Circle-based · 2D · v1.0.0
Metatron's Cube
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Mathematics
Geometric construction
The 13 Fruit of Life centres (origin, six at 2R, six at 4R) with all 78 straight segments connecting every pair of centres, plus the 13 circles of radius R.
Symmetry: Dihedral D6.
Culture
Symbolic interpretation
Named after the archangel Metatron in later esoteric tradition; said to contain projections of the Platonic solids.
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 | 1.0000 | 0.0000 |
| P2 | -0.8660 | 0.5000 | 0.0000 |
| P3 | -0.8660 | -0.5000 | 0.0000 |
| P4 | -0.0000 | -1.0000 | 0.0000 |
| P5 | 0.8660 | -0.5000 | 0.0000 |
| P6 | 0.8660 | 0.5000 | 0.0000 |
| P7 | 0.0000 | 2.0000 | 0.0000 |
| P8 | -1.7321 | 1.0000 | 0.0000 |
| P9 | -1.7321 | -1.0000 | 0.0000 |
| P10 | -0.0000 | -2.0000 | 0.0000 |
| P11 | 1.7321 | -1.0000 | 0.0000 |
| P12 | 1.7321 | 1.0000 | 0.0000 |
Distances & angles
|P0P1|1.0000 u
|P0P2|1.0000 u
|P0P3|1.0000 u
|P0P4|1.0000 u
|P0P5|1.0000 u
|P0P6|1.0000 u
|P0P7|2.0000 u
|P0P8|2.0000 u
|P0P9|2.0000 u
|P0P10|2.0000 u
|P0P11|2.0000 u
|P0P12|2.0000 u
|P1P2|1.0000 u
|P1P3|1.7321 u
|P1P4|2.0000 u
|P1P5|1.7321 u
|P1P6|1.0000 u
|P1P7|1.0000 u
|P1P8|1.7321 u
|P1P9|2.6458 u
|P1P10|3.0000 u
|P1P11|2.6458 u
|P1P12|1.7321 u
|P2P3|1.0000 u
|P2P4|1.7321 u
|P2P5|2.0000 u
|P2P6|1.7321 u
|P2P7|1.7321 u
|P2P8|1.0000 u
|P2P9|1.7321 u
|P2P10|2.6458 u
|P2P11|3.0000 u
|P2P12|2.6458 u
|P3P4|1.0000 u
|P3P5|1.7321 u
|P3P6|2.0000 u
|P3P7|2.6458 u
|P3P8|1.7321 u
|P3P9|1.0000 u
|P3P10|1.7321 u
|P3P11|2.6458 u
|P3P12|3.0000 u
|P4P5|1.0000 u
|P4P6|1.7321 u
|P4P7|3.0000 u
|P4P8|2.6458 u
|P4P9|1.7321 u
|P4P10|1.0000 u
|P4P11|1.7321 u
|P4P12|2.6458 u
|P5P6|1.0000 u
|P5P7|2.6458 u
|P5P8|3.0000 u
|P5P9|2.6458 u
|P5P10|1.7321 u
|P5P11|1.0000 u
|P5P12|1.7321 u
|P6P7|1.7321 u
|P6P8|2.6458 u
|P6P9|3.0000 u
|P6P10|2.6458 u
|P6P11|1.7321 u
|P6P12|1.0000 u
|P7P8|2.0000 u
|P7P9|3.4641 u
|P7P10|4.0000 u
|P7P11|3.4641 u
|P7P12|2.0000 u
|P8P9|2.0000 u
|P8P10|3.4641 u
|P8P11|4.0000 u
|P8P12|3.4641 u
|P9P10|2.0000 u
|P9P11|3.4641 u
|P9P12|4.0000 u
|P10P11|2.0000 u
|P10P12|3.4641 u
|P11P12|2.0000 u
r(C0)0.5000 u
r(C1)0.5000 u
r(C2)0.5000 u
r(C3)0.5000 u
r(C4)0.5000 u
r(C5)0.5000 u
r(C6)0.5000 u
r(C7)0.5000 u
r(C8)0.5000 u
r(C9)0.5000 u
r(C10)0.5000 u
r(C11)0.5000 u
r(C12)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|2.0000 u
|C0C8|2.0000 u
|C0C9|2.0000 u
|C0C10|2.0000 u
|C0C11|2.0000 u
|C0C12|2.0000 u
|C1C2|1.0000 u
|C1C3|1.7321 u
|C1C4|2.0000 u
|C1C5|1.7321 u
|C1C6|1.0000 u
|C1C7|1.0000 u
|C1C8|1.7321 u
|C1C9|2.6458 u
|C1C10|3.0000 u
|C1C11|2.6458 u
|C1C12|1.7321 u
|C2C3|1.0000 u
|C2C4|1.7321 u
|C2C5|2.0000 u
|C2C6|1.7321 u
|C2C7|1.7321 u
|C2C8|1.0000 u
|C2C9|1.7321 u
|C2C10|2.6458 u
|C2C11|3.0000 u
|C2C12|2.6458 u
|C3C4|1.0000 u
|C3C5|1.7321 u
|C3C6|2.0000 u
|C3C7|2.6458 u
|C3C8|1.7321 u
|C3C9|1.0000 u
|C3C10|1.7321 u
|C3C11|2.6458 u
|C3C12|3.0000 u
|C4C5|1.0000 u
|C4C6|1.7321 u
|C4C7|3.0000 u
|C4C8|2.6458 u
|C4C9|1.7321 u
|C4C10|1.0000 u
|C4C11|1.7321 u
|C4C12|2.6458 u
|C5C6|1.0000 u
|C5C7|2.6458 u
|C5C8|3.0000 u
|C5C9|2.6458 u
|C5C10|1.7321 u
|C5C11|1.0000 u
|C5C12|1.7321 u
|C6C7|1.7321 u
|C6C8|2.6458 u
|C6C9|3.0000 u
|C6C10|2.6458 u
|C6C11|1.7321 u
|C6C12|1.0000 u
|C7C8|2.0000 u
|C7C9|3.4641 u
|C7C10|4.0000 u
|C7C11|3.4641 u
|C7C12|2.0000 u
|C8C9|2.0000 u
|C8C10|3.4641 u
|C8C11|4.0000 u
|C8C12|3.4641 u
|C9C10|2.0000 u
|C9C11|3.4641 u
|C9C12|4.0000 u
|C10C11|2.0000 u
|C10C12|3.4641 u
|C11C12|2.0000 u
∠P1P0P21.0472 rad
∠P1P0P32.0944 rad
∠P1P0P43.1416 rad
∠P1P0P52.0944 rad
∠P1P0P61.0472 rad
∠P1P0P81.0472 rad
∠P1P0P92.0944 rad
∠P1P0P103.1416 rad
∠P1P0P112.0944 rad
∠P1P0P121.0472 rad
∠P2P0P31.0472 rad
∠P2P0P42.0944 rad
∠P2P0P53.1416 rad
∠P2P0P62.0944 rad
∠P2P0P71.0472 rad
∠P2P0P91.0472 rad
∠P2P0P102.0944 rad
∠P2P0P113.1416 rad
∠P2P0P122.0944 rad
∠P3P0P41.0472 rad
∠P3P0P52.0944 rad
∠P3P0P63.1416 rad
∠P3P0P72.0944 rad
∠P3P0P81.0472 rad
∠P3P0P101.0472 rad
∠P3P0P112.0944 rad
∠P3P0P123.1416 rad
∠P4P0P51.0472 rad
∠P4P0P62.0944 rad
∠P4P0P73.1416 rad
∠P4P0P82.0944 rad
Derived ratios
r(C0)1.000000
∠P6P0P121.000000
|P4P5|2.000000
|P1P8|3.464102
|P10P11|4.000000
|P2P12|5.291503
|P6P9|6.000000
|P7P9|6.928203
|P7P10|8.000000
∠P3P1P912759113.257896
∠P6P5P722379005.764297
∠P5P1P1035138119.022193
∠P3P7P544758011.528594
∠P5P2P1047897232.280089
∠P10P6P1157517124.786490
∠P1P0P270276238.044385
∠P2P4P1283035351.302281
∠P3P7P1292655243.808682
∠P8P4P1295794464.560177
∠P0P5P12105414357.066578
∠P2P1P11118173470.324474
∠P1P0P3140552476.088771
∠P9P4P12153311589.346667
∠P7P1P9162931481.853068
∠P1P6P11175690595.110964
∠P0P6P12210828713.133157
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 | 2.000000000 | 2 | |P0P1| |P0P2| |P0P3| … | 220 × 2 = 440.000 Hz |
| R2 | 3.464101615 | 2√3 | |P1P3| |P1P5| |P1P8| … | 220 × 2√3 = 762.102 Hz |
| R3 | 4.000000000 | 4 | |P0P7| |P0P8| |P0P9| … | 220 × 4 = 880.000 Hz |
| R4 | 5.291502622 | 2√7 | |P1P9| |P1P11| |P2P10| … | 220 × 2√7 = 1164.131 Hz |
| R5 | 6.000000000 | 6 | |P1P10| |P2P11| |P3P12| … | 220 × 6 = 1320.000 Hz |
| R6 | 6.928203230 | 4√3 | |P7P9| |P7P11| |P8P10| … | 220 × 4√3 = 1524.205 Hz |
| R7 | 8.000000000 | 8 | |P7P10| |P8P11| |P9P12| | 220 × 8 = 1760.000 Hz |
Click a row to highlight its source points in the drawing. R = 0.5000 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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