Eight Convex Deltahedra

Theorem
There exist exactly $8$ distinct convex deltahedra:


 * $4$ faces: regular tetrahedron


 * $6$ faces: triangular bipyramid


 * $8$ faces: regular octahedron


 * $10$ faces: pentagonal bipyramid


 * $12$ faces: snub disphenoid (split a regular tetrahedron into two wedges and join them with a band of $8$ equilateral triangles)


 * $14$ faces: triaugmented triangular prism (attach $3$ square pyramids to a triangular prism)


 * $16$ faces: gyroelongated square bipyramid (attach $2$ square pyramids to a square antiprism)


 * $20$ faces: regular icosahedron.