<Tr> <Td> hexoctahedral </Td> <Td> O </Td> <Td> m3m </Td> <Td> * 432 </Td> <Td> (4, 3) </Td> <Td> centrosymmetric </Td> <Td> 48 </Td> <Td> S 4 × Z 2 (\ displaystyle \ mathbb (S) _ (4) \ times \ mathbb (Z) _ (2)) </Td> </Tr> <P> Point symmetry can be thought of in the following fashion: consider the coordinates which make up the structure, and project them all through a single point, so that (x, y, z) becomes (− x, − y, − z). This is the' inverted structure' . If the original structure and inverted structure are identical, then the structure is centrosymmetric . Otherwise it is non-centrosymmetric . Still, even for non-centrosymmetric case, inverted structure in some cases can be rotated to align with the original structure . This is the case of non-centrosymmetric achiral structure . If the inverted structure cannot be rotated to align with the original structure, then the structure is chiral (enantiomorphic) and its symmetry group is enantiomorphic . </P> <P> A direction (meaning a line without an arrow) is called polar if its two directional senses are geometrically or physically different . A polar symmetry direction of a crystal is called a polar axis . Groups containing a polar axis are called polar . A polar crystal possess a "unique" axis (found in no other directions) such that some geometrical or physical property is different at the two ends of this axis . It may develop a dielectric polarization, e.g. in pyroelectric crystals . A polar axis can occur only in non-centrosymmetric structures . There should also not be a mirror plane or twofold axis perpendicular to the polar axis, because they will make both directions of the axis equivalent . </P> <P> The crystal structures of chiral biological molecules (such as protein structures) can only occur in the 65 enantiomorphic space groups (biological molecules are usually chiral). </P>

How do these axes differ to create each type of crystal system