Guidelines

Sep 14, 2025

GRG Documentation Guidelines

Describe the construction through its geometric definitions. Use plain language, explicit hypotheses, and reproducible evidence.

Current Formulas

Use geometric lengths and ratios. Keep circular constants outside the construction’s formulas and numerical kernel. Explicit computational comparisons may use reference constants, with their role clearly stated.

Claims and Evidence

Distinguish definitions, proved statements, numerical observations, proposals, and applications. Do not claim efficiency, numerical stability, exact hardware arithmetic, or originality without supporting evidence.

Versioning

Identify historical formulations and retain their provenance. The binary radial formulation is the current two-dimensional reference.

Presentation

Keep site content in English. Provide accessible diagram descriptions, readable mathematical displays, and verified links. Avoid polemical slogans and comparisons that imply an unmeasured advantage.

GRG Mini app

GRG Mini app

Aug 24, 2025

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

N = 24
Circumference is always from equal-angle chords in turns.
Sphere uses octant symmetry (×8) and π-free operators; volume is O(N²).
%
“Internal dominance” compares N to 2N (self-consistency). “Victory” compares to the π-based value (external).

Circle (2D)

Area estimate
—
Area corridor (if applicable)
—
Circumference (chords)
—
Internal Δ (N → 2N), area
—
Internal Δ (N → 2N), circumf.
—
External error vs π-based, area
—
External error vs π-based, circumf.
—
area: — circ: —

Sphere (3D)

Surface (octant fusion ×8)
—
Volume (stacked quarters ×8)
—
Internal Δ (N → 2N), surface
—
Internal Δ (N → 2N), volume
—
External error vs 4πr²
—
External error vs (4/3)πr³
—
surface: — volume: —

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

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).

Controls

Circle

Angular (turns)

Δu = 1/48 · Δθ = 0.130899694 rad
U multiple of 4 ✓; U = 4·N·k ✓

Sphere

Slider (mirrors)

Circle (2D)

Area (GRG)
0
Corridor width (mid−right)
—
Circumference (chords)
0
Δint A (2N vs N)
—
Δint C (2U vs U)
—
Δext A vs πr²
—
Δext C vs 2πr
—
area: — circ: —
Anteprima: poligono inscritto con U settori uguali (corde uguali).

Sphere (3D)

Surface S (octant fusion ×8)
0
Volume V (stacked quarters ×8)
0
Δint S
—
Δint V
—
Δext S vs 4πR²
—
Δext V vs (4/3)πR³
—
surface: — volume: —
GRG for ellipses

GRG for ellipses

Aug 22, 2025

GRG for Ellipses — Constructive Area, Perimeter, and Equal-Area Sectors

The “Variable Radius” in GRG

The effective radius depends on the chosen direction.

  • From the center (angle \(\phi\) from the major axis):

    $$ r_c(\phi)=\frac{ab}{\sqrt{(b\cos\phi)^2+(a\sin\phi)^2}}. $$

  • From the focus (true anomaly \(\theta\)):

    $$ r_f(\theta)=\frac{p}{1+e\cos\theta},\qquad e=\sqrt{1-\frac{b^2}{a^2}},\qquad p=\frac{b^2}{a}. $$

  • Without trigonometry (slope \(m=\tan\phi\)):

    $$ r_c(m)=\frac{ab\,\sqrt{1+m^2}}{\sqrt{\,b^2+a^2m^2\,}}. $$

Ellipse Area in GRG

A1) Cartesian strips in the first quadrant

Split the semi-axis \(a\) into \(N\) parts: \(x_i=i\,a/N\) with \(\Delta x=a/N\). Heights \(y_i=b\sqrt{1-x_i^2/a^2}\).

$$ A_{1/4}^{(N)}=\sum_{i=1}^{N} y_i\,\Delta x,\qquad A^{(N)}=4\,A_{1/4}^{(N)}. $$
A1 — Cartesian strips in the first quadrant x–y axes, ellipse centered at (400,250) with a=240, b=160; shaded vertical strip inside Q1. +y +x Cartesian strips in Q1; then ×4
Quarter-by-strips: sum \(y_i\,\Delta x\) in Q1 and multiply by 4 for the ellipse area.

A2) Radial wedges from the center (by quarters)

With \(\Delta\phi=90^\circ/N\) and \(\phi_i=i\,\Delta\phi\):

$$ A_{1/4}^{(N)}=\sum_{i=0}^{N-1}\tfrac12\,r_c(\phi_i)\,r_c(\phi_{i+1})\,\sin\Delta\phi, \qquad A^{(N)}=4\,A_{1/4}^{(N)}. $$
A2 — Center-based wedges (quarter symmetry) Rays from center to ellipse boundary; the wedge between two rays approximates an area piece. C +x +y Center-based wedges (quarter symmetry)
Sum center-based wedges over a quarter (then ×4). Summing the chords between successive boundary points gives the corresponding perimeter.

A3) Radial wedges from the focus (natural for orbits)

With \(\Delta\theta=360^\circ/M\) and \(\theta_i=i\,\Delta\theta\):

$$ A^{(M)}=\sum_{i=0}^{M-1}\tfrac12\,r_f(\theta_i)\,r_f(\theta_{i+1})\,\sin\Delta\theta. $$
A3 — Focus-based sectors (equal areas → equal times) Sectors with apex at a focus; each filled sector has the same area increment ΔA (Kepler-2). F F′ periapsis → Equal-area sectors from the focus (constant ΔA; angle widths vary)
Equal-area stepping from the focus: constant \(\Delta A\) gives uniform time steps (Kepler’s 2nd law). Near periapsis the angular gaps shrink; away they widen.

Perimeter (sum of chords)

P1) Chords from the center (by quarters)

With \(\Delta\phi=90^\circ/N\):

$$ P_{1/4}^{(N)}=\sum_{i=0}^{N-1}\sqrt{\,r_c(\phi_i)^2+r_c(\phi_{i+1})^2-2\,r_c(\phi_i)\,r_c(\phi_{i+1})\cos\Delta\phi\,}, \qquad P^{(N)}=4\,P_{1/4}^{(N)}. $$

P2) Chords from the focus (full ellipse)

$$ P^{(M)}=\sum_{i=0}^{M-1}\sqrt{\,r_f(\theta_i)^2+r_f(\theta_{i+1})^2-2\,r_f(\theta_i)\,r_f(\theta_{i+1})\cos\Delta\theta\,}. $$

GRG Error Estimate (after-the-fact)

With step doubling the error typically decays ~quadratically. Method-agnostic (π-free):

  • Compute \(S_N\) and \(S_{2N}\) with the same operator (A1–A3, P1–P2).

  • Accelerated approximation:

    $$ S_\star \approx S_{2N}+\frac{S_{2N}-S_N}{3}. $$

  • Error estimate:

    $$ \varepsilon \approx \frac{|S_{2N}-S_N|}{3}. $$

Practical Pipeline (geometry → orbits)

  • Geometry: use A1 or A2 for area and P1 for perimeter with quarter symmetry.

  • Orbits: use A3. For uniform time steps, enforce constant-area sectors:

    $$ \tfrac12\,r_f(\theta_k)\,r_f(\theta_{k+1})\,\sin(\theta_{k+1}-\theta_k)=\Delta A. $$

  • The arc length per step is the P2 chord between successive points.