Log24

The reference to David Justice at the beginning of
yesterday afternoon's post does not imply an
endorsement of all his writings. For instance, a
Justice post from yesterday contains the following—

Correction—

The above author name and page number are wrong.

Related to the above "fundamental theme" — 

Midsummer Geometry.

(Continued) 

Three Notes on Design

1.  From the Museum of Modern Art  today—

“It’s a very nice gesture of a kind of new ethos:
To make publicly accessible, unticketed space
that is attractive and has cultural programming,”
Glenn D. Lowry, MoMA’s director, said.

2.  From The New York Times  today—

3.  From myself  last December—

Click for a larger image. 

For a different pictorial approach, see Polster's
1998 Geometrical Picture Book , pp. 77-80.

Update:  Added to finitegeometry.org on Jan. 2, 2014.
(The source of the images of the 35 lines was the image
"Geometry of the Six-Element Set," with, in the final two
of the three projective-line parts, the bottom two rows
and the rightmost two columns interchanged.)

The title is taken from a book for ages 8-12 published
on Shakespeare's birthday, April 23, 2013.

Also from that date, a note for older readers—

… Half a dozen of the other —

For further context, see all  posts for the cruelest month of this past year.

* Secrets :  A sometimes dangerous word.

The Miami-Dade County Public Schools math webpage
​now has a link to the Diamond 16 Puzzle.

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 —

​

See also last night's "Pink Champagne on Ice" post.
The "ice" in that post's title refers to the white lines
forming a tesseract in the book cover's background—
"icy white and crystalline," as Johnny Mercer put it.
(A Tune for Josefine, Nov. 25.)

See also the tag Diamond Theory tesseract in this journal.

See my Google Sites page if you would like to
download a zipped copy (31 MB) of my
Finite-Geometry Notes  site
(not zipped, at finitegeometry.org/sc/map.html).
Or you can of course use a website downloader.

(Suggested by a recent NY Times  piece on
a company, Citia, that splits books into pieces 
for easier electronic access. The large zipped
file referred to above is sort of a reverse of this
process.)

On St. Andrew's Day.

A Connery adventure in Kuala Lumpur—

For another Kuala Lumpur adventure, see today's update
to "In Defense of Plato's Realism"—

The July 5, 2007, post linked to
"Plato, Pegasus, and the Evening Star."
For related drama from Kuala Lumpur, see
"Occam's Razor, Plato's Beard."

From the obituary of a Bletchley Park
codebreaker who reportedly died on
Armistice Day (Monday, Nov. 11)—

"The main flaw of the Enigma machine,
seen by the inventors as a security-enhancing
measure, was that it would never encipher
a letter as itself…."

Update of 9 PM ET Nov. 13—

"The rogue’s yarn that will run through much of
the material is the algebraic symmetry to which
the name of Galois is attached…."

— Robert P. Langlands,
     Institute for Advanced Study, Princeton

"All the turmoil, all the emotions of the scenes
have been digested by the mind into
a grave intellectual whole.  It is as though
Bach had written the 1812 Overture."

— Aldous Huxley, "The Best Picture," 1925

The title is from an essay by James C. Nohrnberg—

(Click to enlarge.)

"Just another shake of the kaleidoscope" —

Related material:

Kaleidoscope Puzzle,  
Design Cube 2x2x2, and 
Through the Looking Glass: A Sort of Eternity.

Yesterday's post on epistemology and geometry
suggests an Amazon customer review of Descartes's
Rules for the Direction of the Mind —

Quoted in that review —

"… we must make use of every assistance 
of the intellect, the imagination, the senses,
and the memory" (Descartes, Rules, XII)

One such assistance is the calendar.
See the date of the Blasjo review, Dec. 20, 2009,
in this journal. See also Descartes.

On Monday, October 14, 2013, Jeremy Gray published
an article titled "Epistemology of Geometry" in the online
Stanford Encyclopedia of Philosophy.

Gray's article did not mention the role of finite  geometry
in such epistemology.

For that role, see Finite Geometry and Physical Space 
as a web page and as a Google image search.

See also my papers at Academia.edu.

(Continued)

Josefine Lyche’s large wall version of the twenty-four 2×2 variations
above was apparently offered for sale today in Norway —

Click image for more details and click here for a translation.

(Continued)

Further context: Galois I Ching

"Righty tighty, lefty loosey." — Folk saying

See also a figure from this journal 
on Lee Marvin's birthday in 2011 —

The square root of the former is the latter.

"Take a cube, and write the numbers 1,…,6 on its faces.
Now the pairs of numbers on opposite faces
form a syntheme. (Standard dice, for example, represent
the syntheme 12|34|56.) "
— Peter J. Cameron, weblog post of May 11, 2010 

"For every kind of vampire, there is a kind of cross."
— Gravity's Rainbow

A girl's best friend?

Examples of "the primordial Riß " as ἀρχή  —

For an explanation in terms of mathematics rather than philosophy,
see the diamond theorem. For more on the Riß  as ἀρχή , see
Function Decomposition Over a Finite Field.

For Fans of Bad Movies*

This post was suggested by my viewing last night
the 1995 horror film Species , and by news that 
Scarlett Johansson will be starring in a similar
production at the Venice Film Festival, which
opens tomorrow.

The new Johansson film, Under the Skin , 
is based on a novel by one Michel Faber.

Faber on books that have influenced him—

"Most influential has possibly been John Berger's Ways of Seeing — 
not a novel at all (although Berger has written fiction) but a book of
art criticism. The influence of these wonderfully perceptive and
thought-provoking essays peeps out everywhere in my own work."

An excerpt from the Berger book—

Click image for a better view of the original.

Related material: Johansson in this journal, Sunday's NY Times 
teaser for a piece on Saturday Night Live, and a more serious
approach to the geometry of perspective.

* And of Ben Kingsley, who starred both in Species  and in
  a previous film by  the director of Under the Skin .

(Continued from 24 hours ago and from May 9, 2012)

Quoted 24 hours ago in this journal—

Remark by Aldous Huxley on an artist's work:

"All the turmoil, all the emotions of the scenes
have been digested by the mind into a
grave intellectual whole."

Quoted in a video uploaded on May 9, 2012:

Norway Toilet Scene

Norway dance (as interpreted by an American)

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

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

Today is the dies natalis  of group theorist  Walter Feit.

     "The Steiner systems S (5,6,12) and S (5,8,24) are remarkable combinatorial
configurations unlike any others. Their automorphism groups are the Mathieu
groups M₁₂ and M₂₄. These are the only 5-transitive permutation groups other
than symmetric and alternating groups: (a fact long conjectured but only
proved as a consequence of the classification). The Leech lattice is a blown up
version of S (5,8,24). It is the unique even unimodular lattice in 24 dimensions
with no vectors of weight 2. This uniqueness is an essential reason why it is a
geometric object of fundamental importance. The automorphism group Co.O
of the Leech lattice involves about half of the sporadic groups and generally it
is felt that these are well understood."

— Walter Feit, book review, Bulletin of the American Mathematical Society ,
     Vol. 8 (1983), 120-124, page 123

This is the weekend for Comic-Con International in San Diego.

The convention includes an art show. (Click above image to enlarge.)

Related material from Norway…

Suggested nominations for a Kavli Prize:

1.  Josefine Lyche's highly imaginative catalog page for
the current Norwegian art exhibition I de lange nætter,
​which mentions her interest in sacred geometry

2.  Sacred Geometry:  Drawing a Metatron Cube
 

… and from San Diego—

The Kavli Institutes logo:

Online biography of author Cormac McCarthy—

"… he left America on the liner Sylvania, intending to visit
the home of his Irish ancestors (a King Cormac McCarthy
built Blarney Castle)." 

Two Years Ago:

Blarney in The Harvard Crimson—

Melissa C. Wong, illustration for "Atlas to the Text,"
by Nicholas T. Rinehart:

Thirty Years Ago:

Non-Blarney from a rural outpost—

Illustration for the generalized diamond theorem,
by Steven H. Cullinane: 

See also Barry's Lexicon .

From a weblog post on June 11, 2013, by one Pete Trbovich:

It isn't  supposed to be such a theory.
I do not know why Trbovich thinks it is . 

— Steven H. Cullinane

Update of 11 PM June 16:

For one such central theory of everything, see
the I Ching .  Diamond theory is, unlike that
Chinese classic, pure mathematics, but the larger
of the binary-coordinate structures  it is based on
are clearly isomorphic, simply as structures , to
the I Ching 's 64 hexagrams.

Make of this what you will.

The Mathematical Association of America has a
submit-a-review form that apparently allows readers
to comment on previously reviewed books.

This morning I submitted the following comment on
Alexander Bogomolny's March 16, 2012, review of 
Martin J. Erickson's Beautiful Mathematics :

In section 5.17, pages 106-108, "A Group of Operations,"
Erickson does not acknowledge any source. This section
is based on the Cullinane diamond theorem. See that
theorem (published in an AMS abstract in 1979) at
PlanetMath.org and EncyclopediaOfMath.org, and
elsewhere on the Web. Details of the proof given by
Erickson may be found in "Binary Coordinate Systems,"
a 1984 article on the Web at
http://finitegeometry.org/sc/gen/coord.html.

If and when the comment may be published, I do not know.

Update of about 6:45 PM ET June 7:

The above comment is now online at the MAA review site.

Update of about 7 PM ET July 29:

The MAA review site's web address was changed, and the 
above comment was omitted from the page at the new address.
This has now been corrected.

Continued.

The order-3 affine plane:

Detail from the video in the previous post:

For other permutations of points in the
order-3 affine plane—

See Quaternions in an Affine Galois Plane
and Group Actions, 1984-2009.

See, too, the Mathematics and Narrative post 
from April 28, 2013, and last night's
For Indiana Spielberg.

(Continued from December 30, 2012)

"And let us finally, then, observe the
parallel progress of the formations of thought
across the species of psychical onomatopoeia
of the primitives, and elementary symmetries
and contrasts, to the ideas of substances,
to metaphors, the faltering beginnings of logic,
formalisms, entities, metaphysical existences."

— Paul Valéry, Introduction to the Method of
    Leonardo da Vinci

But first, a word from our sponsor…

Brought to you by two uploads, each from Sept. 11, 2012—

Symmetry and Hierarchy and the above VINCI Genius commercial.

A page with the above title has been created at
the Encyclopedia of Mathematics.

How long it will stay there remains to be seen.