Turns & Precision

Turns & Precision

Sep 6, 2025

Turns & Precision

This page formalises the angle unit used across ERG/GRG. Angles are measured in turns (fractions of a full rotation). We also define a binary fixed-point representation for hardware/software precision.

Angle unit (turns)

A full rotation is \(1\) turn. Quarter-turn \(= \tfrac{1}{4}\), half-turn \(= \tfrac{1}{2}\), etc. Trigonometric functions are period-1 in turns: \(c(u),\, s(u)\) with \(c(0)=1,\ s(0)=0,\ c(\tfrac{1}{4})=0,\ s(\tfrac{1}{4})=1\).

Equal-angle partition (N sectors)

Choose the number of sectors \(N\) and set the step \(\Delta u = \tfrac{1}{N}\). Sampling angles: \(u_k = \tfrac{k}{N}\), \(k=0,\dots,N-1\).

Binary fixed-point turns (B bits)

Store angles as \(u = \tfrac{k}{2^{B}}\) turns with \(k \in \{0,\dots,2^{B}\!-\!1\}\). The angular resolution is \(\Delta u_B = 2^{-B}\). To get exactly \(N\) equal sectors, pick \(N\) dividing \(2^{B}\) (e.g. \(N=2^{B}\)). Then use indices \(k = m\,j\) with \(m = \tfrac{2^{B}}{N}\), \(j=0,\dots,N-1\).

0 1/8 1/4 3/8 1/2 5/8 3/4 7/8
Example with \(B = 3\) bits ⇒ \(2^{3}=8\) ticks. Increasing \(B\) doubles the ticks and the precision.

Why more bits ⇒ more accuracy

  • Circumference (chords): \( C_N \;=\; 2\,N\,r\, s\!\bigl(\tfrac{1}{2N}\bigr) \) is the inscribed \(N\)-gon perimeter; as \(N\) doubles, the step halves and the error typically drops ≈ \(\times 4\).
  • Area (sectors): \( A_N \;=\; \tfrac{N}{2}\,r^{2}\, s\!\bigl(\tfrac{1}{N}\bigr) \) behaves similarly (step-squared).

Quick reference

Inline: \( \Delta u = \tfrac{1}{N},\quad u_k = \tfrac{k}{N},\quad u = \tfrac{k}{2^{B}},\quad \Delta u_{B} = 2^{-B} \).
Display:

$$ C_{N} \;=\; 2\,N\,r\, s\!\Bigl(\frac{1}{2N}\Bigr), \qquad A_{N} \;=\; \frac{N}{2}\,r^{2}\, s\!\Bigl(\frac{1}{N}\Bigr). $$

Implementation notes

  • Keep an angle register \(k\) modulo \(2^{B}\) (wrap-around on \(B\) bits).
  • For LUTs of size \(2^{M}\) with \(M \le B\), index by \(k \gg (B-M)\).
  • Step-doubling check: compare \(N\) vs \(2N\) to gauge the remaining gap.
π (GRG)

π (GRG)

Aug 25, 2025

Historical Comparison Page

This page preserves earlier experiments involving a circular constant. It is not the current GRG construction specification and does not establish a computational advantage. Read the current geometric formulation. The calculators and efficiency claims on this historical page require separate validation.

What this page is about

Here we determine π by looking at the circle from the inside. We rotate a central ray with equal step and sum only internal chords. If you “really want π”, this is the simplest way I know: increase only N. Everything else follows from the circle you already hold in your hands.

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.

The definition is exact and π is not assumed in advance. In turns (period-1 sine s(u)=\sin(2\pi u)) it is fixed by the limit

$$ \pi \;=\; \lim_{N\to\infty}\; 4N\, s\!\Big(\tfrac{1}{8N}\Big) \;=\; \lim_{N\to\infty}\; 2N\, s\!\Big(\tfrac{1}{4N}\Big). $$

For practical work we use the finite version

$$ \pi_N \;=\; 4N\, s\!\Big(\tfrac{1}{8N}\Big), $$

which increases from below and converges to π as N grows. A useful accuracy hint is

$$ \pi - \pi_N \;\approx\; \frac{\pi^3}{96\,N^2}. $$

Consequences you can feel while using the page:

  • Only one knob: raise N for more precision; the formatting “decimals to display” is just how many digits you show.
  • Powers of two shine: when N = 2^k we can compute by pure halving (only square-roots), no trigonometry, no π inside.
  • Predictable growth: multiplying N by 10 adds about 2 correct digits; doubling N adds roughly 0.6 digits.

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.

Interactive: increase only N

Choose N and how many decimals to display. For powers of two, the calculator uses pure halving (no trig). For general N it uses a high-precision series. Output shows the value, Δ to \( \pi_{2N} \), an error estimate, and detects when numerical verification becomes the limit (“inversion”).

Mode: —

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How this π calculator works

Formula (TURNS): \( \pi_N = 4N\, s(\tfrac{1}{8N}) \) with \(s(u)=\sin(2\pi u)\). Increase only N; the value grows from below and converges to π.

Two modes

Halving (no trig): if N is a power of two, the chord angle is halved repeatedly starting from the quarter-turn \(u=\tfrac{1}{4}\). This uses only square-roots; no trig, no π inside.

Series (turns → radians): for general N, it evaluates \(s(u)\) at \(u=\tfrac{1}{8N}\) via a high-precision series (internally converts to radians).

Precision & inversion

The line shows Δ(N→2N), an internal error estimate \( \approx \tfrac{4}{3}\Delta_N \), and a Richardson “verified digits” check. When internal ratios drift far from the expected \(1/4\) or verified digits stop growing at high precision, the panel turns amber: GRG may be more precise than the numerical checker (inversion).

Concrete examples (GRG)

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

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

These numbers come straight from the turn-native formula

$$ \pi_N \;=\; 4N\, s\!\Big(\tfrac{1}{8N}\Big) \quad\text{and}\quad \pi - \pi_N \;\approx\; \frac{\pi^3}{96\,N^2}. $$

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:

$$ s(u) := \sin(2\pi u), \qquad c(u) := \cos(2\pi u). $$

What really happens — three regimes

  • Model error dominates. The inside chord-sum approximation $$ \pi_N \;=\; 4N\,s\!\Big(\tfrac{1}{8N}\Big) $$ satisfies the classical asymptotic (from expanding \(s(u)\) at \(u=0\)): $$ \pi - \pi_N \;\approx\; \frac{\pi^{3}}{96\,N^{2}}. $$ Doubling N quarters the error.
  • Transition zone. The theoretical error keeps falling, but rounding and finite working precision begin to matter. If “shown decimals” or internal precision do not increase accordingly, 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 \( \pi \))

  1. \(\Delta\)–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, numerics have taken over ⇒ inversion.
  2. Internal error estimate (no \( \pi \)). From the series behavior, $$ \pi - \pi_N \;\approx\; \frac{4}{3}\,\Delta_N. $$ 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): one knob N, monotone inside-approximation, predictable error.
  • Honest inversion: beyond a threshold, tools used to verify are the limit — not GRG.
  • Self-certification: via \( \Delta_N \), \( \pi_{\text{ex}} \), and \( E_N \), no external \( \pi \) needed.

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.

Context note (as of 10 Sep 2025)

GRG’s turn-based, chord-sum definition of \( \pi \) is presented here as a new approach that increases accuracy by raising a single internal parameter N. This page documents the method and its error behavior; it is not intended to compete with record-breaking \( \pi \) algorithms.

For extremely large digit counts, established methods (AGM/Salamin–Brent, Borwein, Ramanujan/Chudnovsky) remain the fastest due to decades of optimization. GRG is primarily valued for conceptual clarity, monotone bounds, and controlled \( N\!\to\!\infty \) convergence.

Concrete examples (GRG)

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

  • N = 16 → πN ≈ 3.140331156954753 (error ≈ 1.261×10⁻³)
  • N = 128 → πN ≈ 3.141572940367091 (error ≈ 1.971×10⁻⁵)
  • N = 1024 → πN ≈ 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 \(s(u):=\sin(2\pi u)\)):

$$ \pi_N \;=\; 4N\,s\!\Big(\tfrac{1}{8N}\Big), \qquad \pi - \pi_N \;\approx\; \frac{\pi^3}{96\,N^2}. $$

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.