WEBVTT 00:00:00.000 --> 00:00:03.641 [Submit subtitle corrections at criblate.com] Whenever I'm making one of these videos, there's sometimes a special moment 00:00:03.641 --> 00:00:07.913 where the act of animating involves solving a whole bunch of little technical puzzlers, 00:00:07.913 --> 00:00:11.312 and then the underlying math I'm trying to explain clicks for me in a 00:00:11.313 --> 00:00:14.080 way that it hadn't before once I see it alive on screen. 00:00:14.480 --> 00:00:17.390 The best versions of those moments often tell me when a video is 00:00:17.390 --> 00:00:20.344 going to be one of my favorites, and putting together the end of 00:00:20.344 --> 00:00:23.299 this piece right here was one such time when I got that feeling. 00:00:27.039 --> 00:00:30.140 Our story doesn't actually start in the math classroom today. 00:00:30.460 --> 00:00:31.859 We begin in the art room. 00:00:32.719 --> 00:00:36.804 Imagine standing in a gallery, looking at a picture of a boat in a harbour, 00:00:36.804 --> 00:00:40.724 and the whole world warps as your gaze shifts upwards and to the right, 00:00:40.723 --> 00:00:43.119 where you see a village on this waterfront. 00:00:43.640 --> 00:00:48.109 Among the tightly clustered buildings, the world warps even more as your gaze shifts 00:00:48.109 --> 00:00:52.420 downwards to the entrance of one building, leading to a hallway full of artwork. 00:00:53.200 --> 00:00:56.880 And at the end of this hall, here you are again, staring at a picture of a boat. 00:00:58.520 --> 00:00:59.860 This is M.C. 00:00:59.979 --> 00:01:05.340 Escher's 1956 lithograph, the Print Gallery, or in Dutch, Prentendentunstelling. 00:01:05.959 --> 00:01:10.003 In a letter that he wrote to his son, describing his creation of a young 00:01:10.004 --> 00:01:13.373 man looking with interest at a print that features himself, 00:01:13.373 --> 00:01:16.799 Escher called this the most peculiar thing I have ever done, 00:01:16.799 --> 00:01:18.540 which for him is saying a lot. 00:01:18.900 --> 00:01:21.332 Escher's art is widely loved around the world, 00:01:21.331 --> 00:01:25.666 frequently featuring paradoxical themes or uniquely satisfying geometric patterns 00:01:25.667 --> 00:01:27.200 with some kind of character. 00:01:27.500 --> 00:01:30.659 This love is especially pronounced among mathematicians, 00:01:30.659 --> 00:01:34.665 since his art often touches on surprisingly deep concepts within math, 00:01:34.665 --> 00:01:38.840 despite the fact that Escher himself had no formal training in the field. 00:01:39.519 --> 00:01:43.379 The Print Gallery offers a perfect example of this unexpected depth. 00:01:43.379 --> 00:01:47.259 In 2003, the mathematicians De Smit and Lenstra offered a delightful 00:01:47.260 --> 00:01:50.512 analysis of what's really going on in this piece and the 00:01:50.512 --> 00:01:54.219 mind-bending self-contained loop that Escher managed to achieve. 00:01:54.719 --> 00:01:59.542 One of my main goals with this video is to offer a visual unpacking of their analysis, 00:01:59.542 --> 00:02:04.420 aiming as always to give you a feeling that you could have rediscovered this yourself. 00:02:05.120 --> 00:02:08.575 Something that unexpectedly pops out of this analysis is an answer to 00:02:08.574 --> 00:02:12.079 the question of what exactly should go in the middle of this picture. 00:02:12.580 --> 00:02:15.240 At first, that might sound like an incoherent question. 00:02:15.599 --> 00:02:19.426 If you come at it from the upper right, it feels like it should be featuring buildings in 00:02:19.426 --> 00:02:23.038 the village, but coming at it from the left, it looks more like part of the picture 00:02:23.038 --> 00:02:23.339 frame. 00:02:23.620 --> 00:02:27.740 Come at it from below, though, and it feels more fitting to be part of the gallery itself. 00:02:28.159 --> 00:02:31.960 Somehow all of the ambiguity about where exactly you are in this 00:02:31.960 --> 00:02:35.760 whole scene is compressed into that blank circle in the middle. 00:02:36.219 --> 00:02:38.566 And just for fun, since we're in the 2020s, I let 00:02:38.566 --> 00:02:40.960 a diffusion model take a stab at filling this in. 00:02:41.500 --> 00:02:43.280 Unsurprisingly, it struggles immensely. 00:02:43.860 --> 00:02:45.200 It just doesn't get it at all. 00:02:45.740 --> 00:02:48.080 And giving the poor machine some credit, of course it struggles. 00:02:48.479 --> 00:02:52.319 This feels like an intrinsically ambiguous and ill-defined part of the scene. 00:02:52.780 --> 00:02:57.542 But nevertheless, by the end, I hope you'll agree there is one, and really only one, 00:02:57.542 --> 00:03:01.340 completion that feels like the right puzzle piece you can slot in. 00:03:02.099 --> 00:03:05.963 Before diving into the math, I want to give you a purely intuitive description for 00:03:05.963 --> 00:03:09.780 how Escher actually made this piece, which breaks up into three different steps. 00:03:10.340 --> 00:03:15.092 Step one is to start out with a straightened out version of the same general concept, 00:03:15.092 --> 00:03:17.160 where a man is looking at a picture. 00:03:17.639 --> 00:03:21.153 That picture contains a harbor, which contains a town, 00:03:21.153 --> 00:03:26.620 that contains a print gallery, that contains that same man, and so on and so forth. 00:03:26.719 --> 00:03:28.819 You could zoom in forever to your heart's content. 00:03:30.379 --> 00:03:34.638 This idea of a self-similar image, where the picture is contained inside itself, 00:03:34.638 --> 00:03:36.819 has a special name to graphic designers. 00:03:37.180 --> 00:03:39.557 It's known as the Droste effect, named after a 00:03:39.557 --> 00:03:42.039 cocoa company that featured it in its branding. 00:03:42.560 --> 00:03:45.433 In fact, this seems to have been a somewhat common marketing 00:03:45.433 --> 00:03:48.020 gimmick for all kinds of early 20th century products. 00:03:48.479 --> 00:03:52.344 The self-similar Droste image that Escher was using, though, 00:03:52.344 --> 00:03:55.758 involves a much, much deeper zoom than any of these, 00:03:55.758 --> 00:04:00.139 where the self-similar copy is 256 times smaller than the original. 00:04:00.520 --> 00:04:02.420 You'll see where that number comes from in just a minute. 00:04:03.159 --> 00:04:08.503 The genius of Escher is that he somehow intuitively realized there must be a way to 00:04:08.503 --> 00:04:14.042 take this concept of a picture nested inside itself and turn it into this warped loop 00:04:14.042 --> 00:04:19.579 where the zooming in happens implicitly as a viewer's gaze wanders around the circle. 00:04:21.180 --> 00:04:24.814 By the way, you might be wondering where I got this straightened out version, 00:04:24.814 --> 00:04:28.920 and the answer is that those two mathematicians I referenced generously let us use it. 00:04:28.920 --> 00:04:32.636 This is something they actually reverse engineered from the original Escher 00:04:32.636 --> 00:04:36.600 piece using the help of two Dutch artists, Hans Richter and Jacqueline Hofstra. 00:04:37.139 --> 00:04:39.899 The way they reverse engineered it is actually super interesting. 00:04:40.279 --> 00:04:44.459 In a certain manner of speaking, it involves taking the logarithm of the original piece. 00:04:44.920 --> 00:04:48.039 I recognize that sentence probably sounds like nonsense right now, 00:04:48.038 --> 00:04:51.300 but later on in this video, I promise that will make abundant sense. 00:04:52.220 --> 00:04:55.215 To outline the basic idea for how Escher made this loop, 00:04:55.214 --> 00:04:59.171 I want to use an example that's simpler than the one he was working with, 00:04:59.172 --> 00:05:02.968 so we'll pull up this custom self-similar Drasta image which has a pie 00:05:02.968 --> 00:05:07.460 creature looking at a framed picture of a house where that same pie creature lives. 00:05:08.139 --> 00:05:12.336 In this example, the self-similar copy is only 16 times smaller than the original, 00:05:12.336 --> 00:05:15.560 and this just makes everything much easier to see all at once. 00:05:15.980 --> 00:05:20.354 What I'll do is keep a separate workspace on the right here to sketch out the 00:05:20.353 --> 00:05:24.954 general goal we have in mind, where the way you might think about it is that you 00:05:24.954 --> 00:05:29.839 want to distribute that 16-fold zoom-in factor across the four corners of the square. 00:05:30.560 --> 00:05:33.521 For example, let's say you want the pie creature itself to 00:05:33.521 --> 00:05:36.379 appear about at the same size in the lower left corner. 00:05:36.879 --> 00:05:39.863 Then if you zoom in by a factor of 2 in the original, 00:05:39.863 --> 00:05:44.762 we'll take what would be in the upper left of that zoomed version and then place it in 00:05:44.762 --> 00:05:46.620 the upper left of our workspace. 00:05:47.339 --> 00:05:52.618 Similarly, zoom in by another factor of 2 and then take the upper right of that zoomed-in 00:05:52.619 --> 00:05:57.780 version and now place an expanded version of that in the upper right of our workspace. 00:05:58.600 --> 00:06:01.422 Finally, one more step, zoom in by another factor of 2, 00:06:01.422 --> 00:06:05.014 take what's in the lower right of that zoomed-in version, blow it up, 00:06:05.014 --> 00:06:07.579 and place it in the lower right of our workspace. 00:06:08.279 --> 00:06:12.879 These four cut-out corners give a rough idea of what we want to create here. 00:06:13.180 --> 00:06:17.287 As long as you can find a smooth way to fill in the gaps between them, 00:06:17.286 --> 00:06:21.216 then as the onlooker's gaze wanders around a circle on this image, 00:06:21.216 --> 00:06:26.144 what they're seeing will zoom in and in and in until it elegantly joins up with the 00:06:26.144 --> 00:06:29.019 self-similar part of the image 16 times smaller. 00:06:29.680 --> 00:06:33.079 We could just try joining it all up naively, something like this, 00:06:33.079 --> 00:06:35.588 but frankly that doesn't really look very good, 00:06:35.588 --> 00:06:39.927 and it's certainly not as smooth and as elegant as what Escher managed to achieve, 00:06:39.927 --> 00:06:42.019 so we still have some work ahead of us. 00:06:42.939 --> 00:06:45.920 This book that I've been pawing through by the way, The Magic of M.C. 00:06:46.040 --> 00:06:50.205 Escher, is something that I got on a delightful visit to the Escher Museum in The Hague, 00:06:50.204 --> 00:06:53.423 which, if you're ever in the Netherlands, I would highly recommend, 00:06:53.423 --> 00:06:56.120 and when we turn to the section about the print gallery, 00:06:56.120 --> 00:06:59.339 it offers us a little peek into what Escher's process actually was. 00:06:59.800 --> 00:07:03.439 For him, step two was to create this warped grid. 00:07:03.939 --> 00:07:06.706 For the animations here, I'm going to pull up a slight modification 00:07:06.706 --> 00:07:09.968 of that grid that is basically just a little nicer mathematically to generate, 00:07:09.968 --> 00:07:11.579 but it illustrates the same key point. 00:07:11.980 --> 00:07:16.487 In his case, the self-similar Droste image he was working with has a scaling 00:07:16.487 --> 00:07:20.165 factor of 256, so to distribute that across the four corners, 00:07:20.165 --> 00:07:24.554 for him it would involve scaling by a factor of four as you walk from one 00:07:24.553 --> 00:07:25.739 corner to the next. 00:07:26.540 --> 00:07:30.833 And if you look closely at his grid, say at one of the squares on the lower right, 00:07:30.833 --> 00:07:35.492 notice how if you follow the top and bottom lines bounding that square over to the lower 00:07:35.492 --> 00:07:40.098 left of the image, those same lines end up enclosing a square that is now four times as 00:07:40.098 --> 00:07:40.360 big. 00:07:40.720 --> 00:07:43.843 And then similarly, if you look at the lines bounding the left and 00:07:43.843 --> 00:07:47.155 right of a small square from that region, and you follow them upward, 00:07:47.154 --> 00:07:50.419 they end up nestled around a square that's nicely four times as big. 00:07:50.860 --> 00:07:54.900 So the grid kind of encodes the scaling from one corner to the next that we want. 00:07:55.439 --> 00:07:58.295 Now for our pet project, where we only need to scale up by a 00:07:58.295 --> 00:08:01.293 factor of two from each corner to the next, we're going to use 00:08:01.293 --> 00:08:04.339 a modified version of this, but it'll be the same general idea. 00:08:05.660 --> 00:08:08.922 You might naturally wonder where on earth does this grid come from, 00:08:08.922 --> 00:08:12.621 but I actually want to postpone that question and skip ahead to step three, 00:08:12.620 --> 00:08:16.954 which is how you can actually use this grid together with the self- similar Drosse image 00:08:16.954 --> 00:08:18.560 to create Escher's final effect. 00:08:19.199 --> 00:08:23.850 The way this works is to first lay down an ordinary square grid on the original image, 00:08:23.850 --> 00:08:28.392 and then let's say you want this portion here to end up in this corner of our final 00:08:28.392 --> 00:08:28.879 version. 00:08:29.279 --> 00:08:33.434 What you would do is take each tiny square inside it and then copy over its 00:08:33.434 --> 00:08:37.699 contents to the corresponding tiny square in the warped version of the grid. 00:08:38.100 --> 00:08:42.322 From there, each neighboring square in our original grid is forced to go 00:08:42.322 --> 00:08:46.017 to the corresponding neighboring square in the warped version, 00:08:46.017 --> 00:08:51.060 so you can kind of just keep following where the image must go by following the grid. 00:08:52.000 --> 00:08:56.347 And the fact that the grid lines on the warped version space out by a factor of 00:08:56.347 --> 00:09:00.751 two as you go from the lower left to the upper left means that the scale of our 00:09:00.751 --> 00:09:05.319 scene automatically gets scaled up by that factor of two as you walk up that line. 00:09:06.659 --> 00:09:09.188 The idea of this process is that as an artist, 00:09:09.188 --> 00:09:12.158 it's relatively straightforward to go piece by piece, 00:09:12.158 --> 00:09:15.841 copying over what's inside one tiny square to another tiny square, 00:09:15.841 --> 00:09:18.700 because at this small scale things are undistorted. 00:09:19.179 --> 00:09:23.570 It's certainly way, way easier than trying to dream up and draw the appropriate 00:09:23.571 --> 00:09:27.740 warped final image starting with nothing but a blank page in front of you. 00:09:28.960 --> 00:09:33.742 So basically with this warped grid in hand, the process of copying over each little 00:09:33.741 --> 00:09:38.639 square is pleasantly automatic, and if we let this automatic process run all the way 00:09:38.639 --> 00:09:43.191 around, it nicely closes up with the initial position, and for that to happen, 00:09:43.191 --> 00:09:47.973 it requires that our original image had this self-similarity when you zoom in by a 00:09:47.972 --> 00:09:48.779 factor of 16. 00:09:49.379 --> 00:09:52.215 I hope you'll agree the final result we have is pretty nice, 00:09:52.215 --> 00:09:55.191 it recreates the same general effect, but more than anything I 00:09:55.191 --> 00:09:58.639 think it gives cause to more deeply appreciate Escher's own composition. 00:09:59.080 --> 00:10:03.360 He was actually very deliberate about the choice of imagery at all of the distinct scales. 00:10:03.879 --> 00:10:07.879 To quote, I quite intentionally chose serial types of objects, 00:10:07.879 --> 00:10:13.299 such for instance as a row of prints along the wall, or blocks of houses in a town. 00:10:13.720 --> 00:10:16.153 Without the cyclic elements, it would be all the more 00:10:16.153 --> 00:10:18.679 difficult to get my meaning over to the random viewer. 00:10:20.059 --> 00:10:23.938 This idea where you have a straight reference image and then a warped grid, 00:10:23.938 --> 00:10:26.783 and you use the two together to create a warped scene, 00:10:26.783 --> 00:10:28.800 is a common process in graphic design. 00:10:29.100 --> 00:10:32.300 It's known as a mesh warp, and Escher didn't invent it for this case, 00:10:32.299 --> 00:10:35.639 it's something that he had used multiple times before for other pieces. 00:10:36.179 --> 00:10:40.083 For our story, the point is that all of the logic for turning 00:10:40.083 --> 00:10:44.500 a Drasta zoom into a loop is abstracted and purified into this grid, 00:10:44.500 --> 00:10:48.019 raising the natural question, where does it come from? 00:10:48.179 --> 00:10:49.000 How do you make this? 00:10:49.799 --> 00:10:54.945 You can kind of imagine as an initial naive approach you might try linearly scaling 00:10:54.946 --> 00:11:00.340 everything from one corner to the next, but if you did that right away you would see a 00:11:00.340 --> 00:11:00.960 conflict. 00:11:01.440 --> 00:11:05.874 These two scaling processes are placing distinct pressures on the individual squares, 00:11:05.874 --> 00:11:09.629 where for example this square wants to get flared out in this direction 00:11:09.629 --> 00:11:13.385 based on the zooming in from the right, but it also wants to get flared 00:11:13.385 --> 00:11:16.360 in this direction based on the zooming out as you go up. 00:11:16.940 --> 00:11:19.499 Escher was evidently pulled to resolve this by 00:11:19.499 --> 00:11:22.279 curving all of the lines to relieve this tension. 00:11:23.159 --> 00:11:26.911 But there's one other crucial constraint that it seems he was guided by, 00:11:26.912 --> 00:11:30.038 one that presumably made the image transfer process easier, 00:11:30.038 --> 00:11:32.489 which made the final result look more natural, 00:11:32.489 --> 00:11:36.137 and which will make the ears of any mathematician immersed in complex 00:11:36.136 --> 00:11:37.699 analysis immediately perk up. 00:11:38.019 --> 00:11:43.240 In his final warped grid, the tiny squares are, well, squares. 00:11:43.679 --> 00:11:46.559 To be clear, this is not at all true for most mesh warps. 00:11:46.759 --> 00:11:51.376 In most cases, the warped grid lines that you draw don't necessarily intersect at right 00:11:51.376 --> 00:11:56.099 angles, and the little regions that they bound will in general be little parallelograms. 00:11:56.500 --> 00:12:00.841 This can still be workable for an artist, you can still kind of do the transfer, 00:12:00.841 --> 00:12:05.126 but presumably the act of copying over from the original to the warped version 00:12:05.126 --> 00:12:07.676 is more awkward when you have that distortion, 00:12:07.677 --> 00:12:10.336 so it would be much nicer if in the warped grid, 00:12:10.336 --> 00:12:13.320 little squares remained at least approximately square. 00:12:13.759 --> 00:12:17.212 And when you look closely at the grid that Escher used for his print gallery, 00:12:17.212 --> 00:12:18.199 it has this property. 00:12:18.600 --> 00:12:22.394 All the lines are intersecting at right angles, and at a small enough scale, 00:12:22.394 --> 00:12:25.240 the regions they bound really are approximately squares. 00:12:25.799 --> 00:12:29.362 Artistically, this has the nice effect that even though the whole 00:12:29.363 --> 00:12:32.762 scene is dramatically warped and distorted at a global scale, 00:12:32.761 --> 00:12:36.379 zoomed in at a local scale, everything is relatively undistorted. 00:12:36.759 --> 00:12:40.480 This is what makes each local part of Escher's image easily recognizable. 00:12:41.720 --> 00:12:44.920 And mathematically, this is where the story really gets interesting. 00:12:45.440 --> 00:12:49.864 You see, this idea of a function from two dimensional space to two dimensional space, 00:12:49.864 --> 00:12:52.362 where tiny squares remain approximately square, 00:12:52.361 --> 00:12:55.120 plays a special role and has a special name in math. 00:12:55.379 --> 00:12:59.689 It's called a conformal map, and one area where it comes up all the time is in 00:12:59.690 --> 00:13:04.000 the study of functions with complex number inputs and complex number outputs. 00:13:04.960 --> 00:13:08.110 At this point, we're going to step back and walk out of the art 00:13:08.110 --> 00:13:10.710 classroom and wander over into the math department, 00:13:10.710 --> 00:13:14.860 where I want to offer you a mini lesson on some of the core ideas from this field. 00:13:16.539 --> 00:13:20.188 The basic game plan from here is that I want to first do a little refresher, 00:13:20.188 --> 00:13:23.501 go over some of the basics of complex numbers and complex functions, 00:13:23.501 --> 00:13:26.910 and then I want to spend some meaningful time building up an intuition 00:13:26.910 --> 00:13:30.079 for what logarithms look like in this context of complex numbers. 00:13:30.480 --> 00:13:34.098 And then once you have that in hand, we're going to step through a completely different 00:13:34.097 --> 00:13:37.839 way that you can think about recreating this effect that Escher had in his print gallery. 00:13:38.980 --> 00:13:40.860 Let's warm up with a review of the basics. 00:13:41.259 --> 00:13:44.955 We typically think about real numbers as living on a one dimensional line, 00:13:44.955 --> 00:13:48.100 the real number line, and complex numbers are two dimensional. 00:13:48.639 --> 00:13:51.750 Specifically, we think of the imaginary constant i, 00:13:51.750 --> 00:13:56.387 defined to be the square root of negative one as being one unit above zero, 00:13:56.388 --> 00:14:01.330 end every other point on this plane represents some combination of a real number 00:14:01.330 --> 00:14:03.160 with some real multiple of i. 00:14:03.700 --> 00:14:07.625 It's typical to use the variable z in referring to a general complex number, 00:14:07.625 --> 00:14:11.240 and the game we want to play is to understand various functions of z. 00:14:11.840 --> 00:14:15.840 A very simple but important example is to multiply z by some constant. 00:14:16.320 --> 00:14:19.221 The function f of z equals two times z has the 00:14:19.221 --> 00:14:22.500 effect of scaling everything up by a factor of two. 00:14:23.039 --> 00:14:25.370 But what about multiplying by something imaginary, 00:14:25.370 --> 00:14:27.279 like i, the square root of negative one? 00:14:27.879 --> 00:14:31.275 Well you know that multiplying one by i gives you i, 00:14:31.275 --> 00:14:34.279 and by definition, i times i is negative one. 00:14:34.700 --> 00:14:38.540 Both of these you'll notice are in 90 degree rotation, and more generally, 00:14:38.539 --> 00:14:42.120 multiplying any value z by i has the effect of a 90 degree rotation. 00:14:42.940 --> 00:14:46.662 And more general than that, when you multiply by any complex constant, 00:14:46.662 --> 00:14:49.639 the effect is some combination of scaling and rotating. 00:14:50.159 --> 00:14:53.419 And there's a nice way to think about exactly how much it should scale and rotate. 00:14:54.000 --> 00:14:58.384 Zero times anything is zero, so the origin has to stay fixed in place, 00:14:58.384 --> 00:15:02.080 and then one times any constant c is that same constant c. 00:15:02.620 --> 00:15:05.812 So that means this point at one has to be dragged over to land 00:15:05.812 --> 00:15:08.748 on whatever that constant c is that we're talking about, 00:15:08.748 --> 00:15:11.940 and that fully determines the amount of scaling and rotating. 00:15:12.399 --> 00:15:14.879 From there, the rest of the grid stays as rigid as it can. 00:15:15.440 --> 00:15:18.926 A key point to emphasize for our story is that if all you're doing 00:15:18.926 --> 00:15:22.200 is multiplying by some constant, shapes are always preserved. 00:15:22.659 --> 00:15:24.933 Anything you might want to draw, like a square, 00:15:24.933 --> 00:15:28.319 can get scaled or rotated, but beyond that, there are no distortions. 00:15:29.080 --> 00:15:31.440 Now things get more interesting for more complicated functions. 00:15:32.120 --> 00:15:36.940 A simple but non-trivial example would be mapping each number z to z squared. 00:15:37.580 --> 00:15:40.639 So the input two is going to have to move to two squared, which is four. 00:15:41.100 --> 00:15:44.519 The input i is going to have to move to i squared, which is negative one. 00:15:45.100 --> 00:15:48.040 Negative one itself would have to get mapped to positive one. 00:15:48.480 --> 00:15:52.383 And in its fullness, here's what it looks like if I let every point among 00:15:52.383 --> 00:15:56.500 the grid lines of this input space move over to their corresponding outputs. 00:15:57.159 --> 00:16:00.427 It's a pretty nice effect, and unlike multiplying by a constant, 00:16:00.427 --> 00:16:02.519 shape is absolutely no longer preserved. 00:16:02.940 --> 00:16:04.680 The grid lines get curved and warped. 00:16:05.279 --> 00:16:09.199 However, and this is a key point, pay attention to what happens at a small scale. 00:16:09.519 --> 00:16:12.439 For example, just focusing on this little square from the input space. 00:16:12.940 --> 00:16:16.231 As you watch the transformation happen again, you can see that 00:16:16.230 --> 00:16:19.840 shape is approximately preserved, at least at a small enough scale. 00:16:20.159 --> 00:16:23.201 Little squares from our original grid remain approximately 00:16:23.201 --> 00:16:25.980 square even after getting processed by the function. 00:16:26.500 --> 00:16:29.980 To use the lingo, the function z squared gives a conformal map. 00:16:30.940 --> 00:16:33.180 And the choice of z squared is really not special here. 00:16:33.539 --> 00:16:37.199 Here's what it looks like if you transform each point z over to z cubed. 00:16:37.580 --> 00:16:39.340 Squares remain approximately square. 00:16:40.120 --> 00:16:43.229 Okay, maybe this example square that I'm highlighting doesn't exactly look 00:16:43.229 --> 00:16:46.379 square in the output, but really it's just because it started off too big. 00:16:46.700 --> 00:16:50.320 To be clear, this conformal property that I'm talking about is a limiting one. 00:16:50.659 --> 00:16:53.235 A particular square might not exactly become square, 00:16:53.235 --> 00:16:55.761 but the idea is that as you zoom in more and more, 00:16:55.761 --> 00:16:59.476 choosing square regions from the input space that are smaller and smaller, 00:16:59.476 --> 00:17:03.439 the resulting output will indeed be better and better approximated by a square. 00:17:04.000 --> 00:17:06.111 And there's also nothing all that special about the 00:17:06.111 --> 00:17:08.140 choice of polynomials like z squared or z cubed. 00:17:08.420 --> 00:17:12.191 For just about any function of complex numbers you could think to write down, 00:17:12.191 --> 00:17:13.220 this property holds. 00:17:13.598 --> 00:17:15.598 Shape is preserved at a small enough scale. 00:17:15.940 --> 00:17:16.880 It's almost like magic. 00:17:17.680 --> 00:17:20.220 The use of complex numbers is very relevant here. 00:17:20.380 --> 00:17:25.082 If instead you were thinking of points in 2D space simply as a pair of real numbers with 00:17:25.082 --> 00:17:29.570 some xy coordinates, and you write down some arbitrary function of x and y to get a 00:17:29.569 --> 00:17:34.003 new pair of numbers, what is way, way more typical as you let points in that input 00:17:34.003 --> 00:17:38.331 space get transformed is that the outputs of those tiny squares get squished and 00:17:38.332 --> 00:17:38.920 distorted. 00:17:39.220 --> 00:17:42.593 Even as you zoom in more and more, the resulting limiting shape 00:17:42.593 --> 00:17:46.019 typically looks like a parallelogram, not necessarily a square. 00:17:46.660 --> 00:17:49.620 So, complex functions really are special in this way. 00:17:50.160 --> 00:17:53.880 And the basic reason that tiny squares remain square comes down to calculus. 00:17:54.579 --> 00:17:58.519 What you're looking at is what it means for these functions to have derivatives. 00:17:59.380 --> 00:18:03.880 That might sound a little strange, but the analogy you can think of is that for most 00:18:03.880 --> 00:18:07.470 functions of real numbers, if you visualize them the ordinary way, 00:18:07.470 --> 00:18:11.006 just with a graph of inputs on the x-axis, outputs on the y-axis, 00:18:11.006 --> 00:18:14.649 as you zoom in more and more to any particular point on that graph, 00:18:14.648 --> 00:18:17.059 it looks more and more like a straight line. 00:18:17.500 --> 00:18:21.986 That is, the rate of change, how much delta f you get for a given delta x, 00:18:21.986 --> 00:18:23.380 approaches a constant. 00:18:24.019 --> 00:18:27.181 This is literally what it means for a function to have a derivative, 00:18:27.181 --> 00:18:29.180 but it's not the only way to visualize it. 00:18:29.440 --> 00:18:31.640 Here's a different way to see the same concept. 00:18:32.220 --> 00:18:34.334 Instead of looking at the graph of a function, 00:18:34.334 --> 00:18:37.000 let's think of the same function but as a transformation. 00:18:37.460 --> 00:18:40.817 That is, you're going to let each of these points on the real number 00:18:40.817 --> 00:18:44.420 line move over to their corresponding outputs on this other number line. 00:18:45.140 --> 00:18:49.772 What you'll notice is that evenly spaced dots from the input space can get warped in 00:18:49.771 --> 00:18:54.679 the output space, meaning the rate of change of the function in general is not constant. 00:18:55.019 --> 00:18:58.759 This spacing between our dots can change from one part of the image to the next. 00:18:59.259 --> 00:19:03.238 But we know that as you zoom in more and more to a particular output, 00:19:03.238 --> 00:19:05.660 the rate of change approaches a constant. 00:19:06.099 --> 00:19:08.399 The dots look more and more evenly spaced. 00:19:09.259 --> 00:19:13.782 Specifically, if you take a tiny patch of dots around a particular input and you 00:19:13.782 --> 00:19:17.626 copy them over to the output space around the corresponding output, 00:19:17.626 --> 00:19:22.318 you can approximately line up all the dots just by scaling everything by a certain 00:19:22.318 --> 00:19:23.279 constant factor. 00:19:23.980 --> 00:19:28.026 This is the same analytical fact as what the slope of a graph is telling you, 00:19:28.026 --> 00:19:30.180 it's just in a different visual context. 00:19:30.619 --> 00:19:34.252 But this context carries over much more easily to thinking about 00:19:34.252 --> 00:19:38.000 complex valued functions as transformations in the complex plane. 00:19:38.539 --> 00:19:42.078 There we're going to think of the neighborhood around a given input 00:19:42.078 --> 00:19:44.931 as a tiny little grid of points, like we were before, 00:19:44.931 --> 00:19:48.840 and we let each point on that grid move over to its corresponding output. 00:19:49.799 --> 00:19:52.645 In this case, what it means for the rate of change to 00:19:52.645 --> 00:19:55.759 approach a constant is basically the exact same equation. 00:19:56.359 --> 00:20:00.609 The visual to have in your head is that if you take that tiny patch of squares 00:20:00.609 --> 00:20:04.043 around the input and copy it over to the corresponding output, 00:20:04.044 --> 00:20:08.294 you can approximately match this up with the output grid lines by multiplying 00:20:08.294 --> 00:20:12.326 by a certain constant, which remember, in the setting of complex numbers, 00:20:12.326 --> 00:20:16.358 means rotating and scaling it in some way, depending on the value of that 00:20:16.358 --> 00:20:17.339 complex constant. 00:20:17.940 --> 00:20:21.969 Since rotation and scaling preserve shape, it means that all the tiny squares from 00:20:21.969 --> 00:20:25.900 the input space remain at least approximately square under this transformation. 00:20:27.299 --> 00:20:31.279 So, stepping back, here's the key point, the reason for talking about any of this at all. 00:20:31.559 --> 00:20:35.598 Even though conformal maps like the one that Escher was using for his print 00:20:35.598 --> 00:20:39.744 gallery are incredibly constrained and highly unusual among the general ways 00:20:39.744 --> 00:20:43.945 that you could continuously squish about two-dimensional space, nevertheless, 00:20:43.945 --> 00:20:47.500 as if by magic, simply by speaking a language of complex numbers, 00:20:47.500 --> 00:20:51.592 you can somehow create entire families of these conformal maps without even 00:20:51.592 --> 00:20:52.400 really trying. 00:20:52.859 --> 00:20:56.279 All you do is mix and match standard functions of complex numbers. 00:20:56.640 --> 00:20:59.514 The one constraint is that these functions have to have derivatives, 00:20:59.513 --> 00:21:02.599 but this will be true for most of the functions you think to write down. 00:21:03.619 --> 00:21:06.721 For our story, decoding what's happening with the print gallery, 00:21:06.721 --> 00:21:10.259 this means that we have an entirely new way to reframe the key question. 00:21:10.759 --> 00:21:15.441 Can you construct some deliberately tailored complex function so that the act of 00:21:15.441 --> 00:21:20.180 zooming in around the inputs looks like walking around a loop among the outputs? 00:21:21.900 --> 00:21:25.298 Now, at this point, with only the bare minimum crash course of complex 00:21:25.298 --> 00:21:28.552 analysis under our belts, it's hard to know where to start without 00:21:28.553 --> 00:21:31.709 first building up a larger palette of functions to work with and 00:21:31.709 --> 00:21:35.400 gaining some familiarity with how they actually behave for complex numbers. 00:21:35.740 --> 00:21:40.263 In this case, there are really only two functions that you need to understand, 00:21:40.263 --> 00:21:42.119 e to the z and the natural log. 00:21:42.880 --> 00:21:46.076 I want to settle in and spend some meaningful time understanding both of these, 00:21:46.076 --> 00:21:48.180 and I think you'll agree that it's time well spent. 00:21:48.579 --> 00:21:51.629 Everybody deserves, in my opinion, at least one time in their 00:21:51.630 --> 00:21:54.880 life to experience the joy of understanding a complex logarithm. 00:21:55.279 --> 00:21:59.619 First though, a necessary prerequisite is to understand the complex exponential. 00:22:00.579 --> 00:22:03.918 Regular viewers will be familiar with what it looks like to raise 00:22:03.919 --> 00:22:07.360 e to the power of a complex number, but a review just never hurts. 00:22:07.839 --> 00:22:12.071 We can start scaffolding the transformation just by focusing on the more familiar 00:22:12.071 --> 00:22:16.199 examples of real number inputs and the real number outputs they correspond to. 00:22:16.599 --> 00:22:21.619 For example, e to the 0 is 1, so this point at 0 maps over to this point at 1. 00:22:22.200 --> 00:22:25.394 And then every time you let that input increase by 1, 00:22:25.394 --> 00:22:30.700 the output grows by a factor of e, meaning it runs away from us actually quite quickly. 00:22:31.319 --> 00:22:34.043 And then on the flip side, as you decrease that input, 00:22:34.044 --> 00:22:38.080 letting it get into the negative numbers, every step to the left corresponds to 00:22:38.079 --> 00:22:40.399 shrinking the output, again by a factor of e. 00:22:40.740 --> 00:22:44.359 In particular, you'll notice in this case that output is always a positive number. 00:22:44.859 --> 00:22:47.272 This of course gets much more interesting when we let 00:22:47.272 --> 00:22:49.639 the inputs and the outputs both be complex numbers. 00:22:50.140 --> 00:22:52.931 And here, everything you need to know comes down to what 00:22:52.931 --> 00:22:56.019 happens as you let the imaginary part of that input increase. 00:22:56.380 --> 00:23:00.700 And what happens there is that the corresponding output walks around a circle. 00:23:02.119 --> 00:23:06.100 If you're wondering why imaginary inputs to an exponential walk you around a 00:23:06.101 --> 00:23:10.240 circle like this, we have discussed it many, many other times on this channel. 00:23:10.519 --> 00:23:12.680 See some of the links on screen and in the description. 00:23:13.160 --> 00:23:17.304 A key point that I'll reiterate here is that what makes the function 00:23:17.304 --> 00:23:21.388 e to the z very nice, as opposed to exponentials with other bases, 00:23:21.387 --> 00:23:25.226 is that as your input walks up at a rate of 1 unit per second, 00:23:25.227 --> 00:23:29.920 the output walks around its circle at a rate of exactly 1 radian per second. 00:23:30.400 --> 00:23:33.536 So in particular, increasing the imaginary part by 00:23:33.536 --> 00:23:36.860 exactly 2 pi causes one full rotation in the output. 00:23:37.640 --> 00:23:42.465 Phrased another way, these vertical line segments that I've been drawing with heights of 00:23:42.464 --> 00:23:47.399 exactly 2 pi each get mapped neatly onto one complete circle when you apply the function. 00:23:48.559 --> 00:23:52.589 You'll notice how I've drawn these particular vertical line segments to be spaced out 00:23:52.589 --> 00:23:56.619 evenly in the real direction, where the real part from 1 to the next increases by 1. 00:23:57.000 --> 00:24:01.396 The corresponding circles on the right each differ by a constant scaling factor, 00:24:01.396 --> 00:24:03.319 specifically the scaling factor e. 00:24:04.220 --> 00:24:08.376 Earlier for the simpler function z squared, I showed it as a transformation, 00:24:08.376 --> 00:24:10.619 moving all of the inputs to the outputs. 00:24:11.019 --> 00:24:14.923 And in this case, for e to the z, if you're curious how that same idea looks, 00:24:14.923 --> 00:24:19.182 it's easiest to take a subset of the grid, like this one here from the input space, 00:24:19.182 --> 00:24:23.440 and here's what it looks like to move each square over to the corresponding output. 00:24:24.079 --> 00:24:27.147 As we actually use this function for our print gallery goals, 00:24:27.147 --> 00:24:31.472 it'll be very helpful to anchor your mind by thinking of these vertical lines and the 00:24:31.472 --> 00:24:32.680 circles they turn into. 00:24:33.500 --> 00:24:37.422 In fact, there's a very playful way that I like to think about how these vertical lines 00:24:37.422 --> 00:24:41.480 turn into concentric circles and how they can carry the full input space along with them. 00:24:41.779 --> 00:24:46.502 What I like to imagine is sort of rolling up that entire z-plane into a tube, 00:24:46.502 --> 00:24:50.000 such that all of those vertical lines end up as circles. 00:24:50.480 --> 00:24:53.180 Specifically, each circle would have a circumference of 2 pi. 00:24:54.240 --> 00:24:58.878 Next, imagine taking this tube, lining it up above the origin of the output space, 00:24:58.878 --> 00:25:02.215 and then kind of squishing it down onto that output space, 00:25:02.215 --> 00:25:06.967 turning all the circles from that tube into these concentric rings of exponentially 00:25:06.968 --> 00:25:07.759 growing size. 00:25:08.700 --> 00:25:11.663 That's just what I like, but however you choose to think about it, 00:25:11.663 --> 00:25:15.211 what I want to be etched into your brain is the idea of vertical lines turning 00:25:15.211 --> 00:25:15.839 into circles. 00:25:16.339 --> 00:25:19.554 Now, the other very important point to emphasize here is 00:25:19.555 --> 00:25:23.000 that multiple different inputs can land on the same output. 00:25:23.819 --> 00:25:29.159 For example, e to the zero is one, but e to the 2 pi i is also one. 00:25:29.539 --> 00:25:34.279 So is e to the negative 2 pi i and e to the 4 pi i and so on. 00:25:34.539 --> 00:25:39.608 In fact, the infinite sequence along any given vertical line spaced out by 2 pi will 00:25:39.608 --> 00:25:44.859 all get collapsed together as that vertical line gets kind of rolled up into a circle. 00:25:45.259 --> 00:25:47.843 We say that the exponential map is many to one, 00:25:47.843 --> 00:25:51.634 although it might not be obvious how this repetition in the vertical 00:25:51.634 --> 00:25:56.140 direction is going to be key to our final recreation of the print gallery effect. 00:25:56.900 --> 00:25:59.576 Okay, so exponentials give us the first ingredient, 00:25:59.576 --> 00:26:03.302 characterized by turning lines into circles, and the second ingredient 00:26:03.301 --> 00:26:07.131 we need is the inverse of such an exponential, known as the natural log, 00:26:07.132 --> 00:26:11.120 where basically the idea is that it unravels those circles back into lines. 00:26:11.500 --> 00:26:15.532 Now, this will be especially fun to visualize and especially relevant to 00:26:15.532 --> 00:26:20.349 our story if we imagine painting this complex plane on the right with a Droste image, 00:26:20.349 --> 00:26:24.494 say the example we were working with earlier with the pi creature looking 00:26:24.494 --> 00:26:27.799 at a picture of a house where that same pi creature lives. 00:26:28.299 --> 00:26:31.639 In other words, it is finally time for you and me to answer that 00:26:31.640 --> 00:26:34.980 question of what it means to take the natural log of a picture. 00:26:36.400 --> 00:26:37.980 Okay, so think about this for a second. 00:26:38.460 --> 00:26:42.693 You already know that a vertical line segment like this one on the left with 00:26:42.693 --> 00:26:47.039 a height of 2 pi gets turned into a circle when you apply the map e to the z. 00:26:47.539 --> 00:26:51.096 So the natural log is going to take a circle of points on this 00:26:51.096 --> 00:26:54.539 image and then straighten them out into one of those lines. 00:26:55.440 --> 00:26:59.439 Similarly, if you looked at a circle which was exactly e times smaller, 00:26:59.439 --> 00:27:04.226 that would also get straightened out into a line with the same height positioned one 00:27:04.226 --> 00:27:05.240 unit to the left. 00:27:05.819 --> 00:27:10.258 And then similarly, every circle in between these two is going to get unwrapped into 00:27:10.258 --> 00:27:14.062 one of these vertical lines between those last two, and more generally, 00:27:14.061 --> 00:27:18.605 smaller and smaller rings from the picture will all get unwrapped into these vertical 00:27:18.605 --> 00:27:21.089 line segments, each one with a height of 2 pi, 00:27:21.089 --> 00:27:23.519 farther and farther to the left in the image. 00:27:24.079 --> 00:27:27.203 The result we get is a bit trippy, but it's pretty cool when you think about it, 00:27:27.203 --> 00:27:29.859 and there's a number of important things that I want you to notice. 00:27:30.059 --> 00:27:33.157 You've probably noticed that it repeats as you move to the left, 00:27:33.157 --> 00:27:37.127 and I'll invite you to ponder why that might be the case in the back of your mind 00:27:37.127 --> 00:27:41.242 for a minute, but before that repetition, I want to talk about a different direction 00:27:41.242 --> 00:27:42.259 in which it repeats. 00:27:42.799 --> 00:27:46.719 The way I've drawn it so far, the imaginary part for the values on the left 00:27:46.719 --> 00:27:50.900 are ranging from 0 up to 2 pi, but that's actually kind of an arbitrary choice. 00:27:51.400 --> 00:27:56.382 Remember, if you keep letting that value z on the left walk up by another 2 pi units, 00:27:56.382 --> 00:28:01.304 the corresponding value e to the z on the right would just keep walking around that 00:28:01.304 --> 00:28:05.993 same circle again, so I hope you'll agree it feels at least enticing to let our 00:28:05.992 --> 00:28:10.740 image repeat in this vertical direction along every one of those vertical lines. 00:28:12.799 --> 00:28:17.290 And it goes the other way too, as you let the imaginary part on the left get smaller 00:28:17.290 --> 00:28:21.621 going down, the corresponding value on that right just keeps walking around that 00:28:21.622 --> 00:28:25.900 same circle, so the pattern that you see should perhaps repeat in that way too. 00:28:26.559 --> 00:28:30.792 To be more explicit, the rule that I'm using to draw this image on the left 00:28:30.792 --> 00:28:33.783 is that for every point z in that plane on the left, 00:28:33.784 --> 00:28:37.226 you look at the corresponding value e to the z on the right, 00:28:37.226 --> 00:28:39.540 and then you assign it a matching color. 00:28:40.059 --> 00:28:45.433 So for example, this warped pi creature and this one and this one all really correspond 00:28:45.433 --> 00:28:50.559 to the same part of the image on the right, the big pi creature in the lower left. 00:28:51.039 --> 00:28:56.049 Another way to think about this is that because the function e to the z is many to one, 00:28:56.049 --> 00:28:59.678 the feeling we get is that the natural logarithm, its inverse, 00:28:59.679 --> 00:29:04.286 wants to be a multi-valued function, something where one input maps to multiple 00:29:04.286 --> 00:29:05.380 different outputs. 00:29:06.319 --> 00:29:10.442 Now in practice, many times you don't want a function to have multiple outputs, 00:29:10.442 --> 00:29:14.252 sometimes that even defies the definition of a function in your context, 00:29:14.252 --> 00:29:18.635 so people will often just choose a band of this plane on the left to be the outputs 00:29:18.635 --> 00:29:19.680 of the natural log. 00:29:20.059 --> 00:29:24.059 In complex analysis, this is called choosing a branch cut for the function. 00:29:24.559 --> 00:29:28.845 For our purposes though, where we want to recreate and understand Escher's piece, 00:29:28.845 --> 00:29:32.548 it's actually nice as to think of the log as a multi-valued function, 00:29:32.548 --> 00:29:37.256 where each point on the right corresponds to a repeating sequence of points on the left, 00:29:37.256 --> 00:29:40.800 spaced out two pi vertically, going infinitely in both directions. 00:29:41.579 --> 00:29:46.224 Now in our special case, you also see this repeating pattern as you move to the left, 00:29:46.224 --> 00:29:48.519 but that is something different entirely. 00:29:48.819 --> 00:29:50.519 You would not see this for most images. 00:29:51.019 --> 00:29:55.823 It arises specifically because we're working with a self-similar Droste image, 00:29:55.823 --> 00:29:59.580 one that looks identical as you zoom in by a certain factor. 00:29:59.759 --> 00:30:04.319 This falls straight out of a core property of logarithms and exponentials. 00:30:04.880 --> 00:30:08.142 Exponentials turn addition into multiplication, 00:30:08.142 --> 00:30:11.960 and logarithms turn multiplication back into addition. 00:30:12.859 --> 00:30:17.333 For example, imagine taking some small value w on this plane on the right, 00:30:17.334 --> 00:30:22.653 and also considering 16 times that value, which is scaled up 16 times farther away from 00:30:22.653 --> 00:30:23.380 the origin. 00:30:24.279 --> 00:30:28.458 Now if we look at the corresponding value, log of w on the left, 00:30:28.458 --> 00:30:32.048 that act of multiplying by 16 now looks like addition, 00:30:32.048 --> 00:30:36.030 specifically shifting to the right by the natural log of 16, 00:30:36.030 --> 00:30:38.119 which is just some real number. 00:30:39.339 --> 00:30:44.952 In fact, this rectangle here in our bizarre warped log image that has a width of log 00:30:44.952 --> 00:30:50.500 16 and a height of 2 pi contains all the information about the image on the right. 00:30:50.900 --> 00:30:54.126 It corresponds to this annulus of the Droste image, 00:30:54.125 --> 00:30:58.997 and if you were to shift that rectangle exactly log of 16 units to the left, 00:30:58.997 --> 00:31:04.182 you would get a scaled-down version of that annulus exactly 16 times smaller that 00:31:04.182 --> 00:31:06.839 nestles in perfectly like a puzzle piece. 00:31:07.359 --> 00:31:10.599 And if you repeat that infinitely many times, it gives you 00:31:10.599 --> 00:31:13.839 this infinite nesting that characterizes the Droste zoom. 00:31:14.720 --> 00:31:19.120 The way I've drawn things so far, we have this cutoff to the log image on the right side, 00:31:19.119 --> 00:31:23.371 and at this point you know well that vertical lines on the left correspond to circles 00:31:23.372 --> 00:31:27.625 on the right, so you can probably guess that this corresponds to the fact that I gave 00:31:27.625 --> 00:31:31.579 a maximum radius to that Droste image, but of course you don't need to do that. 00:31:31.720 --> 00:31:35.451 In principle, it can extend out infinitely far in all directions, 00:31:35.451 --> 00:31:38.722 following the same self-similar pattern as you scale up, 00:31:38.722 --> 00:31:43.544 and the result for the log image on the left would be to extend as far rightward as 00:31:43.545 --> 00:31:46.300 you want, again with a repeated tiling pattern. 00:31:47.400 --> 00:31:51.534 So to conclude, the logarithm image on the left is periodic vertically, 00:31:51.534 --> 00:31:54.620 basically because rotation on the right is periodic. 00:31:54.980 --> 00:31:56.259 This would happen for any image. 00:31:56.559 --> 00:32:01.444 And then in this special case of a Droste image, it's also periodic horizontally, 00:32:01.444 --> 00:32:04.399 because the Droste image repeats as you zoom in. 00:32:04.640 --> 00:32:07.696 This doubly periodic property is what we'll ultimately 00:32:07.695 --> 00:32:09.959 take advantage of for the final result. 00:32:10.819 --> 00:32:12.799 And we now have all the foundation we need. 00:32:13.000 --> 00:32:16.333 I can now finally describe for you the function that turns 00:32:16.333 --> 00:32:19.779 the Droste zoom into this Escher-style self-contained loop. 00:32:20.319 --> 00:32:23.301 Before just jumping right into it, we have been kind of covering a lot, 00:32:23.301 --> 00:32:25.779 so it might be worth giving some space to let this digest. 00:32:26.180 --> 00:32:29.505 And I was meaning to take 30 quick seconds at some point in this video 00:32:29.505 --> 00:32:32.640 to talk about 3b1b talent, so now might be as good a time as any. 00:32:32.940 --> 00:32:36.456 This is the virtual career fair that I'm experimenting with this year, 00:32:36.455 --> 00:32:40.623 and the basic idea is that you, a person who spends their free time learning about 00:32:40.624 --> 00:32:44.893 things like complex logarithms, are probably interested in working with like-minded, 00:32:44.893 --> 00:32:46.400 curious, and technical teams. 00:32:46.779 --> 00:32:51.639 So if you're seeking or open to a new job, check out the organizations at 3b1b.co. 00:32:52.160 --> 00:32:52.560 talent. 00:32:52.940 --> 00:32:56.705 When you explore that page, you'll find interviews between me and the relevant teams, 00:32:56.704 --> 00:32:59.938 technical puzzles and challenges that they've chosen to feature for you, 00:32:59.939 --> 00:33:03.660 and various other things aimed at giving you a feel for the technical team culture. 00:33:04.160 --> 00:33:07.223 And I just kind of like the idea of exposing this audience to 00:33:07.222 --> 00:33:10.687 aligned career opportunities, so whenever you are looking for a job, 00:33:10.688 --> 00:33:13.400 whether that's now or later, be sure to check it out. 00:33:14.299 --> 00:33:15.859 Okay, so back to our main goal. 00:33:16.140 --> 00:33:20.275 How is it that we can use everything that we've built up about complex functions 00:33:20.275 --> 00:33:24.048 and transformations that give conformal maps and everything like that to 00:33:24.048 --> 00:33:28.079 construct some kind of function that recreates Escher's print gallery effect? 00:33:28.680 --> 00:33:31.605 Maybe it's easiest if I just throw down the whole outline of the function, 00:33:31.605 --> 00:33:33.700 and then we can go through each step in more detail. 00:33:34.440 --> 00:33:38.969 First, take a logarithm, giving this bizarre doubly periodic tiling pattern, 00:33:38.969 --> 00:33:43.320 and then you rotate and scale that tiling pattern in just the right way, 00:33:43.319 --> 00:33:47.075 and then from there you take an exponential, which unworps it, 00:33:47.075 --> 00:33:49.220 but this time with a certain twist. 00:33:50.079 --> 00:33:52.820 This might seem a little bizarre, but there's actually a really 00:33:52.820 --> 00:33:55.779 nice way to motivate and to understand what's really going on here. 00:33:56.299 --> 00:33:59.134 Take a look back at that original Droste image, 00:33:59.134 --> 00:34:02.694 and remember how we want to take this big pie creature and 00:34:02.694 --> 00:34:07.279 somehow identify it with the smaller self-similar copy, 16 times zoomed in. 00:34:07.539 --> 00:34:10.659 I want you to think about a line connecting both of those. 00:34:11.219 --> 00:34:14.900 One way to frame the goal that we have is that we want the final 00:34:14.900 --> 00:34:19.099 function to transform such a line into a closed loop in the final space. 00:34:19.539 --> 00:34:24.025 The endpoints of the line on the top represent the big and the small pie creatures, 00:34:24.025 --> 00:34:28.077 so whatever function identifies those creatures should, at the very least, 00:34:28.077 --> 00:34:29.159 close up this line. 00:34:29.599 --> 00:34:33.599 Now think about what this line looks like in the logarithm of the image. 00:34:33.599 --> 00:34:36.184 The big pie creature corresponds to all these 00:34:36.184 --> 00:34:39.000 multiple copies here next to the imaginary line. 00:34:39.639 --> 00:34:42.675 As we talked about, when you scale down the image, 00:34:42.675 --> 00:34:45.771 it corresponds to shifting to the left in the log, 00:34:45.771 --> 00:34:49.900 so these copies in the log image represent that small pie creature. 00:34:50.300 --> 00:34:52.682 And actually, instead of using this horizontal line, 00:34:52.681 --> 00:34:56.437 we're going to want to take advantage of the fact that this log image is periodic 00:34:56.438 --> 00:34:57.720 in two separate directions. 00:34:58.360 --> 00:35:03.097 So instead, what we'll work with is a diagonal line that connects this 00:35:03.097 --> 00:35:07.699 copy of the big character to this lower left copy of the small one. 00:35:07.940 --> 00:35:10.302 That might seem unmotivated at the moment, and it is, 00:35:10.302 --> 00:35:13.512 but the best way to explain why is just to show you how this plays out, 00:35:13.512 --> 00:35:16.989 and afterwards, if you want, we can contrast with what would have happened if 00:35:16.989 --> 00:35:18.639 you tried using the horizontal line. 00:35:19.199 --> 00:35:22.539 Now this new downward component in the log image corresponds to 00:35:22.539 --> 00:35:26.039 adding some clockwise rotation to the path in the original image. 00:35:26.460 --> 00:35:30.538 Remember, our goal is to turn this into a loop in the final image, 00:35:30.538 --> 00:35:34.677 but you and I just spent like 10 minutes talking extensively about 00:35:34.677 --> 00:35:38.940 a function that turns line segments into circles, namely e to the z. 00:35:39.320 --> 00:35:43.660 What you need to do is get that line segment to end up perfectly vertical 00:35:43.659 --> 00:35:47.880 with a height of 2 pi, and doing this basically comes down to rotating 00:35:47.880 --> 00:35:52.339 and scaling it in just the right way to give it that length and direction. 00:35:52.940 --> 00:35:56.298 Now if you remember, as we were warming up with complex numbers, 00:35:56.297 --> 00:36:00.443 we talked about how multiplication by a constant gives you some combination of 00:36:00.443 --> 00:36:04.536 rotation and scaling, and in this case, one artistic feature we might like is 00:36:04.536 --> 00:36:08.419 for our big pi creature on the lower left to stay fixed in that position, 00:36:08.420 --> 00:36:12.775 and that would mean that this point in our log image needs to stay fixed in place, 00:36:12.775 --> 00:36:16.815 so instead of pivoting around the origin, we really want everything to pivot 00:36:16.815 --> 00:36:17.760 about that point. 00:36:18.280 --> 00:36:21.104 Let's say we label that point something like z naught, 00:36:21.103 --> 00:36:25.182 then here's how the updated formula would look for that rotation and scaling, 00:36:25.182 --> 00:36:29.105 but really most of the content comes down to choosing this constant c that 00:36:29.105 --> 00:36:32.034 you're going to multiply by, and if you like exercises, 00:36:32.034 --> 00:36:35.851 you might enjoy actually taking a moment to work out what specific value 00:36:35.851 --> 00:36:37.420 that constant should take on. 00:36:38.199 --> 00:36:42.102 But right here, since we're being very visual and I want to have a little fun, 00:36:42.103 --> 00:36:45.255 let me show you that constant in its own little complex plane, 00:36:45.255 --> 00:36:49.559 and then also show what happens if I kind of grab it and move it around a little bit. 00:36:50.059 --> 00:36:53.143 Different choices give you different scaling and rotation, 00:36:53.143 --> 00:36:57.820 which after exponentiating gives you all these completely bizarre kaleidoscopic images. 00:36:58.420 --> 00:37:01.787 One way you could think about the whole operation is as a game where 00:37:01.786 --> 00:37:05.301 you're trying to find just the right value that corresponds everything 00:37:05.302 --> 00:37:08.519 to line up as you need them to in that image of the lower right. 00:37:09.119 --> 00:37:12.949 When it is set to that appropriate value and we get this recreated Escher effect, 00:37:12.949 --> 00:37:16.875 in that final image on the lower right, I've been showing this hole in the middle, 00:37:16.875 --> 00:37:19.760 analogous to the hole that Escher had in his original piece. 00:37:20.639 --> 00:37:22.420 But that's actually entirely artificial. 00:37:22.780 --> 00:37:25.507 The output image of the function naturally fills 00:37:25.507 --> 00:37:28.119 that in with its own repeating spiral inward. 00:37:28.119 --> 00:37:31.324 After all, the rotated tiling pattern on the left extends 00:37:31.324 --> 00:37:35.261 infinitely through the whole plane, so when you take its exponential, 00:37:35.262 --> 00:37:38.580 it also fills in everything, except for the point at zero. 00:37:39.880 --> 00:37:40.400 And that's it! 00:37:40.539 --> 00:37:42.039 That is the basic operation. 00:37:42.519 --> 00:37:47.153 Translate to this bizarre looking log space, rotate and scale to realign things, 00:37:47.153 --> 00:37:49.760 and then translate back with an exponential. 00:37:51.500 --> 00:37:54.771 At this point, having recreated our own variation on Escher, 00:37:54.771 --> 00:37:59.623 it might be nice to step through what that same process looks like for Escher's specific 00:37:59.623 --> 00:38:04.365 example, that seaside Maltese town containing a print gallery with a person looking at 00:38:04.365 --> 00:38:05.619 a picture of the town. 00:38:06.420 --> 00:38:10.641 As before, step number one is to place this scene on its own little complex plane, 00:38:10.641 --> 00:38:15.119 where the infinite limiting point about which everything scales is positioned at zero. 00:38:16.340 --> 00:38:20.541 Step two, take a logarithm of this, which in this case gives us a similarly bizarre 00:38:20.541 --> 00:38:24.539 transformation of the whole scene, but there is a little difference this time. 00:38:24.980 --> 00:38:29.692 Because Escher was working with a much deeper zoom, with a scale factor of 256, 00:38:29.692 --> 00:38:35.000 the repeating tiles in this new example actually span a wider part of the complex plane. 00:38:36.280 --> 00:38:38.920 Step three, again, is to rotate and scale this, 00:38:38.920 --> 00:38:43.300 which depends on multiplying by the appropriate choice of a complex constant. 00:38:43.760 --> 00:38:48.221 The constraint is to ensure that this rotated image still repeats every 2 pi 00:38:48.221 --> 00:38:52.800 units vertically, but along a part of the image that was previously diagonal. 00:38:53.300 --> 00:38:56.588 And then step four, plug this all through e to the z, 00:38:56.588 --> 00:39:00.931 and what you get is something very similar to Escher's final picture, 00:39:00.931 --> 00:39:06.079 an image where walking around a loop corresponds to zooming in by a factor of 256. 00:39:07.719 --> 00:39:10.432 And again, when we frame it this way with complex functions, 00:39:10.432 --> 00:39:11.880 there is no hole in the middle. 00:39:12.159 --> 00:39:16.616 That final result is itself a Droste image, albeit this time with a twist, 00:39:16.617 --> 00:39:21.736 where there's a kind of self-similarity spiraling down infinitely far as much as you 00:39:21.735 --> 00:39:22.759 want to zoom in. 00:39:24.000 --> 00:39:26.880 There are a couple things worth noticing about this whole process. 00:39:27.480 --> 00:39:31.141 First off, if you write this idea down as a formula, where you take a log, 00:39:31.141 --> 00:39:33.615 multiply it by a constant, and then exponentiate, 00:39:33.615 --> 00:39:37.476 that can actually be simplified to simply look like raising your input to the 00:39:37.476 --> 00:39:39.059 power of some complex constant. 00:39:39.539 --> 00:39:43.199 And if you reintroduce that offset factor, it just introduces another constant. 00:39:43.980 --> 00:39:46.947 On the one hand, it's very pleasing that this entire complicated 00:39:46.947 --> 00:39:49.635 process can be boiled down to so few symbols on the page, 00:39:49.635 --> 00:39:52.880 but I do think this risks kind of obscuring what's actually going on. 00:39:53.739 --> 00:39:55.738 And then, I had mentioned things would not work 00:39:55.739 --> 00:39:57.780 if you only tried using those horizontal lines. 00:39:58.039 --> 00:40:00.746 If you're curious, here's what it looks like if you try that, 00:40:00.746 --> 00:40:04.739 which would mean rotating everything by 90 degrees and scaling it the appropriate amount. 00:40:05.320 --> 00:40:08.880 You do get something kind of interesting, but it's just not what we're going for. 00:40:09.179 --> 00:40:11.656 And if you think about it, what's basically going on is 00:40:11.657 --> 00:40:14.360 that you are swapping the roles of rotation and of scaling. 00:40:16.679 --> 00:40:20.528 Stepping back from all this, this second perspective where the piece is 00:40:20.528 --> 00:40:24.594 all about rotating in a log space feels completely different from Escher's 00:40:24.594 --> 00:40:28.280 more intuitive approach using his distorted grid and the mesh warp. 00:40:28.760 --> 00:40:31.180 But there is a through line connecting both perspectives. 00:40:31.900 --> 00:40:34.675 For one thing, you can now understand how exactly we've 00:40:34.675 --> 00:40:37.500 been recreating and modifying the grid that Escher had. 00:40:37.940 --> 00:40:40.927 You basically take that same function that we've built up, 00:40:40.927 --> 00:40:45.099 but instead of applying it to an image, you apply it to an ordinary square grid. 00:40:45.780 --> 00:40:47.760 Well, actually not quite an ordinary square grid. 00:40:48.019 --> 00:40:50.764 It's nicest to work with one that gets more dense as 00:40:50.764 --> 00:40:53.559 you zoom in so that it looks the same at all scales. 00:40:54.199 --> 00:40:58.179 When you do this, the logarithm gives you this very beautiful curved tiling pattern. 00:40:58.719 --> 00:41:02.159 And then from there, you do the same trick of rotating things in just 00:41:02.159 --> 00:41:05.350 the right way and then taking an exponential with the result of 00:41:05.351 --> 00:41:09.140 something quite close to what Escher spent many arduous hours constructing. 00:41:09.679 --> 00:41:12.922 And remember, the entire reason that we're using the language of 00:41:12.922 --> 00:41:16.213 complex numbers is the motivation for me having you jump through 00:41:16.213 --> 00:41:19.759 those hoops is that this conformal property falls out as a byproduct. 00:41:20.179 --> 00:41:23.818 Tiny squares from the original grid remain, at least approximately, 00:41:23.818 --> 00:41:25.340 square in the final result. 00:41:26.000 --> 00:41:29.500 Escher said that making this piece gave him some almighty headaches, 00:41:29.500 --> 00:41:33.413 and though maybe you and I had to endure a few headaches ourselves for very 00:41:33.413 --> 00:41:37.478 different reasons, the payoff is that this property is one that we get with no 00:41:37.478 --> 00:41:38.199 added effort. 00:41:39.000 --> 00:41:42.724 Now to be clear, you certainly don't need to understand complex derivatives or 00:41:42.724 --> 00:41:46.400 logarithms to enjoy Escher's work, and I wouldn't want to imply that you do. 00:41:46.739 --> 00:41:49.614 However, when you understand that more mathematical side, 00:41:49.614 --> 00:41:53.498 it gives you the capacity for a very different kind of appreciation for what 00:41:53.498 --> 00:41:54.759 Escher was really doing. 00:41:55.179 --> 00:41:57.717 And this is what I find fascinating about all of this, 00:41:57.717 --> 00:42:00.679 the real connection I want to give between the two storylines. 00:42:01.139 --> 00:42:05.185 If you look at Escher's career, throughout it he was drawn to certain concepts, 00:42:05.186 --> 00:42:07.900 things like representing infinity in a finite space. 00:42:08.280 --> 00:42:11.907 And at the same time, he seemed to be guided by a certain aesthetic, 00:42:11.907 --> 00:42:16.600 often some implicit rigid rule like the idea of little squares remaining little squares 00:42:16.599 --> 00:42:17.400 for this mesh. 00:42:17.760 --> 00:42:21.257 The end result when concept and aesthetic are combined like this is 00:42:21.257 --> 00:42:25.014 that a given piece from Escher often feels like a solution to a puzzle, 00:42:25.014 --> 00:42:28.460 but one where it's not even obvious that a solution should exist. 00:42:29.039 --> 00:42:32.784 Now what I find so thought-provoking is that these visual puzzles 00:42:32.784 --> 00:42:36.759 that he landed on are not merely analogous to the act of doing math. 00:42:36.840 --> 00:42:40.190 The specific structures that he was intuitively drawn to 00:42:40.190 --> 00:42:43.780 often hide within them very real and very deep mathematics. 00:42:44.260 --> 00:42:46.992 In our example, the deeper structure is not just that 00:42:46.992 --> 00:42:49.620 the piece can be described with complex functions. 00:42:50.119 --> 00:42:53.190 The ideas that we've touched on in this storyline are actually 00:42:53.190 --> 00:42:56.260 a lot closer than you might expect to the research frontiers. 00:42:56.840 --> 00:42:59.664 If you look back at our approach in the second half, 00:42:59.664 --> 00:43:04.171 remember how it relied on using this doubly periodic pattern in the complex plane, 00:43:04.170 --> 00:43:06.939 something that repeats in two separate directions. 00:43:07.860 --> 00:43:10.048 There's actually a special name for functions of 00:43:10.048 --> 00:43:12.420 complex numbers that are doubly periodic like this. 00:43:12.639 --> 00:43:14.559 They're known as elliptic functions. 00:43:14.900 --> 00:43:19.384 It would be way too much to explain right here exactly why these functions are so useful, 00:43:19.384 --> 00:43:23.565 but one thing I want to highlight is that those two mathematicians I referenced at 00:43:23.565 --> 00:43:27.697 the start, the ones who provided the analysis of this piece, De Smit and Lenstra, 00:43:27.697 --> 00:43:31.728 are both number theorists, and elliptic functions play a very prominent role in 00:43:31.728 --> 00:43:35.759 modern number theory, providing a kind of bridge to other parts of mathematics. 00:43:36.400 --> 00:43:39.869 This may at least partially explain how it is that they could look at 00:43:39.869 --> 00:43:43.440 this piece and think to construct that function that we laid out here. 00:43:44.659 --> 00:43:48.824 When I look at Escher's work, whether it's the print gallery or many other favorites, 00:43:48.824 --> 00:43:52.204 the reason I love these pieces so much is that they awaken within me 00:43:52.204 --> 00:43:54.900 a very specific feeling that's hard to find elsewhere. 00:43:55.280 --> 00:43:57.740 It's a feeling of things perfectly fitting into place. 00:43:58.139 --> 00:43:59.579 But that's not exactly it. 00:43:59.619 --> 00:44:02.319 It's something more than just the pleasure of seeing a puzzle solved. 00:44:02.539 --> 00:44:05.311 It's an appreciation for the creative genius required 00:44:05.311 --> 00:44:07.820 to even dream up the puzzle in the first place. 00:44:08.139 --> 00:44:11.599 The only other place where I get that feeling is in doing math. 00:44:11.940 --> 00:44:16.364 So the fact that an artist and a mathematician can be drawn to the same structures, 00:44:16.364 --> 00:44:20.414 but for completely different reasons, suggests to me that there's something 00:44:20.414 --> 00:44:24.199 universal in what exactly it is that both of them seem to be drawn to.