A convex deltahedron in $\mathbb{R}^3$ is a convex polyhedron whose faces are all equilateral triangles. There exist precisely 8 convex deltahedra. Some examples are the regular tetrahedron, the regular octahedron, and the regular icosahedron. As regular polyhedra, these three can be inscribed in the sphere $\mathbb{S}^2$, meaning all of their vertices lie in the sphere. Are there any other inscribable deltahedra?
Which convex deltahedra are inscribable in the sphere
convex-geometryconvex-hullsdiscrete geometrypolyhedrapolytopes
Best Answer
Assuming that you ask about being inscribed (rather than being inscribable) the answer is: no, there aren't any others.
You can take a look at the non-regular deltahedra over here (all of them are Johnson solids). And over here you will find all the Johnson solids that are inscribed, and none of these is a deltahedron.
Asking about inscribable, i.e. having an inscribed realization, I would say all of them are inscribable (just my intuition, given the pictures), though I am not absolutely sure for the snub disphenoid.