# Solids that are platonic apart from faces being irregular polyhedra.

Scouring through Wikipedia, I've found the following analogs to platonic solids that are composed of irregular faces.

Cube = Trigonal Trapezohedron

Dodecahedron = Tetartoid

Tetrahedron = Disphenoid

I couldn't find the analogs for the Octahedron and Icosahedron. Does that mean they don't exist? How do I prove either way?

EDIT: My bad, Octahedron exists as well. That leaves Icosahedron.

Why, of course it can be done. Start with the net, assign the lengths so that all triangles will be equal and scalene: Then fold it and glue everything together. True, the figure looks god-awfully ugly. This is a consequence of the fact that it can't be made face-transitive, unlike your three examples. (To make 20 faces symmetrically equivalent, we'd have to keep at least one 5-fold symmetry axis, which would keep passing through a vertex, which would make its five edges equal, which in turn doesn't sit well with the triangles being scalene.)

If you agree on isosceles faces, then you may keep a 5-fold axis and arrive at Aretino's example.

I suspect there are multiple ways to connect the scalene triangles (this is the case with octahedron, after all), but this post is getting too long, so let's call it a day.

• Is this figure convex? Aug 16 '18 at 23:30
• Also, how did you make the animations? Aug 16 '18 at 23:31
• No. Geogebra.$\,$ Aug 16 '18 at 23:37
• Ok, then it doesn't quite fit my requirement "platonic solids with irregular faces". I meant they should have all other properties of platonic solids. Still cool shape :) Aug 17 '18 at 1:13
• You don't have to distort the triangle as much as I did. With smaller distortion it will be convex all right. Aug 17 '18 at 3:29

An icosahedron can be viewed as a pentagonal antiprism with two pyramids built on its bases, hence it can also be made with isosceles triangles (see image below).

And I suspect that other more exotic topologies are possible. 