Spirals · 2D · v1.0.0
Logarithmic Spiral
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Mathematics
Geometric construction
r(θ) = R·e^{b(θ−θmax)} with b = 0.2. Quarter-turn points have constant ratio e^{bπ/2}.
Symmetry: Self-similar under scaling + rotation.
Culture
Symbolic interpretation
Bernoulli's spira mirabilis; seen in shells and galaxies.
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 | 0.0316 | 0.0000 |
| P2 | -0.0432 | 0.0000 | 0.0000 |
| P3 | -0.0000 | -0.0592 | 0.0000 |
| P4 | 0.0810 | -0.0000 | 0.0000 |
| P5 | 0.0000 | 0.1109 | 0.0000 |
| P6 | -0.1518 | 0.0000 | 0.0000 |
| P7 | -0.0000 | -0.2079 | 0.0000 |
| P8 | 0.2846 | -0.0000 | 0.0000 |
| P9 | 0.0000 | 0.3897 | 0.0000 |
| P10 | -0.5335 | 0.0000 | 0.0000 |
| P11 | -0.0000 | -0.7304 | 0.0000 |
| P12 | 1.0000 | -0.0000 | 0.0000 |
Distances & angles
|P1P2|0.0535 u
|P2P3|0.0733 u
|P3P4|0.1003 u
|P4P5|0.1373 u
|P5P6|0.1880 u
|P6P7|0.2574 u
|P7P8|0.3524 u
|P8P9|0.4825 u
|P9P10|0.6606 u
|P10P11|0.9045 u
|P11P12|1.2383 u
∠P1P2P31.5708 rad
∠P2P3P41.5708 rad
∠P3P4P51.5708 rad
∠P4P5P61.5708 rad
∠P5P6P71.5708 rad
∠P6P7P81.5708 rad
∠P7P8P91.5708 rad
∠P8P9P101.5708 rad
∠P9P10P111.5708 rad
∠P10P11P121.5708 rad
Derived ratios
|P1P2|1.000000
∠P3P4P51.000000
|P2P3|1.369108
|P3P4|1.874456
|P4P5|2.566332
|P5P6|3.513586
|P6P7|4.810477
|P7P8|6.586062
|P8P9|9.017029
|P9P10|12.345284
|P10P11|16.902024
|P11P12|23.140693
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