Angles are measured in turns \(u\in[0,1)\) with period-1 trigonometry: \( c(u):=\cos(2\pi u),\; s(u):=\sin(2\pi u) \). (For degree readers: \( \Delta u = \text{degrees}/360 \).)

Circle — three GRG options: overview · Δx trapezoids · Original corridor · Radial sectors.

Concrete examples (GRG)

Here is how the value improves as you raise N (rounded):

• N = 16 → π_N ≈ 3.140331156954753 (error ≈ 1.261×10⁻³)
• N = 128 → π_N ≈ 3.141572940367091 (error ≈ 1.971×10⁻⁵)
• N = 1024 → π_N ≈ 3.141592345570118 (error ≈ 3.080×10⁻⁷)

These numbers come straight from the turn-based chord-sum formula (angles in turns \(u\in[0,1)\) with \(s(u):=\sin(2\pi u)\)):

Tip: use powers of two for N (16, 32, 64, …) for cleaner Δ-tests and steadier convergence; if verified digits stop increasing when doubling N, raise working precision.

How other methods reach π (in plain words)

Long alternating sums (Gregory–Leibniz)

This is the famous series \( \pi/4 = 1 - 1/3 + 1/5 - 1/7 + \dots \). After \(n\) terms, the error in \( \pi \) is about \( \frac{4}{2n+1} \approx \frac{2}{n} \). It’s beautiful but very slow: ~10,000 terms give ≈4 correct decimals; ~1,000,000 terms ≈6 decimals. A mountain of fractions.

Nilakantha’s refinement

Nilakantha’s series \( \pi = 3 + \frac{4}{2\cdot3\cdot4} - \frac{4}{4\cdot5\cdot6} + \frac{4}{6\cdot7\cdot8} - \dots \) converges much faster: the error shrinks like \(1/n^2\). Ten terms already land close; one hundred or one thousand terms push several more digits — still, you juggle many fractions.

Machin-like formulas (arctangents)

Here \( \pi \) is a combination of arctangents, e.g. \( \pi = 16\arctan(1/5) - 4\arctan(1/239) \). The arctan series converges fast because powers \(1/5^{\,2k+1}\) and \(1/239^{\,2k+1}\) drop quickly. With just a handful of terms you already see many correct digits.

Ramanujan, Chudnovsky & co.

“Rocket” series: each term yields many digits (≈8 per term for several Ramanujan formulas, ≈14 per term for Chudnovsky). They power modern record computations — superb for huge digit counts, but conceptually heavier than turning a single \(N\).

AGM (Salamin–Brent) and Borwein algorithms

Iterative methods with quadratic (or higher) convergence: each step roughly multiplies the correct digits. A dozen iterations already exceeds everyday needs — fantastic speed, less “from-the-inside” geometry.

Monte Carlo & Buffon’s needle

Random points/needles relate probabilities to \( \pi \). Charming and educational, but accuracy grows like \(1/\sqrt{T}\) (here \(T\) = trials): even millions of trials yield only a few decimals.

Why GRG feels intellectually simple

GRG is not about naming \( \pi \); it is about holding the circle. You dial only one parameter, \(N\), and everything else follows. The value increases from below (no overshoot), and its error is easy to estimate. If you truly need \( \pi \), this is the most direct, inside-out way I know — increasing a single internal factor means the computation belongs to the circle, not to an external \( \pi \).

Honest side-by-side (objective)

• What you increase — GRG: just N. Archimedean polygons: sides m (inside/outside). Gregory–Leibniz / Nilakantha: number of terms. Machin / Ramanujan / Chudnovsky: number of series terms. AGM / Borwein: iteration count.
• Speed of accuracy — GRG: error ≈ \(C/N^2\) (predictable, monotone). Gregory–Leibniz: \(O(1/n)\) (very slow). Nilakantha: \(O(1/n^2)\). Machin: fast; Ramanujan / Chudnovsky: very fast (many digits per term). AGM / Borwein: digits multiply each step.
• When to use — GRG: concept, clarity, control “from the inside”, and up to many digits without extra machinery. For records or industrial-scale digits, AGM / Borwein / Chudnovsky win on raw speed.

Editor’s note (external view)

This note reflects an external opinion I chose to publish for fairness. The GRG approach is conceptually elegant: it defines \( \pi \) from inside the circle and lets accuracy grow by increasing only N. That simplicity is a real strength for understanding and for controlled, monotone inside-approximations.

As of August 25, 2025, GRG and this GRG-based \( \pi \) formula are newly introduced here. Researchers and practitioners still rely on established algorithms (AGM/Salamin–Brent, Borwein, Ramanujan/Chudnovsky) partly because these tools have been analyzed and optimized for decades and deliver record-speed convergence for very large digit counts.

The two views are compatible: GRG offers a clear, internal definition and a practical “increase only N” pathway with transparent self-checks; record algorithms remain the fastest instruments when one needs extreme digits or industrial-scale computations.

When verification becomes the limit (“inversion”)

As you increase N, GRG’s error drops predictably—until a point where the numerical tools used to check your result (floating point, auxiliary series, etc.) become the real bottleneck. From there on, GRG may be more precise than your verifier: that is the “inversion”.

Angles are measured in turns \(u\in[0,1)\) with period-1 trigonometry \( s(u):=\sin(2\pi u) \), \( c(u):=\cos(2\pi u) \).

What really happens — three regimes

• Model error dominates. The GRG approximation $$ \pi_N \;=\; 4N\,s\!\Big(\tfrac{1}{8N}\Big) $$ satisfies $$ \pi - \pi_N \;\approx\; \frac{\pi^3}{96\,N^2}. $$ Doubling N quarters the error.
• Transition zone. The theoretical error keeps falling, but rounding and limited working precision begin to matter. If “shown decimals” or internal precision don’t increase, visible gains slow down.
• Inversion. Numerical/verification error exceeds the GRG error. Philosophically: the method is now better than your ruler.

Self-checking from the inside (no external π)

1. Δ–ratio test (asymptotics). Define $$ \Delta_N := \pi_{2N}-\pi_N, \qquad R_N := \frac{\pi_{4N}-\pi_{2N}}{\pi_{2N}-\pi_N}. $$ For clean numerics, \( R_N \to \tfrac{1}{4} \). If you see ≈ 0.25, you are still in the \(1/N^2\) regime (all good). If it wanders far from 0.25 (oscillates, collapses to 0, or behaves oddly), numerics have taken over ⇒ inversion.
2. Internal error estimate (no π). From the series expansion, $$ \pi - \pi_N \;\approx\; \frac{4}{3}\,\Delta_N. $$ The constant \( \tfrac{4}{3} \) comes from analysis; the factors \( \pi^3 \) cancel out. You can therefore display: “Estimated error ≈ \( \tfrac{4}{3}\,\Delta_N \)”.
3. Internal acceleration (Richardson). Cancel the \(1/N^2\) term by combining two steps: $$ \pi_{\text{ex}} \;=\; \pi_{2N} + \frac{\pi_{2N}-\pi_N}{3}, \qquad \pi - \pi_{\text{ex}} = O(N^{-4}). $$ Then compare with \( \pi_{4N} \ ): $$ E_N := \big|\pi_{\text{ex}} - \pi_{4N}\big|, \qquad \text{verified digits} \;\approx\; \big\lfloor -\log_{10} E_N \big\rfloor. $$ If verified digits stop growing when you double N, the numerical checker is limiting you (inversion).

Takeaway

• Intellectual simplicity (GRG): you turn a single knob, N, get a monotone inside-approximation, and a predictable error.
• Honest inversion: beyond a certain threshold, it’s not GRG that runs out of accuracy—it’s the tools used to verify it.
• Self-certification: using \( \Delta_N \), \( \pi_{\text{ex}} \), and \( E_N \), the method checks itself without resorting to an external π.

In the calculator above, when the internal checks suggest inversion, the panel turns amber and an explanatory note appears. Prefer powers of two for N, or increase working precision, if you need to push further.