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