assume the points on the Bloch sphere of a POVM to coincide with the vertices of the platonic solid corresponding to the symmetry group.

The vertices of the regular polygons and the platonic solids depend on the orientation of the polygons and platonic solids in the Bloch sphere. Each orientation leads to another POVM. In order to simplify the constructions in the following sections we choose specific orientations of the regular polygons and platonic solids. To obtain the implementation of a POVM corresponding to the same polygon or platonic solid with another orientation, it suffices to implement a single qubit operation on the qubit to be measured.

Cyclic groups {#Section 2.1}
=============

The simplest finite symmetry groups of points on the Bloch sphere are the cyclic groups. For a fixed $m\geq 2$ we consider the rotations of an $m$-sided regular polygon with a common axis perpendicular to the face of the polygon. The rotations form a group that is isomorphic to the group $C_m=
\langle r \rangle$ with $r^m=1$. The implementation of POVMs corresponding to a single orbit of points under the action of a cyclic symmetry group is the basis of all constructions in the following sections.

In principle, 