Rigor of GRG

Rigor of GRG

Aug 22, 2025

Earlier Formulation

This page preserves a historical GRG method. Its notation and formulas belong to that version. The current binary radial formulation and verification are available here. Historical computational claims require separate validation.

Generative Radial Geometry (GRG)

A Constructive Alternative to Classical Circle Measurement

Introduction

Generative Radial Geometry (GRG) is a digital, constructive approach: it does not assume π in the build step. Circles are generated from the radius with finite operators (Pythagoras, chords, tangents); classical values (e.g. π) then emerge as limits or comparisons.

Two complementary builds

  • Quarter / Cartesian (Δx): work on one quarter of the disk, sum rectangles/trapezoids with \(\Delta x=R/N\), then ×4. Certified right/midpoint corridor.
  • Full turn / Radial (Δu): work on the full loop in turns \(u\in[0,1)\), step \(Δu=1/U\); equal-angle chords/tangents give in/out polygons (inscribed/circumscribed).

Notation in turns: \(c(u)=\cos(2\pi u),\ s(u)=\sin(2\pi u)\). π appears only when comparing to classical formulas.

Quarter coordinates (Δx, no trig in the step)

On the quarter, sample \(x_k=k\,\Delta x\) with \(\Delta x=R/N\) and heights by Pythagoras:

$$ y_k=\sqrt{R^2-x_k^2},\qquad k=0,\dots,N. $$

Area — Cartesian quarter (GRG)

Δx Trapezoidal (lower)

$$ A_{\text{quarter}}^{\text{trap}}(R,N)=\sum_{k=1}^{N}\frac{y_{k-1}+y_k}{2}\,\Delta x,\quad A_{\text{GRG}}^{\text{trap}}=4\,A_{\text{quarter}}^{\text{trap}}. $$

Projection corridor (certified bounds)

$$ A_{\text{quarter}}^{\mathrm{right}}=\!\sum_{k=1}^{N} y_k\,\Delta x,\qquad A_{\text{quarter}}^{\mathrm{mid}}=\!\sum_{k=1}^{N} y\!\left(x_{k-\tfrac12}\right)\Delta x, $$
$$ A_{\text{quarter}}^{\mathrm{right}}\le A_{\text{quarter}}\le A_{\text{quarter}}^{\mathrm{mid}},\qquad 4A_{\text{quarter}}^{\mathrm{right}}\le A_{\text{GRG}}\le 4A_{\text{quarter}}^{\mathrm{mid}}. $$

Circumference — Radial full turn (GRG)

With \(U\) equal sectors per full turn (step \(Δu=1/U\)), all chords are equal. The inscribed polygon length is:

$$ C_U \;=\; 2\,U\,R\,s\!\Big(\tfrac{1}{2U}\Big) \;=\; 2\,U\,R\,\sin\!\Big(\tfrac{\pi}{U}\Big)\quad\text{(classical cross-check)}. $$

On a quarter with \(N\) steps set \(U=4N\) and get \(C=8NR\,s\!\big(\tfrac{1}{8N}\big)\).

Radial area — in/out corridor on the full turn

Inscribed polygon area (lower, shoelace)

Let \(P_j=(x_j,y_j)\) be border points in CCW order, \(j=0,\dots,U-1\) and \(P_U=P_0\) with \(x_j^2+y_j^2=R^2\).

$$ A_{\text{in}}(R,U)=\tfrac12\sum_{j=0}^{U-1}\bigl(x_jy_{j+1}-x_{j+1}y_j\bigr)\ \le\ A. $$

Circumscribed polygon area (upper, tangents)

The tangent at \(P_j\) is \(x\,x_j+y\,y_j=R^2\). Intersecting tangents at \(P_j,P_{j+1}\) yields \(T_j\):

$$ \det_j=x_jy_{j+1}-y_jx_{j+1},\qquad T_j=\Bigl(\frac{R^2\,(y_{j+1}-y_j)}{\det_j},\ \frac{R^2\,(x_j-x_{j+1})}{\det_j}\Bigr), $$
$$ A_{\text{out}}(R,U)=\tfrac12\sum_{j=0}^{U-1}\bigl(T_j^xT_{j+1}^y-T_{j+1}^xT_j^y\bigr)\ \ge\ A. $$

Hence \(A_{\text{in}}\le A \le A_{\text{out}}\) and the corridor collapses as \(U\) increases.

Answering common objections

“This is nothing new”

GRG constructs perimeter and area from the radius alone in two certified ways: quarter/Δx (right–mid corridor) and full-turn/Δu (inscribed–circumscribed corridor).

“It is only an approximation”

Accuracy is controlled by \(N\) or \(U\); both builds provide bounding pairs that converge.

“It lacks rigor”

Only algebraic steps (Pythagoras, chords, tangents, polygon sums). π is used solely for external comparison, not as an input.

“It is less practical”

Finite sums with elementary operations are ideal for digital computation and come with a transparent error model.

Conclusion

GRG complements classical geometry with a generative language. On the quarter, Δx gives fast areas with bounds; on the full turn, equal-angle chords/tangents give circumference and radial area—letting π emerge as a limit.

Circle with inscribed (lower) and circumscribed (upper) polygons; arrows suggest convergence. A_in A_out U ↑ ⇒ A_in ↑, A_out ↓ inscribed (lower) circumscribed (upper)