Twenty-one circles
The source radii move from 0.0379 to 11.6414 drawing units, spanning more than two orders of magnitude.
Digital artifact 003 · Source study / harmonicexpansion.dxf
Harmonic Expansion
The π Cascade.
A droplet becomes a field through the fourth root of π. This interactive exhibit preserves the twenty-one measured source radii, then reveals the repeating harmonic rule beneath them.
21 concentric radii36 angular sectors1,152 field links
Infinity Envisioned · Water-Bearer study
Harmonic Expansion A droplet becomes a field through the fourth root of π.rn = 1 / (2π) · (⁴√π)n
Live construction
Expansion1.331335 Active step01 / 21 Scale0.038 Circles21 Sectors36Exact mode preserves the twenty-one measured radii from the source drawing. Double-click the field for fullscreen.
01 / Harmonic construction
One interval.
Repeated with care.
The starting radius is 1/(2π). Each succeeding step multiplies that radius by the fourth root of π—approximately 1.331335.
Because four quarter-power steps equal one full power of π, each group of four expansions multiplies the radius by π. Read as diameter, the principal landmarks resolve to 1, π, and π².
Recurrencern+1 = rn · π1/4
Step 4D = 1
Step 8D = π
Step 12D = π²
02 / Reading the field
Measure becomes
visual rhythm.
Switch between the measured CAD radii and a normalized equal-step construction. The first view respects the supplied drawing; the second makes the generating interval easier to compare.
Thirty-six sectors
Ten-degree divisions turn scalar radii into an addressable field and provide the scaffold for radial motion.
Half-step phase
Successive rings alternate by five degrees, producing the woven diamond field visible during the expansion.
From drop to field
Animation stages a contemplative interpretation: condensation, impact, radial growth, and sustained flow.
03 / Evidence boundary
Source geometry
and interpretation.
The radii, circle count, sector count, recurrence, π landmarks, and 1,152 line segments are mathematical or directly derived from the supplied source study.
The droplet, water-field, cosmological, and contemplative language is exhibition framing. It suggests ways to experience the construction without presenting metaphor as physical proof.