The Mathematics of Twisted Waves: How signals are decomposed into waves — and what happens when the world of the signal is a Möbius strip. (High-school level)

Abstract

This document explains the mathematics of a real research project using high-school tools: trigonometric functions, addition formulas and the trigonometric form of complex numbers. The starting point is the Fourier decomposition as a "basis and coordinates" idea, followed by an observation: many signals in nature are not periodic but antiperiodic — after one full loop they go over into their negative, and only after two loops do they return to themselves. Half-wave symmetric electrical signals, orientation fields and the electron itself are of this kind. The document derives, step by step, the basis suited to them (waves of half-integer frequency), presents the sector decomposition (the signal splits into two channels by itself), the discrete version (as fast as the ordinary FFT), and finally the pairing rule of the Klein bottle. Eight figures, worked examples and exercises support understanding; a separate section connects the material to high-school mathematics (Euler formula, addition formulas, vector bases, parity, geometric series) and physics (oscillations, alternating current, polarization, interference, spin, standing waves). The closing inventory honestly separates what comes from classical mathematics and what is the contribution of the present work.