Caltrops, when resting on a horizontal surface, have a sharp, narrow point sticking straight up. Stepping on such objects is painful. Most polyhedra do not have such a shape; the most well-known example of an exception to this is the tetrahedron. This fact is well-known to many players of role-playing games, who often use the term “d4” for tetrahedral dice, and who usually try to avoid stepping on them. Here are some other polyhedra which resemble caltrops. All were made using Stella 4d, software available at this website. The first two images may be made larger by simply clicking on them.
The third example, made with the same program, varies this idea somewhat: in physical form, resting on a floor, this caltrop-polyhedron would have three, not just one, potentially foot-damaging “spikes” sticking straight up.
I made this by stellating a dodecahedron repeatedly, but doing so with Stella 4d, the polyhedral-manipulation software I use (available here), set to use tetrahedral symmetry, rather than the higher-order icosahedral symmetry (which I often call “icosidodecahedral” symmetry) inherent to Platonic dodecahedra.
The same polyhedron appears below, but with the coloring-scheme, rotational direction, and rotational speed all set differently.
Dodecahedra have icosahedral (also called icosidodecahedral) symmetry. In the figure above, this symmetry is changed to tetrahedral, by truncation of four vertices with positions corresponding to the vertices (or, instead, faces) of a tetrahedron. The interchangeability of vertices and faces for the tetrahedron is related to the fact that the tetrahedron is self-dual.
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.
In the last post on this blog, there were three images, and the first of these was a rotating icosahedron, rendered in three face-colors. After making it, I decided to see what I could build, using these tri-colored icosahedra as building blocks. Augmenting the central icosahedron’s red and blue faces with identical icosahedra creates this cubic cluster of nine icosahedra:
If, on the other hand, this augmentation is performed only on the blue faces of the central icosahedron, the result is a tetrahedral cluster of five icosahedra:
The next augmentation I performed started with this tetrahedral cluster of five icosahedra, and added twelve more of these icosahedra, one on each of the blue faces of the four outer icosahedra. The result is a cluster of 17 icosahedra, with an overall icosahedral shape.