Geodesic Dome Frequency

Icosahedron subdivision: V=10ν²+2, E=30ν², F=20ν²

3D Geodesic Sphere

Frequency ν:

Strut Length Distribution

Theory

Vertices: V = 10ν² + 2
Edges: E = 30ν²
Faces: F = 20ν²
Euler characteristic: V − E + F = 2 ✓

A geodesic dome is constructed by subdividing each triangular face of an icosahedron into ν² smaller triangles, then projecting all vertices onto the circumscribed sphere.


Frequency ν: each edge is divided into ν equal segments. Higher ν → more triangles → more spherical, but more strut types. The ν=3 dome was popularized by Buckminster Fuller for the 1967 Montreal Expo.


Most subdivision methods produce 2 or 3 distinct strut lengths for low ν, rising with ν. The structure distributes loads efficiently — geodesic domes have the highest volume-to-surface-area ratio of any man-made structure.