Peabodies of Constant Width

TL;DR

This paper introduces a new 3D family of bodies of constant width called peabodies, constructed from Reuleaux tetrahedra and related to confocal quadrics, including known shapes like Meissner solids.

Contribution

It presents the novel construction of peabodies, expanding the class of bodies of constant width using confocal quadrics and envelope techniques.

Findings

Peabodies have constant width property.

Includes known bodies like Meissner solids as special cases.

Uses confocal quadrics to prove constant width.

Abstract

The purpose of this paper is to describe a new $3$-dimensional family of bodies of constant width that we have called peabodies, obtained from the Reuleaux tetrahedron by replacing a small neighborhood of all six edges with sections of an envelope of spheres. This family contains, in particular, the two Meissner solids and a body with tetrahedral symmetry that we have called Robert's body. Behind the construction of this family lies the classical notion of confocal quadrics discussed, for example, by Hilbert in his famous book. We study confocal quadrics and prove that the distances of an alternating sequence of four points in two confocal quadrics always satisfies a simple equation and use this equation to prove that our bodies have constant width.