WEBVTT 00:00:02.120 --> 00:00:05.759 Hi. So far this week, we've explored infinity 00:00:05.759 --> 00:00:09.660 through numbers and infinite processes. In a 00:00:09.660 --> 00:00:14.560 sense, we chose infinity as our input. We chose 00:00:14.560 --> 00:00:18.879 the prime numbers. We chose infinite sums. But 00:00:18.879 --> 00:00:23.000 today, infinity appears in a different way. This 00:00:23.000 --> 00:00:26.620 time, it was already there waiting to be discovered. 00:00:27.140 --> 00:00:32.259 Today, we witness infinity appearing inside geometry 00:00:32.259 --> 00:00:36.679 itself. In a weird sort of way, I think of it 00:00:36.679 --> 00:00:42.380 as infinity will impose itself on us and we must 00:00:42.380 --> 00:00:46.280 decide how to deal with it. Let's begin with 00:00:46.280 --> 00:00:50.119 a simple square. Imagine a square with side length 00:00:50.119 --> 00:00:54.219 one. Now draw a diagonal from one corner to the 00:00:54.219 --> 00:00:58.079 opposite corner. What is the length of that diagonal? 00:00:59.140 --> 00:01:02.200 Using the Pythagorean theorem, The answer is 00:01:02.200 --> 00:01:05.120 the square root of the sum of 1 squared plus 00:01:05.120 --> 00:01:09.260 1 squared. So the diagonal has length square 00:01:09.260 --> 00:01:13.959 root of 2. Now at first, that may not seem unusual. 00:01:14.260 --> 00:01:17.040 We have a calculator button for this problem 00:01:17.040 --> 00:01:21.140 for Pete's sake. But the intriguing part is we 00:01:21.140 --> 00:01:25.540 cannot write square root of 2 as a fraction of 00:01:25.540 --> 00:01:34.959 integers. Forever. Not only do the digits continue 00:01:34.959 --> 00:01:40.560 without end, they never repeat. An infinite decimal 00:01:40.560 --> 00:01:46.799 hidden inside a simple square. Cool. Okay, now 00:01:46.799 --> 00:01:49.280 let's consider another simple shape, a circle. 00:01:49.900 --> 00:01:53.019 If you take the circumference of any circle and 00:01:53.019 --> 00:01:56.159 divide it by its diameter, you always get the 00:01:56.159 --> 00:02:01.340 same number. And that number begins 3 .14159 00:02:01.340 --> 00:02:05.840 dot dot dot. And it continues forever. And we 00:02:05.840 --> 00:02:09.819 call this number pi. And just like square root 00:02:09.819 --> 00:02:14.139 of 2, the decimal expansion of pi never ends 00:02:14.139 --> 00:02:20.259 and never repeats. So today, infinity is not 00:02:20.259 --> 00:02:23.800 showing up because we invented an infinite process. 00:02:24.620 --> 00:02:28.169 It emerges naturally. from shapes themselves, 00:02:28.550 --> 00:02:34.530 from space itself. And historically, this was 00:02:34.530 --> 00:02:38.689 shocking. The ancient Greeks believed that all 00:02:38.689 --> 00:02:42.629 lengths could be written as fractions. And then 00:02:42.629 --> 00:02:46.590 square root of two appeared and shattered that 00:02:46.590 --> 00:02:50.509 assumption. Infinity revealed that reality was 00:02:50.509 --> 00:02:54.430 deeper than they imagined. And that's what makes 00:02:54.430 --> 00:02:57.569 today... Such an important step in our journey. 00:02:58.110 --> 00:03:02.810 Infinity is no longer just an abstract idea that 00:03:02.810 --> 00:03:07.370 is connected to counting or adding forever. Now 00:03:07.370 --> 00:03:13.689 it appears woven into geometry itself, into circles, 00:03:13.930 --> 00:03:18.449 into squares, into the structure of the world 00:03:18.449 --> 00:03:23.590 around us. And once again, humanity did not invent 00:03:23.590 --> 00:03:28.610 this. We discovered it. People across centuries 00:03:28.610 --> 00:03:32.569 and cultures kept encountering these same strange 00:03:32.569 --> 00:03:37.889 truths hidden inside mathematics. And every time 00:03:37.889 --> 00:03:42.129 they looked deeper, more structure appeared beneath 00:03:42.129 --> 00:03:48.030 the surface. Infinity keeps doing that. It takes 00:03:48.030 --> 00:03:51.349 something simple, a square, a circle, a diagonal, 00:03:51.509 --> 00:03:54.580 and reveals... There was something far deeper 00:03:54.580 --> 00:04:00.520 hidden all along. So when I think about infinity 00:04:00.520 --> 00:04:05.259 appearing inside basic shapes, always existing 00:04:05.259 --> 00:04:09.419 and never created, I just cannot help but wonder, 00:04:09.560 --> 00:04:13.060 where did this come from?