The Reuleaux triangle of Figure 1(a) is obtained by drawing three circular arcs of radius d with centres at the vertices of an equilateral triangle of side d. It can be rolled between a pair of parallel lines at distance d apart and is said to be of constant breadth d.

Applying an analogous construction in three dimensions to a regular tetrahedron of side d, we obtain a solid figure bounded by four spherical triangles intersecting in six circular arcs, the centre of each spherical portion being at the opposite vertex of the tetrahedron and its radius of curvature being d. This figure has two modes of rolling between parallel planes. In the first mode a vertex touches one of the planes and the opposite spherical surface rolls on the other; in the second mode, opposite circular edges touch the planes. The distance separating the planes in the first mode is d; in the second mode, the separation is d when the points of contact with the tetrahedron are at the opposite ends of the circular arcs, but this distance increases to d(√3 − ½√2), or l·025d, when the centres of the circular edges touch the planes. By slightly rounding one edge in each of the three opposed pairs, a solid of constant breadth may be produced. The geometry of this process is illustrated at Figure 1(b).