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.
Special right triangle than 256: Isosceles right-angled triangles cannot have sides with integer values, because the ratio of the hypotenuse to either other side is , but cannot be expressed as a ratio of two integers. However, infinitely many "almost-isosceles" right triangles do exist. These are right-angled triangles with integral sides for which the lengths of the non-hypotenuse edges differ by one. Such almost-isosceles right-angled triangles can be obtained recursively, "a" is length of hypotenuse, "n" = 1, 2, 3, ... Equivalently, where {"x", "y"} are the solutions to the Pell equation , with the hypotenuse "y" being the odd terms of the