The five fold rotation axis exists on the regular pentagon. On the other hand, it is impossible to tile the plane with regular pentagons, as showed by the above animation.

Twelve equal regular pentagons fit perfectly to close the Platon polyhedron named pentagonal dodecahedron. But there is no way of space tiling with pentagonal dodecahedra.

Only proper rotation 2, 3, 4 and 6 fold axes exist in crystallography.

Exercise

Show why 7 fold and 8 fold axes does not exist in crystallography.

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