Step 2 of 4
Explore the Structure
Look at the construction, its points and its distances. Ratios between distances are what become frequencies.
Scroll or pinch to zoom, drag to move.
Geometry structure
The complete construction, whatever is shown in the drawing.
13
Points
shown
0
Circles
none in this shape
11
Lines
shown
Measurements & ratios 23
Each length divided by the reference length gives a ratio r. These ratios become frequencies in the next step.
| Measurement | Length | Ratio r |
|---|---|---|
| ρ(P1) | 0.0316 | 1.0000 |
| ρ(P2) | 0.0432 | 1.3691 |
| |P1P2| | 0.0535 | 1.6954 |
| ρ(P3) | 0.0592 | 1.8745 |
| |P2P3| | 0.0733 | 2.3212 |
| ρ(P4) | 0.0810 | 2.5663 |
| |P3P4| | 0.1003 | 3.1780 |
| ρ(P5) | 0.1109 | 3.5136 |
| |P4P5| | 0.1373 | 4.3510 |
| ρ(P6) | 0.1518 | 4.8105 |
| |P5P6| | 0.1880 | 5.9570 |
| ρ(P7) | 0.2079 | 6.5861 |
| |P6P7| | 0.2574 | 8.1558 |
| ρ(P8) | 0.2846 | 9.0170 |
| |P7P8| | 0.3524 | 11.1662 |
| ρ(P9) | 0.3897 | 12.3453 |
| |P8P9| | 0.4825 | 15.2877 |
| ρ(P10) | 0.5335 | 16.9020 |
| |P9P10| | 0.6606 | 20.9305 |
| ρ(P11) | 0.7304 | 23.1407 |
| |P10P11| | 0.9045 | 28.6561 |
| ρ(P12) | 1.0000 | 31.6821 |
| |P11P12| | 1.2383 | 39.2332 |
Mathematical details
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.
Symbolic interpretation
Bernoulli's spira mirabilis; seen in shells and galaxies.