By Steven Gubkin


2012-03-01 21:01:17 8 Comments

A first stab at a definition of surface area might go like this:

Let S be a surface. Select finitely many points from S and make a bunch of triangles having these points as vertexes. Add up the areas of all of the triangles. This is a finite approximation to the surface area. Now to get the actual surface area, we increase the number of points so that they "densely cover" the surface (Maybe a good working definition would be that any open set of S should eventually contain some vertexes of triangles).

I seem to remember reading about a counterexample to this naive definition, but I can't find a reference. I believe there is even a natural looking polygonal approximation to the cylinder whose surface area diverges to infinity. Can anyone help me out?

1 comments

@Joseph O'Rourke 2012-03-01 21:26:29

Perhaps you are thinking of the Schwartz Lantern? It converges to the cylinder in the Hausdorff metric but its area can be arranged to head toward $\infty$. It was mentioned in the earlier MO question, "Convergence of finite element method: counterexamples." There is nice applet here showing the lantern rotating. Here is an image from Conan Wu's blog:
          

@Will Jagy 2012-03-01 21:51:23

page 129, Calculus on Manifolds, by Michael Spivak.

@Steven Gubkin 2012-03-01 22:22:00

@Joseph Excellent! Thanks a lot. @Will Yes that is where I must have seen it.

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