Radial · 2D · v1.0.0
Radial Mandala
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Mathematics
Geometric construction
m concentric rings of n points (radius R·k/m), alternate rings rotated by π/n, joined to neighbours on adjacent rings.
Symmetry: Dihedral Dn.
Culture
Symbolic interpretation
Mandala patterns are central to Hindu and Buddhist practice; this is a geometric abstraction.
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 | -0.0957 | 0.2310 | 0.0000 |
| P2 | -0.2310 | 0.0957 | 0.0000 |
| P3 | -0.2310 | -0.0957 | 0.0000 |
| P4 | -0.0957 | -0.2310 | 0.0000 |
| P5 | 0.0957 | -0.2310 | 0.0000 |
| P6 | 0.2310 | -0.0957 | 0.0000 |
| P7 | 0.2310 | 0.0957 | 0.0000 |
| P8 | 0.0957 | 0.2310 | 0.0000 |
| P9 | 0.0000 | 0.5000 | 0.0000 |
| P10 | -0.3536 | 0.3536 | 0.0000 |
| P11 | -0.5000 | 0.0000 | 0.0000 |
| P12 | -0.3536 | -0.3536 | 0.0000 |
| P13 | -0.0000 | -0.5000 | 0.0000 |
| P14 | 0.3536 | -0.3536 | 0.0000 |
| P15 | 0.5000 | -0.0000 | 0.0000 |
| P16 | 0.3536 | 0.3536 | 0.0000 |
| P17 | -0.2870 | 0.6929 | 0.0000 |
| P18 | -0.6929 | 0.2870 | 0.0000 |
| P19 | -0.6929 | -0.2870 | 0.0000 |
| P20 | -0.2870 | -0.6929 | 0.0000 |
| P21 | 0.2870 | -0.6929 | 0.0000 |
| P22 | 0.6929 | -0.2870 | 0.0000 |
| P23 | 0.6929 | 0.2870 | 0.0000 |
| P24 | 0.2870 | 0.6929 | 0.0000 |
| P25 | 0.0000 | 1.0000 | 0.0000 |
| P26 | -0.7071 | 0.7071 | 0.0000 |
| P27 | -1.0000 | 0.0000 | 0.0000 |
| P28 | -0.7071 | -0.7071 | 0.0000 |
| P29 | -0.0000 | -1.0000 | 0.0000 |
| P30 | 0.7071 | -0.7071 | 0.0000 |
| P31 | 1.0000 | -0.0000 | 0.0000 |
| P32 | 0.7071 | 0.7071 | 0.0000 |
Distances & angles
|P1P2|0.1913 u
|P2P3|0.1913 u
|P3P4|0.1913 u
|P4P5|0.1913 u
|P5P6|0.1913 u
|P6P7|0.1913 u
|P7P8|0.1913 u
|P8P1|0.1913 u
|P0P1|0.2500 u
|P0P2|0.2500 u
|P0P3|0.2500 u
|P0P4|0.2500 u
|P0P5|0.2500 u
|P0P6|0.2500 u
|P0P7|0.2500 u
|P0P8|0.2500 u
|P9P10|0.3827 u
|P10P11|0.3827 u
|P11P12|0.3827 u
|P12P13|0.3827 u
|P13P14|0.3827 u
|P14P15|0.3827 u
|P15P16|0.3827 u
|P16P9|0.3827 u
|P9P1|0.2855 u
|P9P8|0.2855 u
|P10P2|0.2855 u
|P10P1|0.2855 u
|P11P3|0.2855 u
|P11P2|0.2855 u
|P12P4|0.2855 u
|P12P3|0.2855 u
|P13P5|0.2855 u
|P13P4|0.2855 u
|P14P6|0.2855 u
|P14P5|0.2855 u
|P15P7|0.2855 u
|P15P6|0.2855 u
|P16P8|0.2855 u
|P16P7|0.2855 u
|P17P18|0.5740 u
|P18P19|0.5740 u
|P19P20|0.5740 u
|P20P21|0.5740 u
|P21P22|0.5740 u
|P22P23|0.5740 u
|P23P24|0.5740 u
|P24P17|0.5740 u
|P17P9|0.3458 u
|P17P10|0.3458 u
|P18P10|0.3458 u
|P18P11|0.3458 u
|P19P11|0.3458 u
|P19P12|0.3458 u
|P20P12|0.3458 u
|P20P13|0.3458 u
|P21P13|0.3458 u
|P21P14|0.3458 u
|P22P14|0.3458 u
|P22P15|0.3458 u
|P23P15|0.3458 u
|P23P16|0.3458 u
|P24P16|0.3458 u
|P24P9|0.3458 u
|P25P26|0.7654 u
|P26P27|0.7654 u
|P27P28|0.7654 u
|P28P29|0.7654 u
|P29P30|0.7654 u
|P30P31|0.7654 u
|P31P32|0.7654 u
|P32P25|0.7654 u
|P25P17|0.4203 u
|P25P24|0.4203 u
|P26P18|0.4203 u
|P26P17|0.4203 u
|P27P19|0.4203 u
|P27P18|0.4203 u
|P28P20|0.4203 u
|P28P19|0.4203 u
|P29P21|0.4203 u
|P29P20|0.4203 u
|P30P22|0.4203 u
|P30P21|0.4203 u
|P31P23|0.4203 u
|P31P22|0.4203 u
|P32P24|0.4203 u
|P32P23|0.4203 u
r(C0)0.2500 u
r(C1)0.5000 u
r(C2)0.7500 u
r(C3)1.0000 u
∠P2P1P82.3562 rad
∠P2P1P01.1781 rad
∠P2P1P92.6979 rad
∠P2P1P101.2291 rad
∠P8P1P01.1781 rad
∠P8P1P91.2291 rad
∠P8P1P102.6979 rad
∠P0P1P92.4072 rad
∠P0P1P102.4072 rad
∠P9P1P101.4687 rad
∠P1P2P32.3562 rad
∠P1P2P01.1781 rad
∠P1P2P101.2291 rad
∠P1P2P112.6979 rad
∠P3P2P01.1781 rad
∠P3P2P102.6979 rad
∠P3P2P111.2291 rad
∠P0P2P102.4072 rad
∠P0P2P112.4072 rad
∠P10P2P111.4687 rad
∠P2P3P42.3562 rad
∠P2P3P01.1781 rad
∠P2P3P111.2291 rad
∠P2P3P122.6979 rad
∠P4P3P01.1781 rad
∠P4P3P112.6979 rad
∠P4P3P121.2291 rad
∠P0P3P112.4072 rad
∠P0P3P122.4072 rad
∠P11P3P121.4687 rad
∠P3P4P52.3562 rad
∠P3P4P01.1781 rad
∠P3P4P121.2291 rad
∠P3P4P132.6979 rad
∠P5P4P01.1781 rad
∠P5P4P122.6979 rad
∠P5P4P131.2291 rad
∠P0P4P122.4072 rad
∠P0P4P132.4072 rad
∠P12P4P131.4687 rad
∠P4P5P62.3562 rad
∠P4P5P01.1781 rad
∠P4P5P131.2291 rad
∠P4P5P142.6979 rad
∠P6P5P01.1781 rad
∠P6P5P132.6979 rad
∠P6P5P141.2291 rad
∠P0P5P132.4072 rad
∠P0P5P142.4072 rad
∠P13P5P141.4687 rad
∠P5P6P72.3562 rad
∠P5P6P01.1781 rad
∠P5P6P141.2291 rad
∠P5P6P152.6979 rad
∠P7P6P01.1781 rad
∠P7P6P142.6979 rad
∠P7P6P151.2291 rad
∠P0P6P142.4072 rad
∠P0P6P152.4072 rad
∠P14P6P151.4687 rad
∠P6P7P82.3562 rad
∠P6P7P01.1781 rad
∠P6P7P151.2291 rad
∠P6P7P162.6979 rad
∠P8P7P01.1781 rad
∠P8P7P152.6979 rad
∠P8P7P161.2291 rad
∠P0P7P152.4072 rad
∠P0P7P162.4072 rad
∠P15P7P161.4687 rad
∠P7P8P12.3562 rad
∠P7P8P01.1781 rad
∠P7P8P92.6979 rad
∠P7P8P161.2291 rad
∠P1P8P01.1781 rad
∠P1P8P91.2291 rad
∠P1P8P162.6979 rad
∠P0P8P92.4072 rad
∠P0P8P162.4072 rad
∠P9P8P161.4687 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
Derived ratios
|P5P6|1.000000
∠P32P25P241.000000
|P0P1|1.306563
∠P17P24P91.387576
|P14P6|1.492277
∠P6P14P51.602262
|P20P12|1.807334
∠P5P0P61.841577
∠P17P24P251.920788
∠P16P9P81.961235
|P10P11|2.000000
|P32P24|2.196772
∠P16P9P242.308364
r(C1)2.613126
∠P11P19P122.749579
∠P5P4P02.762365
∠P1P8P92.882023
|P21P22|3.000000
∠P9P24P253.308364
∠P12P4P133.443838
∠P20P28P193.524731
∠P11P12P43.563496
∠P2P0P43.683154
r(C2)3.919689
|P26P27|4.000000
∠P21P20P124.137155
∠P8P9P244.269599
∠P27P28P204.524731
∠P21P14P224.591156
r(C3)5.226252
∠P25P17P265.366307
∠P5P0P85.524731
∠P0P4P135.644388
∠P1P9P245.871860
∠P16P24P256.057943
∠P2P1P96.325861
∠P15P14P216.899520
∠P17P18P277.287096
∠P1P0P57.366307
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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