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.
Euclidean geometry equal (SAS), or two angles and a side equal (ASA) (Book I, propositions 4, 8, and 26). Triangles with three equal angles (AAA) are similar, but not necessarily congruent. Also, triangles with two equal sides and an adjacent angle are not necessarily equal or congruent. The sum of the angles of a triangle is equal to a straight angle (180 degrees). This causes an equilateral triangle to have three interior angles of 60 degrees. Also, it causes every triangle to have at least two acute angles and up to one obtuse or right angle. The celebrated Pythagorean theorem (book I,