Log24

Friday, December 4, 2020

Here’s to Efficient Packing!

Filed under: General — Tags: , — m759 @ 4:19 PM

"Rock music from car . . ." 

http://www.log24.com/log/pix11A/110614-TarantinoCar.jpg

Monday, September 13, 2021

Cube Space Revisited

Filed under: General — Tags: , , , , — m759 @ 3:02 PM

The above Quanta  article mentions

"Maryna Viazovska’s 2016 discovery of the most efficient
ways of packing spheres in dimensions eight and 24."

From a course to be taught by Viazovska next spring:

The Lovasz reference suggests a review of my own webpage
Cube Space, 1984-2003.

See as well a review of Log24 posts on Packing.

Saturday, September 4, 2021

Number, Time, and Dan Brown*

Filed under: General — Tags: — m759 @ 2:12 PM

The title is a nod to the book Number and Time  ( not  by Dan Brown ).

An illustration —

This journal 10 years ago today

"Dan Brown certainly packed a lot into
the 500-plus pages of The Lost Symbol ."

— DailyGrail.com

That sentence suggests a review of Efficient Packing

Wednesday, August 4, 2021

“Old men ought to be explorers” — T. S. Eliot

Filed under: General — Tags: — m759 @ 2:38 AM

My own lucubrations from April 12, 2012 include . . .

From the target of the above link —

Image of MOMA Chess Set cover.

"Here's to efficient packing.

Friday, April 16, 2021

Schoolgirl Problem (Kirkman’s, not Epstein’s)

Filed under: General — m759 @ 2:30 PM

 

As for Epstein’s problem . . .

Sophia Lillis in Stephen King's IT (2017)— 'Right stuff' question

Friday, December 4, 2020

German Lesson: Untergang

Filed under: General — Tags: — m759 @ 11:37 PM

The recent posts “Bunker Bingo” and “Here’s to Efficient Packing!
suggest a review.

Alex Ross in The New Yorker  on Dec. 2, 2020, on the German
word “Untergang ” —

“The usual translation is ‘downfall,’ although
the various implications of the word—
literally, “going-under”—are difficult to capture
in English. In some contexts, Untergang  simply
means descent: a sunset is a Sonnenuntergang .

Lauren  German in a 2005 film —

See as well . . .

Thursday, December 3, 2020

Bunker Bingo

Filed under: General — Tags: — m759 @ 11:59 AM

See as well 5×5The Matrix of Abraham,  and Deutsche Schule Montevideo .

IMAGE- Right 3-4-5 triangle with squares on sides and hypotenuse as base

“If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”

— Henry David Thoreau

Thursday, August 27, 2020

The Complete Extended Binary Golay Code

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

All 4096 vectors in the code are at . . .

http://neilsloane.com/oadir/oa.4096.12.2.7.txt.

Sloane’s list* contains the 12 generating vectors
listed in 2011 by Adlam —

As noted by Conway in Sphere Packings, Lattices and Groups ,
these 4096 vectors, constructed lexicographically, are exactly
the same vectors produced by using the Conway-Sloane version
of the Curtis Miracle Octad Generator (MOG). Conway says this
lexico-MOG equivalence was first discovered by M. J. T. Guy.

(Of course, any  permutation of the 24 columns above produces
a version of the code qua  code. But because the lexicographic and
the MOG constructions yield the same result, that result is in
some sense canonical.)

See my post of July 13, 2020 —

The lexicographic Golay code
contains, embedded within it,
the Miracle Octad Generator.

For some related results, Google the twelfth generator:

* Sloane’s list is of the codewords as the rows of  an orthogonal array

See also http://neilsloane.com/oadir/.

Sunday, July 7, 2019

Schoolgirl Problem

Filed under: General — Tags: , , — m759 @ 11:18 PM

Anonymous remarks on the schoolgirl problem at Wikipedia —

"This solution has a geometric interpretation in connection with 
Galois geometry and PG(3,2). Take a tetrahedron and label its
vertices as 0001, 0010, 0100 and 1000. Label its six edge centers
as the XOR of the vertices of that edge. Label the four face centers
as the XOR of the three vertices of that face, and the body center
gets the label 1111. Then the 35 triads of the XOR solution correspond
exactly to the 35 lines of PG(3,2). Each day corresponds to a spread
and each week to a packing
."

See also Polster + Tetrahedron in this  journal.

There is a different "geometric interpretation in connection with
Galois geometry and PG(3,2)" that uses a square  model rather
than a tetrahedral  model. The square  model of PG(3,2) last
appeared in the schoolgirl-problem article on Feb. 11, 2017, just
before a revision that removed it.

Tuesday, December 25, 2018

Simply

Filed under: General — m759 @ 11:42 AM
 

"So to obtain the isomorphism from L2(7) onto L3(2) we simply
multiply any given permutation of L2(7) by the affine translation
that restores to its rightful place."

— Sphere Packings, Lattices and Groups ,
by John H. Conway and N. J. A. Sloane.
First edition, 1988, published by Springer-Verlag New York, Inc.
Chapter 11 (by J. H. Conway), "The Golay Codes and the Mathieu Groups," 
Section 12, "The trio group 26:(S3×L2(7))"

Compare and contrast —

Why PSL(2,7) is isomorphic to GL(3.2)

This post was suggested by a New York Times  headline today —

Thursday, November 29, 2018

The Fortune Cookie

Filed under: General — m759 @ 10:54 AM

The mathematician Chuanming Zong in the previous post
has also written about  Aristotle's pyramid scheme

Monday, May 2, 2016

Subjective Quality

Filed under: General,Geometry — m759 @ 6:01 AM

The previous post deals in part with a figure from the 1988 book
Sphere Packings, Lattices and Groups , by J. H. Conway and
N. J. A. Sloane.

Siobhan Roberts recently wrote a book about the first of these
authors, Conway.  I just discovered that last fall she also had an
article about the second author, Sloane, published:

"How to Build a Search Engine for Mathematics,"
Nautilus , Oct 22, 2015.

Meanwhile, in this  journal

Log24 on that same date, Oct. 22, 2015 —

Roberts's remarks on Conway and later on Sloane are perhaps
examples of subjective  quality, as opposed to the objective  quality
sought, if not found, by Alexander, and exemplified by the
above bijection discussed here  last October.

Sunday, May 1, 2016

Sunday Appetizer from 1984

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

Judith Shulevitz in The New York Times
on Sunday, July 18, 2010
(quoted here Aug. 15, 2010) —

“What would an organic Christian Sabbath look like today?”

The 2015 German edition of Beautiful Mathematics ,
a 2011 Mathematical Association of America (MAA) book,
was retitled Mathematische Appetithäppchen —
Mathematical Appetizers . The German edition mentions
the author's source, omitted in the original American edition,
for his section 5.17, "A Group of Operations" (in German,
5.17, "Eine Gruppe von Operationen") —  

Mathematische Appetithäppchen:
Faszinierende Bilder. Packende Formeln. Reizvolle Sätze

Autor: Erickson, Martin —

"Weitere Informationen zu diesem Themenkreis finden sich
unter http://​www.​encyclopediaofma​th.​org/​index.​php/​
Cullinane_​diamond_​theorem
und http://​finitegeometry.​org/​sc/​gen/​coord.​html ."

That source was a document that has been on the Web
since 2002. The document was submitted to the MAA
in 1984 but was rejected. The German edition omits the
document's title, and describes it as merely a source for
"further information on this subject area."

The title of the document, "Binary Coordinate Systems,"
is highly relevant to figure 11.16c on page 312 of a book
published four years after the document was written: the 
1988 first edition of Sphere Packings, Lattices and Groups
by J. H. Conway and N. J. A. Sloane —

A passage from the 1984 document —

Thursday, March 24, 2016

Many Dimensions…

Filed under: General — m759 @ 2:00 PM

For Women's History Month —

Friday, January 17, 2014

The 4×4 Relativity Problem

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

The sixteen-dot square array in yesterday’s noon post suggests
the following remarks.

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

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

The Galois tesseract  appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

The 1977 matrix Q is echoed in the following from 2002—

IMAGE- Dolgachev and Keum, coordinatization of the 4x4 array in 'Birational Automorphisms of Quartic Hessian Surfaces,' AMS Transactions, 2002

A different representation of Cullinane’s 1977 square model of the
16-point affine geometry over the two-element Galois field GF(2)
is supplied by Conway and Sloane in Sphere Packings, Lattices and Groups   
(first published in 1988) :

IMAGE- The Galois tesseract as a four-dimensional vector space, from a diagram by Conway and Sloane in 'Sphere Packings, Lattices, and Groups'

Here a, b, c, d   are basis vectors in the vector 4-space over GF(2).
(For a 1979 version of this vector space, see AMS Abstract 79T-A37.)

See also a 2011 publication of the Mathematical Association of America —

From 'Beautiful Mathematics,' by Martin Erickson, an excerpt on the Cullinane diamond theorem (with source not mentioned)

Sunday, December 15, 2013

Sermon

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

Odin's Jewel

Jim Holt, the author of remarks in yesterday's
Saturday evening post

"It turns out that the Kyoto school of Buddhism
makes Heidegger seem like Rush Limbaugh—
it’s so rarified, I’ve never been able to
understand it at all. I’ve been knocking my head
against it for years."

Vanity Fair Daily , July 16, 2012

Backstory Odin + Jewel in this journal.

See also Odin on the Kyoto school —

For another version of Odin's jewel, see Log24
on the date— July 16, 2012— that Holt's Vanity Fair
remarks were published. Scroll to the bottom of the
"Mapping Problem continued" post for an instance of
the Galois tesseract —

IMAGE- The Galois tesseract as a four-dimensional vector space, from a diagram by Conway and Sloane in 'Sphere Packings, Lattices, and Groups'

Saturday, December 14, 2013

Beautiful Mathematics

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

The title, which I dislike, is taken from a 2011 publication
of the MAA, also sold by Cambridge University Press.

Some material relevant to the title adjective:

"For those who have learned something of higher mathematics, nothing could be more natural than to use the word 'beautiful' in connection with it. Mathematical beauty, like the beauty of, say, a late Beethoven quartet, arises from a combination of strangeness and inevitability. Simply defined abstractions disclose hidden quirks and complexities. Seemingly unrelated structures turn out to have mysterious correspondences. Uncanny patterns emerge, and they remain uncanny even after being underwritten by the rigor of logic."— Jim Holt, opening of a book review in the Dec. 5, 2013, issue of The New York Review of Books

Some relevant links—

The above list was updated on Jan. 31, 2014, to include the
"Strangeness" and "Hidden quirks" links.  See also a post of
​Jan. 31, 2014.

Update of March 9, 2014 —

The link "Simply defined abstractions" is to the construction of the Steiner
system S(5, 8, 24) described by R. T. Curtis in his 1976 paper defining the
Miracle Octad Generator. It should be noted that this construction is due
to Richard J. Turyn, in a 1967 Sylvania research report. (See Emily Jennings's
talk of 1 Nov. 2012.) Compare  the Curtis construction, written in 1974,
with the Turyn construction of 1967 as described in Sphere Packings, Lattices
and Groups , by J. H. Conway and N. J. A. Sloane (first published in 1988).

Sunday, May 19, 2013

Sermon

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

Best vs. Bester

The previous post ended with a reference mentioning Rosenhain.

For a recent application of Rosenhain's work, see
Desargues via Rosenhain (April 1, 2013).

From the next day, April 2, 2013:

"The proof of Desargues' theorem of projective geometry
comes as close as a proof can to the Zen ideal.
It can be summarized in two words: 'I see!' "

– Gian-Carlo Rota in Indiscrete Thoughts (1997)

Also in that book, originally from a review in Advances in Mathematics ,
Vol. 84, Number 1, Nov. 1990, p. 136:
IMAGE- Rota's review of 'Sphere Packings, Lattices and Groups'-- in a word, 'best'

See, too, in the Conway-Sloane book, the Galois tesseract  
and, in this journal, Geometry for Jews and The Deceivers , by Bester.

Tuesday, April 2, 2013

Rota in a Nutshell

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

"The proof of Desargues' theorem of projective geometry
comes as close as a proof can to the Zen ideal.
It can be summarized in two words: 'I see!' "

— Gian-Carlo Rota in Indiscrete Thoughts (1997)

Also in that book, originally from a review in Advances in Mathematics,
Vol. 84, Number 1, Nov. 1990, p. 136:

IMAGE- Rota's review of 'Sphere Packings, Lattices and Groups'-- in a word, 'best'

Related material:

Pascal and the Galois nocciolo ,
Conway and the Galois tesseract,
Gardner and Galois.

See also Rota and Psychoshop.

Saturday, January 5, 2013

Vector Addition in a Finite Field

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

The finite (i.e., Galois) field GF(16),
according to J. J. Seidel in 1974—

The same field according to Steven H. Cullinane in 1986,
in its guise as the affine 4-space over GF(2)—


The same field, again disguised as an affine 4-space,
according to John H. Conway and N.J.A. Sloane in
Sphere Packings, Lattices, and Groups , first published in 1988—

The above figure by Conway and Sloane summarizes, using
a 4×4 array, the additive vector-space structure of the finite
field GF(16).

This structure embodies what in Euclidean space is called
the parallelogram rule for vector addition—

(Thanks to June Lester for the 3D (uvw) part of the above figure.)

For the transition from this colored Euclidean hypercube
(used above to illustrate the parallelogram rule) to the
4×4 Galois space (illustrated by Cullinane in 1979 and
Conway and Sloane in 1988— or later… I do not have
their book’s first edition), see Diamond Theory in 1937,
Vertex Adjacency in a Tesseract and in a 4×4 Array,
Spaces as Hypercubes, and The Galois Tesseract.

For some related narrative, see tesseract  in this journal.

(This post has been added to finitegeometry.org.)

Update of August 9, 2013—

Coordinates for hypercube vertices derived from the
parallelogram rule in four dimensions were better
illustrated by Jürgen Köller in a web page archived in 2002.

Update of August 13, 2013—

The four basis vectors in the 2002 Köller hypercube figure
are also visible at the bottom of the hypercube figure on
page 7 of “Diamond Theory,” excerpts from a 1976 preprint
in Computer Graphics and Art , Vol. 2, No. 1, February 1977.
A predecessor:  Coxeter’s 1950 hypercube figure from
Self-Dual Configurations and Regular Graphs.”

Wednesday, October 31, 2012

The Malfunctioning TARDIS

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

(Continued from previous TARDIS posts)

Summary: A review of some  posts from last August is suggested by the death,
reportedly during the dark hours early on October 30, of artist Lebbeus Woods.

An (initially unauthorized) appearance of his work in the 1995 film
Twelve Monkeys 

 … suggests a review of three posts from last August.

Wednesday, August 1, 2012

Defining Form

Filed under: Uncategorized — m759 @ 11:01 AM 

Continued from July 29 in memory of filmmaker Chris Marker,
who reportedly* died on that date at 91 at his home in Paris.

See Slides and Chantingand Where Madness Lies.

See also Sherrill Grace on Malcolm Lowry.

Washington PostOther sources say Marker died on July 30.

 These notably occur in Marker's masterpiece
     La Jetée  (review with spoilers).

 

Wednesday, August 1, 2012

Triple Feature

Filed under: Uncategorized — m759 @ 11:11 PM

IMAGE- Triple Feature: 'Twelve Monkeys,' Reagan National Airport on July 31, 2012, and 'Die Hard 2'

For related material, see this morning's post Defining Form.

 

Sunday, August 12, 2012

Doctor Who

Filed under: Uncategorized — m759 @ 2:00 PM

On Robert A. Heinlein's novel Glory Road

"Glory Road  (1963) included the foldbox , a hyperdimensional packing case that was bigger inside than outside. It is unclear if Glory Road  was influenced by the debut of the science fiction television series Doctor Who  on the BBC that same year. In Doctor Who , the main character pilots a time machine called a TARDIS, which is built with technology which makes it 'dimensionally transcendental,' that is, bigger inside than out."

— Todd, Tesseract article at exampleproblems.com

From the same exampleproblems.com article—

"The connection pattern of the tesseract's vertices is the same as that of a 4×4 square array drawn on a torus; each cell (representing a vertex of the tesseract) is adjacent to exactly four other cells. See geometry of the 4×4 square."

For further details, see today's new page on vertex adjacency at finitegeometry.org.

 

"It was a dark and stormy night."— A Wrinkle in Time

Sunday, August 12, 2012

Doctor Who

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

On Robert A. Heinlein's novel Glory Road

"Glory Road  (1963) included the foldbox , a hyperdimensional packing case that was bigger inside than outside. It is unclear if Glory Road  was influenced by the debut of the science fiction television series Doctor Who  on the BBC that same year. In Doctor Who , the main character pilots a time machine called a TARDIS, which is built with technology which makes it 'dimensionally transcendental,' that is, bigger inside than out."

— Todd, Tesseract article at exampleproblems.com

From the same exampleproblems.com article—

"The connection pattern of the tesseract's vertices is the same as that of a 4×4 square array drawn on a torus; each cell (representing a vertex of the tesseract) is adjacent to exactly four other cells. See geometry of the 4×4 square."

For further details, see today's new page on vertex adjacency at finitegeometry.org.

Sunday, July 29, 2012

The Galois Tesseract

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

(Continued)

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

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

— share the same vector-space structure:

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

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

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

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

Source:

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

Monday, July 16, 2012

Mapping Problem continued

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

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

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

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

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

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

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

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

— share the following vector-space structure —

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

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

Thursday, September 1, 2011

The Galois Tesseract

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

Click to enlarge

IMAGE- The Galois Tesseract, 1979-1999

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

Wednesday, June 1, 2011

The Schwartz Notes

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

A Google search today for material on the Web that puts the diamond theorem
in context yielded a satisfyingly complete list. (See the first 21 results.)
(Customization based on signed-out search activity was disabled.)

The same search limited to results from only the past month yielded,
in addition, the following—

http://www.log24.com/log/pix11A/110601-Search.jpg

This turns out to be a document by one Richard Evan Schwartz,
Chancellor’s Professor of Mathematics at Brown University.

Pages 12-14 of the document, which is untitled, undated, and
unsigned, discuss the finite-geometry background of the R.T.
Curtis Miracle Octad Generator (MOG) . As today’s earlier search indicates,
this is closely related to the diamond theorem. The section relating
the geometry to the MOG is titled “The MOG and Projective Space.”
It does not mention my own work.

See Schwartz’s page 12, page 13, and page 14.

Compare to the web pages from today’s earlier search.

There are no references at the end of the Schwartz document,
but there is this at the beginning—

These are some notes on error correcting codes. Two good sources for
this material are
From Error Correcting Codes through Sphere Packings to Simple Groups ,
by Thomas Thompson.
Sphere Packings, Lattices, and Simple Groups  by J. H. Conway and N.
Sloane
Planet Math (on the internet) also some information.

It seems clear that these inadequate remarks by Schwartz on his sources
can and should be expanded.

Wednesday, October 27, 2010

Review

Filed under: General — m759 @ 12:00 AM

A mathematical review—

IMAGE- Rota's review of 'Sphere Packings, Lattices and Groups'-- in a word, 'best'

      — Gian-Carlo Rota

A science fiction—

Tenser, said the tensor

      — Alfred Bester

Saturday, May 29, 2010

Packed

Filed under: General — m759 @ 7:11 AM

Significant Passage:
On the Writing Style of Visual Thinkers

"The words are filled with unstated meaning.
They are (the term is Ricoeur's) 'packed'
and need unpacking." —Gerald Grow

From the date of Ricoeur's death,
May 20, 2005

“Plato’s most significant passage
    may be found in Phaedrus  265b…."

From Sept. 30, 2004

With a little effort,
anything can be shown
to connect with anything else:
existence is infinitely

cross-referenced."Image-- 8-rayed asterisk

— Opening sentence
of Martha Cooley's
The Archivist

Image-- 8-rayed asterisk Example:
Mozart's K 265,
the page number 265,
and a story by George MacDonald.

Mozart's K 265 is variations on the theme
now known as "Twinkle, Twinkle, Little Star."

For darker variations on the Twinkle theme,
see the film "Joshua" and Martin Gardner's
Annotated Alice  (Norton, 2000, pp. 73-75).

Image-- From the film 'Joshua,' Joshua with the Alice statue in Central Park

Joshua

For darker variations on the asterisk theme,
see Darkness Visible (May 25)
and Vonnegut's Asterisk.

Friday, May 28, 2010

Multispeech for Oxford

Filed under: General — m759 @ 7:00 PM

Happy Birthday,
Carey Mulligan

http://www.log24.com/log/pix10A/100528-Mulligan.jpg

Star of "An Education"

In "An Education," Mulligan's character
applies for admission to Oxford.

Today's New York Times:

Education »

Oxford Tradition
Comes to This:
‘Death’ (Expound)

Related material:

Such words arrive on the page like suitcases at the baggage claim: You know there is something in them and they have travelled far, but you cannot tell what the writer means. The words are filled with unstated meaning. They are (the term is Ricoeur's) "packed" and need unpacking.

This method of using language, however, is not always a defect; radiantly evocative words have long been the language of myth, mysticism, and love. Also, in earlier centuries, educated readers expected to interpret writing on several different levels at once (e.g., literal, allegorical, moral, and anagogical or spiritual), so that multiple meanings were the norm. This was before the era of clear, expository, fully-explicit prose.

Visual thinkers are accustomed to their own kind of interpreting; the very act of visual perception, as Gregory (1966, 1970) and Gombrich (1959) have shown, is interpretive. When oral thinkers leave you to guess at something they have written, it is usually something that would have been obvious had the writing been a conversation. Such is not the case with visual thinkers, even whose spoken words can be mysterious references to visual thoughts invisible to anyone but the thinker.

Writing done in this "packed" manner makes more sense when read as poetry than when read as prose.

References:

Gombrich, E. H. (1959). Art and Illusion: A Study in the Psychology of Pictorial Representation. London: Phaidon.

Gregory, R. L. (1966). Eye and Brain: The Psychology of Seeing. New York: McGraw-Hill.

Gregory, R. L. (1970). The Intelligent Eye. New York: McGraw-Hill.

"Stacking, Packing, and Enfolding Words," by Gerald Grow in "The Writing Problems of Visual Thinkers"

Those wishing to emulate Mulligan's
character in "An Education" might,
having read the Times article above,
consult this journal's post of May 17,
"Rolling the Stone."

That post contains the following
image from the Times

http://www.log24.com/log/pix10A/100517-NYT-Stone.jpg

May 17 was, by the way, the day
that R. L. Gregory, author of
The Intelligent Eyedied.

Wednesday, November 30, 2005

Wednesday November 30, 2005

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

For St. Andrew’s Day

The miraculous enters…. When we investigate these problems, some fantastic things happen….”

— John H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, preface to first edition (1988)

The remarkable Mathieu group M24, a group of permutations on 24 elements, may be studied by picturing its action on three interchangeable 8-element “octads,” as in the “Miracle Octad Generator” of R. T. Curtis.

A picture of the Miracle Octad Generator, with my comments, is available online.


 Cartoon by S.Harris

Related material:
Mathematics and Narrative.

Friday, November 18, 2005

Friday November 18, 2005

Filed under: General — m759 @ 2:56 AM
It’s still the same old story,
a fight for love and…

Glory

Wikipedia on the tesseract:

Glory Road (1963) included the foldbox, a hyperdimensional packing case that was bigger inside than outside.”

Robert A. Heinlein in Glory Road:

    “Rufo’s baggage turned out to be a little black box about the size and shape of a portable typewriter. He opened it.
    And opened it again.
    And kept on opening it– And kept right on unfolding its sides and letting them down until the durn thing was the size of a small moving van and even more packed….
    … Anyone who has studied math knows that the inside does not have to be smaller than the outside, in theory….  Rufo’s baggage just carried the principle further.”

Johnny Cash: “And behold, a white horse.”

On The Last Battle
, a book in the Narnia series by C. S. Lewis:

“… there is much glory in this wonderfully written apocalypse.  Tirian, looking into the stable through the hole in the door, says, ‘The stable seen from within and the stable seen from without are two different places.’ Digory answers, ‘Its inside is bigger than its outside.’  It is the perceptive Lucy who voices the hope that is in us, ‘In our world, too, a stable once had something inside it that was bigger than our whole world.'”


Lewis said in “The Weight of Glory”

 

“Do you think I am trying to weave a spell? Perhaps I am; but remember your fairy tales. Spells are used for breaking enchantments as well as for inducing them.”

On enchantments that need to be broken:

See the description of the Eater of Souls in Glory Road and of Scientism in

Tuesday, September 14, 2004

Tuesday September 14, 2004

Filed under: General,Geometry — Tags: , — m759 @ 3:00 PM

The Square Wheel

Harmonic analysis may be based either on the circular (i.e., trigonometric) functions or on the square (i. e., Walsh) functions.  George Mackey's masterly historical survey showed that the discovery of Fourier analysis, based on the circle, was of comparable importance (within mathematics) to the discovery (within general human history) of the wheel.  Harmonic analysis based on square functions– the "square wheel," as it were– is also not without its importance.

For some observations of Stephen Wolfram on square-wheel analysis, see pp. 573 ff. in Wolfram's magnum opus, A New Kind of Science (Wolfram Media, May 14, 2002).  Wolfram's illustration of this topic is closely related, as it happens, to a note on the symmetry of finite-geometry hyperplanes that I wrote in 1986.  A web page pointing out this same symmetry in Walsh functions was archived on Oct. 30, 2001.

That web page is significant (as later versions point out) partly because it shows that just as the phrase "the circular functions" is applied to the trigonometric functions, the phrase "the square functions" might well be applied to Walsh functions– which have, in fact, properties very like those of the trig functions.  For details, see Symmetry of Walsh Functions, updated today.

"While the reader may draw many a moral from our tale, I hope that the story is of interest for its own sake.  Moreover, I hope that it may inspire others, participants or observers, to preserve the true and complete record of our mathematical times."

From Error-Correcting Codes
Through Sphere Packings
To Simple Groups
,
by Thomas M. Thompson,
Mathematical Association of America, 1983

Friday, May 23, 2003

Friday May 23, 2003

Filed under: General — m759 @ 7:23 PM

Mental Health Month, Day 23:

The Prime Cut Gospel

On Christmas Day, 1949,
Mary Elizabeth Spacek was born in Texas.

Lee Marvin, Sissy Spacek in “Prime Cut”

Exercises for Mental Health Month:

Read this discussion of the phrase, suggested by Spacek’s date of birth, “God’s gift to men.”

Read this discussion of the phrase “the same yesterday, today, and forever,” suggested by the previous reading.

Read the more interesting of these discussions of the phrase “the eternal in the temporal.”

Read this discussion of eternal, or “necessary,” truths versus other sorts of alleged “truths.”

Read this discussion of unimportant mathematical properties of the prime number 23.

Read these discussions of important properties of 23:

  • R. D. Carmichael’s 1937 discussion of the linear fractional group modulo 23 in 

Introduction to the Theory of Groups of Finite Order, Ginn, Boston, 1937 (reprinted by Dover in 1956), final chapter, “Tactical Configurations,” and

  • Conway’s 1969 discussion of the same group in    

J. H. Conway, “Three Lectures on Exceptional Groups,” pp. 215-247 in Finite Simple Groups (Oxford, 1969), edited by M. B. Powell and G. Higman, Academic Press, London, 1971….. Reprinted as Ch. 10 in Sphere Packings, Lattices, and Groups 

Read this discussion of what might be called “contingent,” or “literary,” properties of the number 23. 

Read also the more interesting of  these discussions of the phrase “the 23 enigma.”

Having thus acquired some familiarity with both contingent and necessary properties of 23…

Read this discussion of Aquinas’s third proof of the existence of God.

Note that the classic Spacek film “Prime Cut” was released in 1972, the year that Spacek turned 23:

1949
+ 23


1972
 
Essay question:  
 
If Jesus was God’s gift to man, and (as many men would agree) so was the young Sissy Spacek (also born on Christmas Day), was young Sissy’s existence in her 23rd year contingent or necessary?  If the latter, should she be recognized as a Person of the Trinity? Quaternity? N-ity?
 
Talk amongst yourselves.

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