By the way, can anyone give me a reference for the generalization of
the Pythagorean theorem that says that squared k-dimensional volume
of a flat k-dimensional object sitting in n-space is the sum of the
squared k-dimensional volumes of the k-dimensional shadows that the
object casts in the n-choose-k different directions associated with
an orthogonal coordinate frame?
For instance, spin a coin in the air; the shadows that it casts on
the floor and two perpendicular walls will rapidly change, but their
squared areas will add up to a constant, namely the squared area of
(one side of!) the coin.
I seem to recall that this is due to Cayley or Sylvester or someone
like that. But I'm more interested in where an interested math major
might read about this result, rather than who first noticed it.
(Surely there's an American Mathematical Monthly or Mathematics Magazine
article about this!)
Jim Propp
