GRG Mini app
GRG Mini-App
Explore Generative Radial Geometry (GRG): figures are generated from the center at a chosen resolution, without assuming π a priori. Angles are measured in turns \(u\in[0,1)\) with period-1 trig \(c(u)=\cos(2\pi u),\ s(u)=\sin(2\pi u)\).
Controls
Circle (2D)
Sphere (3D)
How the GRG Mini-App works — Circle & Sphere
What you control
Inputs. Two modules:
- Circle (2D): choose radius r, subdivisions N and the Area method:
- Δx Trapezoidal — quarter → ×4 (concave arc, under-tiling, fast O(N⁻²)).
- Original Projection — midpoint (upper) / right (lower) corridor, quarter → ×4.
- Radial Sectors (turns) — full turn, equal-angle sectors (inscribed, product form).
Note: for Δx/Original the slider N refers to the quarter; for Radial it refers to the full turn. Circumference always uses turns with equal-angle chords. - Sphere (3D): choose radius R, vertical slices Nz, radial trapezoids for quarter-areas Mr, and quarter-circumference subdivisions Mc.
- Show traditional comparison toggles all external (vs π) quantities and the grey series in charts.
- Show π estimates reveals π computed internally (from A/C in 2D; S/V in 3D).
- Download CSV exports the full range (1…Nmax); Copy link preserves the current settings.
Status badges & thresholds
🟢 Internal dominance The internal change between the current resolution and the doubled one is small relative to the value:
\(\displaystyle \frac{\Delta_{\mathrm{int}}}{\text{value}} \le 0.15\%\).
🏆 Victory vs classical The external error is also small:
\(\displaystyle \frac{\Delta_{\mathrm{ext}}}{\text{value}_\pi} \le 0.10\%\).
Formulas used — Circle (2D)
- Area (Δx Trapezoidal): quarter sum of trapezoids with uniform \(\Delta x=r/N\) → ×4 (Kahan summation for stability).
- Area (Original Projection): midpoint (upper) / right (lower) corridor on the quarter → ×4 (Kahan).
- Area (Radial Sectors, turns): \(A_N=\tfrac{N}{2}r^2\,s(\tfrac{1}{N})\) with \(s(u)=\sin(2\pi u)\).
- Circumference (turns): equal-angle chords on the full turn: \(C_{N_{\text{full}}}=2N_{\text{full}}\,r\,s(\tfrac{1}{2N_{\text{full}}})\). For Δx/Original we use \(N_{\text{full}}=4N\); for Radial, \(N_{\text{full}}=N\).
Formulas used — Sphere (3D)
- Surface: octant fusion of quarter-circumferences along meridian (\(\Delta s\)), then ×8.
- Volume: octant stacking of quarter-areas along \(z\) with trapezoid in \(\Delta z\), then ×8.
What this app demonstrates
GRG reconstructs circle and sphere without assuming π. With angles in turns and quarter/octant symmetries, internal stabilization (dominance) precedes external agreement (victory). Construct first, constants emerge.
GRG Mini-App — 3 righe
Layout a righe: Controls → Circle (2D) → Sphere (3D). Tutti gli slider sono two-way con i numeri; i turns (U) hanno preview live (corde uguali).