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My textbook states

"Cubic crystals contain four three-fold axes, converting the ellipsoid to a sphere."

What indeed is the ellipsoid and sphere in the lattice? I know the meaning of three fold axes but unable to figure out what and how ellipsoid is converted to sphere.

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From the limited excerpt of your book (title? author?) it is likely it just deals with the interaction of linearly polarized light with condensed matter. If there is near- and far range order (as regular consecution of building elements), the solid is crystalline. Among the crystaline states, there are (Bravais) lattice systems, that differ in symmetry, i.e. because they possess different symmetry elements, some are of higher symmetry (like cubic) than others (like monoclinic). And vectorial physical properties of the material are coherent with the crystalline symmetry met.

source

(image source).

By means of an index ellipsoid, you may map the orientational dependance of physical properties, in other words, anisotropy. In the case of mineral- and more general in polarized light microscopy, such an ellipsoid may be used as representation to map refractive indices along three directions orthogonal to each other. Now

  • either the three axes of the indicatrix differ from each other: optical biaxial system, characteristic Bravais lattice of lower symmetry: orthorhombic, monoclinic and triclinic.

  • two of the three axes of the indicatrix are identic, the third is different: optical mono- or uniaxial system, characteristic for hexagonal, trigonal and tetragonal Bravais lattice.

  • all three axes of the optical indicatrix are identic, in other words, the material is optical isotropic: this is characteristic for the cubic Bravais lattice. (And: for the state of randomness, the glasses).

So in short: it is the number, the degree, and the relative orientation of the symmetry elements of the cubic Bravais latice that renders the more general case of an ellipse into the special case of a sphere. And this means, the question should be "cubic crystal's isotropy".

Further information may be found for example here.

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The ellipsoid refers to the size of the anisotropic parameters in different directions at right angles to one another and along some axis, say refractive index, being different in two directions compared to the third. This 'ellipsoid' of refractive index may be elongated like a rugby ball (prolate ellipsoid) or squashed like a doughnut (oblate ellipsoid). In a cubic crystal the value is the same in all three directions hence a sphere. A cube has four 3-fold axes, look down in a diagonal direction along opposite corners.

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