What's in between a tetrahedron and a sphere? You could smooth the tetrahedron partially but this no longer has any flat sides, or you could go to higher solids like an octahedron then an icosahedron, but these still have sharp edges, unlike a sphere.
A polyhedron has C0 continuity and a sphere has Cinfinity continuity. It makes sense that an intermediate could have C1 continuity, which is what this shape has:
We start with a tetrahedron:
I am just showing the edges, but this is a convex solid with flat surfaces in all the circles. In this one the corners are cut off so the slices touch, and the edges cut off one third of the way along the sliced edges. The result has octahedral symmetry, with 8 largest circle faces.
It is C1 continuous because its face angles (the first derivative) follow a devil's staircase function across an edge, this is a continuous function.
Here's the solid object:
and here's the same thing but starting with a cube:
Here is the Octahedral rendered with transparency and refraction:
