Scientific correction — 13 October 2026
For the complete quarter-circle profile sampled uniformly along x, midpoint and trapezoidal area errors have order N^(-3/2), not N^(-2), because the endpoint derivative is singular. The right-endpoint error and midpoint/right corridor width have order N^(-1). The geometric upper/lower corridor remains valid in exact arithmetic. These rates concern the historical Cartesian rules, not the current weighted binary radial kernels. Both modules now compare the actual resolution with its doubled value, without silently capping the second calculation. The first sphere-volume module uses right-endpoint rectangles in both directions; the second uses trapezoids. Historical trigonometric measurements are not implementations of the current geometric kernels. Resolution differences are diagnostics, not certified error bounds.
Current reference: Circle and Area from Binary Radial Modules.
GRG Mini-App
Explore Generative Radial Geometry (GRG): figures are generated from the center at a chosen resolution, using the historical operators documented below. 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, area error of order N^(-3/2)).
- 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:
\(\displaystyle \frac{\Delta_{\mathrm{int}}}{\text{value}} \le 0.15\%\).
🏆 Agreement with the reference 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 (ordinary floating-point summation).
- Area (Original Projection): midpoint (upper) / right (lower) corridor on the quarter → ×4 (ordinary floating-point summation).
- 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: the first module uses right-endpoint rectangles for quarter-areas and vertical stacking; the second uses trapezoids in both directions. Both use octant symmetry, then multiply by eight.
What this app demonstrates
This historical app approximates circle and sphere measures using numerical operators. With angles in turns and quarter/octant symmetries, internal resolution differences and external reference agreement are distinct diagnostics. These tolerance checks are diagnostics, not evidence of computational superiority.
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).