Angular structure of Reuleaux cones

Abstract

In this note we exhibit some examples of proper cones that have the property of being of constant opening angle. In particular, we analyze the class of Reuleaux cones in \(\mathbb {R}^n\) with \(n\ge 3\). Such cones are constructed as intersection of n revolutions cones \(\textrm{Rev}(g_1,\psi ),\ldots , \textrm{Rev}(g_n,\psi )\) whose incenters \(g_1,\ldots , g_n\) are unit vectors forming a common angle. The half-aperture angle \(\psi \) of each revolution cone corresponds to the common angle between the incenters. A major result of this work is that a Reuleaux cone in \(\mathbb {R}^n\) is of constant opening angle if and only if \(n= 3\). Reuleaux cones in dimension higher than 3 are not of constant opening angle, but such mathematical objects are still of interest. In the same way that a Reuleaux triangle is a “rounded” version of an equilateral triangle, a Reuleaux cone can be viewed as a rounded version of an equiangular simplicial cone and, therefore, it has a lot of symmetry in it.