paradoxically, if the index ellipsoid of a medium is triaxial, the medium itself is called biaxial. If two of the principal semiaxes of the index ellipsoid are equal (in which case their common length is called the ordinary index, and the third length the extraordinary index), the ellipsoid reduces to a spheroid (ellipsoid of revolution), and the two optic axes merge, so that the medium is said to be uniaxial. As the index ellipsoid reduces to a spheroid, the two-sheeted index surface constructed therefrom reduces to a sphere and a spheroid touching at opposite ends of their common axis, which is parallel to that of the index ellipsoid; but the principal axes of the spheroidal index ellipsoid and the spheroidal sheet of the index surface are interchanged. In the well-known case of calcite, for example, the index ellipsoid is an oblate spheroid, so that one sheet of the index surface is a sphere touching that oblate spheroid at the equator, while the other sheet of the index surface is a prolate spheroid touching the sphere at the poles, with an equatorial radius (extraordinary index) equal to the polar radius of the oblate spheroidal index ellipsoid. If all three principal semi-axes of the index ellipsoid are equal, it reduces to a sphere: all diametral sections of the index ellipsoid are circular, whence all polarizations are permitted for all directions of propagation, with the same refractive index for all directions, and the index surface merges with the (spherical) index ellipsoid; in short, the medium is optically isotropic. Cubic crystals exhibit this property as well as amorphous transparent media such as glass and water. History A surface analogous to the index ellipsoid can be defined for the wave speed (normal to the wavefront) instead of the refractive index. Let denote the length of the radius vector from the origin to a general point on the index ellipsoid. Then dividing equation () by gives where , , and are the direction cosines of the radius vector. But is also the refractive index for a wavefront parallel to a diametral section of which the radius vector is major or minor semiaxis. If that wavefront has speed , we have , where is the speed of light in a vacuum. For the principal semiaxes of the index ellipsoid, for which takes the values let take the values respectively, so that and . Making these substitutions in