How many convex solids have the same faces?

Question

The five platonic solids have all faces regular Polyhedra. They are also all convex. But there are other properties they satisfy. Like each vertex should have the same number of edges meeting there. Let's say we relax all other criterion and only require that the solid be convex and that it have all faces regular. How many such solids can exist? One example is pasting two tetrahedra together along a face to get a solid with 6 sides and triangular faces.

Answer 1

Convex polyhedra with regular faces which are not Platonic are called Johnson solids. There are 5 Johnson solids with the same faces. It happens they are all triangular faced (can you see why?)

• J12 Triangular bipyramid
• J13 Pentagonal bipyramid
• J17 Gyroelongated square bipyramid
• J51 Triaugmented triangular prism
• J84 Snub disphenoid

So together with 5 Platonic solids, you have 10.