Regular Polyhedra in £ 4 and E N
So, I state that there is no another solid, besides five mentioned solids, which is contained between equilateral and equiangular polygons equal to each other.
There are five regular polyhedra in £ 3 . In Euclid's "Origins" the 13th book is devoted to the theory of regular polyhedra, which are often called Platonic solids because in a famous dialog "Thimey" the four of them were described: tetrahedron, icosahedron, octahedron and cube. But there is the background to suppose that the dodecahedron was unknown to Plato at that time. Morduchaj-Boltowski wrote about this in his comments to Euclid's 13th book. It seems that the dodecahedron was discovered considerably later than the other four regular solids. The science historians (see, for instance, Zmud' "Pythagoras and his school") relate the discovery of the dodecahedron by Hyppas, the Pythagorean, and by the octahedron and the icosahedron by Thiatet.