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Thursday, March 10, 2011

Paradigms Lost

Filed under: General,Geometry — Tags: — m759 @ 5:48 PM

(Continued from February 19)

The cover of the April 1, 1970 second edition of The Structure of Scientific Revolutions , by Thomas S. Kuhn—

http://www.log24.com/log/pix11/110310-KuhnCover.jpg

This journal on January 19, 2011

IMAGE- A Galois cube: model of the 27-point affine 3-space

If Galois geometry is thought of as a paradigm shift from Euclidean geometry,
both images above— the Kuhn cover and the nine-point affine plane—
may be viewed, taken together, as illustrating the shift. The nine subcubes
of the Euclidean  3×3 cube on the Kuhn cover do not  form an affine plane
in the coordinate system of the Galois  cube in the second image, but they
at least suggest  such a plane. Similarly, transformations of a
non-mathematical object, the 1974 Rubik  cube, are not Galois  transformations,
but they at least suggest  such transformations.

See also today's online Harvard Crimson  illustration of problems of translation
not unrelated to the problems of commensurability  discussed by Kuhn.

http://www.log24.com/log/pix11/110310-CrimsonSm.jpg

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