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Transcript
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[Submit subtitle corrections at criblate.com]
Whenever I'm making one of these videos, there's sometimes a special moment
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where the act of animating involves solving a whole bunch of little technical puzzlers,
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and then the underlying math I'm trying to explain clicks for me in a
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way that it hadn't before once I see it alive on screen.
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The best versions of those moments often tell me when a video is
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going to be one of my favorites, and putting together the end of
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this piece right here was one such time when I got that feeling.
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Our story doesn't actually start in the math classroom today.
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We begin in the art room.
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Imagine standing in a gallery, looking at a picture of a boat in a harbour,
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and the whole world warps as your gaze shifts upwards and to the right,
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where you see a village on this waterfront.
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Among the tightly clustered buildings, the world warps even more as your gaze shifts
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downwards to the entrance of one building, leading to a hallway full of artwork.
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And at the end of this hall, here you are again, staring at a picture of a boat.
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This is M.C.
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Escher's 1956 lithograph, the Print Gallery, or in Dutch, Prentendentunstelling.
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In a letter that he wrote to his son, describing his creation of a young
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man looking with interest at a print that features himself,
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Escher called this the most peculiar thing I have ever done,
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which for him is saying a lot.
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Escher's art is widely loved around the world,
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frequently featuring paradoxical themes or uniquely satisfying geometric patterns
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with some kind of character.
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This love is especially pronounced among mathematicians,
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since his art often touches on surprisingly deep concepts within math,
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despite the fact that Escher himself had no formal training in the field.
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The Print Gallery offers a perfect example of this unexpected depth.
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In 2003, the mathematicians De Smit and Lenstra offered a delightful
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analysis of what's really going on in this piece and the
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mind-bending self-contained loop that Escher managed to achieve.
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One of my main goals with this video is to offer a visual unpacking of their analysis,
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aiming as always to give you a feeling that you could have rediscovered this yourself.
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Something that unexpectedly pops out of this analysis is an answer to
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the question of what exactly should go in the middle of this picture.
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At first, that might sound like an incoherent question.
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If you come at it from the upper right, it feels like it should be featuring buildings in
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the village, but coming at it from the left, it looks more like part of the picture
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frame.
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Come at it from below, though, and it feels more fitting to be part of the gallery itself.
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Somehow all of the ambiguity about where exactly you are in this
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whole scene is compressed into that blank circle in the middle.
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And just for fun, since we're in the 2020s, I let
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a diffusion model take a stab at filling this in.
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Unsurprisingly, it struggles immensely.
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It just doesn't get it at all.
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And giving the poor machine some credit, of course it struggles.
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This feels like an intrinsically ambiguous and ill-defined part of the scene.
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But nevertheless, by the end, I hope you'll agree there is one, and really only one,
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completion that feels like the right puzzle piece you can slot in.
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Before diving into the math, I want to give you a purely intuitive description for
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how Escher actually made this piece, which breaks up into three different steps.
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Step one is to start out with a straightened out version of the same general concept,
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where a man is looking at a picture.
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That picture contains a harbor, which contains a town,
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that contains a print gallery, that contains that same man, and so on and so forth.
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You could zoom in forever to your heart's content.
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This idea of a self-similar image, where the picture is contained inside itself,
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has a special name to graphic designers.
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It's known as the Droste effect, named after a
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cocoa company that featured it in its branding.
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In fact, this seems to have been a somewhat common marketing
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gimmick for all kinds of early 20th century products.
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The self-similar Droste image that Escher was using, though,
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involves a much, much deeper zoom than any of these,
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where the self-similar copy is 256 times smaller than the original.
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You'll see where that number comes from in just a minute.
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The genius of Escher is that he somehow intuitively realized there must be a way to
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take this concept of a picture nested inside itself and turn it into this warped loop
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where the zooming in happens implicitly as a viewer's gaze wanders around the circle.
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By the way, you might be wondering where I got this straightened out version,
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and the answer is that those two mathematicians I referenced generously let us use it.
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This is something they actually reverse engineered from the original Escher
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piece using the help of two Dutch artists, Hans Richter and Jacqueline Hofstra.
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The way they reverse engineered it is actually super interesting.
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In a certain manner of speaking, it involves taking the logarithm of the original piece.
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I recognize that sentence probably sounds like nonsense right now,
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but later on in this video, I promise that will make abundant sense.
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To outline the basic idea for how Escher made this loop,
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I want to use an example that's simpler than the one he was working with,
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so we'll pull up this custom self-similar Drasta image which has a pie
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creature looking at a framed picture of a house where that same pie creature lives.
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In this example, the self-similar copy is only 16 times smaller than the original,
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and this just makes everything much easier to see all at once.
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What I'll do is keep a separate workspace on the right here to sketch out the
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general goal we have in mind, where the way you might think about it is that you
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want to distribute that 16-fold zoom-in factor across the four corners of the square.
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For example, let's say you want the pie creature itself to
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appear about at the same size in the lower left corner.
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Then if you zoom in by a factor of 2 in the original,
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we'll take what would be in the upper left of that zoomed version and then place it in
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the upper left of our workspace.
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Similarly, zoom in by another factor of 2 and then take the upper right of that zoomed-in
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version and now place an expanded version of that in the upper right of our workspace.
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Finally, one more step, zoom in by another factor of 2,
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take what's in the lower right of that zoomed-in version, blow it up,
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and place it in the lower right of our workspace.
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These four cut-out corners give a rough idea of what we want to create here.
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As long as you can find a smooth way to fill in the gaps between them,
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then as the onlooker's gaze wanders around a circle on this image,
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what they're seeing will zoom in and in and in until it elegantly joins up with the
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self-similar part of the image 16 times smaller.
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We could just try joining it all up naively, something like this,
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but frankly that doesn't really look very good,
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and it's certainly not as smooth and as elegant as what Escher managed to achieve,
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so we still have some work ahead of us.
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This book that I've been pawing through by the way, The Magic of M.C.
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Escher, is something that I got on a delightful visit to the Escher Museum in The Hague,
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which, if you're ever in the Netherlands, I would highly recommend,
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and when we turn to the section about the print gallery,
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it offers us a little peek into what Escher's process actually was.
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For him, step two was to create this warped grid.
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For the animations here, I'm going to pull up a slight modification
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of that grid that is basically just a little nicer mathematically to generate,
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but it illustrates the same key point.
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In his case, the self-similar Droste image he was working with has a scaling
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factor of 256, so to distribute that across the four corners,
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for him it would involve scaling by a factor of four as you walk from one
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corner to the next.
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And if you look closely at his grid, say at one of the squares on the lower right,
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notice how if you follow the top and bottom lines bounding that square over to the lower
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left of the image, those same lines end up enclosing a square that is now four times as
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big.
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And then similarly, if you look at the lines bounding the left and
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right of a small square from that region, and you follow them upward,
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they end up nestled around a square that's nicely four times as big.
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So the grid kind of encodes the scaling from one corner to the next that we want.
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Now for our pet project, where we only need to scale up by a
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factor of two from each corner to the next, we're going to use
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a modified version of this, but it'll be the same general idea.
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You might naturally wonder where on earth does this grid come from,
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but I actually want to postpone that question and skip ahead to step three,
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which is how you can actually use this grid together with the self- similar Drosse image
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to create Escher's final effect.
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The way this works is to first lay down an ordinary square grid on the original image,
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and then let's say you want this portion here to end up in this corner of our final
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version.
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What you would do is take each tiny square inside it and then copy over its
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contents to the corresponding tiny square in the warped version of the grid.
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From there, each neighboring square in our original grid is forced to go
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to the corresponding neighboring square in the warped version,
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so you can kind of just keep following where the image must go by following the grid.
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And the fact that the grid lines on the warped version space out by a factor of
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two as you go from the lower left to the upper left means that the scale of our
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scene automatically gets scaled up by that factor of two as you walk up that line.
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The idea of this process is that as an artist,
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it's relatively straightforward to go piece by piece,
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copying over what's inside one tiny square to another tiny square,
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because at this small scale things are undistorted.
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It's certainly way, way easier than trying to dream up and draw the appropriate
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warped final image starting with nothing but a blank page in front of you.
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So basically with this warped grid in hand, the process of copying over each little
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square is pleasantly automatic, and if we let this automatic process run all the way
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around, it nicely closes up with the initial position, and for that to happen,
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it requires that our original image had this self-similarity when you zoom in by a
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factor of 16.
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I hope you'll agree the final result we have is pretty nice,
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it recreates the same general effect, but more than anything I
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think it gives cause to more deeply appreciate Escher's own composition.
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He was actually very deliberate about the choice of imagery at all of the distinct scales.
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To quote, I quite intentionally chose serial types of objects,
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such for instance as a row of prints along the wall, or blocks of houses in a town.
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Without the cyclic elements, it would be all the more
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difficult to get my meaning over to the random viewer.
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This idea where you have a straight reference image and then a warped grid,
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and you use the two together to create a warped scene,
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is a common process in graphic design.
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It's known as a mesh warp, and Escher didn't invent it for this case,
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it's something that he had used multiple times before for other pieces.
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For our story, the point is that all of the logic for turning
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a Drasta zoom into a loop is abstracted and purified into this grid,
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raising the natural question, where does it come from?
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How do you make this?
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You can kind of imagine as an initial naive approach you might try linearly scaling
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everything from one corner to the next, but if you did that right away you would see a
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conflict.
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These two scaling processes are placing distinct pressures on the individual squares,
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where for example this square wants to get flared out in this direction
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based on the zooming in from the right, but it also wants to get flared
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in this direction based on the zooming out as you go up.
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Escher was evidently pulled to resolve this by
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curving all of the lines to relieve this tension.
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But there's one other crucial constraint that it seems he was guided by,
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one that presumably made the image transfer process easier,
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which made the final result look more natural,
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and which will make the ears of any mathematician immersed in complex
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analysis immediately perk up.
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In his final warped grid, the tiny squares are, well, squares.
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To be clear, this is not at all true for most mesh warps.
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In most cases, the warped grid lines that you draw don't necessarily intersect at right
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angles, and the little regions that they bound will in general be little parallelograms.
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This can still be workable for an artist, you can still kind of do the transfer,
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but presumably the act of copying over from the original to the warped version
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is more awkward when you have that distortion,
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so it would be much nicer if in the warped grid,
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little squares remained at least approximately square.
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And when you look closely at the grid that Escher used for his print gallery,
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it has this property.
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All the lines are intersecting at right angles, and at a small enough scale,
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the regions they bound really are approximately squares.
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Artistically, this has the nice effect that even though the whole
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scene is dramatically warped and distorted at a global scale,
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zoomed in at a local scale, everything is relatively undistorted.
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This is what makes each local part of Escher's image easily recognizable.
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And mathematically, this is where the story really gets interesting.
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You see, this idea of a function from two dimensional space to two dimensional space,
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where tiny squares remain approximately square,
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plays a special role and has a special name in math.
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It's called a conformal map, and one area where it comes up all the time is in
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the study of functions with complex number inputs and complex number outputs.
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At this point, we're going to step back and walk out of the art
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classroom and wander over into the math department,
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where I want to offer you a mini lesson on some of the core ideas from this field.
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The basic game plan from here is that I want to first do a little refresher,
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go over some of the basics of complex numbers and complex functions,
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and then I want to spend some meaningful time building up an intuition
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for what logarithms look like in this context of complex numbers.
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And then once you have that in hand, we're going to step through a completely different
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way that you can think about recreating this effect that Escher had in his print gallery.
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Let's warm up with a review of the basics.
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We typically think about real numbers as living on a one dimensional line,
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the real number line, and complex numbers are two dimensional.
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Specifically, we think of the imaginary constant i,
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defined to be the square root of negative one as being one unit above zero,
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end every other point on this plane represents some combination of a real number
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with some real multiple of i.
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It's typical to use the variable z in referring to a general complex number,
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and the game we want to play is to understand various functions of z.
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A very simple but important example is to multiply z by some constant.
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The function f of z equals two times z has the
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effect of scaling everything up by a factor of two.
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But what about multiplying by something imaginary,
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like i, the square root of negative one?
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Well you know that multiplying one by i gives you i,
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and by definition, i times i is negative one.
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Both of these you'll notice are in 90 degree rotation, and more generally,
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multiplying any value z by i has the effect of a 90 degree rotation.
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And more general than that, when you multiply by any complex constant,
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the effect is some combination of scaling and rotating.
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And there's a nice way to think about exactly how much it should scale and rotate.
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Zero times anything is zero, so the origin has to stay fixed in place,
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and then one times any constant c is that same constant c.
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So that means this point at one has to be dragged over to land
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on whatever that constant c is that we're talking about,
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and that fully determines the amount of scaling and rotating.
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From there, the rest of the grid stays as rigid as it can.
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A key point to emphasize for our story is that if all you're doing
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is multiplying by some constant, shapes are always preserved.
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Anything you might want to draw, like a square,
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can get scaled or rotated, but beyond that, there are no distortions.
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Now things get more interesting for more complicated functions.
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A simple but non-trivial example would be mapping each number z to z squared.
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So the input two is going to have to move to two squared, which is four.
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The input i is going to have to move to i squared, which is negative one.
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Negative one itself would have to get mapped to positive one.
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And in its fullness, here's what it looks like if I let every point among
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the grid lines of this input space move over to their corresponding outputs.
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It's a pretty nice effect, and unlike multiplying by a constant,
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shape is absolutely no longer preserved.
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The grid lines get curved and warped.
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However, and this is a key point, pay attention to what happens at a small scale.
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For example, just focusing on this little square from the input space.
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As you watch the transformation happen again, you can see that
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shape is approximately preserved, at least at a small enough scale.
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Little squares from our original grid remain approximately
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square even after getting processed by the function.
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To use the lingo, the function z squared gives a conformal map.
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And the choice of z squared is really not special here.
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Here's what it looks like if you transform each point z over to z cubed.
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Squares remain approximately square.
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Okay, maybe this example square that I'm highlighting doesn't exactly look
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square in the output, but really it's just because it started off too big.
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To be clear, this conformal property that I'm talking about is a limiting one.
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A particular square might not exactly become square,
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but the idea is that as you zoom in more and more,
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choosing square regions from the input space that are smaller and smaller,
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the resulting output will indeed be better and better approximated by a square.
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And there's also nothing all that special about the
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choice of polynomials like z squared or z cubed.
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For just about any function of complex numbers you could think to write down,
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this property holds.
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Shape is preserved at a small enough scale.
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It's almost like magic.
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The use of complex numbers is very relevant here.
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If instead you were thinking of points in 2D space simply as a pair of real numbers with
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some xy coordinates, and you write down some arbitrary function of x and y to get a
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new pair of numbers, what is way, way more typical as you let points in that input
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space get transformed is that the outputs of those tiny squares get squished and
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distorted.
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Even as you zoom in more and more, the resulting limiting shape
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typically looks like a parallelogram, not necessarily a square.
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So, complex functions really are special in this way.
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And the basic reason that tiny squares remain square comes down to calculus.
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What you're looking at is what it means for these functions to have derivatives.
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That might sound a little strange, but the analogy you can think of is that for most
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functions of real numbers, if you visualize them the ordinary way,
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just with a graph of inputs on the x-axis, outputs on the y-axis,
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as you zoom in more and more to any particular point on that graph,
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it looks more and more like a straight line.
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That is, the rate of change, how much delta f you get for a given delta x,
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approaches a constant.
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This is literally what it means for a function to have a derivative,
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but it's not the only way to visualize it.
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Here's a different way to see the same concept.
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Instead of looking at the graph of a function,
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let's think of the same function but as a transformation.
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That is, you're going to let each of these points on the real number
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line move over to their corresponding outputs on this other number line.
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What you'll notice is that evenly spaced dots from the input space can get warped in
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the output space, meaning the rate of change of the function in general is not constant.
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This spacing between our dots can change from one part of the image to the next.
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But we know that as you zoom in more and more to a particular output,
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the rate of change approaches a constant.
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The dots look more and more evenly spaced.
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Specifically, if you take a tiny patch of dots around a particular input and you
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copy them over to the output space around the corresponding output,
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you can approximately line up all the dots just by scaling everything by a certain
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constant factor.
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This is the same analytical fact as what the slope of a graph is telling you,
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it's just in a different visual context.
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But this context carries over much more easily to thinking about
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complex valued functions as transformations in the complex plane.
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There we're going to think of the neighborhood around a given input
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as a tiny little grid of points, like we were before,
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and we let each point on that grid move over to its corresponding output.
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In this case, what it means for the rate of change to
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approach a constant is basically the exact same equation.
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The visual to have in your head is that if you take that tiny patch of squares
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around the input and copy it over to the corresponding output,
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you can approximately match this up with the output grid lines by multiplying
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by a certain constant, which remember, in the setting of complex numbers,
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means rotating and scaling it in some way, depending on the value of that
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complex constant.
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Since rotation and scaling preserve shape, it means that all the tiny squares from
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the input space remain at least approximately square under this transformation.
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So, stepping back, here's the key point, the reason for talking about any of this at all.
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Even though conformal maps like the one that Escher was using for his print
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gallery are incredibly constrained and highly unusual among the general ways
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that you could continuously squish about two-dimensional space, nevertheless,
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as if by magic, simply by speaking a language of complex numbers,
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you can somehow create entire families of these conformal maps without even
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really trying.
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All you do is mix and match standard functions of complex numbers.
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The one constraint is that these functions have to have derivatives,
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but this will be true for most of the functions you think to write down.
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For our story, decoding what's happening with the print gallery,
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this means that we have an entirely new way to reframe the key question.
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Can you construct some deliberately tailored complex function so that the act of
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zooming in around the inputs looks like walking around a loop among the outputs?
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Now, at this point, with only the bare minimum crash course of complex
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analysis under our belts, it's hard to know where to start without
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first building up a larger palette of functions to work with and
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gaining some familiarity with how they actually behave for complex numbers.
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In this case, there are really only two functions that you need to understand,
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e to the z and the natural log.
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I want to settle in and spend some meaningful time understanding both of these,
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and I think you'll agree that it's time well spent.
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Everybody deserves, in my opinion, at least one time in their
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life to experience the joy of understanding a complex logarithm.
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First though, a necessary prerequisite is to understand the complex exponential.
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Regular viewers will be familiar with what it looks like to raise
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e to the power of a complex number, but a review just never hurts.
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We can start scaffolding the transformation just by focusing on the more familiar
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examples of real number inputs and the real number outputs they correspond to.
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For example, e to the 0 is 1, so this point at 0 maps over to this point at 1.
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And then every time you let that input increase by 1,
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the output grows by a factor of e, meaning it runs away from us actually quite quickly.
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And then on the flip side, as you decrease that input,
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letting it get into the negative numbers, every step to the left corresponds to
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shrinking the output, again by a factor of e.
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In particular, you'll notice in this case that output is always a positive number.
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This of course gets much more interesting when we let
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the inputs and the outputs both be complex numbers.
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And here, everything you need to know comes down to what
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happens as you let the imaginary part of that input increase.
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And what happens there is that the corresponding output walks around a circle.
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If you're wondering why imaginary inputs to an exponential walk you around a
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circle like this, we have discussed it many, many other times on this channel.
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See some of the links on screen and in the description.
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A key point that I'll reiterate here is that what makes the function
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e to the z very nice, as opposed to exponentials with other bases,
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is that as your input walks up at a rate of 1 unit per second,
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the output walks around its circle at a rate of exactly 1 radian per second.
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So in particular, increasing the imaginary part by
23:33
exactly 2 pi causes one full rotation in the output.
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Phrased another way, these vertical line segments that I've been drawing with heights of
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exactly 2 pi each get mapped neatly onto one complete circle when you apply the function.
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You'll notice how I've drawn these particular vertical line segments to be spaced out
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evenly in the real direction, where the real part from 1 to the next increases by 1.
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The corresponding circles on the right each differ by a constant scaling factor,
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specifically the scaling factor e.
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Earlier for the simpler function z squared, I showed it as a transformation,
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moving all of the inputs to the outputs.
24:11
And in this case, for e to the z, if you're curious how that same idea looks,
24:14
it's easiest to take a subset of the grid, like this one here from the input space,
24:19
and here's what it looks like to move each square over to the corresponding output.
24:24
As we actually use this function for our print gallery goals,
24:27
it'll be very helpful to anchor your mind by thinking of these vertical lines and the
24:31
circles they turn into.
24:33
In fact, there's a very playful way that I like to think about how these vertical lines
24:37
turn into concentric circles and how they can carry the full input space along with them.
24:41
What I like to imagine is sort of rolling up that entire z-plane into a tube,
24:46
such that all of those vertical lines end up as circles.
24:50
Specifically, each circle would have a circumference of 2 pi.
24:54
Next, imagine taking this tube, lining it up above the origin of the output space,
24:58
and then kind of squishing it down onto that output space,
25:02
turning all the circles from that tube into these concentric rings of exponentially
25:06
growing size.
25:08
That's just what I like, but however you choose to think about it,
25:11
what I want to be etched into your brain is the idea of vertical lines turning
25:15
into circles.
25:16
Now, the other very important point to emphasize here is
25:19
that multiple different inputs can land on the same output.
25:23
For example, e to the zero is one, but e to the 2 pi i is also one.
25:29
So is e to the negative 2 pi i and e to the 4 pi i and so on.
25:34
In fact, the infinite sequence along any given vertical line spaced out by 2 pi will
25:39
all get collapsed together as that vertical line gets kind of rolled up into a circle.
25:45
We say that the exponential map is many to one,
25:47
although it might not be obvious how this repetition in the vertical
25:51
direction is going to be key to our final recreation of the print gallery effect.
25:56
Okay, so exponentials give us the first ingredient,
25:59
characterized by turning lines into circles, and the second ingredient
26:03
we need is the inverse of such an exponential, known as the natural log,
26:07
where basically the idea is that it unravels those circles back into lines.
26:11
Now, this will be especially fun to visualize and especially relevant to
26:15
our story if we imagine painting this complex plane on the right with a Droste image,
26:20
say the example we were working with earlier with the pi creature looking
26:24
at a picture of a house where that same pi creature lives.
26:28
In other words, it is finally time for you and me to answer that
26:31
question of what it means to take the natural log of a picture.
26:36
Okay, so think about this for a second.
26:38
You already know that a vertical line segment like this one on the left with
26:42
a height of 2 pi gets turned into a circle when you apply the map e to the z.
26:47
So the natural log is going to take a circle of points on this
26:51
image and then straighten them out into one of those lines.
26:55
Similarly, if you looked at a circle which was exactly e times smaller,
26:59
that would also get straightened out into a line with the same height positioned one
27:04
unit to the left.
27:05
And then similarly, every circle in between these two is going to get unwrapped into
27:10
one of these vertical lines between those last two, and more generally,
27:14
smaller and smaller rings from the picture will all get unwrapped into these vertical
27:18
line segments, each one with a height of 2 pi,
27:21
farther and farther to the left in the image.
27:24
The result we get is a bit trippy, but it's pretty cool when you think about it,
27:27
and there's a number of important things that I want you to notice.
27:30
You've probably noticed that it repeats as you move to the left,
27:33
and I'll invite you to ponder why that might be the case in the back of your mind
27:37
for a minute, but before that repetition, I want to talk about a different direction
27:41
in which it repeats.
27:42
The way I've drawn it so far, the imaginary part for the values on the left
27:46
are ranging from 0 up to 2 pi, but that's actually kind of an arbitrary choice.
27:51
Remember, if you keep letting that value z on the left walk up by another 2 pi units,
27:56
the corresponding value e to the z on the right would just keep walking around that
28:01
same circle again, so I hope you'll agree it feels at least enticing to let our
28:05
image repeat in this vertical direction along every one of those vertical lines.
28:12
And it goes the other way too, as you let the imaginary part on the left get smaller
28:17
going down, the corresponding value on that right just keeps walking around that
28:21
same circle, so the pattern that you see should perhaps repeat in that way too.
28:26
To be more explicit, the rule that I'm using to draw this image on the left
28:30
is that for every point z in that plane on the left,
28:33
you look at the corresponding value e to the z on the right,
28:37
and then you assign it a matching color.
28:40
So for example, this warped pi creature and this one and this one all really correspond
28:45
to the same part of the image on the right, the big pi creature in the lower left.
28:51
Another way to think about this is that because the function e to the z is many to one,
28:56
the feeling we get is that the natural logarithm, its inverse,
28:59
wants to be a multi-valued function, something where one input maps to multiple
29:04
different outputs.
29:06
Now in practice, many times you don't want a function to have multiple outputs,
29:10
sometimes that even defies the definition of a function in your context,
29:14
so people will often just choose a band of this plane on the left to be the outputs
29:18
of the natural log.
29:20
In complex analysis, this is called choosing a branch cut for the function.
29:24
For our purposes though, where we want to recreate and understand Escher's piece,
29:28
it's actually nice as to think of the log as a multi-valued function,
29:32
where each point on the right corresponds to a repeating sequence of points on the left,
29:37
spaced out two pi vertically, going infinitely in both directions.
29:41
Now in our special case, you also see this repeating pattern as you move to the left,
29:46
but that is something different entirely.
29:48
You would not see this for most images.
29:51
It arises specifically because we're working with a self-similar Droste image,
29:55
one that looks identical as you zoom in by a certain factor.
29:59
This falls straight out of a core property of logarithms and exponentials.
30:04
Exponentials turn addition into multiplication,
30:08
and logarithms turn multiplication back into addition.
30:12
For example, imagine taking some small value w on this plane on the right,
30:17
and also considering 16 times that value, which is scaled up 16 times farther away from
30:22
the origin.
30:24
Now if we look at the corresponding value, log of w on the left,
30:28
that act of multiplying by 16 now looks like addition,
30:32
specifically shifting to the right by the natural log of 16,
30:36
which is just some real number.
30:39
In fact, this rectangle here in our bizarre warped log image that has a width of log
30:44
16 and a height of 2 pi contains all the information about the image on the right.
30:50
It corresponds to this annulus of the Droste image,
30:54
and if you were to shift that rectangle exactly log of 16 units to the left,
30:58
you would get a scaled-down version of that annulus exactly 16 times smaller that
31:04
nestles in perfectly like a puzzle piece.
31:07
And if you repeat that infinitely many times, it gives you
31:10
this infinite nesting that characterizes the Droste zoom.
31:14
The way I've drawn things so far, we have this cutoff to the log image on the right side,
31:19
and at this point you know well that vertical lines on the left correspond to circles
31:23
on the right, so you can probably guess that this corresponds to the fact that I gave
31:27
a maximum radius to that Droste image, but of course you don't need to do that.
31:31
In principle, it can extend out infinitely far in all directions,
31:35
following the same self-similar pattern as you scale up,
31:38
and the result for the log image on the left would be to extend as far rightward as
31:43
you want, again with a repeated tiling pattern.
31:47
So to conclude, the logarithm image on the left is periodic vertically,
31:51
basically because rotation on the right is periodic.
31:54
This would happen for any image.
31:56
And then in this special case of a Droste image, it's also periodic horizontally,
32:01
because the Droste image repeats as you zoom in.
32:04
This doubly periodic property is what we'll ultimately
32:07
take advantage of for the final result.
32:10
And we now have all the foundation we need.
32:13
I can now finally describe for you the function that turns
32:16
the Droste zoom into this Escher-style self-contained loop.
32:20
Before just jumping right into it, we have been kind of covering a lot,
32:23
so it might be worth giving some space to let this digest.
32:26
And I was meaning to take 30 quick seconds at some point in this video
32:29
to talk about 3b1b talent, so now might be as good a time as any.
32:32
This is the virtual career fair that I'm experimenting with this year,
32:36
and the basic idea is that you, a person who spends their free time learning about
32:40
things like complex logarithms, are probably interested in working with like-minded,
32:44
curious, and technical teams.
32:46
So if you're seeking or open to a new job, check out the organizations at 3b1b.co.
32:52
talent.
32:52
When you explore that page, you'll find interviews between me and the relevant teams,
32:56
technical puzzles and challenges that they've chosen to feature for you,
32:59
and various other things aimed at giving you a feel for the technical team culture.
33:04
And I just kind of like the idea of exposing this audience to
33:07
aligned career opportunities, so whenever you are looking for a job,
33:10
whether that's now or later, be sure to check it out.
33:14
Okay, so back to our main goal.
33:16
How is it that we can use everything that we've built up about complex functions
33:20
and transformations that give conformal maps and everything like that to
33:24
construct some kind of function that recreates Escher's print gallery effect?
33:28
Maybe it's easiest if I just throw down the whole outline of the function,
33:31
and then we can go through each step in more detail.
33:34
First, take a logarithm, giving this bizarre doubly periodic tiling pattern,
33:38
and then you rotate and scale that tiling pattern in just the right way,
33:43
and then from there you take an exponential, which unworps it,
33:47
but this time with a certain twist.
33:50
This might seem a little bizarre, but there's actually a really
33:52
nice way to motivate and to understand what's really going on here.
33:56
Take a look back at that original Droste image,
33:59
and remember how we want to take this big pie creature and
34:02
somehow identify it with the smaller self-similar copy, 16 times zoomed in.
34:07
I want you to think about a line connecting both of those.
34:11
One way to frame the goal that we have is that we want the final
34:14
function to transform such a line into a closed loop in the final space.
34:19
The endpoints of the line on the top represent the big and the small pie creatures,
34:24
so whatever function identifies those creatures should, at the very least,
34:28
close up this line.
34:29
Now think about what this line looks like in the logarithm of the image.
34:33
The big pie creature corresponds to all these
34:36
multiple copies here next to the imaginary line.
34:39
As we talked about, when you scale down the image,
34:42
it corresponds to shifting to the left in the log,
34:45
so these copies in the log image represent that small pie creature.
34:50
And actually, instead of using this horizontal line,
34:52
we're going to want to take advantage of the fact that this log image is periodic
34:56
in two separate directions.
34:58
So instead, what we'll work with is a diagonal line that connects this
35:03
copy of the big character to this lower left copy of the small one.
35:07
That might seem unmotivated at the moment, and it is,
35:10
but the best way to explain why is just to show you how this plays out,
35:13
and afterwards, if you want, we can contrast with what would have happened if
35:16
you tried using the horizontal line.
35:19
Now this new downward component in the log image corresponds to
35:22
adding some clockwise rotation to the path in the original image.
35:26
Remember, our goal is to turn this into a loop in the final image,
35:30
but you and I just spent like 10 minutes talking extensively about
35:34
a function that turns line segments into circles, namely e to the z.
35:39
What you need to do is get that line segment to end up perfectly vertical
35:43
with a height of 2 pi, and doing this basically comes down to rotating
35:47
and scaling it in just the right way to give it that length and direction.
35:52
Now if you remember, as we were warming up with complex numbers,
35:56
we talked about how multiplication by a constant gives you some combination of
36:00
rotation and scaling, and in this case, one artistic feature we might like is
36:04
for our big pi creature on the lower left to stay fixed in that position,
36:08
and that would mean that this point in our log image needs to stay fixed in place,
36:12
so instead of pivoting around the origin, we really want everything to pivot
36:16
about that point.
36:18
Let's say we label that point something like z naught,
36:21
then here's how the updated formula would look for that rotation and scaling,
36:25
but really most of the content comes down to choosing this constant c that
36:29
you're going to multiply by, and if you like exercises,
36:32
you might enjoy actually taking a moment to work out what specific value
36:35
that constant should take on.
36:38
But right here, since we're being very visual and I want to have a little fun,
36:42
let me show you that constant in its own little complex plane,
36:45
and then also show what happens if I kind of grab it and move it around a little bit.
36:50
Different choices give you different scaling and rotation,
36:53
which after exponentiating gives you all these completely bizarre kaleidoscopic images.
36:58
One way you could think about the whole operation is as a game where
37:01
you're trying to find just the right value that corresponds everything
37:05
to line up as you need them to in that image of the lower right.
37:09
When it is set to that appropriate value and we get this recreated Escher effect,
37:12
in that final image on the lower right, I've been showing this hole in the middle,
37:16
analogous to the hole that Escher had in his original piece.
37:20
But that's actually entirely artificial.
37:22
The output image of the function naturally fills
37:25
that in with its own repeating spiral inward.
37:28
After all, the rotated tiling pattern on the left extends
37:31
infinitely through the whole plane, so when you take its exponential,
37:35
it also fills in everything, except for the point at zero.
37:39
And that's it!
37:40
That is the basic operation.
37:42
Translate to this bizarre looking log space, rotate and scale to realign things,
37:47
and then translate back with an exponential.
37:51
At this point, having recreated our own variation on Escher,
37:54
it might be nice to step through what that same process looks like for Escher's specific
37:59
example, that seaside Maltese town containing a print gallery with a person looking at
38:04
a picture of the town.
38:06
As before, step number one is to place this scene on its own little complex plane,
38:10
where the infinite limiting point about which everything scales is positioned at zero.
38:16
Step two, take a logarithm of this, which in this case gives us a similarly bizarre
38:20
transformation of the whole scene, but there is a little difference this time.
38:24
Because Escher was working with a much deeper zoom, with a scale factor of 256,
38:29
the repeating tiles in this new example actually span a wider part of the complex plane.
38:36
Step three, again, is to rotate and scale this,
38:38
which depends on multiplying by the appropriate choice of a complex constant.
38:43
The constraint is to ensure that this rotated image still repeats every 2 pi
38:48
units vertically, but along a part of the image that was previously diagonal.
38:53
And then step four, plug this all through e to the z,
38:56
and what you get is something very similar to Escher's final picture,
39:00
an image where walking around a loop corresponds to zooming in by a factor of 256.
39:07
And again, when we frame it this way with complex functions,
39:10
there is no hole in the middle.
39:12
That final result is itself a Droste image, albeit this time with a twist,
39:16
where there's a kind of self-similarity spiraling down infinitely far as much as you
39:21
want to zoom in.
39:24
There are a couple things worth noticing about this whole process.
39:27
First off, if you write this idea down as a formula, where you take a log,
39:31
multiply it by a constant, and then exponentiate,
39:33
that can actually be simplified to simply look like raising your input to the
39:37
power of some complex constant.
39:39
And if you reintroduce that offset factor, it just introduces another constant.
39:43
On the one hand, it's very pleasing that this entire complicated
39:46
process can be boiled down to so few symbols on the page,
39:49
but I do think this risks kind of obscuring what's actually going on.
39:53
And then, I had mentioned things would not work
39:55
if you only tried using those horizontal lines.
39:58
If you're curious, here's what it looks like if you try that,
40:00
which would mean rotating everything by 90 degrees and scaling it the appropriate amount.
40:05
You do get something kind of interesting, but it's just not what we're going for.
40:09
And if you think about it, what's basically going on is
40:11
that you are swapping the roles of rotation and of scaling.
40:16
Stepping back from all this, this second perspective where the piece is
40:20
all about rotating in a log space feels completely different from Escher's
40:24
more intuitive approach using his distorted grid and the mesh warp.
40:28
But there is a through line connecting both perspectives.
40:31
For one thing, you can now understand how exactly we've
40:34
been recreating and modifying the grid that Escher had.
40:37
You basically take that same function that we've built up,
40:40
but instead of applying it to an image, you apply it to an ordinary square grid.
40:45
Well, actually not quite an ordinary square grid.
40:48
It's nicest to work with one that gets more dense as
40:50
you zoom in so that it looks the same at all scales.
40:54
When you do this, the logarithm gives you this very beautiful curved tiling pattern.
40:58
And then from there, you do the same trick of rotating things in just
41:02
the right way and then taking an exponential with the result of
41:05
something quite close to what Escher spent many arduous hours constructing.
41:09
And remember, the entire reason that we're using the language of
41:12
complex numbers is the motivation for me having you jump through
41:16
those hoops is that this conformal property falls out as a byproduct.
41:20
Tiny squares from the original grid remain, at least approximately,
41:23
square in the final result.
41:26
Escher said that making this piece gave him some almighty headaches,
41:29
and though maybe you and I had to endure a few headaches ourselves for very
41:33
different reasons, the payoff is that this property is one that we get with no
41:37
added effort.
41:39
Now to be clear, you certainly don't need to understand complex derivatives or
41:42
logarithms to enjoy Escher's work, and I wouldn't want to imply that you do.
41:46
However, when you understand that more mathematical side,
41:49
it gives you the capacity for a very different kind of appreciation for what
41:53
Escher was really doing.
41:55
And this is what I find fascinating about all of this,
41:57
the real connection I want to give between the two storylines.
42:01
If you look at Escher's career, throughout it he was drawn to certain concepts,
42:05
things like representing infinity in a finite space.
42:08
And at the same time, he seemed to be guided by a certain aesthetic,
42:11
often some implicit rigid rule like the idea of little squares remaining little squares
42:16
for this mesh.
42:17
The end result when concept and aesthetic are combined like this is
42:21
that a given piece from Escher often feels like a solution to a puzzle,
42:25
but one where it's not even obvious that a solution should exist.
42:29
Now what I find so thought-provoking is that these visual puzzles
42:32
that he landed on are not merely analogous to the act of doing math.
42:36
The specific structures that he was intuitively drawn to
42:40
often hide within them very real and very deep mathematics.
42:44
In our example, the deeper structure is not just that
42:46
the piece can be described with complex functions.
42:50
The ideas that we've touched on in this storyline are actually
42:53
a lot closer than you might expect to the research frontiers.
42:56
If you look back at our approach in the second half,
42:59
remember how it relied on using this doubly periodic pattern in the complex plane,
43:04
something that repeats in two separate directions.
43:07
There's actually a special name for functions of
43:10
complex numbers that are doubly periodic like this.
43:12
They're known as elliptic functions.
43:14
It would be way too much to explain right here exactly why these functions are so useful,
43:19
but one thing I want to highlight is that those two mathematicians I referenced at
43:23
the start, the ones who provided the analysis of this piece, De Smit and Lenstra,
43:27
are both number theorists, and elliptic functions play a very prominent role in
43:31
modern number theory, providing a kind of bridge to other parts of mathematics.
43:36
This may at least partially explain how it is that they could look at
43:39
this piece and think to construct that function that we laid out here.
43:44
When I look at Escher's work, whether it's the print gallery or many other favorites,
43:48
the reason I love these pieces so much is that they awaken within me
43:52
a very specific feeling that's hard to find elsewhere.
43:55
It's a feeling of things perfectly fitting into place.
43:58
But that's not exactly it.
43:59
It's something more than just the pleasure of seeing a puzzle solved.
44:02
It's an appreciation for the creative genius required
44:05
to even dream up the puzzle in the first place.
44:08
The only other place where I get that feeling is in doing math.
44:11
So the fact that an artist and a mathematician can be drawn to the same structures,
44:16
but for completely different reasons, suggests to me that there's something
44:20
universal in what exactly it is that both of them seem to be drawn to.
— end of transcript —
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