The first rectangle dissection was known to Lewis Carroll who carried it around for young people, and knew it was based on a property of Fibonacci numbers. The Banach-Tarski dissection is more difficult to demonstrate in real life - the pieces are complicated and composed of an infinite number of sub pieces, and the proof involves the axiom of choice. There is a good book on it by Stan Wagon. Stan Isaacs Sent from my iPad
On Jun 1, 2019, at 9:37 PM, Bill Gosper <billgosper@gmail.com> wrote:
n 2019-06-01 18:08, Dan Asimov wrote:
1. There is a lovely dissection proof of the fact that a rectangle of area 65 can be dissected into one of area 64. Hence 1 = 0.
http://gosper.org/8x8=5x13.gif —rwg
2. Also, Banach-Tarski showed that a unit ball B = {p in R^3 | ||p|| <= 1} can be dissected into 5 pieces that can be reassembled to comprise a partition of *two* unit balls. A likely story!
Hence 1 = 0.
3. Furthermore, the vector field on the complex plane given by
V(z) = i(z^3 - z)
is holomorphic, so the fact that the flow {phi_t} of V satisfies that the times t for which
phi_t(z) = z
for all z form a discrete subgroup G_z of the reals.
Also note that phi_t(z) is jointly holomorphic in both z and t.
But it's not the same subgroup for all z (!) Which clearly contradicts the principle of permanence for holomorphic functions.=
—Dan
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