Which has more volume, a dodecahedron or an icosahedron, both having the same edge length? (The same question can be asked of the cube and octahedron, and the following discussion applies just as well to them.) It’s tempting to think that the icosahedron is bigger, because it has more faces (20 to the dodecahedron’s 12). I think it is also natural to think that they are very close in size. In fact, the dodecahedron has about 3.5 times more volume.
People sometimes think that math is about manipulating strings of symbols according to formal rules, but really it’s about figuring out which intuitions are right and why others are wrong. Before we figure out some good intuitions for the relative volumes of polyhedra, let’s look at the numbers for the Platonic solids, assuming an edge length of 1. (If the edge length is actually , multiply the given volume by .)
One reason we might think the icosahedron and dodecahedron are roughly the same volume is that we might picture them with the same diameter. Since they are both pretty spherical, they would have roughly the same volume in that case. For example, see Wikipedia’s stock photos of the two solids:
However, when the two shapes are the same volume, the dodecahedron’s edges are about 34% shorter. If we expand the dodecahedron to make its edges the same length as the icosahedron’s, that increases the volume by a factor of .
In fact, this turns out to be a pretty good way to think about volumes of polyhedra in general. Instead of asking what the relative volumes are for the same edge length, ask what the relative edge lengths are for the same volume. Smaller edge length for the same volume means larger volume for the same edge length.
So how can we get a handle on relative edge lengths for the same volume? One useful intuition is to picture two polyhedra that are the same volume as same-sized spheres. Which one will have shorter edges? Well, the vertices are pretty evenly spaced over the surface, so perhaps the one with more vertices will have more closely packed vertices, and therefore shorter edges. This suggests that holding the edge length constant, the solids with more vertices will tend to have larger volumes.
Looking back to the table above, this appears to be fairly accurate. Holding edge length constant, the number of vertices seems to correspond pretty well with volume, certainly better than the number of faces or edges.
Incidentally, why does the number of faces suggest the wrong volume ordering? This is because different faces are different sizes. Sure, the dodecahedron only has 12 faces, but they are pentagons — much bigger than the icosahedron’s 20 triangular faces. A regular pentagon has about four times as much area as an equilateral triangle with the same edge length. So the 12 pentagons are worth about 48 equilateral triangles. Now we also have a strong face-based intuition that points in the right direction: the dodecahedron is significantly larger than the icosahedron.
For Platonic solids, the ordering by number of vertices is the same as the volume ordering. Let’s take a look at a more general class of solids. Archimedean solids have faces that are not all the same, but the faces are still regular polygons, and there is the same configuration of faces at each vertex. For example a cuboctahedron has two squares and two triangles meeting at each vertex. Here are the 13 Archimedean solids in volume order.
Well it’s not perfect, but vertices still correspond to volumes far better than faces or edges. Let’s look at the relationship between number of vertices and volume a little more closely.
As you can see, the volume does correspond pretty well with the number of vertices, but there are some glaring problems. Most notably, look at the four with 60 vertices; there is over a factor of two between the volumes of the biggest (truncated dodecahedron) and smallest (snub dodecahedron). Above we speculated that for the same volume, more vertices would be more densely packed, leading to shorter edges. Why does the truncated dodecahedron have shorter edges than the snub dodecahedron for the same volume and same number of vertices? Let’s take a look.
The snub dodecahedron (left) has 80 triangles and 12 pentagons, and the vertices look pretty well spaced out. The truncated dodecahedron (right) has 20 triangles and 12 decagons. The vertices are not well spaced out at all; they’re basically all lined up to make those big empty decagons. This explains why the truncated dodecahedron has so much volume for a given edge length and number of vertices. For a given volume, the vertices are very unevenly spaced, making the edges short.
Four-Dimensional Platonic Solids
Finally, do the intuitions we developed above for three-dimensional solids carry over to four-dimensional solids? Absolutely. Six data points isn’t much, but here are the volumes of the four-dimensional Platonic solids (4-d volumes in this case, so multiply by if the side length is ).
Looks good to me. The volumes are in the same order as the number of vertices, and the relatively large volume increases correspond to the relatively large increases in the number of vertices.