Archimedean hypersurfaces, catenoids, and the grazing goat
This section describes some old joint work with Vincent Coll and Jeff Dodd, which comprised my Master's thesis and a few papers.
The following discovery dates back to Archimedes: if a 2-dimensional sphere is sliced by two parallel planes, the surface area of the zone of the sphere between the planes is proportional to the distance between the planes. So the shaded areas to the right are equal.
Although this property appears in standard calculus texts, I have seen audience members in serious math seminars pleasantly surprised by this fact (I see it mentioned sometimes as it has implications in symplectic geometry).
An analogous property can be defined and studied for n-dimensional hypersurfaces of revolution in Rn+1. Following Coll and Dodd, we say that an n-dimensional hypersurface of revolution H in Rn+1 has the equal area zones property if the n-volume of the zone of H, which lies between parallel hyperplanes orthogonal to the axis of revolution, is proportional to the distance between the hyperplanes. Coll and Dodd showed that for each dimension n > 1, there exists a unique closed, C2-smooth, n-dimensional hypersurface of revolution satisfying the equal area zones property.
The uniqueness here holds modulo scaling and shifting, leading Coll and Dodd to distinguish the specific hypersurface which is appropriately centered and whose largest spherical cross-section has radius 1; this object is aptly named the (unit) equizonal ovaloid EOn.
Coll, Dodd, and I systematically studied the equizonal ovaloids from a variety of perspectives. For example, we used a novel application of Bernstein's theory of absolutely-monotonic functions and completely-monotonic functions to show that even-dimensional equizonal ovaloids are analytic.
In a separate paper, Coll and I characterized the equizonal ovaloids based on a property exhibited by their principal curvatures.
Fix n > 1 and 1 < k < n+1, and let p : Rn+1 to Rn+1-k be a codimension-k coordinate projection. A hypersurface H in Rn+1 satisfies the Archimedean projection property (with respect to p) if, for every measurable U in p(H), Volume(p-1(U)) = Volume(Sk-1)*Volume(U).
Coll, Dodd, and I studied the APP on a class of hypersurfaces which are topologically sphere bundles over a base U, and geometrically warped products; we called these spherical arrays. Given a base U, we showed that one can always generate a spherical array with the APP, though it may not be smooth. Moreover, given n and k, there is a unique C^2, closed, strictly convex spherical array satisfying the APP; this is the sphere when k = 2 and the equizonal ovaloid when k = n. We studied the smoothness and volumes of these objects.
Similar considerations led to pedagogical articles in the Monthly and Math Magazine:
1. Gabriel’s Horn: A Revolutionary Tale (with V. Coll), Mathematics Magazine Vol. 87, No. 4 (October 2014), pp. 263-275.
2. Two Generalizations of a Property of the Catenary (with V. Coll), The American Mathematical Monthly Vol. 121, No. 2 (February 2014), pp. 109-119.
3. The Grazing Goat and Spherical Curiosities, Mathematics Magazine Vol 94 No. 5 (October 2021), pp. 369-382.
Summary of 1: We give a gentle introduction to hypersurfaces of revolution and their associated volume formulas, using the geometric features of Gabriel's Horn to motivate the excursion to higher dimensions. We include some history of Gabriel's Horn and the furor caused by its initial discovery.
Summary of 2: The catenary curve is characterized by the following property: the ratio of the area under the curve to the arc length of the curve is independent of the interval over which these quantities are concurrently measured. We develop two higher-dimensional generalizations of this ratio, and we find that each ratio identifies a class of hypersurfaces connected to classical objects from differential geometry.
Summary of 3: We consider a variation of the classical Grazing Goat Problem and we explore connections to a number of surprising properties of spheres, including the curious volume-accumulation in higher-dimensional spheres and the Equal Area Zones property of the 2-sphere.