Enneacontahedron Again

Finding omni-symmetrical polyhedrons with volumes close to the sphere's volume, while having the same radius in some important direction, has been a preoccupation of some geometers, David Koski among them.

Sir David works in tetravolumes, a little known standard, wherein the unit sphere is rt2(2)*pi in volume. That's (4/3)*pi*r**2 times S3, the Synergetics Constant. Either way, the ratios come out the same, so even in XYZ you will find the enneacontahedron comes within a fraction of a percent of a sphere's volume, hugging it even more closely than the rhombic triacontahedrons you may have encountered in this neighborhood: the 5 and the 5+.

Below is a rendering developed in POV-Ray using the export feature from vZome. I've jiggered with the camera and added a textured sphere inside, still growing to become tangent to the enneacontahedron's narrow diamonds, embedded in our rhombic triacontahedron's.