Chakra · 2D · v1.0.0

Crown Chakra — Sahasrara

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Mathematics

Geometric construction

Central circle of radius R; 20 touching petal circles of radius R·sin(π/20) centred on it (an abstraction of the thousand-petalled lotus); an inner circle of radius R/2.

Symmetry: Dihedral D20.

Culture

Symbolic interpretation

Thousand-petalled lotus, here abstracted to twenty petals, associated with unity and pure awareness. Experimental chakra-inspired construction defined in this product; traditional chakra iconography varies and this is one reproducible reading of 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
P0	0.0000	0.0000
P1	0.0000	1.0000
P2	-0.3090	0.9511
P3	-0.5878	0.8090
P4	-0.8090	0.5878
P5	-0.9511	0.3090
P6	-1.0000	0.0000
P7	-0.9511	-0.3090
P8	-0.8090	-0.5878
P9	-0.5878	-0.8090
P10	-0.3090	-0.9511
P11	-0.0000	-1.0000
P12	0.3090	-0.9511
P13	0.5878	-0.8090
P14	0.8090	-0.5878
P15	0.9511	-0.3090
P16	1.0000	-0.0000
P17	0.9511	0.3090
P18	0.8090	0.5878
P19	0.5878	0.8090
P20	0.3090	0.9511

Distances & angles

r(C0)1.0000 u

r(C1)0.1564 u

r(C2)0.1564 u

r(C3)0.1564 u

r(C4)0.1564 u

r(C5)0.1564 u

r(C6)0.1564 u

r(C7)0.1564 u

r(C8)0.1564 u

r(C9)0.1564 u

r(C10)0.1564 u

r(C11)0.1564 u

r(C12)0.1564 u

r(C13)0.1564 u

r(C14)0.1564 u

r(C15)0.1564 u

r(C16)0.1564 u

r(C17)0.1564 u

r(C18)0.1564 u

r(C19)0.1564 u

r(C20)0.1564 u

r(C21)0.5000 u

|C0C1|1.0000 u

|C0C2|1.0000 u

|C0C3|1.0000 u

|C0C4|1.0000 u

|C0C5|1.0000 u

|C0C6|1.0000 u

|C0C7|1.0000 u

|C0C8|1.0000 u

|C0C9|1.0000 u

|C0C10|1.0000 u

|C0C11|1.0000 u

|C0C12|1.0000 u

|C0C13|1.0000 u

|C0C14|1.0000 u

|C0C15|1.0000 u

|C0C16|1.0000 u

|C0C17|1.0000 u

|C0C18|1.0000 u

|C0C19|1.0000 u

|C0C20|1.0000 u

|C1C2|0.3129 u

|C1C3|0.6180 u

|C1C4|0.9080 u

|C1C5|1.1756 u

|C1C6|1.4142 u

|C1C7|1.6180 u

|C1C8|1.7820 u

|C1C9|1.9021 u

|C1C10|1.9754 u

|C1C11|2.0000 u

|C1C12|1.9754 u

|C1C13|1.9021 u

|C1C14|1.7820 u

|C1C15|1.6180 u

|C1C16|1.4142 u

|C1C17|1.1756 u

|C1C18|0.9080 u

|C1C19|0.6180 u

|C1C20|0.3129 u

|C1C21|1.0000 u

|C2C3|0.3129 u

|C2C4|0.6180 u

|C2C5|0.9080 u

|C2C6|1.1756 u

|C2C7|1.4142 u

|C2C8|1.6180 u

|C2C9|1.7820 u

|C2C10|1.9021 u

|C2C11|1.9754 u

|C2C12|2.0000 u

|C2C13|1.9754 u

|C2C14|1.9021 u

|C2C15|1.7820 u

|C2C16|1.6180 u

|C2C17|1.4142 u

|C2C18|1.1756 u

|C2C19|0.9080 u

|C2C20|0.6180 u

|C2C21|1.0000 u

|C3C4|0.3129 u

|C3C5|0.6180 u

|C3C6|0.9080 u

|C3C7|1.1756 u

|C3C8|1.4142 u

|C3C9|1.6180 u

|C3C10|1.7820 u

|C3C11|1.9021 u

|C3C12|1.9754 u

|C3C13|2.0000 u

|C3C14|1.9754 u

|C3C15|1.9021 u

|C3C16|1.7820 u

|C3C17|1.6180 u

|C3C18|1.4142 u

|C3C19|1.1756 u

|C3C20|0.9080 u

|C3C21|1.0000 u

|C4C5|0.3129 u

|C4C6|0.6180 u

|C4C7|0.9080 u

|C4C8|1.1756 u

|C4C9|1.4142 u

|C4C10|1.6180 u

|C4C11|1.7820 u

|C4C12|1.9021 u

|C4C13|1.9754 u

|C4C14|2.0000 u

|C4C15|1.9754 u

|C4C16|1.9021 u

|C4C17|1.7820 u

|C4C18|1.6180 u

|C4C19|1.4142 u

|C4C20|1.1756 u

|C4C21|1.0000 u

|C5C6|0.3129 u

|C5C7|0.6180 u

|C5C8|0.9080 u

|C5C9|1.1756 u

|C5C10|1.4142 u

|C5C11|1.6180 u

|C5C12|1.7820 u

|C5C13|1.9021 u

|C5C14|1.9754 u

|C5C15|2.0000 u

|C5C16|1.9754 u

|C5C17|1.9021 u

|C5C18|1.7820 u

|C5C19|1.6180 u

|C5C20|1.4142 u

|C5C21|1.0000 u

|C6C7|0.3129 u

|C6C8|0.6180 u

|C6C9|0.9080 u

|C6C10|1.1756 u

|C6C11|1.4142 u

|C6C12|1.6180 u

|C6C13|1.7820 u

|C6C14|1.9021 u

|C6C15|1.9754 u

|C6C16|2.0000 u

|C6C17|1.9754 u

|C6C18|1.9021 u

|C6C19|1.7820 u

|C6C20|1.6180 u

|C6C21|1.0000 u

|C7C8|0.3129 u

|C7C9|0.6180 u

|C7C10|0.9080 u

|C7C11|1.1756 u

|C7C12|1.4142 u

|C7C13|1.6180 u

|C7C14|1.7820 u

|C7C15|1.9021 u

|C7C16|1.9754 u

|C7C17|2.0000 u

|C7C18|1.9754 u

|C7C19|1.9021 u

|C7C20|1.7820 u

|C7C21|1.0000 u

|C8C9|0.3129 u

|C8C10|0.6180 u

|C8C11|0.9080 u

|C8C12|1.1756 u

|C8C13|1.4142 u

|C8C14|1.6180 u

|C8C15|1.7820 u

|C8C16|1.9021 u

|C8C17|1.9754 u

|C8C18|2.0000 u

|C8C19|1.9754 u

|C8C20|1.9021 u

|C8C21|1.0000 u

|C9C10|0.3129 u

|C9C11|0.6180 u

|C9C12|0.9080 u

|C9C13|1.1756 u

|C9C14|1.4142 u

|C9C15|1.6180 u

|C9C16|1.7820 u

|C9C17|1.9021 u

|C9C18|1.9754 u

|C9C19|2.0000 u

|C9C20|1.9754 u

|C9C21|1.0000 u

|C10C11|0.3129 u

|C10C12|0.6180 u

|C10C13|0.9080 u

|C10C14|1.1756 u

|C10C15|1.4142 u

|C10C16|1.6180 u

|C10C17|1.7820 u

|C10C18|1.9021 u

|C10C19|1.9754 u

|C10C20|2.0000 u

|C10C21|1.0000 u

|C11C12|0.3129 u

|C11C13|0.6180 u

|C11C14|0.9080 u

Derived ratios

r(C1)1.000000

|C3C4|2.000000

r(C21)3.196227

|C1C3|3.950753

|C9C12|5.804226

|C0C3|6.392453

|C3C7|7.514779

|C1C6|9.040294

|C1C7|10.343207

|C1C8|11.391435

|C1C13|12.159169

|C2C13|12.627503

|C3C13|12.784906

f_i = f_0 × r_i · v1.0.0 · user-configured

Frequency derivation — centre distances

Reference frequency f₀ (Hz)

id	r = d / R	exact (reference)	sources	f = f₀ × r
R1	0.312868930	—	\|P1P2\| \|P1P20\| \|P2P3\| …	220 × 0.312869 = 68.831 Hz
R2	0.618033989	φ^-1	\|P1P3\| \|P1P19\| \|P2P4\| …	220 × φ^-1 = 135.967 Hz
R3	0.907980999	—	\|P1P4\| \|P1P18\| \|P2P5\| …	220 × 0.907981 = 199.756 Hz
R4	1.000000000	1	\|P0P1\| \|P0P2\| \|P0P3\| …	220 × 1 = 220.000 Hz
R5	1.175570505	—	\|P1P5\| \|P1P17\| \|P2P6\| …	220 × 1.175571 = 258.626 Hz
R6	1.414213562	√2	\|P1P6\| \|P1P16\| \|P2P7\| …	220 × √2 = 311.127 Hz
R7	1.618033989	φ	\|P1P7\| \|P1P15\| \|P2P8\| …	220 × φ = 355.967 Hz
R8	1.782013048	—	\|P1P8\| \|P1P14\| \|P2P9\| …	220 × 1.782013 = 392.043 Hz
R9	1.902113033	—	\|P1P9\| \|P1P13\| \|P2P10\| …	220 × 1.902113 = 418.465 Hz
R10	1.975376681	—	\|P1P10\| \|P1P12\| \|P2P11\| …	220 × 1.975377 = 434.583 Hz
R11	2.000000000	2	\|P1P11\| \|P2P12\| \|P3P13\| …	220 × 2 = 440.000 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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