The Differential Geometry of the Sphericon

6. The total curvature of the Sphericon

According to the Gauss-Bonnet Theorem, the total curvature of a smooth convex surface is 4. We can check that this statement holds for the more exotic curvature of the Sphericon.

The Sphericon has four cone-points and two arcs of zip-loci.Otherwise it has no curvature, since it can be assembled fromflat pieces without stretching.

• At each cone-point there is a concentration of curvatureequal to -. Total contribution from cone-points:4 - 4.

• If the common radius of the cones is R, then each componentof the zip-locus has length R. Since a length L of zip-locuscarries a concentration 2L/R of curvature, each component of thezip-locus contributes 2L/R x R = 2 of curvature. Totalcurvature carried by the zip-locus: 4.

• Total curvature of the Sphericon: 4.

Back to Sphericon page 5.