Log24

See The Echo in Plato's Cave and
a four-color decomposition theorem.

An illustration —

The previous post discussed some art related to the
deceptively simple concept of "four colors."

For other related material, see posts that contain a link 
to "…mapsys.html."

My webpage "The Order-4 Latin Squares" has a rival—

"Latin squares of order 4: Enumeration of the
 24 different 4×4 Latin squares. Symmetry and
 other features."

The author — Yp de Haan, a professor emeritus of
materials science at Delft University of Technology —

The main difference between de Haan's approach and my own
is my use of the four-color decomposition theorem, a result that
I discovered in 1976.  This would, had de Haan known it, have
added depth to his "symmetry and other features" remarks.

(Continued from July 19, 2008)

From the Diamond 16 Puzzle —

The resemblance between the "quadrants" part of
the above picture and the new Microsoft symbol—

— is of course purely coincidental, as is the fact
that the new symbol illustrates four colors.

A search tonight for material related to the four-color
decomposition theorem yielded the Wikipedia article
Functional decomposition.

The article, of more philosophical than mathematical
interest, is largely due to one David Fass at Rutgers.

(See the article's revision history for mid-August 2007.)

Fass's interest in function decomposition may or may not
be related to the above-mentioned theorem, which 
originated in the investigation of functions into the
four-element Galois field from a 4×4 square domain.

Some related material involving Fass and 4×4 squares—

A 2003 paper he wrote with Jacob Feldman—

(Click to enlarge.)

"Design is how it works." — Steve Jobs

An assignment for Jobs in the afterlife—

Discuss the Fass-Feldman approach to "categorization under
complexity" in the context of the Wikipedia article's
philosophical remarks on "reductionist tradition."

The Fass-Feldman paper was assigned in an MIT course
for a class on Walpurgisnacht 2003.

Some background—

Richard J. Trudeau, a mathematics professor and Unitarian minister, published in 1987 a book, The Non-Euclidean Revolution , that opposes what he calls the Story Theory of truth [i.e., Quine, nominalism, postmodernism] to what he calls the traditional Diamond Theory of truth [i.e., Plato, realism, the Roman Catholic Church]. This opposition goes back to the medieval "problem of universals" debated by scholastic philosophers.

(Trudeau may never have heard of, and at any rate did not mention, an earlier 1976 monograph on geometry, "Diamond Theory," whose subject and title are relevant.)

From yesterday's Sunday morning New York Times—

"Stories were the primary way our ancestors transmitted knowledge and values. Today we seek movies, novels and 'news stories' that put the events of the day in a form that our brains evolved to find compelling and memorable. Children crave bedtime stories…."

— Drew Westen, professor at Emory University

From May 22, 2009—

Towards a Philosophy of Real Mathematics, by David Corfield, Cambridge U. Press, 2003, p. 206:

"Now, it is no easy business defining what one means by the term conceptual…. I think we can say that the conceptual is usually expressible in terms of broad principles. A nice example of this comes in the form of harmonic analysis, which is based on the idea, whose scope has been shown by George Mackey (1992) to be immense, that many kinds of entity become easier to handle by decomposing them into components belonging to spaces invariant under specified symmetries."

For a simpler example of this idea, see the entities in The Diamond Theorem, the decomposition in A Four-Color Theorem, and the space in Geometry of the 4×4 Square.  The decomposition differs from that of harmonic analysis, although the subspaces involved in the diamond theorem are isomorphic to Walsh functions— well-known as discrete analogues of the trigonometric functions of traditional harmonic analysis.

Old Year, Raus!

Also in today’s New York Times obituaries index:

 John T. Elson, Editor Who Asked
“Is God Dead?” at Time, Dies at 78

Wikipedia article on George Polya:

• Look for a pattern
• Draw a picture
• Solve a simpler problem
• Use a model
• Work backward

From the date of Elson’s death:

A Crystal Block —

Related material:
“A Four-Color Theorem.”