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sphericon 5

The Differential Geometry of the Sphericon

5. Curvature on the Sphericon: the cone-points

The cone-points on the Sphericon are of a special type, because they are on the ends of the zip-locus. The Sphericon geographers can still analyze them using Gauss' Theorem.

In this figure they have drawn a right triangle around one of the cone-points.

  • One side of the triangle is drawn in blue using parts of radii. Since radii are perpendicular to the zip-locus, the two pieces fit together without forming a corner in the surface and form a geodesic edge.
  • One side of the triangle is drawn in red using segments perpendicular to the green line along which the initial gluing is done, so they also fit together to give another geodesic edge.
  • The third side is perpendicular to the red side.

What are the other angles in this triangle? Here a little plane geometry leads to the answer. The black-blue angle is gamma + pi - beta; the red-blue angle is gamma + pi/2.

So the sum of the angles in the triangle is 2pi-beta+2gamma.

Gauss'Theorem tells us that the total curvature enclosed in the triangle is pi-beta+2gamma. We can see that the gamma curvature is coming from the zip-locus. As we take smaller and smaller such triangles about the cone point, the gamma contribution will go to zero, leaving us with a point-concentration of curvature at the cone-point equal to pi-beta.

On to Sphericon page 6.

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