sphericon 4

$e-MATH$
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 the zip-locus. (They can do this without actually crossing that locus themselves!). Its red and blue edges are drawn using straight line segments from the cone-points to the zip-circles, so they meet those circles at right angles. In that way, when the zipping is done, each pair of edges fits together without forming a corner as seen in the surface. Since they are both straight line segments before the zipping, they will form a single geodesic edge. The resulting figure 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 is L. If the sides are extended to the cone-points, they will meet at an angle = L/R, in radian measure.
If the quadrilateral is drawn symmetrically, its angles will all be equal, and equal to ( + )/2. The sum of the interior angles is therefore 2( + ).

By Gauss' Theorem, the total enclosed curvature is equal to this sum minus 2

(here n=4), so the total enclosed curvature is 2 gamma

= 2L/R.

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

On to Sphericon page 5.

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