What is crystallographic equivalence? I’ve learnt that in crystallography, there are many crystallographic directions that are “equivalent.” For example, in a cubic crystal system, the [011] and [110] and [101] directions are “crystallographically equivalent.” See the image below:



I cannot begin to fathom how they are equivalent. These are vectors after all, and from what I can tell with my rudimentary knowledge of maths, these vectors are not equivalent in direction. Yet all materials science textbooks and powerpoints insist that, for cubic systems, any permutation of positive and negative indices will yield a same equivalent direction. That is, the [011], [110], and [101] vectors are… the same vector?

I share the same confusion with planes, even if they aren’t vectors. For cubic systems, just as for directions, any arrangement of positive and negative indices produces an equivalent plane. For example, (100) and (010) are the same plane, when to me they clearly look different.

So what exactly is “crystallographic equivalence”?
Directions &amp; sites are said to be crystallographically equivalent, because there are symmetry operators that relate them.  So 


learn about simpler symmetry operations (proper axes, mirror planes, centres of inversion) or advanced ones (like screw axes, glide planes, etc.)
learn how to recognize them in every-day's life.  It may be helpful for crystallography, and in other fields too: IR / Raman spectroscopy, for example, not only for understanding (descriptive), but for forcasting (predictive).  Tutorials like this represent an entry.