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The polyhedron, upon which most geodesic domes are based, is the icosahedron.

The icosahedron has 20 equilateral triangle faces. By subdividing the icosahedron face into smaller triangles, then "pushing" the triangle vertices outward (to the surface of a circumscribing sphere,) a more-complex triangulated polyhedron can be produced -- a "geodesic" sphere, or dome.

By subdividing the icosahedron face into greater-and-greater numbers of smaller-and-smaller triangles, more complex geodesic spheres/domes are produced.

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