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Wednesday, May 23, 2007

Wednesday May 23, 2007

Filed under: General,Geometry — m759 @ 7:00 AM
 
Strong Emergence Illustrated:
 
The Beauty Test
 
"There is no royal road
to geometry"

— Attributed to Euclid

There are, however, various non-royal roads.  One of these is indicated by yesterday's Pennsylvania lottery numbers:

PA Lottery May 22, 2007: Mid-day 515, Evening 062

The mid-day number 515 may be taken as a reference to 5/15. (See the previous entry, "Angel in the Details," and 5/15.)

The evening number 062, in the context of Monday's entry "No Royal Roads" and yesterday's "Jewel in the Crown," may be regarded as naming a non-royal road to geometry: either U. S. 62, a major route from Mexico to Canada (home of the late geometer H.S.M. Coxeter), or a road less traveled– namely, page 62 in Coxeter's classic Introduction to Geometry (2nd ed.):

The image “http://www.log24.com/log/pix07/070523-Coxeter62.jpg” cannot be displayed, because it contains errors.

The illustration (and definition) is
of regular tessellations of the plane.

This topic Coxeter offers as an
illustration of remarks by G. H. Hardy
that he quotes on the preceding page:

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One might argue that such beauty is strongly emergent because of the "harmonious way" the parts fit together: the regularity (or fitting together) of the whole is not reducible to the regularity of the parts.  (Regular triangles, squares, and hexagons fit together, but regular pentagons do not.)

The symmetries of these regular tessellations of the plane are less well suited as illustrations of emergence, since they are tied rather closely to symmetries of the component parts.

But the symmetries of regular tessellations of the sphere— i.e., of the five Platonic solids– do emerge strongly, being apparently independent of symmetries of the component parts.

Another example of strong emergence: a group of 322,560 transformations acting naturally on the 4×4 square grid— a much larger group than the group of 8 symmetries of each component (square) part.

The lottery numbers above also supply an example of strong emergence– one that nicely illustrates how it can be, in the words of Mark Bedau, "uncomfortably like magic."

(Those more comfortable with magic may note the resemblance of the central part of Coxeter's illustration to a magical counterpart– the Ojo de Dios of Mexico's Sierra Madre.)

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