The skeleton of the 120-cell is not 5-gonal Original Research Article

The edge-graphs of m-polygons and of 5 platonic solids admit (unique) isometric embedding into a half cube (of dimension m and 6, resp.). The skeletons of αn, βn, γn are also ℓ1-graphs but it was proved in Assouad (1991) that 24-cell and 600-cell are not (since not 5- and not 7-gonal, resp.). We show here that the last remaining regular polytope, 120-cell, is also not an ℓ1-graph.