Below is an image of a biaxial mineral. Inside of the mineral is a biaxial indicatrix, which is slightly more complicated than a uniaxial indicatrix.
Optically speaking, biaxial minerals are more complicated than uniaxial minerals. Biaxial mineral have far less symmetry in their crystaline form than uniaxial minerals, and is it this lack of symmetry which yields the two optic axes and two isogyres when we take an interference figure.
Biaxial minerals have three allowed vibration directions-- light can travel as the alpha ray, the beta ray, or the gamma ray. These three vibration directions correspond to the three crstallographic axes. Whereas in the uniaxial system the relative refractive indices can change, the relative refractive indices are fixed in the biaxial system. The alpha ray, corresponding to the X-axis, always has the lowest refractive index. The beta ray, corresponding to the Y-axis, has the middle-value refractive index. The gamma ray, corresponding to the Z-axis, always has the greatest refractive index.
Determining the sign of a biaxial mineral using an idicatrix is more difficult than a uniaxial mineral. Since the relative refractive indices are fixed, we can no longer base sign on refractive indices. Instead, we must look at the optic axes. The optic axes are at some angle from the Z-axis and their position relative to this axis. If we make a circular section which is perpendicular to the optic axes, and has a radius equal to the Y-axis, we can think of this as being equal to the omega vibration direction in the uniaxial system.
We look at angle of inclination of the optic axes from the Z-axis and call this the 2V. When the optic axes are within 45 degrees from the Z-axis, the mineral is positive. When the optic axes are within 45 degrees from the Y-axis, the mineral is negative.