The golden ratio is represented by the Greek letter φ and satisfies:
φ = (1 + √5) / 2 = 1.618033988749895
1 / φ = φ - 1 = (√5 - 1) / 2 = 0.618033988749895In the current dodecahedron vertex set:
V0 = (1, 1, 1)
V8 = (0, 1 / φ, φ)Therefore, the edge length from V0 to V8 is:
d = √((1 - 0)^2 + (1 - 1/φ)^2 + (1 - φ)^2)
= √(1 + 0.3819660113^2 + (-0.6180339887)^2)
= 1.2360679774997898
= 2 / φ
= √5 - 1Move the slider to observe the 12 pentagonal faces unfold into one connected flat net and merge back toward the solid.
Select a method, then move the step slider to inspect the construction.