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

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 .


Introduction
It is well -known that the axes of symmetry of the crystal can merely exist C1, C2, C3, C4 and C6, this is the so -called crystallographic restriction theorem [1][2][3][4] .Among various proofs of this theorem, there is a famous proof which is generally concurred by those people who are familiar with the solid state physics.Although the well -known proof of the theorem has been applied in many famous books [2][3] , it is not satisfactory at least due to the following two reasons.First, considering an n-fold (n is an integer) rotation of the crystal in the twodimensional space, as shown as Fig .1 [2] , Fig. 1, The point B' is one of the points generated by an n-fold rotation axis through point A operating on lattice point B with angle δ, and the point A' by a similar axis through point B inversely operating on lattice point A with angle δ.The value of angle δis equal to n in terms of the proof , because of the periodicity of lattice structure , the length of B'A' must be equal to the integral multiples of that of AB, namely, B'A'= m t .
(1) However, in the viewpoint of mathematics, eq. ( 1) is not clear as an argument for the 14 different Bravais lattice structures of real crystals , but not the supposing Bravais lattice, all of the 14 lattices should be respectively demonstrated in order to support eq. ( 1).
Second, the calculation from eq. ( 1) in accordance with Fig . 1 demonstrates that the possible values of m are -1, 0, 1, 2, and 3, nevertheless, if m takes the value of -1, neither of the lengths of B'A' and t in eq. ( 1) can be significant to be negative .
Therefore, it is necessary to find a rigorous proof of the crystallographic restriction theorem .

II. Review of Literature
The inexistence of C5 axis is equivalent of that pentagons are impossible to fill all the space with a connected array of pentagons [ 3] , and this can be easily generalized to all the cases of Cn (n≥7 ) axis .At first , let us consider two congruent regular pentagons such as A5 and B5 in the two -dimensional space, as shown as Fig. 2 .(3) It is clear from Fig. 2 that if the other one or more pentagons just can fill the space within the scope of θ5 with no "gaps " between pentagons, it requires that one or more interior angles can just fill θ5 angle with no "gaps" between them, or the size of θ5 must be just equal to the integral multiples ( positive ) of the size of an interior angle, namely, θ5= mα5 (m = 1, 2, 3……).( 4) Nevertheless, using eq.( 2) , the size of θ5 is given by θ5= 2β5=2×  72 =  144 , (5) from eq. ( 5) and eq.( 3), it can be found in terms of eq. ( 4), C5 axis can not exist .Fig .3 depicts the "gaps" between pentagons in the scope of θ5 in a close packing of pentagons in the two-dimensional space [ 3 ] .
In the viewpoint of mathematics, C1, C2, C3, C4 and C6 axes must also be discussed .At first, it is clear that C1 axis represents an one-fold rotation with the rotation angle  0 or  360 , and will certainly remain the crystal invariant .
We separately consider an oblique Bravias lattice in the two-dimensional space, if a two-fold rotation with the rotation angle of  180 through any lattice point in a primitive cell, the primitive cell will remain invariant, this is also true for equivalent points in other primitive cells [ 3 ] .Therefore, C2 axis for a crystal based on such a primitive cell can exist .

IV. Conclusion
With respect to the crystallographic restriction theorem, the paper proposed different opinions on a proof applied in many famous books.Due to the periodicity of the lattice structure, the inexistence of C5 axis is equivalent of that pentagons are impossible to fill all the space with a connected array of pentagons, for example, in Fig. 2, if one or more other congruent regular pentagons can fill all the space within the scope of θ5 with no "gaps" between them, the value of θ5 must be the integral multiples of the value of an interior angle of the pentagon.Nevertheless, from the calculation it can be found that the value of θ5 is not an integral multiple of that of α5, therefore, C5 axis do not exist.
Similarly, if assuming to substitute two congruent regular n-sided (n≥7 ) polygons for the two pentagons in Fig. 2, the present calculation demonstrates that the value of θn (n≥7 ) is smaller than that of the interior angle of the n-sided (n≥7 ) polygon, no possible to be its integral multiple.But differing from these cases, if assuming to substitute the two pentagons in Fig . 2 with two congruent regular triangles, or tetragons, or hexagons, it is easy to calculate out thatθ3= 4α3, θ4 =2α4, θ6 =α6 , they are consistent with eq. ( 4) .Moreover, it is clear that C1 and C2 are compatible with translational symmetry .Therefore, the possible axes of rotation of the crystal are merely C1, C2, C3, C4 and C6 .