Polyhedra · 3D · v1.0.0
Icosahedron
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Mathematics
Geometric construction
Vertices: cyclic permutations of (0, ±1, ±φ), φ = (1+√5)/2, scaled to circumradius R.
Symmetry: Icosahedral Ih (order 120).
Culture
Symbolic interpretation
Plato associated it with water.
Formulas
Available frequency mappings
- A · Direct ratio fᵢ = f₀ · rᵢ
- B · Power fᵢ = f₀ · rᵢ^α
- C · Logarithmic fᵢ = f₀ · 2^(k · log₂ rᵢ)
- D · Reference distance fᵢ = f₀ · (dᵢ / d_ref)^α
- E · Angular rᵢ = θᵢ / θ_ref ; fᵢ = f₀ · rᵢ^α
Units u
Coordinates
| id | x | y | z |
|---|---|---|---|
| P0 | 0.0000 | 0.5257 | 0.8507 |
| P1 | 0.0000 | 0.5257 | -0.8507 |
| P2 | 0.0000 | -0.5257 | 0.8507 |
| P3 | 0.0000 | -0.5257 | -0.8507 |
| P4 | 0.8507 | 0.0000 | 0.5257 |
| P5 | 0.8507 | 0.0000 | -0.5257 |
| P6 | -0.8507 | 0.0000 | 0.5257 |
| P7 | -0.8507 | 0.0000 | -0.5257 |
| P8 | 0.5257 | 0.8507 | 0.0000 |
| P9 | 0.5257 | -0.8507 | 0.0000 |
| P10 | -0.5257 | 0.8507 | 0.0000 |
| P11 | -0.5257 | -0.8507 | 0.0000 |
| P12 | 0.0000 | 0.0000 | 0.0000 |
Distances & angles
|P0P2|1.0515 u
|P0P4|1.0515 u
|P0P6|1.0515 u
|P0P8|1.0515 u
|P0P10|1.0515 u
|P1P3|1.0515 u
|P1P5|1.0515 u
|P1P7|1.0515 u
|P1P8|1.0515 u
|P1P10|1.0515 u
|P2P4|1.0515 u
|P2P6|1.0515 u
|P2P9|1.0515 u
|P2P11|1.0515 u
|P3P5|1.0515 u
|P3P7|1.0515 u
|P3P9|1.0515 u
|P3P11|1.0515 u
|P4P5|1.0515 u
|P4P8|1.0515 u
|P4P9|1.0515 u
|P5P8|1.0515 u
|P5P9|1.0515 u
|P6P7|1.0515 u
|P6P10|1.0515 u
|P6P11|1.0515 u
|P7P10|1.0515 u
|P7P11|1.0515 u
|P8P10|1.0515 u
|P9P11|1.0515 u
r(C0)1.0000 u
∠P2P0P41.0472 rad
∠P2P0P61.0472 rad
∠P2P0P81.8850 rad
∠P2P0P101.8850 rad
∠P4P0P61.8850 rad
∠P4P0P81.0472 rad
∠P4P0P101.8850 rad
∠P6P0P81.8850 rad
∠P6P0P101.0472 rad
∠P8P0P101.0472 rad
∠P0P2P41.0472 rad
∠P0P2P61.0472 rad
∠P0P2P91.8850 rad
∠P0P2P111.8850 rad
∠P4P2P61.8850 rad
∠P4P2P91.0472 rad
∠P4P2P111.8850 rad
∠P6P2P91.8850 rad
∠P6P2P111.0472 rad
∠P9P2P111.0472 rad
∠P0P4P21.0472 rad
∠P0P4P51.8850 rad
∠P0P4P81.0472 rad
∠P0P4P91.8850 rad
∠P2P4P51.8850 rad
∠P2P4P81.8850 rad
∠P2P4P91.0472 rad
∠P5P4P81.0472 rad
∠P5P4P91.0472 rad
∠P8P4P91.8850 rad
∠P0P6P21.0472 rad
∠P0P6P71.8850 rad
∠P0P6P101.0472 rad
∠P0P6P111.8850 rad
∠P2P6P71.8850 rad
∠P2P6P101.8850 rad
∠P2P6P111.0472 rad
∠P7P6P101.0472 rad
∠P7P6P111.0472 rad
∠P10P6P111.8850 rad
∠P0P8P11.8850 rad
∠P0P8P41.0472 rad
∠P0P8P51.8850 rad
∠P0P8P101.0472 rad
∠P1P8P41.8850 rad
∠P1P8P51.0472 rad
∠P1P8P101.0472 rad
∠P4P8P51.0472 rad
∠P4P8P101.8850 rad
∠P5P8P101.8850 rad
∠P0P10P11.8850 rad
∠P0P10P61.0472 rad
∠P0P10P71.8850 rad
∠P0P10P81.0472 rad
∠P1P10P61.8850 rad
∠P1P10P71.0472 rad
∠P1P10P81.0472 rad
∠P6P10P71.0472 rad
∠P6P10P81.8850 rad
∠P7P10P81.8850 rad
∠P3P1P51.0472 rad
∠P3P1P71.0472 rad
∠P3P1P81.8850 rad
∠P3P1P101.8850 rad
∠P5P1P71.8850 rad
∠P5P1P81.0472 rad
∠P5P1P101.8850 rad
∠P7P1P81.8850 rad
∠P7P1P101.0472 rad
∠P8P1P101.0472 rad
∠P1P3P51.0472 rad
∠P1P3P71.0472 rad
∠P1P3P91.8850 rad
∠P1P3P111.8850 rad
∠P5P3P71.8850 rad
∠P5P3P91.0472 rad
∠P5P3P111.8850 rad
∠P7P3P91.8850 rad
∠P7P3P111.0472 rad
∠P9P3P111.0472 rad
∠P1P5P31.0472 rad
∠P1P5P41.8850 rad
∠P1P5P81.0472 rad
∠P1P5P91.8850 rad
∠P3P5P41.8850 rad
∠P3P5P81.8850 rad
∠P3P5P91.0472 rad
∠P4P5P81.0472 rad
∠P4P5P91.0472 rad
∠P8P5P91.8850 rad
∠P1P7P31.0472 rad
∠P1P7P61.8850 rad
∠P1P7P101.0472 rad
∠P1P7P111.8850 rad
∠P3P7P61.8850 rad
∠P3P7P101.8850 rad
∠P3P7P111.0472 rad
∠P6P7P101.0472 rad
∠P6P7P111.0472 rad
∠P10P7P111.8850 rad
∠P2P9P31.8850 rad
∠P2P9P41.0472 rad
∠P2P9P51.8850 rad
∠P2P9P111.0472 rad
∠P3P9P41.8850 rad
∠P3P9P51.0472 rad
∠P3P9P111.0472 rad
∠P4P9P51.0472 rad
∠P4P9P111.8850 rad
∠P5P9P111.8850 rad
∠P2P11P31.8850 rad
∠P2P11P61.0472 rad
∠P2P11P71.8850 rad
∠P2P11P91.0472 rad
∠P3P11P61.8850 rad
∠P3P11P71.0472 rad
∠P3P11P91.0472 rad
∠P6P11P71.0472 rad
∠P6P11P91.8850 rad
∠P7P11P91.8850 rad
Derived ratios
r(C0)1.000000
∠P2P0P41.000000
|P0P2|1.051462
∠P2P0P81.800000
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