WEBVTT 00:00:03.529 --> 00:00:06.910 Hi, I have a friend who has transitioned through 00:00:06.910 --> 00:00:10.449 many stages of life, from being married to raising 00:00:10.449 --> 00:00:13.029 children and eventually becoming a grandparent. 00:00:13.449 --> 00:00:16.829 She once described those transitions as moving 00:00:16.829 --> 00:00:20.649 from being the picture to becoming the frame. 00:00:21.129 --> 00:00:26.030 I think that perspective is beautiful. Ironically, 00:00:26.109 --> 00:00:28.870 I see something similar illustrated in math. 00:00:29.769 --> 00:00:34.530 Think of a simple circle. with radius one. Now 00:00:34.530 --> 00:00:37.409 we could spend all day exploring the wonderful 00:00:37.409 --> 00:00:41.070 properties of the circle itself, its circumference, 00:00:41.210 --> 00:00:44.390 its area, and eventually the beautiful relationship 00:00:44.390 --> 00:00:49.509 involving pi. But instead, let's allow the circle 00:00:49.509 --> 00:00:54.369 to become the frame. Let it become the boundary 00:00:54.369 --> 00:00:58.450 of a little mathematical world where new ideas 00:00:58.450 --> 00:01:02.939 can grow. Imagine placing the circle on an x 00:01:02.939 --> 00:01:05.980 -y coordinate plane with its center at the origin. 00:01:07.140 --> 00:01:10.579 Now, draw a right triangle inside the circle 00:01:10.579 --> 00:01:14.420 so that the hypotenuse stretches from the center 00:01:14.420 --> 00:01:19.400 to a point on the circle. Now, because the radius 00:01:19.400 --> 00:01:22.719 of the circle is 1, the hypotenuse of the triangle 00:01:22.719 --> 00:01:26.780 is also 1. And if the point on the circle is 00:01:26.780 --> 00:01:31.170 x -y, Then the Pythagorean theorem tells us that 00:01:31.170 --> 00:01:37.709 x squared plus y squared equals 1. Already, we 00:01:37.709 --> 00:01:42.530 see geometry, algebra, and circles all connecting 00:01:42.530 --> 00:01:46.790 together. But now let's add another dimension. 00:01:47.390 --> 00:01:51.430 Suppose we know the angle between the hypotenuse 00:01:51.430 --> 00:01:57.010 and the x -axis. Suddenly, we need trigonometry. 00:01:58.030 --> 00:02:02.469 the height of the triangle becomes the sine of 00:02:02.469 --> 00:02:07.709 the angle. But now, let's really open things 00:02:07.709 --> 00:02:11.370 up. Rather than studying one triangle at a time, 00:02:11.530 --> 00:02:16.009 imagine every possible triangle that is formed 00:02:16.009 --> 00:02:21.030 as the angle changes continuously around the 00:02:21.030 --> 00:02:25.150 circle. And if we graph the height of each triangle, 00:02:26.030 --> 00:02:32.430 we create a sine wave. Now a static triangle 00:02:32.430 --> 00:02:38.729 suddenly becomes a flowing curve. Studying a 00:02:38.729 --> 00:02:42.330 single triangle is only the starting point, but 00:02:42.330 --> 00:02:46.030 what transforms our thinking is placing the triangle 00:02:46.030 --> 00:02:49.870 inside the circle and letting the two shapes 00:02:49.870 --> 00:02:55.300 dance together. And out of that dance, emerges 00:02:55.300 --> 00:02:59.099 one of the foundational patterns in all of math. 00:03:00.479 --> 00:03:03.939 And of course this sine wave is not just an abstract 00:03:03.939 --> 00:03:09.340 mathematical curiosity. It is a theory born for 00:03:09.340 --> 00:03:13.800 reality. It helps describe sound waves and light 00:03:13.800 --> 00:03:18.979 waves, vibrations, electricity, and much more. 00:03:20.099 --> 00:03:24.219 It is literally... woven into the invisible structure 00:03:24.219 --> 00:03:28.979 behind many of our senses. When Eddie Wu says 00:03:28.979 --> 00:03:33.039 mathematics develops a new sense within us, I 00:03:33.039 --> 00:03:36.599 think this is part of what he means. We begin 00:03:36.599 --> 00:03:40.819 seeing how simple shapes connect and grow into 00:03:40.819 --> 00:03:44.979 entirely new dimensions of understanding. A circle 00:03:44.979 --> 00:03:50.240 is no longer just a circle. It becomes a framework 00:03:50.240 --> 00:03:55.400 for discovering triangles, waves, motions, and 00:03:55.400 --> 00:04:00.620 patterns hidden underneath reality itself. Now, 00:04:00.680 --> 00:04:04.460 in my creative world, I think we are all wired 00:04:04.460 --> 00:04:08.860 a bit like that circle. It is good to understand 00:04:08.860 --> 00:04:12.719 who we are and how we are uniquely and wonderfully 00:04:12.719 --> 00:04:17.100 made. But life grows into new dimensions. when 00:04:17.100 --> 00:04:20.819 what we have been given becomes a blessing to 00:04:20.819 --> 00:04:25.899 others. Over time, we begin realizing that our 00:04:25.899 --> 00:04:29.819 role is not only to be the picture. Sometimes, 00:04:30.100 --> 00:04:34.279 we are also becoming the frame, creating space 00:04:34.279 --> 00:04:38.759 for others to grow, to flourish, and to discover 00:04:38.759 --> 00:04:44.199 beauty themselves. Yesterday, we talked about 00:04:44.199 --> 00:04:47.790 how transformation often begins with small beginnings. 00:04:48.850 --> 00:04:51.910 So, maybe one small step in this transformation 00:04:51.910 --> 00:04:55.930 is simply realizing that we can become a frame 00:04:55.930 --> 00:04:59.629 for others. Not to control them, of course, but 00:04:59.629 --> 00:05:02.990 to support them, to encourage them, and help 00:05:02.990 --> 00:05:05.230 their unique shape come alive.