Polyhedra · 3D · v1.0.0
Dodecahedron
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Mathematics
Geometric construction
Vertices (±1,±1,±1) and cyclic permutations of (0, ±1/φ, ±φ), scaled to circumradius R.
Symmetry: Icosahedral Ih (order 120).
Culture
Symbolic interpretation
Plato associated it with the cosmos / aether.
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 | 0.5774 | 0.5774 | 0.5774 |
| P1 | 0.5774 | 0.5774 | -0.5774 |
| P2 | 0.5774 | -0.5774 | 0.5774 |
| P3 | 0.5774 | -0.5774 | -0.5774 |
| P4 | -0.5774 | 0.5774 | 0.5774 |
| P5 | -0.5774 | 0.5774 | -0.5774 |
| P6 | -0.5774 | -0.5774 | 0.5774 |
| P7 | -0.5774 | -0.5774 | -0.5774 |
| P8 | 0.0000 | 0.3568 | 0.9342 |
| P9 | 0.0000 | 0.3568 | -0.9342 |
| P10 | 0.0000 | -0.3568 | 0.9342 |
| P11 | 0.0000 | -0.3568 | -0.9342 |
| P12 | 0.9342 | 0.0000 | 0.3568 |
| P13 | 0.9342 | 0.0000 | -0.3568 |
| P14 | -0.9342 | 0.0000 | 0.3568 |
| P15 | -0.9342 | 0.0000 | -0.3568 |
| P16 | 0.3568 | 0.9342 | 0.0000 |
| P17 | 0.3568 | -0.9342 | 0.0000 |
| P18 | -0.3568 | 0.9342 | 0.0000 |
| P19 | -0.3568 | -0.9342 | 0.0000 |
| P20 | 0.0000 | 0.0000 | 0.0000 |
Distances & angles
|P0P8|0.7136 u
|P0P12|0.7136 u
|P0P16|0.7136 u
|P1P9|0.7136 u
|P1P13|0.7136 u
|P1P16|0.7136 u
|P2P10|0.7136 u
|P2P12|0.7136 u
|P2P17|0.7136 u
|P3P11|0.7136 u
|P3P13|0.7136 u
|P3P17|0.7136 u
|P4P8|0.7136 u
|P4P14|0.7136 u
|P4P18|0.7136 u
|P5P9|0.7136 u
|P5P15|0.7136 u
|P5P18|0.7136 u
|P6P10|0.7136 u
|P6P14|0.7136 u
|P6P19|0.7136 u
|P7P11|0.7136 u
|P7P15|0.7136 u
|P7P19|0.7136 u
|P8P10|0.7136 u
|P9P11|0.7136 u
|P12P13|0.7136 u
|P14P15|0.7136 u
|P16P18|0.7136 u
|P17P19|0.7136 u
r(C0)1.0000 u
∠P8P0P121.8850 rad
∠P8P0P161.8850 rad
∠P12P0P161.8850 rad
∠P0P8P41.8850 rad
∠P0P8P101.8850 rad
∠P4P8P101.8850 rad
∠P0P12P21.8850 rad
∠P0P12P131.8850 rad
∠P2P12P131.8850 rad
∠P0P16P11.8850 rad
∠P0P16P181.8850 rad
∠P1P16P181.8850 rad
∠P9P1P131.8850 rad
∠P9P1P161.8850 rad
∠P13P1P161.8850 rad
∠P1P9P51.8850 rad
∠P1P9P111.8850 rad
∠P5P9P111.8850 rad
∠P1P13P31.8850 rad
∠P1P13P121.8850 rad
∠P3P13P121.8850 rad
∠P10P2P121.8850 rad
∠P10P2P171.8850 rad
∠P12P2P171.8850 rad
∠P2P10P61.8850 rad
∠P2P10P81.8850 rad
∠P6P10P81.8850 rad
∠P2P17P31.8850 rad
∠P2P17P191.8850 rad
∠P3P17P191.8850 rad
∠P11P3P131.8850 rad
∠P11P3P171.8850 rad
∠P13P3P171.8850 rad
∠P3P11P71.8850 rad
∠P3P11P91.8850 rad
∠P7P11P91.8850 rad
∠P8P4P141.8850 rad
∠P8P4P181.8850 rad
∠P14P4P181.8850 rad
∠P4P14P61.8850 rad
∠P4P14P151.8850 rad
∠P6P14P151.8850 rad
∠P4P18P51.8850 rad
∠P4P18P161.8850 rad
∠P5P18P161.8850 rad
∠P9P5P151.8850 rad
∠P9P5P181.8850 rad
∠P15P5P181.8850 rad
∠P5P15P71.8850 rad
∠P5P15P141.8850 rad
∠P7P15P141.8850 rad
∠P10P6P141.8850 rad
∠P10P6P191.8850 rad
∠P14P6P191.8850 rad
∠P6P19P71.8850 rad
∠P6P19P171.8850 rad
∠P7P19P171.8850 rad
∠P11P7P151.8850 rad
∠P11P7P191.8850 rad
∠P15P7P191.8850 rad
Derived ratios
|P8P10|1.000000
∠P0P8P41.000000
r(C0)1.401259
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