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Whenever I'm making one of these videos, there's sometimes a special moment
[00:03] where the act of animating involves solving a whole bunch of little technical puzzlers,
[00:07] and then the underlying math I'm trying to explain clicks for me in a
[00:11] way that it hadn't before once I see it alive on screen.
[00:14] The best versions of those moments often tell me when a video is
[00:17] going to be one of my favorites, and putting together the end of
[00:20] this piece right here was one such time when I got that feeling.
[00:27] Our story doesn't actually start in the math classroom today.
[00:30] We begin in the art room.
[00:32] Imagine standing in a gallery, looking at a picture of a boat in a harbour,
[00:36] and the whole world warps as your gaze shifts upwards and to the right,
[00:40] where you see a village on this waterfront.
[00:43] Among the tightly clustered buildings, the world warps even more as your gaze shifts
[00:48] downwards to the entrance of one building, leading to a hallway full of artwork.
[00:53] And at the end of this hall, here you are again, staring at a picture of a boat.
[00:58] This is M.C.
[00:59] Escher's 1956 lithograph, the Print Gallery, or in Dutch, Prentendentunstelling.
[01:05] In a letter that he wrote to his son, describing his creation of a young
[01:10] man looking with interest at a print that features himself,
[01:13] Escher called this the most peculiar thing I have ever done,
[01:16] which for him is saying a lot.
[01:18] Escher's art is widely loved around the world,
[01:21] frequently featuring paradoxical themes or uniquely satisfying geometric patterns
[01:25] with some kind of character.
[01:27] This love is especially pronounced among mathematicians,
[01:30] since his art often touches on surprisingly deep concepts within math,
[01:34] despite the fact that Escher himself had no formal training in the field.
[01:39] The Print Gallery offers a perfect example of this unexpected depth.
[01:43] In 2003, the mathematicians De Smit and Lenstra offered a delightful
[01:47] analysis of what's really going on in this piece and the
[01:50] mind-bending self-contained loop that Escher managed to achieve.
[01:54] One of my main goals with this video is to offer a visual unpacking of their analysis,
[01:59] aiming as always to give you a feeling that you could have rediscovered this yourself.
[02:05] Something that unexpectedly pops out of this analysis is an answer to
[02:08] the question of what exactly should go in the middle of this picture.
[02:12] At first, that might sound like an incoherent question.
[02:15] If you come at it from the upper right, it feels like it should be featuring buildings in
[02:19] the village, but coming at it from the left, it looks more like part of the picture
[02:23] frame.
[02:23] Come at it from below, though, and it feels more fitting to be part of the gallery itself.
[02:28] Somehow all of the ambiguity about where exactly you are in this
[02:31] whole scene is compressed into that blank circle in the middle.
[02:36] And just for fun, since we're in the 2020s, I let
[02:38] a diffusion model take a stab at filling this in.
[02:41] Unsurprisingly, it struggles immensely.
[02:43] It just doesn't get it at all.
[02:45] And giving the poor machine some credit, of course it struggles.
[02:48] This feels like an intrinsically ambiguous and ill-defined part of the scene.
[02:52] But nevertheless, by the end, I hope you'll agree there is one, and really only one,
[02:57] completion that feels like the right puzzle piece you can slot in.
[03:02] Before diving into the math, I want to give you a purely intuitive description for
[03:05] how Escher actually made this piece, which breaks up into three different steps.
[03:10] Step one is to start out with a straightened out version of the same general concept,
[03:15] where a man is looking at a picture.
[03:17] That picture contains a harbor, which contains a town,
[03:21] that contains a print gallery, that contains that same man, and so on and so forth.
[03:26] You could zoom in forever to your heart's content.
[03:30] This idea of a self-similar image, where the picture is contained inside itself,
[03:34] has a special name to graphic designers.
[03:37] It's known as the Droste effect, named after a
[03:39] cocoa company that featured it in its branding.
[03:42] In fact, this seems to have been a somewhat common marketing
[03:45] gimmick for all kinds of early 20th century products.
[03:48] The self-similar Droste image that Escher was using, though,
[03:52] involves a much, much deeper zoom than any of these,
[03:55] where the self-similar copy is 256 times smaller than the original.
[04:00] You'll see where that number comes from in just a minute.
[04:03] The genius of Escher is that he somehow intuitively realized there must be a way to
[04:08] take this concept of a picture nested inside itself and turn it into this warped loop
[04:14] where the zooming in happens implicitly as a viewer's gaze wanders around the circle.
[04:21] By the way, you might be wondering where I got this straightened out version,
[04:24] and the answer is that those two mathematicians I referenced generously let us use it.
[04:28] This is something they actually reverse engineered from the original Escher
[04:32] piece using the help of two Dutch artists, Hans Richter and Jacqueline Hofstra.
[04:37] The way they reverse engineered it is actually super interesting.
[04:40] In a certain manner of speaking, it involves taking the logarithm of the original piece.
[04:44] I recognize that sentence probably sounds like nonsense right now,
[04:48] but later on in this video, I promise that will make abundant sense.
[04:52] To outline the basic idea for how Escher made this loop,
[04:55] I want to use an example that's simpler than the one he was working with,
[04:59] so we'll pull up this custom self-similar Drasta image which has a pie
[05:02] creature looking at a framed picture of a house where that same pie creature lives.
[05:08] In this example, the self-similar copy is only 16 times smaller than the original,
[05:12] and this just makes everything much easier to see all at once.
[05:15] What I'll do is keep a separate workspace on the right here to sketch out the
[05:20] general goal we have in mind, where the way you might think about it is that you
[05:24] want to distribute that 16-fold zoom-in factor across the four corners of the square.
[05:30] For example, let's say you want the pie creature itself to
[05:33] appear about at the same size in the lower left corner.
[05:36] Then if you zoom in by a factor of 2 in the original,
[05:39] we'll take what would be in the upper left of that zoomed version and then place it in
[05:44] the upper left of our workspace.
[05:47] Similarly, zoom in by another factor of 2 and then take the upper right of that zoomed-in
[05:52] version and now place an expanded version of that in the upper right of our workspace.
[05:58] Finally, one more step, zoom in by another factor of 2,
[06:01] take what's in the lower right of that zoomed-in version, blow it up,
[06:05] and place it in the lower right of our workspace.
[06:08] These four cut-out corners give a rough idea of what we want to create here.
[06:13] As long as you can find a smooth way to fill in the gaps between them,
[06:17] then as the onlooker's gaze wanders around a circle on this image,
[06:21] what they're seeing will zoom in and in and in until it elegantly joins up with the
[06:26] self-similar part of the image 16 times smaller.
[06:29] We could just try joining it all up naively, something like this,
[06:33] but frankly that doesn't really look very good,
[06:35] and it's certainly not as smooth and as elegant as what Escher managed to achieve,
[06:39] so we still have some work ahead of us.
[06:42] This book that I've been pawing through by the way, The Magic of M.C.
[06:46] Escher, is something that I got on a delightful visit to the Escher Museum in The Hague,
[06:50] which, if you're ever in the Netherlands, I would highly recommend,
[06:53] and when we turn to the section about the print gallery,
[06:56] it offers us a little peek into what Escher's process actually was.
[06:59] For him, step two was to create this warped grid.
[07:03] For the animations here, I'm going to pull up a slight modification
[07:06] of that grid that is basically just a little nicer mathematically to generate,
[07:09] but it illustrates the same key point.
[07:11] In his case, the self-similar Droste image he was working with has a scaling
[07:16] factor of 256, so to distribute that across the four corners,
[07:20] for him it would involve scaling by a factor of four as you walk from one
[07:24] corner to the next.
[07:26] And if you look closely at his grid, say at one of the squares on the lower right,
[07:30] notice how if you follow the top and bottom lines bounding that square over to the lower
[07:35] left of the image, those same lines end up enclosing a square that is now four times as
[07:40] big.
[07:40] And then similarly, if you look at the lines bounding the left and
[07:43] right of a small square from that region, and you follow them upward,
[07:47] they end up nestled around a square that's nicely four times as big.
[07:50] So the grid kind of encodes the scaling from one corner to the next that we want.
[07:55] Now for our pet project, where we only need to scale up by a
[07:58] factor of two from each corner to the next, we're going to use
[08:01] a modified version of this, but it'll be the same general idea.
[08:05] You might naturally wonder where on earth does this grid come from,
[08:08] but I actually want to postpone that question and skip ahead to step three,
[08:12] which is how you can actually use this grid together with the self- similar Drosse image
[08:16] to create Escher's final effect.
[08:19] The way this works is to first lay down an ordinary square grid on the original image,
[08:23] and then let's say you want this portion here to end up in this corner of our final
[08:28] version.
[08:29] What you would do is take each tiny square inside it and then copy over its
[08:33] contents to the corresponding tiny square in the warped version of the grid.
[08:38] From there, each neighboring square in our original grid is forced to go
[08:42] to the corresponding neighboring square in the warped version,
[08:46] so you can kind of just keep following where the image must go by following the grid.
[08:52] And the fact that the grid lines on the warped version space out by a factor of
[08:56] two as you go from the lower left to the upper left means that the scale of our
[09:00] scene automatically gets scaled up by that factor of two as you walk up that line.
[09:06] The idea of this process is that as an artist,
[09:09] it's relatively straightforward to go piece by piece,
[09:12] copying over what's inside one tiny square to another tiny square,
[09:15] because at this small scale things are undistorted.
[09:19] It's certainly way, way easier than trying to dream up and draw the appropriate
[09:23] warped final image starting with nothing but a blank page in front of you.
[09:28] So basically with this warped grid in hand, the process of copying over each little
[09:33] square is pleasantly automatic, and if we let this automatic process run all the way
[09:38] around, it nicely closes up with the initial position, and for that to happen,
[09:43] it requires that our original image had this self-similarity when you zoom in by a
[09:47] factor of 16.
[09:49] I hope you'll agree the final result we have is pretty nice,
[09:52] it recreates the same general effect, but more than anything I
[09:55] think it gives cause to more deeply appreciate Escher's own composition.
[09:59] He was actually very deliberate about the choice of imagery at all of the distinct scales.
[10:03] To quote, I quite intentionally chose serial types of objects,
[10:07] such for instance as a row of prints along the wall, or blocks of houses in a town.
[10:13] Without the cyclic elements, it would be all the more
[10:16] difficult to get my meaning over to the random viewer.
[10:20] This idea where you have a straight reference image and then a warped grid,
[10:23] and you use the two together to create a warped scene,
[10:26] is a common process in graphic design.
[10:29] It's known as a mesh warp, and Escher didn't invent it for this case,
[10:32] it's something that he had used multiple times before for other pieces.
[10:36] For our story, the point is that all of the logic for turning
[10:40] a Drasta zoom into a loop is abstracted and purified into this grid,
[10:44] raising the natural question, where does it come from?
[10:48] How do you make this?
[10:49] You can kind of imagine as an initial naive approach you might try linearly scaling
[10:54] everything from one corner to the next, but if you did that right away you would see a
[11:00] conflict.
[11:01] These two scaling processes are placing distinct pressures on the individual squares,
[11:05] where for example this square wants to get flared out in this direction
[11:09] based on the zooming in from the right, but it also wants to get flared
[11:13] in this direction based on the zooming out as you go up.
[11:16] Escher was evidently pulled to resolve this by
[11:19] curving all of the lines to relieve this tension.
[11:23] But there's one other crucial constraint that it seems he was guided by,
[11:26] one that presumably made the image transfer process easier,
[11:30] which made the final result look more natural,
[11:32] and which will make the ears of any mathematician immersed in complex
[11:36] analysis immediately perk up.
[11:38] In his final warped grid, the tiny squares are, well, squares.
[11:43] To be clear, this is not at all true for most mesh warps.
[11:46] In most cases, the warped grid lines that you draw don't necessarily intersect at right
[11:51] angles, and the little regions that they bound will in general be little parallelograms.
[11:56] This can still be workable for an artist, you can still kind of do the transfer,
[12:00] but presumably the act of copying over from the original to the warped version
[12:05] is more awkward when you have that distortion,
[12:07] so it would be much nicer if in the warped grid,
[12:10] little squares remained at least approximately square.
[12:13] And when you look closely at the grid that Escher used for his print gallery,
[12:17] it has this property.
[12:18] All the lines are intersecting at right angles, and at a small enough scale,
[12:22] the regions they bound really are approximately squares.
[12:25] Artistically, this has the nice effect that even though the whole
[12:29] scene is dramatically warped and distorted at a global scale,
[12:32] zoomed in at a local scale, everything is relatively undistorted.
[12:36] This is what makes each local part of Escher's image easily recognizable.
[12:41] And mathematically, this is where the story really gets interesting.
[12:45] You see, this idea of a function from two dimensional space to two dimensional space,
[12:49] where tiny squares remain approximately square,
[12:52] plays a special role and has a special name in math.
[12:55] It's called a conformal map, and one area where it comes up all the time is in
[12:59] the study of functions with complex number inputs and complex number outputs.
[13:04] At this point, we're going to step back and walk out of the art
[13:08] classroom and wander over into the math department,
[13:10] where I want to offer you a mini lesson on some of the core ideas from this field.
[13:16] The basic game plan from here is that I want to first do a little refresher,
[13:20] go over some of the basics of complex numbers and complex functions,
[13:23] and then I want to spend some meaningful time building up an intuition
[13:26] for what logarithms look like in this context of complex numbers.
[13:30] And then once you have that in hand, we're going to step through a completely different
[13:34] way that you can think about recreating this effect that Escher had in his print gallery.
[13:38] Let's warm up with a review of the basics.
[13:41] We typically think about real numbers as living on a one dimensional line,
[13:44] the real number line, and complex numbers are two dimensional.
[13:48] Specifically, we think of the imaginary constant i,
[13:51] defined to be the square root of negative one as being one unit above zero,
[13:56] end every other point on this plane represents some combination of a real number
[14:01] with some real multiple of i.
[14:03] It's typical to use the variable z in referring to a general complex number,
[14:07] and the game we want to play is to understand various functions of z.
[14:11] A very simple but important example is to multiply z by some constant.
[14:16] The function f of z equals two times z has the
[14:19] effect of scaling everything up by a factor of two.
[14:23] But what about multiplying by something imaginary,
[14:25] like i, the square root of negative one?
[14:27] Well you know that multiplying one by i gives you i,
[14:31] and by definition, i times i is negative one.
[14:34] Both of these you'll notice are in 90 degree rotation, and more generally,
[14:38] multiplying any value z by i has the effect of a 90 degree rotation.
[14:42] And more general than that, when you multiply by any complex constant,
[14:46] the effect is some combination of scaling and rotating.
[14:50] And there's a nice way to think about exactly how much it should scale and rotate.
[14:54] Zero times anything is zero, so the origin has to stay fixed in place,
[14:58] and then one times any constant c is that same constant c.
[15:02] So that means this point at one has to be dragged over to land
[15:05] on whatever that constant c is that we're talking about,
[15:08] and that fully determines the amount of scaling and rotating.
[15:12] From there, the rest of the grid stays as rigid as it can.
[15:15] A key point to emphasize for our story is that if all you're doing
[15:18] is multiplying by some constant, shapes are always preserved.
[15:22] Anything you might want to draw, like a square,
[15:24] can get scaled or rotated, but beyond that, there are no distortions.
[15:29] Now things get more interesting for more complicated functions.
[15:32] A simple but non-trivial example would be mapping each number z to z squared.
[15:37] So the input two is going to have to move to two squared, which is four.
[15:41] The input i is going to have to move to i squared, which is negative one.
[15:45] Negative one itself would have to get mapped to positive one.
[15:48] And in its fullness, here's what it looks like if I let every point among
[15:52] the grid lines of this input space move over to their corresponding outputs.
[15:57] It's a pretty nice effect, and unlike multiplying by a constant,
[16:00] shape is absolutely no longer preserved.
[16:02] The grid lines get curved and warped.
[16:05] However, and this is a key point, pay attention to what happens at a small scale.
[16:09] For example, just focusing on this little square from the input space.
[16:12] As you watch the transformation happen again, you can see that
[16:16] shape is approximately preserved, at least at a small enough scale.
[16:20] Little squares from our original grid remain approximately
[16:23] square even after getting processed by the function.
[16:26] To use the lingo, the function z squared gives a conformal map.
[16:30] And the choice of z squared is really not special here.
[16:33] Here's what it looks like if you transform each point z over to z cubed.
[16:37] Squares remain approximately square.
[16:40] Okay, maybe this example square that I'm highlighting doesn't exactly look
[16:43] square in the output, but really it's just because it started off too big.
[16:46] To be clear, this conformal property that I'm talking about is a limiting one.
[16:50] A particular square might not exactly become square,
[16:53] but the idea is that as you zoom in more and more,
[16:55] choosing square regions from the input space that are smaller and smaller,
[16:59] the resulting output will indeed be better and better approximated by a square.
[17:04] And there's also nothing all that special about the
[17:06] choice of polynomials like z squared or z cubed.
[17:08] For just about any function of complex numbers you could think to write down,
[17:12] this property holds.
[17:13] Shape is preserved at a small enough scale.
[17:15] It's almost like magic.
[17:17] The use of complex numbers is very relevant here.
[17:20] If instead you were thinking of points in 2D space simply as a pair of real numbers with
[17:25] some xy coordinates, and you write down some arbitrary function of x and y to get a
[17:29] new pair of numbers, what is way, way more typical as you let points in that input
[17:34] space get transformed is that the outputs of those tiny squares get squished and
[17:38] distorted.
[17:39] Even as you zoom in more and more, the resulting limiting shape
[17:42] typically looks like a parallelogram, not necessarily a square.
[17:46] So, complex functions really are special in this way.
[17:50] And the basic reason that tiny squares remain square comes down to calculus.
[17:54] What you're looking at is what it means for these functions to have derivatives.
[17:59] That might sound a little strange, but the analogy you can think of is that for most
[18:03] functions of real numbers, if you visualize them the ordinary way,
[18:07] just with a graph of inputs on the x-axis, outputs on the y-axis,
[18:11] as you zoom in more and more to any particular point on that graph,
[18:14] it looks more and more like a straight line.
[18:17] That is, the rate of change, how much delta f you get for a given delta x,
[18:21] approaches a constant.
[18:24] This is literally what it means for a function to have a derivative,
[18:27] but it's not the only way to visualize it.
[18:29] Here's a different way to see the same concept.
[18:32] Instead of looking at the graph of a function,
[18:34] let's think of the same function but as a transformation.
[18:37] That is, you're going to let each of these points on the real number
[18:40] line move over to their corresponding outputs on this other number line.
[18:45] What you'll notice is that evenly spaced dots from the input space can get warped in
[18:49] the output space, meaning the rate of change of the function in general is not constant.
[18:55] This spacing between our dots can change from one part of the image to the next.
[18:59] But we know that as you zoom in more and more to a particular output,
[19:03] the rate of change approaches a constant.
[19:06] The dots look more and more evenly spaced.
[19:09] Specifically, if you take a tiny patch of dots around a particular input and you
[19:13] copy them over to the output space around the corresponding output,
[19:17] you can approximately line up all the dots just by scaling everything by a certain
[19:22] constant factor.
[19:23] This is the same analytical fact as what the slope of a graph is telling you,
[19:28] it's just in a different visual context.
[19:30] But this context carries over much more easily to thinking about
[19:34] complex valued functions as transformations in the complex plane.
[19:38] There we're going to think of the neighborhood around a given input
[19:42] as a tiny little grid of points, like we were before,
[19:44] and we let each point on that grid move over to its corresponding output.
[19:49] In this case, what it means for the rate of change to
[19:52] approach a constant is basically the exact same equation.
[19:56] The visual to have in your head is that if you take that tiny patch of squares
[20:00] around the input and copy it over to the corresponding output,
[20:04] you can approximately match this up with the output grid lines by multiplying
[20:08] by a certain constant, which remember, in the setting of complex numbers,
[20:12] means rotating and scaling it in some way, depending on the value of that
[20:16] complex constant.
[20:17] Since rotation and scaling preserve shape, it means that all the tiny squares from
[20:21] the input space remain at least approximately square under this transformation.
[20:27] So, stepping back, here's the key point, the reason for talking about any of this at all.
[20:31] Even though conformal maps like the one that Escher was using for his print
[20:35] gallery are incredibly constrained and highly unusual among the general ways
[20:39] that you could continuously squish about two-dimensional space, nevertheless,
[20:43] as if by magic, simply by speaking a language of complex numbers,
[20:47] you can somehow create entire families of these conformal maps without even
[20:51] really trying.
[20:52] All you do is mix and match standard functions of complex numbers.
[20:56] The one constraint is that these functions have to have derivatives,
[20:59] but this will be true for most of the functions you think to write down.
[21:03] For our story, decoding what's happening with the print gallery,
[21:06] this means that we have an entirely new way to reframe the key question.
[21:10] Can you construct some deliberately tailored complex function so that the act of
[21:15] zooming in around the inputs looks like walking around a loop among the outputs?
[21:21] Now, at this point, with only the bare minimum crash course of complex
[21:25] analysis under our belts, it's hard to know where to start without
[21:28] first building up a larger palette of functions to work with and
[21:31] gaining some familiarity with how they actually behave for complex numbers.
[21:35] In this case, there are really only two functions that you need to understand,
[21:40] e to the z and the natural log.
[21:42] I want to settle in and spend some meaningful time understanding both of these,
[21:46] and I think you'll agree that it's time well spent.
[21:48] Everybody deserves, in my opinion, at least one time in their
[21:51] life to experience the joy of understanding a complex logarithm.
[21:55] First though, a necessary prerequisite is to understand the complex exponential.
[22:00] Regular viewers will be familiar with what it looks like to raise
[22:03] e to the power of a complex number, but a review just never hurts.
[22:07] We can start scaffolding the transformation just by focusing on the more familiar
[22:12] examples of real number inputs and the real number outputs they correspond to.
[22:16] For example, e to the 0 is 1, so this point at 0 maps over to this point at 1.
[22:22] And then every time you let that input increase by 1,
[22:25] the output grows by a factor of e, meaning it runs away from us actually quite quickly.
[22:31] And then on the flip side, as you decrease that input,
[22:34] letting it get into the negative numbers, every step to the left corresponds to
[22:38] shrinking the output, again by a factor of e.
[22:40] In particular, you'll notice in this case that output is always a positive number.
[22:44] This of course gets much more interesting when we let
[22:47] the inputs and the outputs both be complex numbers.
[22:50] And here, everything you need to know comes down to what
[22:52] happens as you let the imaginary part of that input increase.
[22:56] And what happens there is that the corresponding output walks around a circle.
[23:02] If you're wondering why imaginary inputs to an exponential walk you around a
[23:06] circle like this, we have discussed it many, many other times on this channel.
[23:10] See some of the links on screen and in the description.
[23:13] A key point that I'll reiterate here is that what makes the function
[23:17] e to the z very nice, as opposed to exponentials with other bases,
[23:21] is that as your input walks up at a rate of 1 unit per second,
[23:25] the output walks around its circle at a rate of exactly 1 radian per second.
[23:30] So in particular, increasing the imaginary part by
[23:33] exactly 2 pi causes one full rotation in the output.
[23:37] Phrased another way, these vertical line segments that I've been drawing with heights of
[23:42] exactly 2 pi each get mapped neatly onto one complete circle when you apply the function.
[23:48] You'll notice how I've drawn these particular vertical line segments to be spaced out
[23:52] evenly in the real direction, where the real part from 1 to the next increases by 1.
[23:57] The corresponding circles on the right each differ by a constant scaling factor,
[24:01] specifically the scaling factor e.
[24:04] Earlier for the simpler function z squared, I showed it as a transformation,
[24:08] 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.