1 00:00:00,000 --> 00:00:04,920 Take any wave, square, sawtooth, triangle, any shape at all. 2 00:00:05,540 --> 00:00:08,700 Fourier proved you can build it from simple sine waves. 3 00:00:09,140 --> 00:00:13,440 Add enough of them together and you can approximate any periodic function you like. 4 00:00:14,040 --> 00:00:16,240 This seems like a mathematical curiosity. 5 00:00:16,960 --> 00:00:23,780 But it unlocked the physics of heat, the engineering of electronics, and the mathematics of infinity itself. 6 00:00:23,780 --> 00:00:30,180 Last time, we met Gaspar Monge, the revolutionary geometer who invented descriptive geometry 7 00:00:30,180 --> 00:00:32,740 and founded the École Polytechnique. 8 00:00:33,400 --> 00:00:38,260 We saw how a military secret became the foundation of modern technical education. 9 00:00:38,920 --> 00:00:41,220 Welcome back to Men of Mathematics. 10 00:00:41,800 --> 00:00:47,640 Today, we meet Joseph Fourier, orphan, revolutionary, Egyptian adventurer, 11 00:00:47,880 --> 00:00:50,940 and the mathematician who was arrested, but survived. 12 00:00:50,940 --> 00:00:54,760 Perhaps his mathematical abilities made him too valuable 13 00:01:33,965 --> 00:01:40,145 schools and nearly losing his head during the terror. He was arrested, but survived. 14 00:01:40,725 --> 00:01:44,525 Perhaps his mathematical abilities made him too valuable to guillotine. 15 00:01:45,645 --> 00:01:52,165 In 1798, Napoleon invited Fourier to join the Egyptian expedition. For three years, 16 00:01:52,625 --> 00:01:58,465 Fourier served as secretary of the Institut d'Egypte, organizing the scientific survey of Egypt 17 00:01:58,465 --> 00:02:02,265 and surviving adventure, plague, and military defeat. 18 00:02:02,825 --> 00:02:04,665 The experience marked him forever. 19 00:02:05,805 --> 00:02:08,805 Back in France, he kept his rooms uncomfortably hot. 20 00:02:09,285 --> 00:02:11,205 He could never get warm enough after Egypt. 21 00:02:12,085 --> 00:02:14,845 As prefect of Iser, a regional administrator, 22 00:02:15,365 --> 00:02:17,425 Fourier had time to think about heat. 23 00:02:18,025 --> 00:02:19,945 How does temperature change over time? 24 00:02:20,505 --> 00:02:22,425 How does heat flow through a metal bar? 25 00:02:23,045 --> 00:02:24,685 He derived the heat equation. 26 00:02:24,685 --> 00:02:30,645 The rate of temperature change over time equals a constant times the curvature of the temperature distribution. 27 00:02:31,325 --> 00:02:34,665 To solve the heat equation, Fourier made a stunning claim. 28 00:02:35,185 --> 00:02:39,825 Any function can be expressed as an infinite sum of sines and cosines. 29 00:02:40,385 --> 00:02:45,185 A square wave from smooth sines, a sawtooth from periodic curves. 30 00:02:45,985 --> 00:02:46,945 It seemed impossible. 31 00:02:47,685 --> 00:02:49,265 Lagrange himself objected. 32 00:02:49,725 --> 00:02:51,065 But Fourier was right. 33 00:02:51,065 --> 00:02:55,345 Each sine component decays at a different rate under the heat equation. 34 00:02:56,205 --> 00:03:00,225 High-frequency components, sharp features, decay fastest. 35 00:03:00,945 --> 00:03:05,345 Low-frequency components, gradual variations, persist longest. 36 00:03:06,245 --> 00:03:09,605 This explains why a heated rod smooths out its temperature. 37 00:03:10,245 --> 00:03:27,310 The mathematics of heat and the mathematics of waves are secretly the same For non functions Fourier extended his idea to the Fourier transform which converts any signal between the time domain when things happen and the frequency domain 38 00:03:27,810 --> 00:03:35,170 which frequencies are present. Fourier analysis is everywhere. MP3 audio compression stores 39 00:03:35,170 --> 00:03:44,090 frequencies, not samples. JPEG image compression uses Fourier cousins in 2D. MRI machines reconstruct 40 00:03:44,090 --> 00:03:50,650 images from frequency data. Telecommunications separates radio stations by frequency. And 41 00:03:50,650 --> 00:03:56,770 quantum mechanics uses wave-particle duality with Fourier. Fourier's results were correct, 42 00:03:57,170 --> 00:04:02,490 but his proofs weren't rigorous by modern standards. Making Fourier series mathematically 43 00:04:02,490 --> 00:04:09,170 precise, occupied the greatest minds of the 19th century. Dirichlet established conditions for 44 00:04:09,170 --> 00:04:17,670 convergence. Riemann developed integration theory. Cantor invented set theory from studying Fourier 45 00:04:17,670 --> 00:04:25,650 series. And Labesco created modern integration. The quest to understand Fourier series drove the 46 00:04:25,650 --> 00:04:32,450 development of modern analysis, Fourier made another discovery. The Earth's atmosphere acts 47 00:04:32,450 --> 00:04:39,590 as an insulating layer, trapping heat from the sun. This was the first recognition of what we now call 48 00:04:39,590 --> 00:04:46,590 the greenhouse effect. Fourier also insisted that physical equations must have consistent dimensions. 49 00:04:46,590 --> 00:05:05,495 You can add meters to seconds This principle of dimensional homogeneity now standard in physics was pioneering in 1822 Fourier masterwork published in 1822 is considered one of the great scientific books of all time 50 00:05:06,295 --> 00:05:11,795 Lord Kelvin called Fourier's theorem, one of the most beautiful results of modern analysis. 51 00:05:13,055 --> 00:05:17,655 Joseph Fourier discovered that any function can be decomposed into simple waves, 52 00:05:17,655 --> 00:05:20,415 signs and cosines at different frequencies. 53 00:05:21,355 --> 00:05:24,095 This insight, born from studying heat flow, 54 00:05:24,515 --> 00:05:27,415 transformed mathematics, physics, and engineering, 55 00:05:27,875 --> 00:05:30,295 from audio compression to medical imaging, 56 00:05:30,955 --> 00:05:34,295 from telecommunications to quantum mechanics. 57 00:05:35,615 --> 00:05:38,055 Fourier's idea echoes everywhere. 58 00:05:39,035 --> 00:05:42,155 There's something poetic about Fourier's hit the notification bell. 59 00:05:42,815 --> 00:05:44,795 New episodes release every week. 60 00:05:45,315 --> 00:05:46,235 I'll see you next time.