A Rigorous Proof on the Crystallographic Restriction Theorem to Establish Human Being

Abstract

It is significant to find a more rigorous and satisfactory proof of the crystallographic restriction theorem. The inexistence of C5 axis of symmetry is equivalent of that pentagons are impossible to fill all the space with a connected array of pentagons, on the basis of this viewpoint, using a purely mathematical approach the paper rigorously proves that C5 and Cn (n ≥7 ) axes of symmetry cannot exist , and one –, two –, three –, four – and six – fold axes of rotational symmetry are allowable. Therefore, the axes of symmetry of the crystal can merely exist C1, C2, C3, C4 and C6 .