sphericon 6

$e-MATH$
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 from flat pieces without stretching.

At each cone-point there is a concentration of curvature equal to -. Total contribution from cone-points: 4 - 4.
If the common radius of the cones is R, then each component of the zip-locus has length R. Since a length L of zip-locus carries a concentration 2L/R of curvature, each component of the zip-locus contributes 2L/R x R = 2 of curvature. Total curvature carried by the zip-locus: 4.
Total curvature of the Sphericon: 4.

On to Sphericon page 7.

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