WEBVTT Kind: captions Language: en 00:00:01.280 --> 00:00:07.460 Okay, uh, hello there everyone I'm Parker Glynn Adey I'm an assistant professor at the University of Toronto Mississauga 00:00:08.080 --> 00:00:11.219 And I want to tell you a little bit about my work on geometries 00:00:11.820 --> 00:00:15.200 mathematics for the eyes and hands, uh in this presentation 00:00:15.200 --> 00:00:21.600 We usually hand out physical objects to play with and then we sort of have a bit of a conversation about what thoughts those 00:00:22.240 --> 00:00:24.000 uh objects 00:00:24.000 --> 00:00:25.840 Bring to mind 00:00:25.840 --> 00:00:27.920 but uh, in this case, we'll be we'll be doing with 00:00:28.720 --> 00:00:33.840 Geogebra apps, so I'm happy that you're here. I'm happy you've come out to watch this talk 00:00:34.480 --> 00:00:38.399 And uh, I hope you are sparked with creativity and joy 00:00:38.960 --> 00:00:43.840 And if you do wind up watching this, please uh, tweet out drop me a message on Twitter 00:00:44.480 --> 00:00:46.480 where I can be found 00:00:47.980 --> 00:00:52.559 @pgadey, okay. So I look forward to hearing any thoughts you might have coming out of this 00:00:53.840 --> 00:00:55.840 Uh, yeah, let's begin 00:00:56.800 --> 00:00:58.800 So I'd like to tell you a little bit about 00:00:59.660 --> 00:01:01.759 Geometry, uh, this is in the context 00:01:02.399 --> 00:01:09.838 Of a fourth year mathematics course that I teach at the University of Toronto Mississauga, uh, but it can be done 00:01:10.640 --> 00:01:15.839 At any level all of these activities can be brought down to the high school level brought down to the public school level 00:01:16.320 --> 00:01:19.519 Um, it's really about meeting your students where they're at 00:01:21.040 --> 00:01:22.400 So 00:01:22.400 --> 00:01:25.199 I'm going to talk a little bit about inquiry-based learning the sort of the 00:01:25.740 --> 00:01:31.680 Pedagogical framework that I come from is it's inquiry-based learning where students engage 00:01:32.320 --> 00:01:34.639 with meaningful mathematical tasks, so 00:01:35.520 --> 00:01:38.000 we mean meaningful here in sort of two senses one that 00:01:38.720 --> 00:01:40.959 It should mean something for the students 00:01:42.159 --> 00:01:45.679 You know, they should understand what's going on. They should feel engaged with what's going on 00:01:46.240 --> 00:01:51.939 And also it should tie into the sort of big ideas of mathematics, right? It should be meaningfully 00:01:53.340 --> 00:01:54.399 mathematical 00:01:54.399 --> 00:02:02.079 And uh should you know have broader connections. So we wanted students to engage deeply with meaningful mathematical tasks 00:02:04.479 --> 00:02:08.559 We want students to work collaboratively to process mathematical ideas 00:02:09.679 --> 00:02:12.079 We want to work in groups and share 00:02:13.280 --> 00:02:17.599 one of the things that I think about is really special about geometry is that we can 00:02:19.040 --> 00:02:21.120 So we can hold the objects in our hands 00:02:21.680 --> 00:02:26.660 We can draw objects for each other. We can pass a mathematical object around 00:02:27.280 --> 00:02:29.360 And uh examine it from different angles 00:02:30.000 --> 00:02:31.360 and uh 00:02:31.360 --> 00:02:38.080 Sort of we can create a kind of a knowing in a small group of people. So I think that working collaboratively is very important 00:02:39.599 --> 00:02:43.599 Uh, we should we should focus on student thinking right we should see where students are at 00:02:44.239 --> 00:02:48.399 Um as teachers, right? This is sort of a key thing, but it's just important to 00:02:49.280 --> 00:02:52.959 To remember that we should ask students what they believe about these things. Um, 00:02:54.000 --> 00:02:56.319 geometry is sort of special and then a lot of people have 00:02:56.860 --> 00:03:00.559 Intuitions about geometry and we can we can point at things and say stuff 00:03:01.200 --> 00:03:07.039 and uh, this makes the conversation between instructors and students very fruitful and engaging 00:03:09.120 --> 00:03:10.400 And then 00:03:10.400 --> 00:03:18.319 We should always foster equity right? So when we're designing our courses, uh, everyone has ideas about geometry, right? It's a sort of a universal 00:03:19.920 --> 00:03:23.360 Given that we can we can see pattern we can see shape 00:03:23.760 --> 00:03:29.679 So I think that uh, there's a lot of room for fostering equity through this and i'm going to talk a little bit about different 00:03:31.340 --> 00:03:33.340 Mathematical approaches to doing that 00:03:34.239 --> 00:03:39.119 So what is special about geometry right take a moment sort of answer this for yourself 00:03:41.120 --> 00:03:46.480 What's what do you think is special about geometry um and then i'll share a couple of my thoughts 00:03:58.879 --> 00:04:00.879 Right so sort of to reiterate those points 00:04:01.840 --> 00:04:06.259 Um, what's special about geometry? We can make meaningful mathematical tasks 00:04:06.959 --> 00:04:10.899 Uh about pictures things that are very in front of us very accessible 00:04:13.340 --> 00:04:17.380 Um students can share their own mathematical thoughts because we have a lot of sort of mathematical intuitions 00:04:19.600 --> 00:04:25.040 Uh the conversations around geometry often end up being very fruitful and we're going to talk a little bit about this equity piece 00:04:25.680 --> 00:04:26.560 as well 00:04:26.560 --> 00:04:27.120 uh 00:04:27.120 --> 00:04:33.359 So those are some some thoughts about why geometry is particularly special. I want to share one thing that gives you the feel 00:04:33.840 --> 00:04:35.840 What I think with geometry 00:04:37.040 --> 00:04:39.040 I think that geometry is going on right here 00:04:41.360 --> 00:04:46.479 Take a moment and kind of look at the different shapes different things that you're seeing in this picture 00:04:46.960 --> 00:04:49.220 Look at different ways that people are engaging 00:04:50.639 --> 00:04:52.639 in this picture 00:05:01.680 --> 00:05:04.639 I always like to draw people's attention over to this corner of the picture 00:05:06.240 --> 00:05:08.720 Where we've got a student volunteer here 00:05:09.440 --> 00:05:15.999 Who's really focusing on this this child right. The child is absolutely absorbed in their coloring task 00:05:16.960 --> 00:05:20.820 This coloring task i'll talk a little bit about later on in the presentation 00:05:21.520 --> 00:05:26.160 But we can see that there's a some kind of a mathematical experience going on here 00:05:27.580 --> 00:05:29.580 another child 00:05:30.080 --> 00:05:32.240 Assembling some blocks. They're trying to make 00:05:33.360 --> 00:05:37.039 their own pattern and then we've got all of these, uh, 00:05:37.979 --> 00:05:40.179 mathematical tangible objects 00:05:40.880 --> 00:05:43.999 Uh, you know, people are holding these things thinking about them 00:05:44.639 --> 00:05:50.559 Um, they're a little bit excited. You know, they've got all these fun colorful things and I think this is one of the most 00:05:51.199 --> 00:05:54.159 Uh exciting things about geometry. It's tangible 00:05:54.960 --> 00:06:00.639 It's playful and it's often colorful, right it uh sort of has some properties that our usual 00:06:01.280 --> 00:06:06.819 Uh, mathematical subjects, um sort of a lack right? We sort of lack these tangible objects 00:06:09.039 --> 00:06:11.039 So, I think this is sort of a very special 00:06:11.199 --> 00:06:15.459 Thing that geometry can do for us now. I want to talk about where geometry is at in the curriculum 00:06:21.840 --> 00:06:25.919 Right geometry has been sort of uh, relegated to the really the 00:06:26.639 --> 00:06:33.059 Flanks of the curriculum. It's uh sort of looked down upon as kind of being 19th century stuff 00:06:34.880 --> 00:06:42.079 You know actual euclidean geometry, we don't see very much of it. we talk about, you know, classifications of different shapes 00:06:42.720 --> 00:06:45.039 Uh, we do some sort of two column proofs 00:06:45.919 --> 00:06:48.478 but there's so much more to geometry than 00:06:49.039 --> 00:06:51.839 uh, this kind of dusty old picture would suggest and I want to 00:06:52.400 --> 00:06:57.039 Come back and talk about why this specific thing would be very interesting to play with with students 00:06:57.360 --> 00:06:59.360 and talk about some of the ways that we 00:06:59.440 --> 00:07:01.440 play with it in our class 00:07:04.400 --> 00:07:08.880 Just uh just as a bit of a sort of historical aside, uh, this model was made in 00:07:09.419 --> 00:07:10.479 1864 00:07:10.479 --> 00:07:12.479 Came to Brown University in 1973 00:07:13.360 --> 00:07:14.639 and uh 00:07:14.639 --> 00:07:16.639 that's sort of when it left the 00:07:17.340 --> 00:07:20.560 university scene it's been sitting in the Smithsonian ever since this was a 00:07:21.280 --> 00:07:22.880 uh 00:07:22.880 --> 00:07:28.719 A model built to demonstrate a particular concept in geometry. And uh 00:07:29.360 --> 00:07:33.440 Since that fell out of the curriculum, it hasn't seen the light of day. Okay 00:07:36.000 --> 00:07:41.539 So I want to show you what we can do with that in modern times what that looks like these days 00:07:59.680 --> 00:08:00.879 Okay 00:08:00.879 --> 00:08:05.439 So i'm going to go over to screen sharing you get this lovely fractal pattern 00:08:05.919 --> 00:08:10.258 Uh, but the object that I was describing moments ago is what's called the Peaucellier-Lipkin 00:08:11.340 --> 00:08:13.679 linkage and if we were 00:08:14.400 --> 00:08:17.679 In the same space, I would hand you a physical copy of these I have 00:08:17.979 --> 00:08:21.598 A 3d printed model of one and I want to show you what it does 00:08:24.160 --> 00:08:26.719 So when we move this point here 00:08:32.159 --> 00:08:37.059 What do you observe going on when you see this thing move 00:08:43.360 --> 00:08:47.539 Or you might not start moving that point you might start moving this red point 00:09:01.120 --> 00:09:08.799 Right, so what do we see happening here? Right. We see sort of some kind of relationship between two fundamental shapes this circle 00:09:10.080 --> 00:09:12.000 And this line. 00:09:12.000 --> 00:09:17.200 And it's very mysterious why this is happening. What what do the red bars have to do with this? What do the 00:09:17.839 --> 00:09:19.839 Yellow bars have to do with this? 00:09:21.339 --> 00:09:23.339 Right this was um 00:09:24.000 --> 00:09:26.000 this linkage, uh 00:09:26.160 --> 00:09:32.880 Was sort of a mathematical construct but it had physical applications. So in the uh 1800s, it was a major 00:09:33.600 --> 00:09:37.040 question in mechanics how to convert circular motion 00:09:37.839 --> 00:09:45.839 Into linear motion to say to take a driving pistol at this, um this red point and convert it into a circular motion 00:09:46.399 --> 00:09:48.559 For a wheel or something at this blue point 00:09:49.680 --> 00:09:50.880 okay, so 00:09:50.880 --> 00:09:52.000 uh 00:09:52.000 --> 00:09:53.680 this was this was 00:09:53.680 --> 00:09:59.039 Open people weren't sure how to solve this. Uh this sort of geometry problem. Um, 00:10:00.080 --> 00:10:07.039 Watt, the inventor of steam engines, uh worked on it and wasn't able to come up with a solution. He was able to convert 00:10:07.760 --> 00:10:12.399 Circular motion into uh elliptical motion, right. So if you move the 00:10:13.120 --> 00:10:15.760 Blue points you get an ellipse at the red point 00:10:17.279 --> 00:10:22.578 But it took a uh, Russian graduate student Lipkin and a French mathematician Peaucellier 00:10:23.279 --> 00:10:25.279 uh to invent this linkage 00:10:25.760 --> 00:10:30.799 So usually in a classroom, I would bring in a physical model of this and we'd play around. Um 00:10:31.440 --> 00:10:32.800 and what's 00:10:32.800 --> 00:10:35.459 Interesting about this. Is that our geometry 00:10:36.240 --> 00:10:43.140 Doesn't get us ready to describe what's going on. I've taken this model to a math seminar of undergraduates 00:10:43.920 --> 00:10:46.719 and uh asked them to try and explain what's going on and 00:10:47.519 --> 00:10:52.559 About two hours later. We fill up a couple chalkboards and we still have no idea. So there's 00:10:53.760 --> 00:10:55.740 We really need some 00:10:55.740 --> 00:10:59.140 Classical euclidean geometry to handle this thing. Um 00:11:00.160 --> 00:11:04.399 Giving this talk on the line now. We we can actually play with the model at a sort of a deeper level 00:11:04.399 --> 00:11:07.999 Which is I think very interesting we can adjust the radii of these. Um 00:11:09.200 --> 00:11:12.559 of these things so we can move the uh, yellow 00:11:13.339 --> 00:11:14.480 bars 00:11:14.480 --> 00:11:16.480 so what happens if we make the 00:11:16.800 --> 00:11:18.800 Yellow bars, very big? 00:11:19.360 --> 00:11:21.360 right 00:11:36.560 --> 00:11:38.560 Right, how does the geometry change now 00:11:41.339 --> 00:11:43.339 All right, what do we notice 00:11:43.920 --> 00:11:49.380 Uh, we can change these relative lengths we can we can increase the red lengths as well 00:11:56.000 --> 00:11:58.500 Right, so one advantage to doing this in geogebra 00:11:59.040 --> 00:12:04.639 Instead of a physical model is that we have a fair bit more control we can play around with the various 00:12:05.519 --> 00:12:07.519 parts of the mechanism 00:12:09.519 --> 00:12:13.999 And see how that affects our geometric outcomes 00:12:18.320 --> 00:12:21.459 Right, so one question that this naturally provokes 00:12:22.160 --> 00:12:25.600 Is how much of the blue circle can we get? 00:12:26.399 --> 00:12:34.078 Right. How big can that blue circle get and what would the relationship between that blue circle and that red line be? 00:12:35.920 --> 00:12:38.880 Um as we as we kept varying things 00:12:40.240 --> 00:12:42.579 Okay, so I want to go back to our our slides 00:13:00.160 --> 00:13:02.959 Okay, and we can see that um 00:13:04.639 --> 00:13:07.599 This Peaucellier-Lipkin linkage it is is much more 00:13:08.300 --> 00:13:11.039 Alive and animated than this old 00:13:11.680 --> 00:13:13.680 model in a 00:13:13.839 --> 00:13:15.919 in the Smithsonian gives us right, so 00:13:16.480 --> 00:13:22.420 There's lots to explore with this. There's rich mathematical tasks that could come out of playing with this mechanism 00:13:23.040 --> 00:13:25.040 and uh 00:13:25.040 --> 00:13:29.519 Bringing in a physical copy of these is one thing we can do we can also bring in a virtual copy 00:13:30.220 --> 00:13:37.600 And the virtual copy is a little bit more flexible that we can adjust various bars various lengths, okay 00:13:40.560 --> 00:13:45.440 Another uh object that's very very satisfying to play with and this is a task that you can do 00:13:46.240 --> 00:13:49.199 with any grade, you know all the way from 00:13:49.920 --> 00:13:54.240 Public to post-secondary is you can play with kaleidoscopes. There's lots to 00:13:54.880 --> 00:13:56.720 um explore 00:13:56.720 --> 00:13:58.720 with regards to kaleidoscopes 00:13:59.600 --> 00:14:01.600 They're very very very engaging 00:14:03.199 --> 00:14:08.819 And they get us thinking about symmetry they get us thinking about geometry they get us thinking about transformations 00:14:11.199 --> 00:14:12.720 So, this is a 00:14:12.720 --> 00:14:16.160 physical kaleidoscope and i've often given the talk where 00:14:16.800 --> 00:14:17.680 we 00:14:17.680 --> 00:14:18.560 uh 00:14:18.560 --> 00:14:20.560 We pass around a kaleidoscope 00:14:20.880 --> 00:14:22.800 Various people look at it 00:14:22.800 --> 00:14:29.839 Then we classify all the kaleidoscopes in the plane. We sort of prove a theorem that says what kaleidoscopes can exist 00:14:31.279 --> 00:14:37.279 We talk about the geometry and symmetry of the kaleidoscope and then we hand the kaleidoscopes around again 00:14:37.839 --> 00:14:39.839 and look at them again with 00:14:40.000 --> 00:14:41.680 fresh eyes 00:14:41.680 --> 00:14:43.680 When we look at things with fresh eyes 00:14:44.800 --> 00:14:51.199 And a little bit more mathematical sophistication we can often see a lot more of what's going on. So 00:14:52.399 --> 00:14:58.638 To take an example. We see that this this has this beautiful pattern, but we uh need to investigate 00:14:58.720 --> 00:15:00.720 You know, what? Are these angles here? 00:15:01.279 --> 00:15:03.439 Right when we prove the classification theorem 00:15:05.600 --> 00:15:07.940 We get talking about what these angles are 00:15:15.040 --> 00:15:17.519 What triangle do we really have 00:15:21.680 --> 00:15:24.659 In this region? and uh 00:15:26.000 --> 00:15:31.039 Right. Does this triangle seem familiar to us? What can we deduce? 00:15:31.759 --> 00:15:35.279 About those angles. Um, it turns out this is a 00:15:35.920 --> 00:15:38.639 30 60 90 triangle one of the special triangles 00:15:39.839 --> 00:15:41.839 We can see that 00:15:41.839 --> 00:15:44.239 If we look here if we look at it in its mirror copy 00:15:47.199 --> 00:15:53.839 When we look the first mirror copy of that triangle 00:15:57.120 --> 00:16:04.419 We see we've got an equilateral triangle, right so this angle here has to be this angle here 00:16:08.880 --> 00:16:16.079 And then that gives us that this has to be the same as well, right, we've got a 60 60 60 triangle 00:16:17.440 --> 00:16:25.139 Okay, so there's lots to there's lots to see and vision and kaleidoscopes going into our virtual environment now 00:16:37.600 --> 00:16:39.600 Right we can do the same kind of thing 00:16:40.320 --> 00:16:42.320 uh online virtually 00:16:42.959 --> 00:16:46.879 Right, we can play with with kaleidoscopes and see what happens 00:16:48.639 --> 00:16:50.719 Right, how do those symmetries repeat 00:16:53.120 --> 00:16:58.799 Uh when we go into a reflection right so notice that these 00:17:01.519 --> 00:17:03.519 Purple swirls 00:17:06.720 --> 00:17:08.720 They get copied into all six 00:17:09.839 --> 00:17:11.839 congruent regions 00:17:12.319 --> 00:17:14.319 And one thing I like about this particular app 00:17:16.000 --> 00:17:20.160 Is that every even reflection is colored the same way right so the 00:17:20.799 --> 00:17:25.839 Uh, the first reflection is colored gray. The second reflection is colored purple 00:17:26.240 --> 00:17:31.679 The third reflection is colored gray. The fourth is colored purple. The fifth is colored gray 00:17:31.760 --> 00:17:35.599 The six is colored purple and so on. Okay, this is this is uh, 00:17:36.160 --> 00:17:37.919 quite nice 00:17:37.919 --> 00:17:41.379 Okay, we can play around with different amounts of symmetry 00:17:46.720 --> 00:17:52.640 Right, so again the the the virtual model gives us a little bit more flexibility than the physical model 00:17:55.600 --> 00:17:57.699 Right we can have four-fold symmetry 00:18:04.400 --> 00:18:08.499 We can have eightfold symmetry 00:18:20.559 --> 00:18:25.519 All right, so there's a lot we can do um with these symmetric regions, um 00:18:26.160 --> 00:18:30.079 And I encourage you to uh play around with these a little bit. Um, 00:18:30.640 --> 00:18:32.640 When we're when we're back in 00:18:32.880 --> 00:18:37.919 Uh physical settings. I highly encourage you to get a pair of mirrors 00:18:38.480 --> 00:18:40.880 and bring those to class and experiment with 00:18:42.960 --> 00:18:49.939 Showing your students how the image changes as you vary the angle between the mirrors 00:18:57.679 --> 00:18:59.260 Okay, so 00:18:59.260 --> 00:19:02.079 Kaleidoscopes are a physical object that we can 00:19:02.640 --> 00:19:09.539 Bring in we can ask students to inquire into them. Uh, there's all sorts of connections throughout the curriculum 00:19:10.080 --> 00:19:11.840 and 00:19:11.840 --> 00:19:16.640 I find that we can really see them with with new mathematical eyes when we 00:19:18.559 --> 00:19:20.559 When we ask students to 00:19:21.200 --> 00:19:24.260 Investigate them and then do some kind of math surrounding 00:19:25.200 --> 00:19:29.280 the geometry and symmetry and then look through them again the the experience of 00:19:29.840 --> 00:19:31.840 Sort of leveling up and noticing that 00:19:32.320 --> 00:19:33.200 um 00:19:33.200 --> 00:19:36.740 Say that equilateral triangle is often very powerful for students 00:19:40.720 --> 00:19:42.720 Okay 00:19:44.480 --> 00:19:46.480 This is a piece of art by 00:19:47.840 --> 00:19:49.840 Escher very famous 00:19:50.320 --> 00:19:53.200 artist not a mathematician. Um, but 00:19:53.840 --> 00:19:56.980 Very very deeply aware of the significance and power of mathematics 00:19:57.760 --> 00:19:58.799 and 00:19:58.799 --> 00:20:00.799 What I love about this painting 00:20:01.520 --> 00:20:02.880 is that 00:20:02.880 --> 00:20:07.119 It suggests something but it is not clear what. 00:20:08.240 --> 00:20:12.079 Right, we see this and we say okay, there is some kind of pattern here 00:20:13.280 --> 00:20:16.159 I feel that there is some kind of structure here 00:20:17.039 --> 00:20:19.039 But I don't know what structure it is 00:20:19.440 --> 00:20:22.000 And that's a really powerful moment for our students 00:20:23.120 --> 00:20:25.120 Right, we look at it we can 00:20:25.260 --> 00:20:27.280 Inquire we can ask questions 00:20:27.919 --> 00:20:30.959 And it's not clear what the answers to those questions are 00:20:32.400 --> 00:20:37.599 There's some kind of structure. I don't know what it is, but I wish I did so 00:20:38.880 --> 00:20:40.880 You know some things that you can point out 00:20:44.000 --> 00:20:47.839 Notice here, uh, we've got a point of four-fold symmetry 00:20:51.280 --> 00:20:53.280 Right, we've got a 00:20:54.000 --> 00:20:56.000 Cross there and we can rotate 00:20:56.159 --> 00:21:00.399 If we ignore colors, we can rotate 90 degrees about that center 00:21:01.039 --> 00:21:02.799 if we 00:21:02.799 --> 00:21:06.079 Care about colors. We can rotate 180 degrees about that center 00:21:07.120 --> 00:21:15.059 Over here. We've got a three-fold reflection point a three-fold rotation point but it's not a rotation point of the whole picture 00:21:16.880 --> 00:21:23.359 But this turns us on to the notion that there are squares and there are triangles right here we've got the outline of a square 00:21:29.520 --> 00:21:31.599 And then here we've got the outline of a triangle 00:21:38.480 --> 00:21:40.480 And then a square 00:21:45.600 --> 00:21:47.600 And then a triangle 00:21:47.919 --> 00:21:51.439 All right, this pattern repeats all the way out to the boundary of the region 00:21:53.679 --> 00:21:55.679 We can talk about 00:21:56.700 --> 00:22:02.080 Particular colors come together right here. We've got brown and yellow 00:22:03.840 --> 00:22:05.919 Here we've got another point with brown and yellow 00:22:07.440 --> 00:22:09.440 Here we've got another point with brown and yellow 00:22:10.559 --> 00:22:16.079 And all the way out to the boundary again, so there's lots of patterns that we can look for in this painting 00:22:16.880 --> 00:22:18.880 Or sorry lithograph. Um 00:22:19.679 --> 00:22:22.239 And the more we look for them the more we find them. Um, 00:22:23.200 --> 00:22:29.600 This painting was famously commented on by the Canadian geometer. Uh, Donald Coxeter who 00:22:30.460 --> 00:22:37.679 initially when he saw it thought oh, this is sort of a, you know, a cute thing, uh, sort of a mathematical mistake somehow but 00:22:38.559 --> 00:22:45.199 We'll let it fly and he uh, he didn't think much of the mathematics of it and then about 10 years later 00:22:45.280 --> 00:22:50.399 He looked at it again, and he saw that some of his research was actually embedded in this painting and he didn't even know 00:22:51.120 --> 00:22:52.480 um 00:22:52.480 --> 00:22:58.719 He uh, he noticed that there was a uniform tiling of the hyperbolic plane that he had discovered, uh that was evidenced in this 00:22:58.880 --> 00:23:00.880 uh, and that Escher 00:23:01.039 --> 00:23:03.519 uh didn't know that he had created so 00:23:04.640 --> 00:23:07.299 There's mathematics to be discovered at a very high level 00:23:08.400 --> 00:23:10.400 inside this the 00:23:11.039 --> 00:23:15.999 the sort of closest connection to contemporary mathematics kind of the deepest connection 00:23:16.559 --> 00:23:20.019 Uh would be to the notion of uh hyperbolic geometry 00:23:32.000 --> 00:23:34.000 Which we can explore 00:23:34.000 --> 00:23:38.579 Again, using software we can use hyperbolic we can use software to investigate 00:23:39.600 --> 00:23:41.919 uh sort of these hyperbolic structures 00:23:44.880 --> 00:23:49.839 Right so looking at um this we see that there are these sort of straight lines 00:23:52.480 --> 00:23:54.480 And there are these curved lines 00:23:58.000 --> 00:24:05.760 And in a hyperbolic geometry sense, these are in fact all straight lines in the hyperbolic metric 00:24:07.279 --> 00:24:10.559 Okay from the perspective of those fish in 00:24:11.840 --> 00:24:13.840 Escher's work 00:24:18.799 --> 00:24:20.799 The line AB 00:24:21.919 --> 00:24:25.359 This curved line here AB is just as straight 00:24:27.840 --> 00:24:31.840 As the line EC okay, so from a hyperbolic perspective 00:24:32.720 --> 00:24:40.720 Uh circles become straight lines and we can talk about what properties uh circles have that are in common with straight lines 00:24:41.919 --> 00:24:44.879 both in the euclidean context and in the hyperbolic context 00:24:45.520 --> 00:24:47.520 so this is something where a 00:24:48.880 --> 00:24:51.599 Computer visualization can really show us a lot 00:24:53.120 --> 00:24:57.520 Right, how do these things morph right we had just said that EC is a straight line 00:24:58.799 --> 00:25:00.799 But now suddenly it's become a circle 00:25:02.400 --> 00:25:04.719 But as we move back towards the center of the disk 00:25:07.200 --> 00:25:09.200 EC becomes a straight line again 00:25:12.159 --> 00:25:16.719 So there's a lot to explore, um, there's lots of questions that can be asked here 00:25:18.720 --> 00:25:22.020 And having a conversation about what's going on in this demonstration 00:25:22.720 --> 00:25:26.319 Would give us a lot of insight into what's happening in 00:25:27.340 --> 00:25:28.880 um 00:25:28.880 --> 00:25:30.799 in those uh 00:25:30.799 --> 00:25:32.799 The universe where those fish live 00:25:41.039 --> 00:25:48.259 Okay, so the universe of the these fish is governed not by our usual 00:25:49.740 --> 00:25:51.740 euclidean notions of straight 00:25:54.159 --> 00:25:58.399 But rather by a hyperbolic geometry notion of straight where 00:25:59.419 --> 00:26:02.799 Circles can become straight lines and straight lines can become circles 00:26:03.520 --> 00:26:08.239 So there's lots of there's lots of mathematics to investigate there. And uh, I think that 00:26:08.799 --> 00:26:11.038 You know with a famous piece of art like this 00:26:11.200 --> 00:26:17.760 we have an opportunity to have something that's both aesthetically satisfying and mathematically satisfying so very very 00:26:18.480 --> 00:26:20.740 deep source of conversation 00:26:25.919 --> 00:26:29.839 To bring this, um back to the equity piece. Here's another 00:26:30.480 --> 00:26:34.079 uh piece by Escher and this is modeled on a 00:26:35.039 --> 00:26:37.039 Piece of art that he found in the Alhambra 00:26:38.400 --> 00:26:39.840 And 00:26:39.840 --> 00:26:43.120 This is something that can be constructed with ruler and compass. This is um 00:26:43.919 --> 00:26:45.919 straight euclidean geometry 00:26:46.000 --> 00:26:49.119 Uh, but it's also a place to talk about 00:26:50.640 --> 00:26:52.799 Culture and mathematics and the role that 00:26:53.500 --> 00:26:55.599 Mathematics can play in culture. Um 00:26:56.640 --> 00:27:01.279 You know much Islamic art is geometric and we can talk about why 00:27:02.080 --> 00:27:06.640 Uh that culture has evolved art in that way and what 00:27:07.520 --> 00:27:10.079 What insight geometry can give us into 00:27:11.039 --> 00:27:13.278 Art from different cultures and the beliefs of different cultures 00:27:14.320 --> 00:27:17.200 I think this is a very very satisfying piece. There's lots to look at 00:27:18.159 --> 00:27:19.360 in it 00:27:19.360 --> 00:27:21.839 Uh to try and hunt down where there are 00:27:22.559 --> 00:27:25.778 straight lines that connect up different parts of the diagram 00:27:29.679 --> 00:27:33.699 Right or what how many sides does this thing have 00:27:37.919 --> 00:27:43.379 Or you know that what kinds of symmetry does this have does this have a four-fold rotational symmetry 00:27:45.440 --> 00:27:50.000 Right what parts of the diagram move when we when we look at the symmetries of this object 00:27:51.440 --> 00:27:53.440 And how many 00:27:53.440 --> 00:27:55.440 of these 00:27:55.919 --> 00:27:57.919 Black regions are there? 00:28:00.000 --> 00:28:03.839 And what angle must be between them? What are these little angles here? 00:28:05.760 --> 00:28:07.760 Is this small yellow angle 00:28:09.120 --> 00:28:14.799 The same as the angle at the center if we were to start at the center and go out to the tips of those regions 00:28:15.440 --> 00:28:17.440 Would we have the same angle? 00:28:17.919 --> 00:28:24.499 So one can produce photocopies of this diagram and go about investigating these things with rulers and protractors 00:28:25.520 --> 00:28:27.520 and engage with the 00:28:28.640 --> 00:28:34.559 Some classical Islamic art and there's also uh constructions that you can you can do based off this so you can 00:28:35.120 --> 00:28:42.260 Get your compass and protractor and ruler and build out some of these things. Um as a geometry lesson 00:28:47.520 --> 00:28:51.439 There's lots and lots and lots of resources on uh teaching inquiry 00:28:52.159 --> 00:28:58.319 Or sorry teaching geometry with inquiry. I highly encourage you to look up these resources. Um 00:28:59.279 --> 00:29:02.319 All of them have activities that could be used in the public school 00:29:03.360 --> 00:29:05.360 setting the high school setting 00:29:06.880 --> 00:29:09.599 There's lots and lots of lots of things to explore 00:29:10.960 --> 00:29:14.399 In geometry, and it can be explored at many different levels. Okay, so 00:29:15.520 --> 00:29:18.179 I want to close off this uh this presentation 00:29:18.880 --> 00:29:25.699 Uh, just with a reminder that you know students come first, uh, there's many many things that we can do with students 00:29:27.279 --> 00:29:31.679 To help them explore geometry to find their way into geometry. Um, 00:29:32.720 --> 00:29:35.360 It's a satisfying exercise to both 00:29:36.240 --> 00:29:38.240 notice how happy these people are 00:29:38.720 --> 00:29:45.839 How much is going on and how many geometric objects there are on the table? Where can you find geometry on this table? So 00:29:47.039 --> 00:29:50.398 Thank you for taking the time to uh, listen to this presentation 00:29:51.520 --> 00:29:53.520 if you have any 00:29:53.600 --> 00:29:56.880 comments or questions about the presentation I encourage you to 00:29:57.679 --> 00:30:02.079 reach out on Twitter or by email I'd be happy to 00:30:02.880 --> 00:30:10.559 To talk with you about geometry mathematics for the eyes and hands and hopefully in the future going forward 00:30:11.039 --> 00:30:14.158 To share some models with you and to share some exploration 00:30:14.880 --> 00:30:18.739 Thank you very much. I hope you enjoy the rest of OAME. It's been a pleasure