon, March 5, 2019).

See also posts tagged Change Arises in this  journal.

Hägglund himself appeared here on June 21, 2014.

The Nothing That Is

Filed under: General — Tags: — m759 @ 2:08 PM

Philosophy professor Agnes Callard on giving advice:

"It’s as though right before I give the advice,
I push a button that sucks all the informational
content out of what I’m about to say, and
I end up saying basically nothing at all."

— https://thepointmag.com/2019/
examined-life/against-advice-agnes-callard

From a University of Chicago description of Callard —

See as well posts before and after the above date, Jan. 3, 2018,
that are now tagged "Lost Horizon."

More generally, see a Log24 search  for "Lost Horizon."

Surrealistic Pillow Talk

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

   "Plan 9 deals with the resurrection of the dead.

IMAGE- Bill Murray explains Ed Wood's 'Plan 9 from Outer Space'- 'Plan 9 deals with the resurrection of the dead.'


"When the men on the chessboard
get up and tell you where to go . . ."

Saturday, May 4, 2019

The Chinese Jars of Shing-Tung Yau

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

The title refers to Calabi-Yau spaces.

T. S. Eliot —

Four Quartets

. . . Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.

A less "cosmic" but still noteworthy code — The Golay code.

This resides in a 12-dimensional space over GF(2).

Related material from Plato and R. T. Curtis

Counting symmetries with the orbit-stabilizer theorem

A related Calabi-Yau "Chinese jar" first described in detail in 1905

Illustration of K3 surface related to Mathieu moonshine

A figure that may or may not be related to the 4x4x4 cube that
holds the classical  Chinese "cosmic code" — the I Ching

ftp://ftp.cs.indiana.edu/pub/hanson/forSha/AK3/old/K3-pix.pdf

Wednesday, May 1, 2019

For the First of May

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

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

 Igor Shafarevitch in 1973

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

— T. S. Eliot in Four Quartets

See also Ultron Cube.

Sunday, February 17, 2019

Sacerdotal K6

Filed under: General,Geometry — Tags: , — m759 @ 7:35 PM

From a recently created Wikipedia article —

See also Sacerdotal Jargon (Log24, Dec. 5, 2002).

See Also …

Filed under: General — Tags: — m759 @ 12:31 PM

"And the Führer digs for trinkets in the desert."

"See also Acht "
— Cambridge German-English Dictionary, article on "Elf "

Dictionary Time

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

Wednesday, January 16, 2019

Permutahedron Dream

Filed under: General — Tags: , , , — m759 @ 3:21 PM

The geometric object of the title appears in a post mentioning Bourgain 
in this journal.  Bourgain appears also in today's online New York Times —

https://www.nytimes.com/2019/01/16/
obituaries/jean-bourgain-dead.html
 .

Bourgain reportedly died on December 22.

An image from this journal on that date

Related poetic meditations —

IMAGE- Herbert John Ryser, 'Combinatorial Mathematics' (1963), page 1

Saturday, September 15, 2018

Sick Framing

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

http://www.log24.com/log/pix18/180915-Marrific-Hungry_I-detail.jpg

Satire on Saturday Night

Filed under: General — Tags: — m759 @ 10:00 PM

http://www.log24.com/log/pix18/180915-Satire_Closes_Saturday_Night.jpg

See as well some other  remarks from March 28, 2014.

Ig Nobel “One Plus One”

Filed under: General — Tags: — m759 @ 1:42 PM

http://www.log24.com/log/pix18/180915-NYT-One_Plus_One-MoMA.jpg

Jean-Luc Godard in this journal on Thursday

"I perceived . . . cinema is that which is between things,
not things [themselves] but between one and another."

Also on Thursday —

"Scientific luminaries gathered at Harvard University’s
historic Sanders Theatre  on Thursday to honor
notable achievements this year…. It was the
28th First Annual Ig Nobel Prize Ceremony …."

Motherboard

Axioms

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 9:50 AM

Tieszen— 'Kurt Godel and Phenomenology' — 1992

Update of 10:18 AM the same day —

See also Logicomix  in this  journal and, at Harvard,

http://www.math.harvard.edu/~mazur/

  • September 6, 2018:  Eric Maskin, Amartya Sen and I
    are giving a course this semester: 'Axiomatic Reasoning'
    (PHIL 273B). Introduction to Axiomatic Reasoning gives a
    general sense of what we intend to cover.

Update of 10:48 AM the same day —

http://www.log24.com/log/pix18/180915-Tieszen_died-March-28-2017.jpg

See Log24 on the date of Tieszen's death.

Eidetic Reduction in Geometry

Filed under: G-Notes,General,Geometry — Tags: , , — m759 @ 1:23 AM
 

"Husserl is not the greatest philosopher of all times.
He is the greatest philosopher since Leibniz."

Kurt Gödel as quoted by Gian-Carlo Rota

Some results from a Google search —

Eidetic reduction | philosophy | Britannica.com

Eidetic reduction, in phenomenology, a method by which the philosopher moves from the consciousness of individual and concrete objects to the transempirical realm of pure essences and thus achieves an intuition of the eidos (Greek: “shape”) of a thing—i.e., of what it is in its invariable and essential structure, apart …

Phenomenology Online » Eidetic Reduction

The eidetic reduction: eidos. Method: Bracket all incidental meaning and ask: what are some of the possible invariate aspects of this experience? The research …

Eidetic reduction – New World Encyclopedia

Sep 19, 2017 – Eidetic reduction is a technique in Husserlian phenomenology, used to identify the essential components of the given phenomenon or experience.

Terminology: Eidos

For example —

The reduction of two-colorings and four-colorings of a square or cubic
array of subsquares or subcubes to lines, sets of lines, cuts, or sets of
cuts between the subsquares or subcubes.

See the diamond theorem and the eightfold cube.

* Cf. posts tagged Interality and Interstice.

Saturday, May 19, 2018

Flux Capacitor

Filed under: General,Geometry — Tags: , — m759 @ 4:13 PM

For Tom Hanks and Dan Brown —

Symbologist Robert Langdon views a corner of Solomon's Cube

From "Raiders of the Lost Images" —

"The cube shape of the lost Mother Box,
also known as the Change Engine,
is shared by the Stone in a novel by
Charles Williams, Many Dimensions .
See the Solomon's Cube webpage."

See as well a Google search for flux philosophy
https://www.google.com/search?q=flux+philosophy.

Sunday, October 15, 2017

The Strength at the Centre

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

The title, a phrase from a poem by Wallace Stevens,
was suggested by the previous post, "Center."

See posts tagged May 19 Gestalt in particular, 
May 19, 2007 — "Point of View."

Monday, October 9, 2017

Still Point for a Dance

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

"At the still point of the turning world. Neither flesh nor fleshless;
Neither from nor towards; at the still point, there the dance is,
But neither arrest nor movement. And do not call it fixity,
Where past and future are gathered. Neither movement from nor towards,
Neither ascent nor decline. Except for the point, the still point,
There would be no dance, and there is only the dance."

— T. S. Eliot, Four Quartets

See also a recurrent image
from this journal —

IMAGE- The ninefold square .

Tuesday, September 12, 2017

Think Different

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

The New York Times  online this evening

"Mr. Jobs, who died in 2011, loomed over Tuesday’s
nostalgic presentation. The Apple C.E.O., Tim Cook,
paid tribute, his voice cracking with emotion, Mr. Jobs’s
steeple-fingered image looming as big onstage as
Big Brother’s face in the classic Macintosh '1984' commercial."

James Poniewozik 

Review —

Thursday, September 1, 2011

How It Works

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

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

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

. . . .

See also 1984 Bricks in this journal.

Chin Music

Filed under: General,Geometry — Tags: — m759 @ 9:45 PM

Related image suggested by "A Line for Frank" (Sept. 30, 2013) —

Thursday, July 27, 2017

Keeping It Simple

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

Michiko Kakutani in The New York Times

"The detective story genre concerns the finding of clues
and the search for hidden designs, and its very form
underscores Mr. Pynchon’s obsession with conspiracies
and the existence of systems too complicated to understand."

Review of Pynchon's Bleeding Edge , Sept. 10, 2013

Background:  "Moss on the Wall," this  journal on that date.

A less complicated system —

"Plan 9 deals with the resurrection of the dead."

— Bill Murray in "Ed Wood"
 

For The Church of Plan 9

(The plan , as well as the elevation ,
of the above structure is a 3×3 grid.)

Monday, June 26, 2017

Four Dots

Analogies — "A : B  ::  C : D"  may be read  "A is to B  as  C is to D."

Gian-Carlo Rota on Heidegger…

"… The universal as  is given various names in Heidegger's writings….

The discovery of the universal as  is Heidegger's contribution to philosophy….

The universal 'as' is the surgence of sense in Man, the shepherd of Being.

The disclosure of the primordial as  is the end of a search that began with Plato….
This search comes to its conclusion with Heidegger."

— "Three Senses of 'A is B' in Heideggger," Ch. 17 in Indiscrete Thoughts
 

See also Four Dots in this journal. 

Some context:  McLuhan + Analogy.

Friday, May 19, 2017

From Algebra to Geometry

Filed under: General,Geometry — Tags: — m759 @ 10:45 PM

Sunday, April 16, 2017

Homily

Filed under: General,Geometry — m759 @ 12:25 PM

See also Plan 9.

Wednesday, April 12, 2017

Contracting the Spielraum

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

The contraction of the title is from group actions on
the ninefold square  (with the center subsquare fixed)
to group actions on the eightfold cube.

From a post of June 4, 2014

At math.stackexchange.com on March 1-12, 2013:

Is there a geometric realization of the Quaternion group?” —

The above illustration, though neatly drawn, appeared under the
cloak of anonymity.  No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note “GL(2,3) actions on a cube” of April 5, 1985).

Wednesday, October 19, 2016

The Crosswicks Curse Continues

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

"There is  such a thing as 1906 "

Thursday, September 15, 2016

The Smallest Perfect Number/Universe

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

The smallest perfect number,* six, meets
"the smallest perfect universe,"** PG(3,2).

IMAGE- Geometry of the Six-Set, Steven H. Cullinane, April 23, 2013

  * For the definition of "perfect number," see any introductory
    number-theory text that deals with the history of the subject.
** The phrase "smallest perfect universe" as a name for PG(3,2),
     the projective 3-space over the 2-element Galois field GF(2),
     was coined by math writer Burkard Polster. Cullinane's square
     model of PG(3,2) differs from the earlier tetrahedral model
     discussed by Polster.

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.

For Harlan Kane

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

(Author of The Abacus Conundrum )

The Galois Box

Pensée II

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

A sequel to the previous post

"I'm in with the in crowd" — Musical motif of "Irrational Man"

Pensée

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

On a stage character named Pascal —

"… the mathematics professor implores Minnie to take
full advantage of her own intelligence and talents,
usually capped off with a non sequitur gift of some kind…."

— Sean T. Collins, " 'I’m better than you, you son of a bitch':
     a review of The Diary of a Teenage Girl: The Play ,
     Comic Book Resources , March 30, 2010

See also, in this  journal, Minnie and The Gift .

Thursday, October 29, 2015

Verhexung

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

(Continued)

When shall we three meet again?

"Carnap, Tarski and Quine met together in 1940-1941 at Harvard
to discuss their views on the nature of language and the differences
between logic, mathematics and science. In this remarkable book
Greg Frost-Arnold presents Carnap’s extensive notes on these
meetings. Frost-Arnold also includes an elegant translation and a
detailed commentary that sets the notes in their philosophical context
and demonstrates their importance for many central debates about
the history of analytic philosophy. This book marks a decisive advance
in our understanding of the philosophical views of Carnap, Tarski and
Quine, and is essential reading for all who work on these topics."

— Christopher Pincock, The Ohio State University

Priest Logic

Filed under: General — Tags: — m759 @ 5:00 PM

Versus the Hobgoblins of Emerson Hall

"A foolish consistency is the hobgoblin of little minds."
Ralph Waldo Emerson

  

A Word in Your Delicate Ear

Filed under: General — Tags: , — m759 @ 11:02 AM

☯

Oslo Halloween

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

(Suggested by the latest Instagram post of Oslo artist Josefine Lyche)

Saturday, September 12, 2015

Ledger

Filed under: General — Tags: , — m759 @ 4:45 PM

Lorrie Moore, in the current New York Review of Books ,
on a detective in a TV series:

He "takes notes in a large ledger and
speaks as if he were the CEO of
a nihilist fortune cookie company."

— "Sympathy for the Devil," NYRB
      issue dated Sept. 24, 2015

See Harvard president Drew Faust as such a CEO.

The Harvard Crimson

UPDATED: September 12, 2015, at 4:22 pm. 

Luke Z. Tang ’18, a Lowell House sophomore,
has died “suddenly and unexpectedly,”
Lowell House Master Diana L. Eck told
House residents in an email Saturday.

Local authorities are investigating the cause
of the death, Eck wrote, and there is “no reason
to believe that foul play was involved.”

At the Still Point (continued)

Filed under: General — Tags: — m759 @ 9:07 AM

"Now I wanna dance, I wanna win.
I want that trophy, so dance good."

"C'est la vie , say the old folks.
It goes to show you never can tell.
"

Monday, May 25, 2015

A Stitch in Time

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

The most recent version of a passage
quoted in posts tagged "May 19 Gestalt" —

"You've got to pick up every stitch." — Donovan

Tuesday, May 19, 2015

The Lindbergh Manifesto

Filed under: General,Geometry — Tags: , — m759 @ 3:24 AM

"Creation is the birth of something, and
something cannot come from nothing."

— Photographer Peter Lindbergh at his website

From a biography of Lindbergh —

" it took Lindbergh awhile to find his true métier.
Born in Krefeld, Germany, in 1944….
Barely out of his teens, he became a painter who
embraced conceptual art and — for reasons he
has since forgotten — adopted the professional
name « Sultan. »   Lindbergh was a few years
short of his 30th birthday when he turned to
photography."

— "The Man Who Loves Women," by Pamela Young,
Toronto Globe & Mail , September 19, 1996

A Lindbergh work (at right below) from his conceptual-art days —

For a connection between the above work by Paul Talman and the
above "Mono Type 1" of Lindbergh, see…

Wednesday, April 29, 2015

Scenes from…

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

An Epic for Drink Boy —

Context:  The post in which the above scenes occur.

Gone

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

Update of 12:25 AM — See, too, The Oxford Murders.

Monday, December 29, 2014

Dodecahedron Model of PG(2,5)

Filed under: General,Geometry — Tags: , , — m759 @ 2:28 PM

Recent posts tagged Sagan Dodecahedron 
mention an association between that Platonic
solid and the 5×5 grid. That grid, when extended
by the six points on a "line at infinity," yields
the 31 points of the finite projective plane of
order five.  

For details of how the dodecahedron serves as
a model of this projective plane (PG(2,5)), see
Polster's A Geometrical Picture Book , p. 120:

For associations of the grid with magic rather than
with Plato, see a search for 5×5 in this journal.

Thursday, December 18, 2014

Platonic Analogy

Filed under: General,Geometry — Tags: , , — m759 @ 2:23 PM

(Five by Five continued)

As the 3×3 grid underlies the order-3 finite projective plane,
whose 13 points may be modeled by
the 13 symmetry axes of the cube,
so the 5×5 grid underlies the order-5 finite projective plane,
whose 31 points may be modeled by
the 31 symmetry axes of the dodecahedron.

See posts tagged Galois-Plane Models.

Wednesday, December 3, 2014

Pyramid Dance

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

Oslo artist Josefine Lyche has a new Instagram post,
this time on pyramids (the monumental kind).

My response —

Wikipedia's definition of a tetrahedron as a
"triangle-based pyramid"

and remarks from a Log24 post of August 14, 2013 :

Norway dance (as interpreted by an American)

IMAGE- 'The geometry of the dance' is that of a tetrahedron, according to Peter Pesic

I prefer a different, Norwegian, interpretation of "the dance of four."

Related material:
The clash between square and tetrahedral versions of PG(3,2).

See also some of Burkard Polster's triangle-based pyramids
and a 1983 triangle-based pyramid in a paper that Polster cites —

(Click image below to enlarge.)

Some other illustrations that are particularly relevant
for Lyche, an enthusiast of magic :

From On Art and Magic (May 5, 2011) —

http://www.log24.com/log/pix11A/110505-ThemeAndVariations-Hofstadter.jpg

http://www.log24.com/log/pix11A/110505-BlockDesignTheory.jpg

Mathematics

http://www.log24.com/log/pix11A/110505-WikipediaFanoPlane.jpg

The Fano plane block design

Magic

http://www.log24.com/log/pix11A/110505-DeathlyHallows.jpg

The Deathly Hallows  symbol—
Two blocks short of  a design.

 

(Updated at about 7 PM ET on Dec. 3.)

Friday, November 28, 2014

Former-Day Saint

Filed under: General — Tags: — m759 @ 4:35 PM

Continued from Friday the 13th of June, 2014 :

"It's going to be accomplished in steps,
this establishment of the Talented
​in the scheme of things."

— To Ride Pegasus ,
     by Anne McCaffrey (Radcliffe '47)

Related material:

Click Zenna Henderson's dates for
an informative essay from April 5, 2009.

See also posts on, or about, that date in this journal.

Wednesday, November 26, 2014

Class Act

Filed under: General,Geometry — Tags: — m759 @ 7:18 AM

Update of Nov. 30, 2014 —

For further information on the geometry in
the remarks by Eberhart below, see
pp. 16-17 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998). Polster
cites a different article by Lemay.

A search for background to the exercise in the previous post
yields a passage from the late Stephen Eberhart:

The first three primes p = 2, 3, and 5 therefore yield finite projective planes with 7, 13, and 31 points and lines, respectively. But these are just the numbers of symmetry axes of the five regular solids, as described in Plato's Timaeus : The tetrahedron has 4 pairs of face planes and comer points + 3 pairs of opposite edges, totalling 7 axes; the cube has 3 pairs of faces + 6 pairs of edges + 4 pairs of comers, totalling 13 axes (the octahedron simply interchanges the roles of faces and comers); and the pentagon dodecahedron has 6 pairs of faces + 15 pairs of edges + 10 pairs of comers, totalling 31 axes (the icosahedron again interchanging roles of faces and comers). This is such a suggestive result, one would expect to find it dealt with in most texts on related subjects; instead, while "well known to those who well know such things" (as Richard Guy likes to quip), it is scarcely to be found in the formal literature [9]. The reason for the common numbers, it turns out, is that the groups of symmetry motions of the regular solids are subgroups of the groups of collineations of the respective finite planes, a face axis being different from an edge axis of a regular solid but all points of a projective plane being alike, so the latter has more symmetries than the former.

[9] I am aware only of a series of in-house publications by Fernand Lemay of the Laboratoire de Didactique, Faculté des Sciences de I 'Éducation, Univ. Laval, Québec, in particular those collectively titled Genèse de la géométrie  I-X.

— Stephen Eberhart, Dept. of Mathematics,
California State University, Northridge, 
"Pythagorean and Platonic Bridges between
Geometry and Algebra," in BRIDGES: Mathematical
Connections in Art, Music, and Science 
, 1998,
archive.bridgesmathart.org/1998/bridges1998-121.pdf

Eberhart died of bone cancer in 2003. A memorial by his
high school class includes an Aug. 7, 2003, transcribed
letter from Eberhart to a classmate that ends…


… I earned MA’s in math (UW, Seattle) and history (UM, Missoula) where a math/history PhD program had been announced but canceled.  So 1984 to 2002 I taught math (esp. non-Euclidean geometry) at C.S.U. Northridge.  It’s been a rich life.  I’m grateful. 
 
Steve
 

See also another informative BRIDGES paper by Eberhart
on mathematics and the seven traditional liberal arts.

Tuesday, November 25, 2014

Euclidean-Galois Interplay

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

For previous remarks on this topic, as it relates to
symmetry axes of the cube, see previous posts tagged Interplay.

The above posts discuss, among other things, the Galois
projective plane of order 3, with 13 points and 13 lines.

Oxley's 2004 drawing of the 13-point projective plane

These Galois points and lines may be modeled in Euclidean geometry
by the 13 symmetry axes and the 13 rotation planes
of the Euclidean cube. They may also be modeled in Galois geometry
by subsets of the 3x3x3 Galois cube (vector 3-space over GF(3)).

http://www.log24.com/log/pix11A/110427-Cube27.jpg

   The 3×3×3 Galois Cube 

Exercise: Is there any such analogy between the 31 points of the
order-5 Galois projective plane and the 31 symmetry axes of the
Euclidean dodecahedron and icosahedron? Also, how may the
31 projective points  be naturally pictured as lines  within the 
5x5x5 Galois cube (vector 3-space over GF(5))?

Update of Nov. 30, 2014 —

For background to the above exercise, see
pp. 16-17 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998), esp.
the citation to a 1983 article by Lemay.

Saturday, September 20, 2014

Contest

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

Once again, Harvard defeats Holy Cross.

IMAGE- Joseph Campbell, 'The Inner Reaches of Outer Space,' meditation on the number nine, the Goddess, and the Angelus

See also a related remark by Norman Mailer, and Plan 9 in this journal.

Presumably the Holy Cross defeat will please art theorist Rosalind Krauss (below).

(Click to enlarge.)

Symplectic Structure

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

(Continued)

The fictional zero theorem  of Terry Gilliam's current film
by that name should not be confused with the zero system
underlying the diamond theorem.

The Metaphysics of Entities

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

Anthony Lane in The New Yorker , issue dated Sept. 22, 2014:

"The hero of 'The Zero Theorem' is a computer genius called Qohen Leth
(Christoph Waltz)…. He is the sole resident of a derelict church, where,
on a crucifix in front of the altar, the head of Christ has been replaced by
a security camera. No prayers are ever said, and none are answered.

In short, the place is deconsecrated, but to claim that it lacks any spark of
sacred yearning would be wrong, because Qohen devotes his days to seeking
the Zero Theorem, which—whatever it may be—lies at the fuzzy limit of
human powers. We crunch entities,” he says, as if that explained anything.
His employer is Mancom, a large corporation that, in Orwellian fashion,
oversees ordinary lives, although it betrays more frantic desperation than
glowering threat."

One approach to the metaphysics of entities was indicated in the previous
post, 'Metaphysics for Gilliam." A different approach:

"Categories, Sets, and the Nature of Mathematical Entities,"
by Jean-Pierre Marquis, Ch. 13, pp. 181-192, in the 2006 book
The Age of Alternative Logics , ed. by van Benthem et al.
(Springer, Netherlands).

From pages 182-183 —

13.2 The nature of mathematical entities

Let us start with the nature of mathematical entities in general and with a
rough and classical distinction that will simply set the stage for the picture we
want to develop. We essentially follow Lowe 1998* for the basic distinctions. We
need to distinguish between abstract and concrete entities, on the one hand, and
universals and particulars on the other hand. For our purpose, it is not necessary
to specify a criterion of demarcation between abstract and concrete entities. We
simply assume that such a distinction can be made, e.g. concrete entities can
change whereas abstract entities cannot. We assume that a universal is an entity
that can be instantiated by entities which themselves are not instantiable, the
latter being of course particulars. Given these distinctions, an entity can be a
concrete particular, a concrete universal, an abstract particular or an abstract
universal.

Our focus here is between the last two possibilities. For we claim that the
current conception of sets makes them abstract particulars whereas for objects
defined within categories, mathematical entities are abstract universals. This,
we claim, is true of category theory as it is.

* Lowe, E.J., 1998, The Possibility of Metaphysics , Oxford: Clarendon Press.

Thursday, September 11, 2014

Portals

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

Part I

Image- Josefine Lyche's work (with 1986 figures by Cullinane) in a 2009 exhibition in Oslo

Part II

Part III

Record-Breaking

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

Blackboard Jungle , 1955 —

IMAGE- Richard Kiley in 'Blackboard Jungle,' with grids and broken records

Today's online Harvard Crimson :

Harvard Crimson, 9/11/2014: 'CS50 Logs Record-Breaking Enrollment Numbers,' by Meg P. Bernhard, Crimson Staff Writer

Oh, Moon of Alabama

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

The mention of Gauss in today's previous post, along with
recent news, suggested this post.

"How do you  get young people excited about space?"

— Megan Garber in The Atlantic , Aug. 16, 2012

Further details:  Child Buyers (July 16, 2013).

A Class by Itself

Filed under: General — Tags: — m759 @ 9:48 AM

The American Mathematical Society yesterday:

Harvey Cohn (1923-2014)
Wednesday September 10th 2014

Cohn, an AMS Fellow and a Putnam Fellow (1942), died May 16 at the age of 90. He served in the Navy in World War II and following the war received his PhD from Harvard University in 1948 under the direction of Lars Ahlfors. He was a member of the faculty at Wayne State University, Stanford University, Washington University in St. Louis, the University of Arizona, and at City College of New York, where he was a distinguished professor. After retiring from teaching, he also worked for the NSA. Cohn was an AMS member since 1942.

Paid death notice from The New York Times , July 27, 2014:

COHN–Harvey. Fellow of the American Mathematical Society and member of the Society since 1942, died on May 16 at the age of 90. He was a brilliant Mathematician, an adoring husband, father and grandfather, and faithful friend and mentor to his colleagues and students. Born in New York City in 1923, Cohn received his B.S. degree (Mathematics and Physics) from CCNY in 1942. He received his M.S. degree from NYU (1943), and his Ph.D. from Harvard (1948) after service in the Navy (Electronic Technicians Mate, 1944-46). He was a member of Phi Beta Kappa (Sigma Chi), won the William Lowell Putnam Prize in 1942, and was awarded the Townsend Harris Medal in 1972. A pioneer in the intensive use of computers in an innovative way in a large number of classical mathematical problems, Harvey Cohn held faculty positions at Wayne State University, Stanford, Washington University Saint Louis (first Director of the Computing Center 1956-58), University of Arizona (Chairman 1958-1967), University of Copenhagen, and CCNY (Distinguished Professor of Mathematics). After his retirement from teaching, he worked in a variety of capacities for the National Security Agency and its research arm, IDA Center for Computing Sciences. He is survived by his wife of 63 years, Bernice, of Laguna Woods, California and Ft. Lauderdale, FL, his son Anthony, daughter Susan Cohn Boros, three grandchildren and one great-granddaughter.

— Published in The New York Times  on July 27, 2014

See also an autobiographical essay found on the web.

None of the above sources mention the following book, which is apparently by this same Harvey Cohn. (It is dedicated to "Tony and Susan.")

From Google Books:

Advanced Number Theory, by Harvey Cohn
Courier Dover Publications, 1980 – 276 pages
(First published by Wiley in 1962 as A Second Course in Number Theory )

Publisher's description:

" 'A very stimulating book … in a class by itself.'— American Mathematical Monthly

Advanced students, mathematicians and number theorists will welcome this stimulating treatment of advanced number theory, which approaches the complex topic of algebraic number theory from a historical standpoint, taking pains to show the reader how concepts, definitions and theories have evolved during the last two centuries. Moreover, the book abounds with numerical examples and more concrete, specific theorems than are found in most contemporary treatments of the subject.

The book is divided into three parts. Part I is concerned with background material — a synopsis of elementary number theory (including quadratic congruences and the Jacobi symbol), characters of residue class groups via the structure theorem for finite abelian groups, first notions of integral domains, modules and lattices, and such basis theorems as Kronecker's Basis Theorem for Abelian Groups.

Part II discusses ideal theory in quadratic fields, with chapters on unique factorization and units, unique factorization into ideals, norms and ideal classes (in particular, Minkowski's theorem), and class structure in quadratic fields. Applications of this material are made in Part III to class number formulas and primes in arithmetic progression, quadratic reciprocity in the rational domain and the relationship between quadratic forms and ideals, including the theory of composition, orders and genera. In a final concluding survey of more recent developments, Dr. Cohn takes up Cyclotomic Fields and Gaussian Sums, Class Fields and Global and Local Viewpoints.

In addition to numerous helpful diagrams and tables throughout the text, appendices, and an annotated bibliography, Advanced Number Theory  also includes over 200 problems specially designed to stimulate the spirit of experimentation which has traditionally ruled number theory."

User Review –

"In a nutshell, the book serves as an introduction to Gauss' theory of quadratic forms and their composition laws (the cornerstone of his Disquisitiones Arithmeticae) from the modern point of view (ideals in quadratic number fields). I strongly recommend it as a gentle introduction to algebraic number theory (with exclusive emphasis on quadratic number fields and binary quadratic forms). As a bonus, the book includes material on Dirichlet L-functions as well as proofs of Dirichlet's class number formula and Dirichlet's theorem in primes in arithmetic progressions (of course this material requires the reader to have the background of a one-semester course in real analysis; on the other hand, this material is largely independent of the subsequent algebraic developments).

Better titles for this book would be 'A Second Course in Number Theory' or 'Introduction to quadratic forms and quadratic fields'. It is not a very advanced book in the sense that required background is only a one-semester course in number theory. It does not assume prior familiarity with abstract algebra. While exercises are included, they are not particularly interesting or challenging (if probably adequate to keep the reader engaged).

While the exposition is *slightly* dated, it feels fresh enough and is particularly suitable for self-study (I'd be less likely to recommend the book as a formal textbook). Students with a background in abstract algebra might find the pace a bit slow, with a bit too much time spent on algebraic preliminaries (the entire Part I—about 90 pages); however, these preliminaries are essential to paving the road towards Parts II (ideal theory in quadratic fields) and III (applications of ideal theory).

It is almost inevitable to compare this book to Borevich-Shafarevich 'Number Theory'. The latter is a fantastic book which covers a large superset of the material in Cohn's book. Borevich-Shafarevich is, however, a much more demanding read and it is out of print. For gentle self-study (and perhaps as a preparation to later read Borevich-Shafarevich), Cohn's book is a fine read."

Tuesday, September 9, 2014

In Memoriam

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

For Loren D. Olson, Harvard '64:

"Even 50 years later, I remember his enthusiasm for a very young
and very gifted Harvard professor named Shlomo Sternberg, one
of whose special areas of interest was Lie groups. I still have no real
understanding of what a Lie group is, but not for want of trying on
Loren’s part. Loren was also quite interested in the thinking of the
theologians Paul Tillich and Reinhold Niebuhr, who were then at
Harvard. He attended some of their lectures, read several of their
books, and enjoyed discussing their ideas."

Harvard classmate David Jackson

See also today's previous post.

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