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sphericon 4

The Differential Geometry of the Sphericon

4. Curvature on the Sphericon: the zip-locus

The Sphericon has more complicated curvature singularities than cone points. Along the curves where the cones are zipped together (the zip-locus) there is a 1-dimensional concentration of curvature. The Sphericon World geographers can analyze this curvature by another application of Gauss' Theorem.

Here they have drawn a quadrilateral that is bisected by thezip-locus. (They can do this without actually crossing thatlocus themselves!). Its red and blue edges aredrawn using straight line segmentsfrom the cone-points to the zip-circles, so they meet thosecircles at right angles.In that way, when the zipping is done, each pair of edgesfits together without forming a corner as seen in the surface.Since they are both straight line segments before the zipping, they willform a single geodesic edge. The resultingfigure has four geodesic edges: one red, one blue and two black.

What are its angles?

  • Suppose the pie-slices have radius R,and that the length of the arc intercepted by the quadrilateral isL. If the sides are extended to the cone-points, they will meetat an angle gamma = L/R, in radian measure.
  • If the quadrilateral is drawnsymmetrically, its angles will all be equal, and equal to(gamma + pi)/2. The sum of the interior angles is therefore2(gamma + pi).

By Gauss' Theorem, the total enclosed curvature is equal to this sum minus 2pi (here n=4), so the total enclosed curvature is 2gamma = 2L/R.

This calculation does not depend on the height of thequadrilateral away from the zip-locus. The only way to explainthe result is to say that the surface curvature is concentratedalong the zip-locus in such a way that any curve intersecting thezip-locus in an arc of length L will enclose total curvature 2L/R.

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