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.
Triangle center if the trilinear coordinates of P can be expressed in the form f(A) : f(B) : f(C) where f(A) is a function of the angle A alone and does not depend on the other angles or on the side lengths. A triangle center P is called a "transcendental triangle center" if P has no trilinear representation using only algebraic functions of a, b and c. Let "f" be a triangle center function. If two sides of a triangle are equal (say "a" = "b") then so two components of the associated triangle center are always equal. Therefore all triangle centers