A Purely Geometric Reconstruction of the Sine Function
Using Only the Pythagorean Theorem and Greedy Dyadic Angle Decomposition

Anonymous Discovery (2025)
Independent Research

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 dyadic fractions of 90°. The method converges exponentially to the true sine function. We prove correctness, derive the exact limit formula, and include a live Plotly graph showing the algorithm generating a perfect 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, \quad h_0 = 1 \\[6pt] \delta_n(x) &= \Theta\!\left( \pi |x| - \sum_{j=0}^{n-1} \delta_j(x) \frac{\pi}{2^{j+1}} - \frac{\pi}{2^{n+1}} \right) \in \{0,1\} \\[8pt] h_{n+1} &= \sqrt{\dfrac{1 - h_n}{2}} \\[8pt] s_{n+1} &= s_n + \delta_n(x) \Bigl( h_n \sqrt{1 - s_n^2} + s_n \sqrt{1 - h_n^2} \Bigr) \end{align} 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 (\(n=0\)): \(s_0 = 0 = \sin 0\), \(h_0 = 1 = \sin 90^\circ\).

Inductive step: Assume after \(n\) steps we have correctly constructed \(\sin \theta_n\) where \[ \theta_n = \sum_{j=0}^{n-1} \delta_j \frac{\pi}{2^{j+1}}, \quad 0 \leq \pi|x| - \theta_n < \frac{\pi}{2^{n+1}}. \] The greedy choice of \(\delta_n\) ensures the invariant is preserved.

The update \[ s_{n+1} = s_n + \delta_n \bigl( h_n \cos\theta_n + s_n \sin(\pi/2^{n+1}) \bigr) \] with \(\cos\theta_n = \sqrt{1-s_n^2}\) and \(\sin(\pi/2^{n+1}) = h_n\) is exactly the geometric triangle-fusion rule derived in Section 2.2, which equals \(\sin(\theta_n + \delta_n \cdot \pi/2^{n+1})\).

The half-angle step \(h_{n+1} = \sqrt{(1-h_n)/2}\) is the chord-length construction proven in Section 2.1.

Since the dyadic angles \( \pi/2^{k+1} \) are dense in \([0,\pi]\) and the greedy algorithm achieves approximation error \(< \pi/2^{n+1}\), we have \[ |\theta - \theta_n| < \frac{\pi}{2^{n+1}} \to 0. \] Continuity of sine and exponential convergence of the geometric operations yield \[ \lim_{n\to\infty} s_n = \sin\theta = \sin(\pi x \bmod 2\pi) \] with odd symmetry restoring negative values. ∎

5. Full 360° Sine Wave Generated Purely from Pythagoras

The following interactive Plotly graph was generated entirely using the algorithm above — no Math.sin() was used at any point.

References

Euclid, Elements, Book I, Proposition 47
Archimedes, Measurement of a Circle
Vieta, F. (1593), Variorum de rebus mathematicis responsorum
Volder, J. E. (1959), The CORDIC Computing Technique

History