WEBVTT 00:00:01.740 --> 00:00:04.919 Hi, have you ever noticed how some of the most 00:00:04.919 --> 00:00:10.259 beautiful things are surprisingly simple? A single 00:00:10.259 --> 00:00:14.320 melody played on piano, a child's drawing, a 00:00:14.320 --> 00:00:19.199 quiet snowfall, a circle drawn in the sand. Beauty 00:00:19.199 --> 00:00:22.100 does not always come from complexity. Sometimes 00:00:22.100 --> 00:00:25.600 beauty comes from simplicity that reaches deeper 00:00:25.600 --> 00:00:28.940 than we expect. And I think mathematics often 00:00:28.940 --> 00:00:32.039 works that way. One of the most famous mathematical 00:00:32.039 --> 00:00:36.000 ideas is the Pythagorean theorem. A squared plus 00:00:36.000 --> 00:00:40.219 B squared equals C squared. Now, at first glance, 00:00:40.299 --> 00:00:44.479 it may not seem beautiful. Just letters and symbols. 00:00:45.299 --> 00:00:49.259 Yet this simple relationship works for every 00:00:49.259 --> 00:00:54.399 right triangle. And how do we know that? Well, 00:00:54.439 --> 00:00:57.200 over thousands of years, mathematicians have 00:00:57.200 --> 00:01:01.530 discovered hundreds of proofs. Some use geometry, 00:01:01.969 --> 00:01:06.769 some use algebra, some use motion. Recently, 00:01:06.890 --> 00:01:09.730 two high school students even found a proof using 00:01:09.730 --> 00:01:14.390 trigonometry. So for fun, I recently reviewed 00:01:14.390 --> 00:01:17.250 three different proofs myself, trying to see 00:01:17.250 --> 00:01:20.409 it from different perspectives. And what surprised 00:01:20.409 --> 00:01:23.069 me was that despite the different perspectives, 00:01:23.510 --> 00:01:28.370 all three proofs depended on the same key point 00:01:28.370 --> 00:01:31.859 inside the triangle. In two of the proves, the 00:01:31.859 --> 00:01:34.540 point was obvious, but in the third, I had to 00:01:34.540 --> 00:01:38.939 dig for it. But once I saw it, it felt almost 00:01:38.939 --> 00:01:43.079 magical. Three completely different approaches, 00:01:43.260 --> 00:01:48.200 yet all centered on the same hidden point. It 00:01:48.200 --> 00:01:50.340 felt like discovering that three different paths 00:01:50.340 --> 00:01:53.340 through a forest all lead to the same clearing. 00:01:54.840 --> 00:01:58.159 So why would so many people spend so much time 00:01:58.159 --> 00:02:02.459 proving the same thing? Because beautiful truths 00:02:02.459 --> 00:02:05.920 invite us to see them from different angles. 00:02:06.879 --> 00:02:10.419 And sometimes the simplest ideas open the door 00:02:10.419 --> 00:02:14.539 to enormous worlds. One of the proofs came from 00:02:14.539 --> 00:02:17.580 Euclid's elements from over 2 ,000 years ago. 00:02:18.740 --> 00:02:21.500 Euclid not only proved the Pythagorean theorem, 00:02:21.639 --> 00:02:25.860 but he built an entire geometry from a small 00:02:25.860 --> 00:02:28.819 collection of assumptions about points, lines, 00:02:29.020 --> 00:02:31.840 angles, and shapes. And one assumption concerned 00:02:31.840 --> 00:02:35.879 parallel lines. So for centuries, mathematicians 00:02:35.879 --> 00:02:38.840 wondered, what if that one assumption changed? 00:02:39.919 --> 00:02:43.919 Surely the whole system would collapse. But rather 00:02:43.919 --> 00:02:48.000 than collapsing, entirely new geometries emerge. 00:02:49.000 --> 00:02:54.259 Beautiful geometries. Coherent geometries. Geometries 00:02:54.259 --> 00:02:57.800 where space curves instead of remaining flat. 00:02:58.919 --> 00:03:02.219 In those worlds, triangles no longer have angles 00:03:02.219 --> 00:03:07.759 that sum to 180 degrees. And today, those geometries 00:03:07.759 --> 00:03:11.159 help us describe the shape of the universe itself. 00:03:12.879 --> 00:03:16.979 I'm drawn to this unique combination of simplicity 00:03:16.979 --> 00:03:22.699 and vastness. First, a hidden point unites many 00:03:22.699 --> 00:03:26.180 different proofs, and then a single changed assumption 00:03:26.909 --> 00:03:32.150 creates an entirely new mathematical world. This 00:03:32.150 --> 00:03:36.110 is why mathematics feels beautiful to me. Sometimes 00:03:36.110 --> 00:03:38.889 beauty comes from discovering that many things 00:03:38.889 --> 00:03:42.210 are secretly connected. And sometimes beauty 00:03:42.210 --> 00:03:45.449 comes from discovering how much we can grow from 00:03:45.449 --> 00:03:46.569 one simple idea.