Crystallographic restriction theorem lattice basis, the rotation operation must map every lattice point into an integer number of lattice vectors, so the entries of the rotation matrix in the lattice basis — and hence the trace — are necessarily integers. Similar as in other proofs, this implies that the only allowed rotational symmetries correspond to 1,2,3,4 or 6-fold invariance. For example, wallpapers and crystals cannot be rotated by 45° and remain invariant, the only possible angles are: 360°, 180°, 120°, 90° or 60°. Example The general crystallographic restriction on rotations does "not" guarantee that a rotation will be compatible with a specific lattice.
Many-angled ones Many-angled ones The many-angled ones are fictional other-dimensional beings linked to the Cthulhu Mythos. They first appeared in Grant Morrison's story "Zenith", which appeared in the British comics anthology "2000 AD". In "Zenith", they are known as the Lloigor, a direct reference to creatures from the Cthulhu Mythos. However, they appear somewhat different from the Cthulhu Mythos entities. In the comic, the many-angled ones plan to impose rigid geometrical order on the whole universe, essentially reducing it to clockwork. The many-angled ones exist in a space with more dimensions than our own; hence, they appear to be "many angled". As