  We analyze the properties of optical cavities contained in spacers with
approximate octahedral symmetry and made of different materials,following the
design of Webster and Gill (S. Webster, P. Gill, Optics Letters 36(18), 3572
(2011)). We show that for isotropic materials with Young's modulus less than
200 GPa, the Poisson's ratio $\nu$ must lie in a "magic" range 0.13<$\nu$<0.23
in order to null the influence of the forces supporting the spacer. This
restriction can be overcome with the use of anisotropic materials such as
silicon. A detailed study aiming at identification of all suitable crystal
orientations of silicon with respect to the resonator body is performed and the
relation to the Poisson's ratio and the Young's modulus along these
orientations is discussed. We also perform an analysis of the sensitivity of
the cavity performance to errors in spacer manufacturing. We find that the
orientation of the [110] or [100] crystallographic directions oriented along
one of the three optical axes of the resonator provides low sensitivities to
imprecise manufacturing and interesting options for fundamental physics
experiments.
