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sphericon 6

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 4pi. 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 pi-beta. Total contribution from cone-points:4pi - 4beta.

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

  • Total curvature of the Sphericon: 4pi.

On to Sphericon page 7.

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