An Excavated Tetrahedron

Polyhedral excavation is the opposite of augmentation. In this excavated tetrahedron, short pyramids have been removed from each face. I made this using Stella 4d, which can be tried for free at this website.

The Compound of the Pyramid-Excavated Dodecahedron and Tetrahedrally-Excavated Icosahedron, Together with Its Interesting Dual

Compound of Augmented Dodeca and Augmented Icosa

This is the familiar dodecahedron/icosahedron compound, but with each face of both components of the compound altered by the excavation of an equal-edge-length pyramid. To make it, as well as the rotating image below, I used Stella 4d, which you can find here.

Also, here is the dual of the compound above:

Compound of Augmented Dodeca and Augmented Icosa

An Excavated Octahedron with Non-Zero Volume, and an Octahedral Nulloid


An Excavated Octahedron, and an Octahedral Nulloid

This is an octahedron which has had short pyramids excavated from each of its faces. If the pyramids are made taller, their vertices coincide at the octahedron’s center. At that point, unlike in the figure above, the polyhedron’s volume reaches zero — turning it into a special type of polyhedron called a nulloid:


You may click on this second picture to enlarge it, if you wish.

Software credit: I made these images using Stella 4d, which you can find at

An Excavated Snub Cube, with Two of Its “Cousins”


An Excavated Snub Cube

In this variation of the snub cube, twenty of the triangular faces have been excavated with short triangular pyramids. Since the snub cube is chiral, it’s possible to make a compound out of it and its mirror-image:

Compound of enantiomorphic pair of excavated snub cubes

A polyhedron which is somewhat similar to the first one shown here can be produced by faceting a snub cube:

Faceted Snub Cube

Stella 4d was used to create these images. You can find this program at

A Modified, Excavated Icosidodecahedron


A Modified, Excavated Icosidodecahedron

I’ve done quite a bit on this blog involving modifiying polyhedra via augmentation, in which polyhedra are attached to faces of another polyhedra. This was made using the opposite process, excavation, where parts of a given polyhedron are removed. First, a pentagonal pyramid was excavated from each pentagonal face of the icosidodecahedron. Next, octahedra were excavated from each triangular face. Finally, the bottom face of each of these octahedra was hidden, making it possible to see all the way through this solid. I constructed this with Stella 4d, software you can try or buy at

Here’s the same polyhedron again, but with a different color-scheme.

Augmented Icosidodeca