Six bilunabirotundae can be augmented around a cube with pyritohedral symmetry.
2.
Pyritohedral symmetry is another doubling of tetrahedral symmetry.
3.
The octahedral group also has a unique subgroup called the pyritohedral symmetry group, [ 3 +, 4 ], of order 12, with a mixture of rotational and reflectional symmetry.
4.
It has chiral tetrahedral symmetry, and so its geometry can be constructed from pyritohedral symmetry of the pseudoicosahedron with 4 faces Dorman Luke dual construction, a unique geometric proportion can be defined.
5.
On the other hand, the group T h of pyritohedral symmetry also has 24 members and a subgroup of index 3 ( this time it is a D 2h prismatic symmetry group, see point groups in three dimensions ), but in this case the whole subgroup is a normal subgroup.
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