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

Zhang Yue

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 .

Keywords


the crystallographic restriction theorem, pentagons, n-sided(n≥7) polygons, proper rotation.

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References


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Animalu A O E. Intermediate Quantum Theory of Crystalline Solids [M]. Englewood Cliffs: Prentice – Hall, Inc., 1977.

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DOI: https://doi.org/10.33258/birci.v1i4.86

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Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.