Buckminster Fuller's frequency subdivision of the icosahedron
Vertices: —
Edges: —
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Euler: —
Frequency (ν)2
Hemisphere Cut0.5
A geodesic dome is constructed by subdividing each triangular face of an icosahedron (20 faces, 12 vertices, 30 edges) into a grid of ν² smaller triangles (frequency ν), then projecting all vertices onto the circumscribed sphere. The result approximates a sphere with flat panels. By Euler's polyhedron formula V − E + F = 2, a frequency-ν geodesic sphere has V = 10ν² + 2 vertices. The dome distributes structural load efficiently — Buckminster Fuller patented it in 1954.