Feature Column

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.

Back to Sphericon page 5.

Welcome to the
Feature Column!

These web essays are designed for those who have already discovered the joys of mathematics as well as for those who may be uncomfortable with mathematics.
Read more . . .

Search Feature Column

Feature Column at a glance

Show Archive

Browse subjects