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No. 93: May-Jun 1994

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Btt And Surreality

If you think that Beloff's "rent" in the fabric of the cosmos is a bizarre concept, it's because you haven't heard about BTT. Named after S. Banach and A. Tarski, the Banach-Tarski Theorems (BTT) were conceived 70 years ago, and they have challenged common sense ever since. Of course, no one expects that all branches of mathematics will mirror physical reality; but BTT is definitely ultra-weird, so much so that we had to invent that adjective! Take, for example, this specific BTT result presented by B. Augenstein:

"In particular, a solid sphere with unit radius can be cut into five pieces in such a way that two of the pieces can be reassembled into one solid sphere with unit radius, while the other three pieces are reassembled into a second solid sphere with unit radius. These are the minimum numbers of pieces required to do the trick, but it can be repeated indefinitely."

Impossible, you say? Can one sphere fragment and be reconstituted as two solid spheres of the same size? Regardless of what the math says, it cannot happen in the real world! Well, BTT actually does mirror just such a phenomenon found in particle physics:

"The magical way in which a proton entering a metal target can produce a swarm of new copies of protons emerging from that target, each identical to the original, is precisely described by the BTT process of cutting spheres into pieces and reassembling them to make pairs of spheres."

(Gribbin, John; "The Prescient Power of Mathematics," New Scientist, p. 14, January 22, 1994. Cr. P. Gunkel.)

Comment. Would it be frivilous to ask that if protons can multiply thus (seemingly magically), why can't fish fall from the sky? Many anomalous phenomena might be explained by BTT and other surreal math, but scientists seem to apply such thinking only to particle physics and cosmology.

From Science Frontiers #93, MAY-JUN 1994. 1994-2000 William R. Corliss