Where Domes Come From

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Geodesic domes possess many advantages. They're cheap, strong, light and graceful. Unfortunately, until very recently, there has been little information available on small domes that could be built by the average person. This article is intended to help fill the information gap by explaining the theory on which domes are based, and to encourage you to try designs of your own.

There is no reason to be intimidated by the apparent complexity of dome patterns. Anyone who can handle simple trigonometry can design a dome to meet his own needs. We'll start by exploring the geometric relations that form the basis for all dome structures.

Every dome ever built ultimately derives from one of the five regular solids shown in figure one. Dome design consists of breaking down these basic frameworks into networks of smaller faces that can be handled more easily.

DOME STARTER

Figure one shows some interesting relationships. Notice that the sum of the faces and the vertices always exceeds the number of edges by two. This relation, V + F = E + 2, is known as Euler's formula, and is true for any solid, regular or not.

Also notice that both cube and octahedron have the same number of edges, while each has as many vertices as the other has faces. This implies that the two solids can be superimposed with the faces of each corresponding to the vertices of the other. This is so, and the relation is known as duality. The dodecahedron and the icosahedron are duals, and the tetrahedron is its own dual. Figure two shows interpenetrating pairs of these dual solids.

Here is another, relationship; if you place a stubby pyramid over each face of one of the regular solids, you'll find that the pyramid sides merge into diamonds. Thus we get a new addition to each of our three pairs of duals, since a solid and its dual will each yield the same form when treated in this way. From the cube and octahedron we get the rhombic dodecahedron. From the dodecahedron and icosahedron, we get the rhombic triacontahedron. From the tetrahedron, we get (surprise!) the cube.

Figure 3a shows how this process generates the cube from the tetrahedron, and the rhombic dodecahedron from the octahedron and the cube.

Comparing figures one and three, we find that these new solids have as many faces as their parents have edges, twice as many edges as their parents, and as many vertices as their parent's faces and vertices combined.

We now have three families of related solids:

I like to call them the three family, the four family, and the five family, after the dominant symmetry in each. In figure four, all three members of each family are superimposed and projected onto a sphere.