Smooth gems
This is an infinite sided polyhedron that has a C(1) continuous surface. This means that it has neither corners nor edges, it is as smooth as a capsule (a cylinder with spheres for ends). It makes a nice gem stone:
Construction is quite simple. Start with a tetrahedron, then repeatedly:
apply a full truncation of its corners (so the new faces touch halfway along the old edges).
truncate the corners formed at those touch points, one third of the way along their edges.
The set of edges forms a fractal-like shape, something like an ellipse packing. However, those edge angles tend to zero as you add iterations, giving the C(1) continuity. This means that normal vectors are continuous over the surface.
The same method can apply to other starting shapes, here it starts with a cube:
