Twelve hexagons

Since ancient times, a figure has been known in which 12 faces are regular pentagons. Such a dodecahedron is one of five Platonic solids and has a fifth-order rotation symmetry . However, in many respects this ideal polyhedron has a drawback. The fact is that regular pentagons cannot cover a plane without gaps. Also, dodecahedrons cannot densely fill the space. This implies the impossibility of the existence of crystals with fifth-order symmetry axes and the impossibility of the existence of crystals in the form of a Platonic dodecahedron. However, viruses and proteins are known in the form of such a dodecahedron, with fifth-order axes of symmetry. It is believed that they have acquired this form in order to avoid crystallization .

Visually very similar to a Platonic solid, but has a completely different symmetry - the central form of symmetry of cubic syngony . Facets are irregular pentagons symmetrical with respect to a plane passing through the center of the figure. The pentagondodecahedron is one of the simplest forms of crystals . The faceting of the crystals by the pentagon dodecahedron is characteristic, for example, for pyrite .

A figure faceted by equal rhombs and being a polyhedron dual to the cuboctahedron .

There is a direct connection between the pentagonondodecahedron and the rhombododecahedron. The pentagondodecahedron is obtained from the rhombic dodecahedron if the face of the rhombic dodecahedron is rejected towards the apex. In this sense, the pentagondodecahedron is the transition form between the cube and the rhombic dodecahedron .

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  face index {10,9,0}

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  face index {2,1,0}

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  face index {7,1,0}