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