A colouring problem for the dodecahedral graph

We consider vertex colourings of the dodecahedral graph with five colours, such that on each face vertices are coloured all colours. show total number these is 240. All can be obtained from any given colouring, by permuting and possibly applying central symmetry respect to centre regular dodecahedron. For colour classes form sets tetrahedra. These tetrahedra together one two compounds tetrahedra, inscribed in give proofs: a combinatorial one, geometrical one. Our result related W. Rouse Ball -- H. S. M. Coxeter, stating there four colourings, as follows. There up an arbitrary orientation-preserving congruence