Polyhedra · 3D · v1.0.0
Merkaba (Stella Octangula)
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Mathematics
Geometric construction
Two regular tetrahedra: T₁ on alternate cube vertices (1,1,1),(1,−1,−1),(−1,1,−1),(−1,−1,1) and T₂ = −T₁. Edges are drawn within each tetrahedron only. Circumradius R.
Symmetry: Octahedral Oh (compound).
Culture
Symbolic interpretation
The term derives from Hebrew merkavah (chariot); its 'light-body' meanings come from modern esoteric writing.
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 | 0.0000 | 0.0000 |
Distances & angles
|P0P1|1.6330 u
|P0P2|1.6330 u
|P0P3|1.6330 u
|P1P2|1.6330 u
|P1P3|1.6330 u
|P2P3|1.6330 u
|P4P5|1.6330 u
|P4P6|1.6330 u
|P4P7|1.6330 u
|P5P6|1.6330 u
|P5P7|1.6330 u
|P6P7|1.6330 u
r(C0)1.0000 u
∠P1P0P21.0472 rad
∠P1P0P31.0472 rad
∠P2P0P31.0472 rad
∠P0P1P21.0472 rad
∠P0P1P31.0472 rad
∠P2P1P31.0472 rad
∠P0P2P11.0472 rad
∠P0P2P31.0472 rad
∠P1P2P31.0472 rad
∠P0P3P11.0472 rad
∠P0P3P21.0472 rad
∠P1P3P21.0472 rad
∠P5P4P61.0472 rad
∠P5P4P71.0472 rad
∠P6P4P71.0472 rad
∠P4P5P61.0472 rad
∠P4P5P71.0472 rad
∠P6P5P71.0472 rad
∠P4P6P51.0472 rad
∠P4P6P71.0472 rad
∠P5P6P71.0472 rad
∠P4P7P51.0472 rad
∠P4P7P61.0472 rad
∠P5P7P61.0472 rad
Derived ratios
r(C0)1.000000
∠P1P0P21.000000
|P0P1|1.632993
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