  We introduce a new discrete system that arises from ellipsoidal billiards and
is closely related to the double reflection nets. The system is defined on the
lattice of a uniform honeycomb consisting of rectified hypercubes and cross
polytopes. In the $2$-dimensional case, the lattice is regular and it
incorporates dynamics both in the original space and its dual. In the
$3$-dimensional case, the lattice consists of tetrahedra and cuboctahedra.
