From: utzoo!decvax!harpo!eagle!wb2!houxz!ihnp4!ihlpb!dap1
Newsgroups: net.math
Title: Re: Squares - (nf)
Article-I.D.: ihlpb.284
Posted: Sat Jan 29 18:01:04 1983
Received: Mon Jan 31 04:10:22 1983

#R:uiucdcs:28200003:ihlpb:6200008:  0:1126
ihlpb!dap1    Jan 29 16:29:00 1983

I believe that the paint in the horn paradox is just a little misleading.
It implies ( to me anyway ) that although you can have a "layer" of
paint "under" the horn, you can't have a layer of paint "on" the horn.
Well, what is being referred to here as a "layer" is described in
mathematics as a 2 dimensional manifold embedded in a 3 dimensional
space.  Such manifolds are intuitively "warped planes", that is,
infinitely thin.  The "paint" could cover the horn if it was spread
out infinitely thin.  That is to say, it can be proven that there is a
many to one correspondence between the points contained in the horn
and the points on the horn (in fact, between any three dimensional
continuum and a two dimensional manifold).  Therefore, the "thickness"
of the layer of paint is the only thing that keeps it from covering
the horn.  Normally, it would take twice as much paint to cover a floor
twice as large but this isn't the case if the paint is allowed to be
placed on in negligibly thin layers.
	Sooooo, while the paint inside could not cover the horn
physically, it can mathematically.
					Darrell Plank
					BTL-IH