Radial · 2D · v1.0.0
Eight-Fold Radial
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Mathematics
Geometric construction
8 points on a circle of radius R joined as the star polygon {8/step}, with spokes to the centre and the star's inner circle.
Symmetry: Dihedral D8.
Culture
Symbolic interpretation
Eight-fold stars appear in Islamic and Buddhist art.
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 | 0.0000 | 0.0000 |
| P1 | 0.0000 | 1.0000 | 0.0000 |
| P2 | -0.7071 | 0.7071 | 0.0000 |
| P3 | -1.0000 | 0.0000 | 0.0000 |
| P4 | -0.7071 | -0.7071 | 0.0000 |
| P5 | -0.0000 | -1.0000 | 0.0000 |
| P6 | 0.7071 | -0.7071 | 0.0000 |
| P7 | 1.0000 | -0.0000 | 0.0000 |
| P8 | 0.7071 | 0.7071 | 0.0000 |
Distances & angles
|P1P4|1.8478 u
|P0P1|1.0000 u
|P2P5|1.8478 u
|P0P2|1.0000 u
|P3P6|1.8478 u
|P0P3|1.0000 u
|P4P7|1.8478 u
|P0P4|1.0000 u
|P5P8|1.8478 u
|P0P5|1.0000 u
|P6P1|1.8478 u
|P0P6|1.0000 u
|P7P2|1.8478 u
|P0P7|1.0000 u
|P8P3|1.8478 u
|P0P8|1.0000 u
r(C0)1.0000 u
r(C1)0.3827 u
∠P4P1P00.3927 rad
∠P4P1P60.7854 rad
∠P0P1P60.3927 rad
∠P1P4P70.7854 rad
∠P1P4P00.3927 rad
∠P7P4P00.3927 rad
∠P1P0P20.7854 rad
∠P1P0P31.5708 rad
∠P1P0P42.3562 rad
∠P1P0P53.1416 rad
∠P1P0P62.3562 rad
∠P1P0P71.5708 rad
∠P1P0P80.7854 rad
∠P2P0P30.7854 rad
∠P2P0P41.5708 rad
∠P2P0P52.3562 rad
∠P2P0P63.1416 rad
∠P2P0P72.3562 rad
∠P2P0P81.5708 rad
∠P3P0P40.7854 rad
∠P3P0P51.5708 rad
∠P3P0P62.3562 rad
∠P3P0P73.1416 rad
∠P3P0P82.3562 rad
∠P4P0P50.7854 rad
∠P4P0P61.5708 rad
∠P4P0P72.3562 rad
∠P4P0P83.1416 rad
∠P5P0P60.7854 rad
∠P5P0P71.5708 rad
∠P5P0P82.3562 rad
∠P6P0P70.7854 rad
∠P6P0P81.5708 rad
∠P7P0P80.7854 rad
∠P5P2P00.3927 rad
∠P5P2P70.7854 rad
∠P0P2P70.3927 rad
∠P2P5P80.7854 rad
∠P2P5P00.3927 rad
∠P8P5P00.3927 rad
∠P6P3P00.3927 rad
∠P6P3P80.7854 rad
∠P0P3P80.3927 rad
∠P3P6P10.7854 rad
∠P3P6P00.3927 rad
∠P1P6P00.3927 rad
∠P4P7P20.7854 rad
∠P4P7P00.3927 rad
∠P2P7P00.3927 rad
∠P5P8P30.7854 rad
∠P5P8P00.3927 rad
∠P3P8P00.3927 rad
Derived ratios
r(C1)1.000000
∠P0P1P61.000000
∠P4P1P62.000000
|P0P1|2.613126
∠P1P0P34.000000
|P2P5|4.828427
∠P1P0P46.000000
∠P4P0P88.000000
f_i = f_0 × r_i · v1.0.0 · user-configured
Frequency derivation — centre distances
| id | r = d / R | exact (reference) | sources | f = f₀ × r |
|---|
Click a row to highlight its source points in the drawing. R = 1.0000 u. Exact forms are recognised reference values; the calculation itself is numeric. This mapping is a chosen design rule, not a law of nature.
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