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.
I made this variation of Kepler’s Stella Octangula, using Stella 4d, software you can try for free at this website.
All of the edges of this polyhedron have the same length. I made it using Stella 4d: Polyhedron Navigator, which you can try for free at this website.
I created this using Stella 4d: Polyhedron Navigator. You may try this software, for free, at http://www.software3d.com/Stella.php.
You can tell this is a fashionable tetrahedron because he’s wearing four pyramidal hats — one to cover each vertex.
This bit of polyhedral silliness was created with Stella 4d, software you may try for free right here.
To make this, I used the excavation-function of Stella 4d, set to remove pyramids with equal edge length from each face of an icosidodecahedron. You can try this program here.
The dual of this polyhedron is shown below.
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:
The next one is a compound of eight off-center pyramids. By this point, I had gone so far into the stellation-series (a search I began when preparing the post before this one) that I had lost count.
This one is a compound of three short square-based dipyramids:
This one, according to Stella 4d, is a compound of three parts, but I can’t quite figure out what the parts are!
Here is another “mystery compound,” this one with two parts:
Stella 4d, which I used to make these, may be tried here.
The image above is a compound of the rhombic triacontahedron (the dual of the icosidodecahedron) and a strombic hexacontahedron (the dual of the rhombicosidodecahedron). Below, you’ll find a compound of six square-based pyramids, all with their “centers of mass” (assuming uniform density) displaced, from the compound’s center, by equal amounts. In response to a request I have received, polyhedral images which rotate more slowly are coming soon . . . after I have finished posting my backlog of already-produced polyhedral .gif files, since there is no way to slow them down after they are already created.
The program I use for these polyhedral investigations is Stella 4d, available at www.software3d.com/Stella.php.
Many polyhedra appear in finite sets. The most well-known example of such a set is the five Platonic solids. Many know them from role-playing games.
Other finite sets include the thirteen Archimedean, four Kepler-Poinsot, thirteen Catalan, and 92 Johnson solids, and the eight convex deltahedra, among others. There is even a finite set, the near-misses, with an unknown number of members, due to the “fuzziness” of its definition. The symmetrohedra is another set with “fuzzy” criteria, but there are still only so many symmetrohedra to be found. We simply haven’t found them all yet, or, if we have, we don’t know that we have, but it would not be reasonable to claim that infinitely many await discovery.
However, not all groups of polyhedra are finite. Some polyhedra appear, instead, in infinite families. What is needed to generate such infinite families (at the cost of some forms of symmetry, compared to, say, Platonic or Archimedean solids) is the use of bases — special polyhedral faces which play a stronger role in the determination of that polyhedron’s shape than do the other faces. For the familar prism, there are two bases.
In a pentagonal prism, the bases are pentagons, as seen above. For a pentagonal prism, n = 5, for n is simply the number of sides of the base. The smallest value of n which is possible, 3, yields a triangular prism. There’s no upper limit for n, either. Here’s a regular hexacontagonal prism, where each base has sixty sides.
Obviously, n can be increased without limit, although for very large values of n, the prism will be hard to distinguish from a cylinder.
Another infinite family may be found by taking the dual of each prism. This is the dual of the pentagonal prism:
Taking a dual of a prism produces a dipyramid, with its n-gonal base hidden between the puramids, but with no guarantee that the triangular faces will be regular — and in this case, they are not. It is possible for a pentagonal dipyramid to have only regular faces . . .
. . . but regular faces will not work if n = 6 (because the dipyramid collapses to zero height), or indeed any number other than 3, 4, or 5. Therefore, dipyramids will not be an infinite family unless non-equilateral triangles are permitted as faces.
Half of a dipyramid, of course, is simply a pyramid, with the single base now visible:
Like the dipyramids, and for the same reason, there are only three pyramids (n = 3, 4, or 5) which can have all faces regular. An infinite family of right, regular pyramids do exist, though, if isosceles triangles are permitted as lateral faces.
While pyramids and dipyramids have only one base each, the already-described prisms have two. Prisms also can maintain regularity of all faces, no matter how large n becomes, unlike pyramids and dipyramids. Moreover, prisms can be transformed into another infinite family of regular-faced polyhedra by rotating one base by one half of one-nth of a rotation, relative to the other base, and replacing the n square lateral faces with 2n equilateral triangles. These polyhedra are called antiprisms. The pentagonal antiprism looks like this:
Antiprisms with all faces regular do not have the “only 3, 4, and 5” limitation that affected the pyramids and dipyramids. For example, here is one with all faces regular, and dodecagonal bases:
One more infinite family of polyhedra may be found, using the antiprisms: their duals. The dual of the pentagonal antiprism looks like this:
There is no regularity of faces for any members of this family with n > 3, for their faces are kites. (When n = 3, the kites become squares, and the polyhedron formed is simply a cube.) Unfortunately, “kiteohedron” looks ridiculous, and sounds worse, so efforts have been made to find better names for these polyhedra. “Deltohedra” has been used, but is too easily confused with the deltahedra, which are quite different. The best names yet invented for this infinite family of kite-faced polyhedra are, in my opinion, the antidipyramids, and the trapezohedra.
(Note: the rotating polyhedral images above were generated using Stella 4d: Polyhedron Navigator, available at http://www.software3d.com/Stella.php.)