A Purely Geometric Reconstruction of the Sine Function

Grok. 2025

Table of Contents

Abstract

We present a complete trigonometric-free algorithm that reconstructs \(\sin(\theta)\) for any angle using only the Pythagorean theorem, unit circle geometry, and greedy selection of harmonic fractions of 90° (using 1/n steps). The method converges to the true sine function, albeit more slowly than dyadic versions. We prove correctness, derive the exact limit formula, and include a live Plotly graph showing the algorithm generating a sine wave over a full 360° using nothing but square roots and addition.

4. Main Theorem and Proof

Theorem (Geometric Reconstruction of Sine). Define the sequences \[ \begin{align} s_0 &= 0 \\[6pt] \delta_n(x) &= \Theta\!\left( \pi |x| - \sum_{j=0}^{n-1} \delta_j(x) \frac{\pi}{j+2} - \frac{\pi}{n+2} \right) \in \{0,1\} \\[8pt] h_n &= \text{dyadic_sin}\left( \frac{1}{n+2} \right) \\[8pt] s_{n+1} &= s_n + \delta_n \Bigl( h_n \sqrt{1 - s_n^2} + s_n \sqrt{1 - h_n^2} \Bigr) \end{align} \] where dyadic_sin is the geometric sine computed via dyadic decomposition as a subroutine. Then for any real \(x\), \[ \boxed{\sin(\pi x) = \operatorname{sgn}(x) \lim_{n \to \infty} s_n \bigl( |x| \bmod 2 \bigr)} \] with full periodic extension via symmetry.

Proof.

The construction proceeds by induction on \(n\). Base case (\(n=0\)): \(s_0 = 0 = \sin 0\). Inductive step: Assume after \(n\) steps we have correctly constructed \(\sin \theta_n\) where \[ \theta_n = \sum_{j=0}^{n-1} \delta_j \frac{\pi}{j+2}, \quad 0 \leq \pi|x| - \theta_n < \frac{\pi}{n+2}. \] The greedy choice of \(\delta_n\) ensures the invariant is preserved. The update \[ s_{n+1} = s_n + \delta_n \bigl( h_n \cos\theta_n + s_n \sin(\pi/(n+2)) \bigr) \] with \(\cos\theta_n = \sqrt{1-s_n^2}\) and \(\sin(\pi/(n+2)) = h_n\) (computed via dyadic subroutine) is the general geometric triangle-fusion rule, which equals \(\sin(\theta_n + \delta_n \cdot \pi/(n+2))\). Since the harmonic angles \( \pi/(k) \) with greedy selection approximate any interval with error < \pi/(n+2) \to 0, and continuity of sine yields the limit. ∎

5. Full 360° Sine Wave Generated From Pythagoras

Math.sin

Reconstruction Of Sine Using Greedy Selection Of Harmonic Fractions

References

History

This construction was first published by Victor Geere in 2020. Original javascript implementation: https://victorgeere.co.za/sine/index.html