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Why curves curve:  Finding the geodesics on a donut

Abstract  Curves on surfaces curve for two reasons: (i) they must, i.e., the surface on which they lie is itself curved; (ii) they feel like it , i.e., they curve more than the surface forces them to.  Curves that only curve because they must are the analogues of straight lines in the plane and are rewarded with distance distance minimizing properties.  Such curves are called  geodesics and their classification on surfaces and higher dimensional manifolds is a huge industry in the field of Differential Geometry.  Even so, the differential geometry of curves and surfaces remains underexposed in most undergraduate curriculums.  This is particularly unfortunate since many of this field's beautiful classical results can be derived without appeal to the complicated language of tensors, Jacobi fields and Christoffel symbols.  This note packages one such result - the classification of the geodesics on the torus - in a form suitable for a standard lecture in vector calculus.

Dissecting the sphere:  slicing, coring and capping in n-dimensions

Abstract  It is a standard calculus exercise to show that if two parallel planes slice through a sphere, the surface area of the resulting zone is proportional to the distance between the planes.  While this slicing property of the sphere was known to Archimedes, it was not until 1921 that Blaschke formally proved that this property characterized the sphere among smooth closed convex surfaces.  The sphere may also be dissected by coring.  That is, a spherical ring is the object that remains when a solid sphere is cored by a cylindrical drill bit centered on an axis of the sphere.  Remarkably, the volume of a spherical ring depends only on its height, and not on the radius of the sphere from which it is cut.  In this presentation, we extend these ideas to higher dimensions and find that the only dimension in which the sphere is so characterized is in ambient dimension there.  We then characterize and mensurate completely the higher dimensional objects for which these slicing and coring properties hold.  Finally, we introduce a little known method of dissection which we have called capping.  We find that coring and capping are equivalent properties.

Archimedean  hypersurfaces

Abstract We develop a new class of hypersurfaces in n-dimensional Euclidean space which are de ned by a single function and contain hypersurfaces of revolution as a proper subclass. These hypersurfaces are Archimedean in a sense suggestedby Rudin in that they satisfy an equizonal type property of a sort well-known to hold for the sphere; that the surface area of a zone between two parallel planes depends only on the distance between the planes. The analogous property for hypersurfaces of revolution in higher dimensional Euclidean spaces has been investigated by Coll and Dodd who show that for each n greater than 2, there is just one smooth n-dimensional hypersurface of revolution in (n + 1)-dimensional Euclidean space that satisfies the equal area zones property. These hypersurfaces of revolution are called equizonal n-ovaloids and generalize the sphere in a previously undiscovered fashion. Making use of these ovaloids, we develop new Archimedean Hypersurfaces, and though not necessarily of revolution, they maintain interesting equizonal type properties. Using special functions, these objects can be fully mensurated.