WEBVTT 00:00:02.380 --> 00:00:06.000 Hi, yesterday we explored symmetry by asking 00:00:06.000 --> 00:00:09.679 what remains the same when something is transformed. 00:00:10.740 --> 00:00:15.199 Today, I want to turn that question around. How 00:00:15.199 --> 00:00:19.300 much can one small change alter the shape of 00:00:19.300 --> 00:00:23.620 something? Think about a familiar U -shaped parabola. 00:00:23.699 --> 00:00:25.980 You know, the kind of curve that you might see 00:00:25.980 --> 00:00:29.839 when a ball is tossed through the air. We can 00:00:29.839 --> 00:00:32.479 change one small part of the equation behind 00:00:32.479 --> 00:00:36.219 that curve and make the parabola taller and narrower. 00:00:36.880 --> 00:00:40.179 Another small change makes it wider and flatter. 00:00:40.679 --> 00:00:44.100 A different change moves the entire curve to 00:00:44.100 --> 00:00:47.820 the right or left. And placing a single negative 00:00:47.820 --> 00:00:50.920 sign in front of the equation turns the whole 00:00:50.920 --> 00:00:55.840 shape upside down. Now, on one hand, this is 00:00:55.840 --> 00:01:00.740 basic math. and it may seem routine and not interesting. 00:01:01.539 --> 00:01:05.299 But when you take a step back, I find that in 00:01:05.299 --> 00:01:09.840 some ways, this really is remarkable. The changes 00:01:09.840 --> 00:01:13.739 within the equation may look small, a number, 00:01:13.879 --> 00:01:17.799 a sign, or a symbol, but they can transform the 00:01:17.799 --> 00:01:22.319 entire shape that appears. There is not just 00:01:22.319 --> 00:01:28.079 one parabola, there is an entire family. of parabolas. 00:01:29.180 --> 00:01:31.760 And this is one of the beautiful connections 00:01:31.760 --> 00:01:36.599 between algebra and geometry. Algebra communicates 00:01:36.599 --> 00:01:40.680 through numbers and symbols. Geometry communicates 00:01:40.680 --> 00:01:44.719 through lines, curves, and shapes. They may appear 00:01:44.719 --> 00:01:48.299 to be two different languages, but a graph allows 00:01:48.299 --> 00:01:52.280 them to speak to one another. The equation tells 00:01:52.280 --> 00:01:56.180 us what is happening beneath the surface. The 00:01:56.180 --> 00:02:00.659 graph allows us to see the result. Without the 00:02:00.659 --> 00:02:04.140 graph, it may be easy to overlook the importance 00:02:04.140 --> 00:02:08.520 of one small change. But once we see the shape, 00:02:08.659 --> 00:02:13.500 its effect becomes obvious. We can play with 00:02:13.500 --> 00:02:16.500 different parts of this equation and literally 00:02:16.500 --> 00:02:20.599 see the family members emerge, all with different 00:02:20.599 --> 00:02:24.840 shapes. I wonder if something similar happens 00:02:24.840 --> 00:02:30.240 outside of math. In his book Creativity, Inc., 00:02:30.240 --> 00:02:34.919 Pixar co -founder Ed Catmull describes the table 00:02:34.919 --> 00:02:38.340 that they use for important meetings. It was 00:02:38.340 --> 00:02:41.039 long, narrow, and large enough to seat everyone. 00:02:41.400 --> 00:02:44.379 In one sense, it served its purpose perfectly. 00:02:45.180 --> 00:02:48.939 But its shape was influencing the conversation 00:02:48.939 --> 00:02:54.259 in a subtle but very real way. The leaders typically 00:02:54.259 --> 00:02:57.120 sat near the middle where they could easily see 00:02:57.120 --> 00:03:00.180 and speak with one another. But people with less 00:03:00.180 --> 00:03:04.159 authority sat towards the ends, further from 00:03:04.159 --> 00:03:08.280 the center of the discussion. Now, nobody had 00:03:08.280 --> 00:03:11.379 deliberately decided that the people at the ends 00:03:11.379 --> 00:03:16.259 were less important. Yet, the shape of the table 00:03:16.259 --> 00:03:20.280 communicated that message. And Catmull finally 00:03:20.280 --> 00:03:23.879 recognized the problem. after attending a meeting 00:03:23.879 --> 00:03:28.400 around a smaller square table. Eye contact was 00:03:28.400 --> 00:03:31.840 easier, the conversation flowed more naturally, 00:03:32.060 --> 00:03:36.960 and everyone felt closer to the center. So, Pixar 00:03:36.960 --> 00:03:41.560 replaced the long table with a square one. Now, 00:03:41.560 --> 00:03:45.139 the people had not changed, the purpose of the 00:03:45.139 --> 00:03:48.259 meeting had not changed, but changing the shape 00:03:48.259 --> 00:03:52.770 of the table help change the shape of the conversation. 00:03:54.090 --> 00:03:58.629 And I find this example powerful because the 00:03:58.629 --> 00:04:03.650 table seemed like a neutral object. Yet beneath 00:04:03.650 --> 00:04:07.949 the surface, it was influencing who felt important, 00:04:08.310 --> 00:04:12.389 who felt included, and who felt free to speak. 00:04:14.490 --> 00:04:17.870 Sometimes the structures shaping our relationships 00:04:18.410 --> 00:04:21.949 are so familiar that we no longer notice them? 00:04:23.629 --> 00:04:27.810 Who sits at the center? Who is pushed toward 00:04:27.810 --> 00:04:33.009 the edge? And what might change if we gave everyone 00:04:33.009 --> 00:04:38.389 a place where they could be seen, heard, and 00:04:38.389 --> 00:04:40.449 most importantly, valued?