The birefringence of a crystal is determined by a three dimensional shape (the indicatrix) describing how refractive index varies with direction within the crystal. You can think of this as a 3d ellipse and the birefringence is given by the difference in length of the two axes of the ellipse 'seen' by light as it passes through the crystal.
The orientation and shape of the indicatrix are constrained by the point group symmetry of the crystal. In the case of cubic crystals, the indicatrix is characterised by four 3-fold axes. The indicatrix for all cubic crystals is thus a sphere and cubic crystals are non-birefringent. Hexagonal, trigonal and tetragonal crystals are uniaxial and the indicatrix is an ellipsoid of revolution - there is one direction in which the crystal appears non-birefringent. Orthorhombic, monoclinic and triclinic systems are biaxial -two axes in which the crystal appears non-birefringent.
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