Circle-based · 2D · v1.0.0
Flower of Life
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Mathematics
Geometric construction
19 circles of radius R centred on a hexagonal lattice of spacing R (all lattice points within two rings of the origin), enclosed by a bounding circle of radius 3R.
Symmetry: Dihedral D6.
Culture
Symbolic interpretation
Found as decoration in many cultures; modern esoteric writers attribute creation symbolism to it.
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 | -2.0000 | 0.0000 | 0.0000 |
| P1 | -1.5000 | 0.8660 | 0.0000 |
| P2 | -1.0000 | 1.7321 | 0.0000 |
| P3 | -1.5000 | -0.8660 | 0.0000 |
| P4 | -1.0000 | 0.0000 | 0.0000 |
| P5 | -0.5000 | 0.8660 | 0.0000 |
| P6 | 0.0000 | 1.7321 | 0.0000 |
| P7 | -1.0000 | -1.7321 | 0.0000 |
| P8 | -0.5000 | -0.8660 | 0.0000 |
| P9 | 0.0000 | 0.0000 | 0.0000 |
| P10 | 0.5000 | 0.8660 | 0.0000 |
| P11 | 1.0000 | 1.7321 | 0.0000 |
| P12 | 0.0000 | -1.7321 | 0.0000 |
| P13 | 0.5000 | -0.8660 | 0.0000 |
| P14 | 1.0000 | 0.0000 | 0.0000 |
| P15 | 1.5000 | 0.8660 | 0.0000 |
| P16 | 1.0000 | -1.7321 | 0.0000 |
| P17 | 1.5000 | -0.8660 | 0.0000 |
| P18 | 2.0000 | 0.0000 | 0.0000 |
Distances & angles
r(C0)1.0000 u
r(C1)1.0000 u
r(C2)1.0000 u
r(C3)1.0000 u
r(C4)1.0000 u
r(C5)1.0000 u
r(C6)1.0000 u
r(C7)1.0000 u
r(C8)1.0000 u
r(C9)1.0000 u
r(C10)1.0000 u
r(C11)1.0000 u
r(C12)1.0000 u
r(C13)1.0000 u
r(C14)1.0000 u
r(C15)1.0000 u
r(C16)1.0000 u
r(C17)1.0000 u
r(C18)1.0000 u
r(C19)3.0000 u
|C0C1|1.0000 u
|C0C2|2.0000 u
|C0C3|1.0000 u
|C0C4|1.0000 u
|C0C5|1.7321 u
|C0C6|2.6458 u
|C0C7|2.0000 u
|C0C8|1.7321 u
|C0C9|2.0000 u
|C0C10|2.6458 u
|C0C11|3.4641 u
|C0C12|2.6458 u
|C0C13|2.6458 u
|C0C14|3.0000 u
|C0C15|3.6056 u
|C0C16|3.4641 u
|C0C17|3.6056 u
|C0C18|4.0000 u
|C0C19|2.0000 u
|C1C2|1.0000 u
|C1C3|1.7321 u
|C1C4|1.0000 u
|C1C5|1.0000 u
|C1C6|1.7321 u
|C1C7|2.6458 u
|C1C8|2.0000 u
|C1C9|1.7321 u
|C1C10|2.0000 u
|C1C11|2.6458 u
|C1C12|3.0000 u
|C1C13|2.6458 u
|C1C14|2.6458 u
|C1C15|3.0000 u
|C1C16|3.6056 u
|C1C17|3.4641 u
|C1C18|3.6056 u
|C1C19|1.7321 u
|C2C3|2.6458 u
|C2C4|1.7321 u
|C2C5|1.0000 u
|C2C6|1.0000 u
|C2C7|3.4641 u
|C2C8|2.6458 u
|C2C9|2.0000 u
|C2C10|1.7321 u
|C2C11|2.0000 u
|C2C12|3.6056 u
|C2C13|3.0000 u
|C2C14|2.6458 u
|C2C15|2.6458 u
|C2C16|4.0000 u
|C2C17|3.6056 u
|C2C18|3.4641 u
|C2C19|2.0000 u
|C3C4|1.0000 u
|C3C5|2.0000 u
|C3C6|3.0000 u
|C3C7|1.0000 u
|C3C8|1.0000 u
|C3C9|1.7321 u
|C3C10|2.6458 u
|C3C11|3.6056 u
|C3C12|1.7321 u
|C3C13|2.0000 u
|C3C14|2.6458 u
|C3C15|3.4641 u
|C3C16|2.6458 u
|C3C17|3.0000 u
|C3C18|3.6056 u
|C3C19|1.7321 u
|C4C5|1.0000 u
|C4C6|2.0000 u
|C4C7|1.7321 u
|C4C8|1.0000 u
|C4C9|1.0000 u
|C4C10|1.7321 u
|C4C11|2.6458 u
|C4C12|2.0000 u
|C4C13|1.7321 u
|C4C14|2.0000 u
|C4C15|2.6458 u
|C4C16|2.6458 u
|C4C17|2.6458 u
|C4C18|3.0000 u
|C4C19|1.0000 u
|C5C6|1.0000 u
|C5C7|2.6458 u
|C5C8|1.7321 u
|C5C9|1.0000 u
|C5C10|1.0000 u
|C5C11|1.7321 u
|C5C12|2.6458 u
|C5C13|2.0000 u
|C5C14|1.7321 u
|C5C15|2.0000 u
|C5C16|3.0000 u
|C5C17|2.6458 u
|C5C18|2.6458 u
|C5C19|1.0000 u
|C6C7|3.6056 u
|C6C8|2.6458 u
|C6C9|1.7321 u
|C6C10|1.0000 u
|C6C11|1.0000 u
|C6C12|3.4641 u
|C6C13|2.6458 u
|C6C14|2.0000 u
|C6C15|1.7321 u
|C6C16|3.6056 u
|C6C17|3.0000 u
|C6C18|2.6458 u
|C6C19|1.7321 u
|C7C8|1.0000 u
|C7C9|2.0000 u
|C7C10|3.0000 u
|C7C11|4.0000 u
|C7C12|1.0000 u
|C7C13|1.7321 u
|C7C14|2.6458 u
|C7C15|3.6056 u
|C7C16|2.0000 u
|C7C17|2.6458 u
|C7C18|3.4641 u
|C7C19|2.0000 u
|C8C9|1.0000 u
|C8C10|2.0000 u
|C8C11|3.0000 u
|C8C12|1.0000 u
|C8C13|1.0000 u
|C8C14|1.7321 u
|C8C15|2.6458 u
|C8C16|1.7321 u
|C8C17|2.0000 u
|C8C18|2.6458 u
|C8C19|1.0000 u
|C9C10|1.0000 u
|C9C11|2.0000 u
|C9C12|1.7321 u
|C9C13|1.0000 u
|C9C14|1.0000 u
|C9C15|1.7321 u
|C9C16|2.0000 u
|C9C17|1.7321 u
|C9C18|2.0000 u
|C10C11|1.0000 u
|C10C12|2.6458 u
|C10C13|1.7321 u
|C10C14|1.0000 u
|C10C15|1.0000 u
|C10C16|2.6458 u
|C10C17|2.0000 u
|C10C18|1.7321 u
|C10C19|1.0000 u
|C11C12|3.6056 u
|C11C13|2.6458 u
|C11C14|1.7321 u
|C11C15|1.0000 u
|C11C16|3.4641 u
|C11C17|2.6458 u
|C11C18|2.0000 u
|C11C19|2.0000 u
|C12C13|1.0000 u
|C12C14|2.0000 u
|C12C15|3.0000 u
|C12C16|1.0000 u
|C12C17|1.7321 u
|C12C18|2.6458 u
|C12C19|1.7321 u
|C13C14|1.0000 u
|C13C15|2.0000 u
|C13C16|1.0000 u
|C13C17|1.0000 u
|C13C18|1.7321 u
|C13C19|1.0000 u
|C14C15|1.0000 u
|C14C16|1.7321 u
|C14C17|1.0000 u
|C14C18|1.0000 u
|C14C19|1.0000 u
|C15C16|2.6458 u
Derived ratios
|C0C1|1.000000
|C0C5|1.732051
|C0C2|2.000000
|C0C6|2.645751
r(C19)3.000000
|C0C11|3.464102
|C0C15|3.605551
|C2C16|4.000000
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 |
|---|---|---|---|---|
| R1 | 1.000000000 | 1 | |P0P1| |P0P3| |P0P4| … | 220 × 1 = 220.000 Hz |
| R2 | 1.732050808 | √3 | |P0P5| |P0P8| |P1P3| … | 220 × √3 = 381.051 Hz |
| R3 | 2.000000000 | 2 | |P0P2| |P0P7| |P0P9| … | 220 × 2 = 440.000 Hz |
| R4 | 2.645751311 | √7 | |P0P6| |P0P10| |P0P12| … | 220 × √7 = 582.065 Hz |
| R5 | 3.000000000 | 3 | |P0P14| |P1P12| |P1P15| … | 220 × 3 = 660.000 Hz |
| R6 | 3.464101615 | 2√3 | |P0P11| |P0P16| |P1P17| … | 220 × 2√3 = 762.102 Hz |
| R7 | 3.605551275 | √13 | |P0P15| |P0P17| |P1P16| … | 220 × √13 = 793.221 Hz |
| R8 | 4.000000000 | 4 | |P0P18| |P2P16| |P7P11| | 220 × 4 = 880.000 Hz |
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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