Space-Filling Bearings in Three Dimensions

We present the first space-filling bearing in three dimensions. It is shown that a packing which contains only loops of an even number of spheres can be constructed in a self-similar way and that it can act as a three-dimensional bearing in which spheres can rotate without slip and with negligible torsion friction.

Figure 1

Bichromatic packing of spheres with octahedral symmetry. No two spheres of the same color touch each other. The image on the left shows the initial configuration and the first generation of inserted spheres.

Figure 2

Plane cut through the center of the unit sphere and four initial spheres. Dashed circles are cuts of the inversion spheres. Sphere $S$ is mapped, by the inversion sphere shown by a thick dashed line, onto sphere $S^{\prime }$. $0^{\prime }$ is mapped by the central inversion sphere onto $0^{\prime \prime }$.

Figure 3

Schematic configuration of a loop of four spheres. The spheres rotate without slip. The centers of spheres need not be in the same plane, although ${\overrightarrow{\omega }}_1$, ${\overrightarrow{\omega }}_2$, and ${\overrightarrow{R}}_{12}$ are coplanar.

Figure 4

Two spheres rolling on each other without slip can be represented by two cones rolling on each other around the common line $S_1{S}_{2}A$. Therefore there is no twist of the spheres against each other.