geodesic sphere

The continuation of a geodesic dome which was invented by by Buckminster Fuller. It is an assemblage of triangular trusses that grows stronger as it grows larger. Spaceship Earth at Disney's EPCOT Center is nearly a complete geodesic sphere.

Geodesic spheres can be described by their "frequency". A geodesic sphere's basic building block is the triangle, and a sphere's frequency is the extent to which its triangles are subdivided into smaller triangles.

A one-frequency geodesic sphere is essentially a regular icosahedron, consisting of 20 faces which are equilateral triangles, with 5 triangles meeting at each of 12 vertices. This figure is fairly pointy and thus not very spherelike.

In a two-frequency geodesic sphere, each edge of each of the 20 triangles is split into two, which serve as the edges of two sub-traingles, so the whole triangle is divided into four. In a three-frequency sphere, each edge of the basic triangle is divided into three, so the whole triangle consists of nine sub-triangles.

 
      /\           /\           /\
     /  \         /  \         /__\
    /    \       /____\       /\  /\
   /      \     /\    /\     /__\/__\
  /        \   /  \  /  \   /\  /\  /\
 /__________\ /____\/____\ /__\/__\/__\

     one          two          three                 ... or more

If the subdivided triangles of a higher-frequency geodesic sphere laid flat, those spheres would be no different from the icosahedron. However, they are convexly curved, so that each vertex where sub-triangles meet is the same distance from the center of the sphere. Because of this, the higher the frequency of a geodesic sphere, the smoother it looks and the closer it approaches the shape of an actual sphere.

Even on a very high-frequency, very rounded geodesic sphere, there are still only 12 vertices where 5 triangles come together. All the other vertices have 6 triangles meeting. If you see a high-frequency dome, such as the Spaceship Earth pavillion at Epcot, try to see the basic icosahedral shape of the dome by picking out the vertices where only 5 triangles meet.

Interested in building a geodesic sphere or dome? www.desertdomes.com has a nifty Dome Calculator which will give you numbers and precise lengths for all the struts.

vivid writes, above:

[Any geodesic sphere has] only 12 vertices where 5 triangles come together. All the other vertices have 6 triangles meeting.

This is a bit odd, surely. Why have exactly 12 vertices of degree 5, and all others of degree 6? Wasn't Buckminster Fuller smart enough to manage to change this?

Actually, it's not Fuller's fault. He couldn't do anything about it; it's mathematics that gets in the way!

Let's set the scenario: we're constructing a "sphere-like" polytope in R³ out of triangles. We want each vertex to have degree 5 or 6.

Sounds like a job for the Euler characteristic, V-E+F=2! Let's call the number of vertices V, the number of edges E, and the number of faces F. Also, let's say we have n vertices of degree 5 (and consequently V-n of degree 6). Every edge participates in 2 triangles, so 3*F=2*E. Every edge has 2 vertices on it, so we have 2*E=5*n+6*(V-n), so 3F=6V-n. And V-E+F=2.

Turns out we can solve for n:

6V-6E+6F=12
(3F+n) - (9F) + 6F = 12
n = 12

So Euler say s we've got to have exactly 12 vertices of degree 5 in this situation.