In December 1807, a French mathematician named Joseph Fourier presented a memoir to the Paris Academy of Sciences claiming that any reasonable signal — any sound, any temperature distribution, any periodic process — could be written as a sum of sines and cosines. Lagrange, who had spent decades on trigonometric series, objected so forcefully that publication was blocked . The manuscript sat for fifteen years before it appeared in book form as Théorie analytique de la chaleur in 1822 . Fourier was trying to model heat flow in a metal bar. Two centuries later, every JPEG image, every MP3 track, every Wi-Fi packet, and every MRI scan in routine clinical use leans on the same idea. Fourier did not aim at any of that. The trick generalized in ways nobody alive in 1807 could have predicted, and the chain from a heat-equation paper to a 5G modem is short enough to walk in a single article. What the trick actually is Take a signal — a string of audio samples, a row of pixel intensities, a slice of MRI sensor data.…