How can there be more dodecahedra?

It's an interesting fact that if you draw a regular pentagon and then extend its sides you will get a regular pentagram (5-pointed star polygon) that surrounds it. Then if you join the vertices of the pentagram you get another pentagon whose extended sides produce yet another pentagram, and so on, each configuration being larger than the previous one. Or, you can go the other way; by beginning with a pentagon you can construct, by joining every other vertex, a pentagram whose sides intersect on the boundary of a smaller regular pentagon. Then the process can be repeated, producing alternately a pentagram, pentagon, pentagram, pentagon, … each inside the previous drawing. See Figure 11.1 where the faces of the special dodecahedra we will describe in this section are labeled.

Hermann Weyl (1885–1955) recalled that when a pentagram has one vertex pointing straight down it is a symbol for evil and when one vertex is pointing straight up it is a symbol for good. It is amusing that in Weyl's book, the pentagram is shown with one side parallel to the vertical side of the page, so that it points neither up nor down. We used the symbol for good in our illustration of Figure 11.1. An interesting feature of any pentagon is that the ratio of its diagonal to its side is the golden ratio.