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.