 So we're delighted to have Ingrid Doboshees to give our third plenary talk of the week. Ingrid Doboshees is a professor at Duke University. She has three titles within Duke University. She is the James B. Duke distinguished professor of mathematics and electrical and computer engineering. She's a professor in the department of mathematics and she's a person in the department of electrical and computer engineering. She is perhaps best known for her work on wavelets and has done work in image compression, digital art restoration and biological morphology. Give some sense of the breadth of Ingrid's work. Her list of accolades is far too long to list. So I'll just mention that she is a member of the National Academy of Engineering, the National Academy of Sciences, the American Academy of Arts and Sciences, and it was also a MacArthur fall. So I'll, that's, that's enough about how amazing Professor Doboshees is. So I'll just hand it over to her. The title of Ingrid's talk is digitalization to research and exposition. Well, first of all, I'd like to thank Henry for this introduction. I also enjoy so much being part of PCMI now in a different mode than I usually have been. I've been there as a lecturer, as a participant, as somebody talking to the teachers and I've enjoyed it all. So second, I would like to beg your forgiveness for the talk which will be a little bit more cobbled together than my talks often are, although they always have some cobbled together for aspect to it but still this one it'll be more so, because until two days we were working very hard. I, as well as Henry Segerman and Sabete Matsumoto and Edmund Harris, all of whom are part of PCMI, on working on a very special presentation that does visualization of math and about which we will, I will talk at the very end of this presentation. So I promised to talk about the role visualization as played in my research, my teaching, exposition and outreach. And although I had planned to really talk about each of these separately, it is and start with research, I realize they're all interwoven somehow. So let me just give you an excerpt of the talk, the introduction talk I usually give on wavelets, which has a lot of visualization in it and which in which I can also help you I mean will show you exposition at the same time as research so. So that's a talk on wavelets, so wavelets illustrated via image analysis. So, well, digital image consist of pixels I'll do it here with black and white grayscale images, because that explains the mathematics I want to visualize but of course, you do it in color. And images have red, green and blue pixels and mixtures of them make the different colors that we see them because of the peculiarities of our eyes. You don't actually do, when you do mathematical image, when you do image processing, you don't do the same in each of these colors that I'm going to describe it turns out that our sensitivity for luminescence or for whether it's dark or light in an image is much higher than for exact tons of color. And so we're much more well most of us models are much more tolerant for slightly change slide changes in scale, although I have worked with artists who are who have an acuity or vision that I find astounding. Okay, small scales squares, each one of them is really one level of gray, and we have a great many of them those are the pixels. The gray levels in typical eight bit images are 256 of them so they're numbered from zero to 255 from pure black, typically zero to 255 pure white. Okay. So, here's an example of a row of self portrait by Van Gogh if I have time I'll talk a little bit about work I've done on images of paintings but. And so if you blow that up and load that up further and further you see all these pixels. I've taken a little portion of it here and to remind myself that the dark numbers are the lower numbers are darker colors I put those under 100 in bold, and we'll just take a very few of those to show the transformation that corresponds to a wavelength transform. I hope you're seeing here that although I'm trying to to bring this concept I make it very visual I think of this in very visual terms always. Okay, so we are one thing that images have in common all natural images, not just photographs of paintings is that an overwhelming number of pixels are very similar to the pixel right next to them. Of course, many pixels are very different from their neighbors, but in any image, the ones that are very different from their neighbors are minority. They are the most important ones in that image because they give us the content of where different objects are and how they behave and so on. And which at which locations those pixels that have this content are situated differs from one image to a next so any pixel in the image can play that role, but only minority play that role in any given image. And so you that's what you exploit when you do image processing with with wavelets. Here, we are exploiting it very simply by saying look if I take things in pairs, then I can compute just the average number, and that will be typically close to each of the two of them, which was an average. And even when I've done that once I can do it again. And still, most of those averages will be close to the four grandparents, it came from. Of course, there are places where they're not the same, and that is expressed very well by the differences. And so, for each of the pairs in this first generation I also compute a difference. But again, the second time. And what is now striking is that in most places those difference are tiny. And where they're not, it really indicates that something was happening in the image. And so, what is going on here is something interesting, we have teased out from by very simple mathematical transformation, where location was where something was happening. In the image. And we can then with just those green circle numbers and a few averages reconstruct back something that's very close to the original. And that's really all that's happening in image compression. There's a very nice. Mathematically what we have done corresponds to the following. Imagine if you just take one line of an image. A reasonable model for that is something that has continuous variation just imagine taking a snapshot of mean where you I'm sitting inside a room you probably are too since you're zooming. I see the ceiling of my kitchen here. It's slowly varying and you see the ball behind me. It's slowly varying in intensity. And so that is a kind of smooth function. But you have a drop transition where you go from the wall to the window there. And so you can have sudden discontinuities and then again slow. The landscape behind has texture and that's more complicated that doesn't correspond to this model. But so if you have something like that the first thing we did is make a piecewise constant approximation that was the pixelization. So we go from what, in fact, was already a model. It's not reality. I mean, because we always make things a little bit more ideal in order to, to, to, to modelize it. So the pixelized version, which is a fine scale approximation. And then what we did to that is go to a course of scale by taking the average of each pairs which is in this lighter blue. And that coarser approximation I will do the same thing to take its averages and, and then its averages and again and again. Now, when we have done that when I went from, from this very fine approximation to a slightly less fine approximation, I made a mistake, and that mistake I've indicated here in red. So for each pair, since I had two levels and I replaced them by their average, I go as much up on one side as I go down on the other side. So if you just look at that difference function. It's an upside or a down up oscillation. How much changes from from space from spot to spot. But if you put them all together all these differences, this is what you get. At this level, a similar thing happens, and so on, and the similar thing. And so the differences I've put together are of this type. And so each of these layers, the differences are just multiples of a basic building block, put at the right amplitude in different places, and do that at different scales. So I have found that I decomposed my whole, my whole function, the different approximations indicated by different indices capital J, J minus one J minus two and so on. And so here are given by linear combinations of one building block scale to the very narrow with and moved around or wider and moved around or even wider and moved around. The loss of detail is expressed in these these wavles. And then, in fact, what happens is that you do a much, a much more smarter averaging than the one I showed, which uses more coefficients, and it's possible to have a better understanding of the detail mean you see if I in what I was doing I was doing a constant approximation. So that's fine if big blocks of your image are just going to have constant intensity. But if they very slowly, then, well, a piece as constant approximation is not going to do so well. There are a few degrees of freedom, but you don't capture them with a large scale constant approximation you would capture them much better with something that has a slope. And so in order to capture that well you have to go to slightly fancier wave. But something else that, and let me show that here on an interactive screen. I should share again share here. Okay. So, something that was striking in what I did was that I was saying one of my building blocks. So I was going from piecewise constant levels. So a function that had this to something that was piecewise constant over larger stretches. This kind of function, I can write as a combination of this top hat function in different locations. So, or if I don't want to write it a and of hat X minus and where H hat is what H is just this function function that's one between zero and one at the next level next lower level I write it as a linear combination. So, I go from here to there. And I write this as a linear combination of functions that are twice as wide. So that makes X over to minus M. I lose something in between, and that's where I introduced these oscillating functions, which a label you to this is a function that if I made the same transition to it would you give me just zero. And that's something that lives only in one of the two representations, and you don't see any more the next, but it's exactly what you need to express the detail. But some of what I had in the earlier one still is expressed in the last one, and that is, because the function age. When I stretch it out by a factor two. So from zero to two. X over to can be expressed as the original function. Plus, it's translated by one. So, the functions the building blocks with which I make my approximation, have this very special relationship that they can be expressed so another way of saying that is that H of T is H of two T plus H of two T minus one. So, more generally, if I don't want to do stupid little averages, I will use building blocks, I use these functions, these functions are called scaling functions and often denoted this five T, but they will have the property that they are linear combinations of the same function squished by a factor two and translated. And that has a very interesting consequence. And let me show that to you in in an animation that I had made some years ago. Let me show you this equation so I have a fight T is h and fight you over and and for reasons that are of convenience. Typically, let me see how in the animation. Yes. So, typically, one puts a square root to here, because that makes it possible, you see this function here that I will have integral of its square equal to one. And that makes this normalization reasonable because if I take these functions that take the integral of their squares, that will be one again so H and are the coefficients that link these two. So, the result of this equation is that you have a very beautiful link between the function so then a very beautiful way of generating their graphs. If you take. Sorry, if I take just functions one between zero and one and elsewhere zero. And I replace I take for this function, this block function, I squish it by a factor two, and I make copies of it with these different coefficients. So if I write four blocks that have heights given by the h ends, then each of those blocks, I will do the same thing to. So at the first block, the second block, the third block, the fourth block. I add them all so I add this to the first division that gives me this then I add the third block to that division gives me this, and then I add the fourth division gives me this, and I get this profile. And I can do that again for each of these blocks, adding them all up gives me this profile. The interesting thing is that I'm combining here to different aspects. I mean, on the one hand, it's clear. I mean I start from a building block. And I do a certain transformation to it. I do this transformation with blocks that are half the width. And each of these blocks is half the width of the original. Then, by doing to each of these blocks, the same thing again. It's clear that if I have a limit of the whole thing. So if this reaches a limit F, each of these blocks here will reach a limit that is a copy of the original F, but just squished and with the right amplitude. I said that the limit function F will be automatically a linear combination of that function squished and translated and with a different aptitude. But the algorithm by which I created is not you see if you look at this question it's an equation in which I relate F with F of 2x. F of 2x minus one and F of 2x minus two. And in this case I even go up to F 2x minus three. So with things that are located at arguments that are a whole integer apart F of 2x 2x minus one and so on. But the algorithm by which I build it is very local. So I'm going to go back to my animation. So now that was not the one I wanted, sorry. Here, what I was doing here was very local. Sorry, was very local. I don't go full unit away. Here I'm actually only a quarter away. And as I go further, I will be even less away here. Every time I become more and more local in my algorithm so my equation follows from the way the algorithm is built, but the algorithm to generate these graphs is even more narrow, and it cascades down very quickly it has exponential convergence to the graph of the of the limit function, which has these kind of fractal properties. So when I was first conceiving of these functions. Oh my God, a long time ago, something like 30 years ago. More than that. It was an essential point for me that this visualization on the one hand the local algorithm, which was an algorithm that was used in computer vision. On the other hand, the fact that things were identical, and therefore you had this two skill equation. It wasn't a key element in in in building these these these bases which then turned out to be very useful for image compression. So, I wanted to give you this this very old example to show how that visualization was important for me in actually doing this this this research. Okay, so that has told you something about research and also already something about teaching about the exposition. And I now just realize that I forgot to prepare to set up that screen for you so let me do that quickly. While I talk about it. So I, I do I teach, I teach courses at many different levels, but I actually regular on a regular basis, teach multivariable calculus to engineers, I really like teaching that course which most of my colleagues do not like to teach. Because it really helps me. I find it a challenge to to explain to them. Exactly why things work. Though I will not give very rigorous apps on Delta proofs. So in all the students who take that class I typically teach in the fall and then I have lots of freshmen, and they have placed out of one variable calculus they are good in in math, but they have by and large had the habit of just formulas in blue boxes in their book by heart, and then just finding when they got a problem which of the blue formulas had to be applied. And I want to win them from this habit because it's a bad habit in doing mathematics, and I also want them to make to really understand what is going on. For this reason I had ages ago, developed a way of flipping the class. And I, I had made little videos in which I explained the material, and I, and I, I in in these videos which I made the students watch ahead of time I explained a little bit and I showed that they had visualizations I made matlab visualizations those they. I told them, I gave them a little tutorial about how to use matlab and their engineering students they had a matlab license, and they could then manipulate a little mobile. And the students really, really liked it they like to flip nature they like the fact that they could review the videos many times, and they like that they could manipulate models, and so I had then told my colleagues about them and by and large colleagues teaching these classes were pure mathematicians they had not use matlab they were not going to use matlab and so it wasn't successful. And I understand, I mean, why would they I don't use the software sage that some of them use and so on so. And then it came around, and we had to teach this class online. I, the Duke asked for ideas of making things I said you know, if somebody could take these visualizations I made and made them something that you didn't have to have matlab for something that was just online and people could play with, and that it's tailored exactly to what we want to teach, then maybe it would be more useful and we did that. And so here, let me start sharing again, I'm going to share a few of those with you. This is part of the sky side for the course. So, and as the student as part of sky they had all these visualizations. This is just to explain a surface plot. I mean, of how do you get a surface from a formula and they could look at points on the surface and what the XYZ components were and so on. So, what did I, what else did I did we make. Okay, partial derivatives and tangent plane. So all these things that I had designed before. We made little visualizations so they got the surface. We can look at where we have a particular X plane, a Y plane. In each of these planes you can write the tangent line and that's something they're familiar with because they remember that from one one variable calculus, and then to these two planes which intersect in the same point, determine these two lines are determined a plane so that's the tension plane, and you can then just look at that tension plane, and the whole thing that's, I mean the whole thing is something where you can play with the figure look at it from different sides and I would have loved something that was more interactive where they could have chosen. And I know there exists software online to do that, but it was important in order to make these visualizations so it was terrible to all my colleagues that it was something for which they would not have to do a lot of legwork all over the place. They were. I know everybody in this PCMI course is completely used and familiar and happy with doing visualizations. So, but if you interact with with with colleagues who are not so prone to do that and I'm afraid that is probably going to take another minute before everybody really really likes this. It is good to have models like this to show and to play with. Okay, what else did we have well sometimes sometimes it was some problems had difficult things to set up. So here they as part of of trying to think of what is it that you define as something bounded by two surfaces. We gave them to parabolic and again they could play with it. And, and then you could just look at the 3d region between them, and they could see that it was this this this weird thing with that when you looked at it from above was bounded by a circle. I mean, well standard things of multivariable calculus but just the fact that they could grab hold of the object on the screen. Made it. Then optimization under constraint. I forget what that one is. Okay, so this one seems to have broken since the last time I looked at it. I'm sorry about that. Next, maybe. Okay, so usually when I teach multivariable calculus. I do a lot of gesturing in front of the class I mean I showed up the subtle when a subtle surface is something that you can put as a saddle on somebody but you see on zoom that doesn't come over in front of the classroom it comes a little bit better. But so the path integral so you have a curve on the plane here. Then you have a surface above it. And then you have that curve on the surface here. And we there's more to this than this visualization that I can't believe that this is already broken. So maybe it's because my screen is too small to big no. Okay, what happens then is that there's a further visualization in which you actually build up the integral of to find the length of the curve or to find a function value on the curve and so on. That's one such thing with visualizations is that they break so quickly. I mean I wish they weren't the case. Maybe that will change, but Okay, so then next outreach. So there are a number of different things I two different things I want to tell you about this here at this stage of this presentation. One is a project in which which had a tremendous impact much more than I expected. And and with in which we worked together with the North Carolina Museum of art. On a rejuvenation of a of a painting so let me quickly tell you about that and then tell you how visualization became important although the whole thing was visualization really. So you can find out more about this in Duke IP a so IPI dot org. I'm pretty typing that in the chat right now. So it's Duke image processing for art investigation was our own joke of how close this was closest we were ever going to get to CSI and so let me share. Let me not forget to share so. Okay, well you probably all are on the screen already but since your visual people the way I know you. So, we did a project with the national with the North Carolina Museum of art, which among its collections has several panels from other piece that I'm showing you now. What I'm showing you now is a mock up. It's actually not a mock up it's a photograph of a the other pieces it was reunited for just six months. So, what happened is that the North Carolina Museum of art had of the panels that you see here, it has in its collection, three panels, the bottom left, the second from left and top row, and then the, the middle, the second from right in the bottom row. These three are in its collection. The curator of Italian art, who retired a year ago, I had discovered that panels from the same piece were in other collections in the US. In fact, they probably were acquired around the same time when the whole outer piece was broken up. It was broken up when church was decommissioned and our dealers in Europe found that yes they could make more money from a whole outer piece, but they could make even more than from a single panel but they could make more money from nine panels from one outer piece. So they typically would divide these things up and that's why other pieces are often divided now among different museums. So the top left was in Portland. The three here the the second from left on on the bottom, and the two at the top right here were in the collection of the Metropolitan Museum are in the collection but the central one is in the Art Institute in Chicago. So when he realized and he could see that from details that they belong to the same outer panel, outer piece, he wanted to bring them together and other museums were not so interested because there's one of them that's missing. This one the bottom right here at the reunion was actually not a painting paint on panel it was a pain it was a print out a print out of what well. It turns out that. So this piece has been missing since the other piece was divided there's no trace of it anywhere in the literature, it may have gone destroyed. We don't know. This outer piece this 14th century altarpiece shows, in fact, the life of john the evangelists in order it's like a little cartoon book, or a graphical novel as they're now called also of the life of john the evangelist according to a medieval bestseller which was called the golden legend, which had a lot of lives of saints, and it followed that exactly so that they knew what scene was missing. And together with the art curator, a reconstruction of this panel was made using elements of the other panel so surely if it ever gets found it's not going to look like this. Because this was just a guess but it would have, if this were had been put in instead of the right panel in the 14th century it would not have shocked anybody because it was the right scene with the right kind of composition. And you can on this website for the beautiful little video of how Charlotte the caspus, the artist who is an expert on reconstructing. So not just copying, but copying in the way things would have been painted by the original artist with pigments original techniques and so on made this new panel, and she made this beautiful, wonderful little panel that looks like it would have looked new because well it was new. And it's only when she had done this that they realized they couldn't put it next to the other ones. Because it would have detracted from these old ones that are faded where the gold is no longer gleaming and polished and burnished, and it would have been the only one that was not authentic. So they just couldn't do that they couldn't age it virtual physically either because that would have defeated the purpose of showing what this thing would have looked like new. I mean curators all over would have been so upset at museum creating something that could be considered a fake, even though they didn't produce it as a fake, it would have easily if it were ever stolen it would. So, what we did is we aged it virtually. That was an interesting. And I didn't find the right figure to show you but I'll do it with my hands in the air on zoom. So the way people paint it in that time. We saw this these these these mantels. So you see here on on the on the right, you see the new panel. And you see how it borrowed composition elements from the left. On the left you see the panel that isn't the collection of museum of art in North Carolina. And you see how the colors have shifted. Now, in that time in Italian, late medieval, early Renaissance painting what they did is they would mix the basic color for instance of the mantle. And then they would prepare something with a little bit more light color in and something with a little bit more dark color in. So what that means in RGB space is that you have the color. And if you have a little bit more light, that means that in RGB space, you move along a certain line of this lighter color and the dark you move in a different line. So in fact, what happens is that you're describing an RGB space, a plane. When you look at all the pixels and I'm so sorry I didn't find that picture again, but when you look at all the pixels. I mean, we have hundreds of pixels because we had very high resolution pictures of this in in in RGB space. They do form a very, very flat pancake like cloud RGB space. The same for the reconstructed one and they follow a simple, a similar pancake. And I thought, oh well that's easy we can do an automatic recoloring by just taking the old pixels and moving and then maybe rotating them the right place. It turned out to our great surprise that the rotation was virtually nonexistent. So now we actually have a whole project where we recolor other paintings we first segment them. Then we with curators talk about what should the average color be, and they have a gooey on which they can choose that. And then we recolored them virtually and so I have a team of undergraduates who are who have recolored other paintings of this type. They were at mean to actually they were students from the multivariable class, because there was social distancing they had no social life. And so they had time on their hands in the fall. I said, could we do a project. And I said well it's more linear algebra project than the multivariable but they were game. So they have for the museum we colored a number of paintings this way. So again visualization played a role because when I could show them that plane and how it moved, it made linear algebra really come alive in a completely different setting than the traditional mechanical throwing tennis ball in the air and making a parabola and so on visualization. So that was that. And then, for the final few minutes before we go to Q&A. Again, I want to talk about visualization of mathematics in a completely different setting. So for the last two years, a team of 24 mathematicians and artists have worked together, first with a lot of discussion and when conceiving and fabricating, all in our own little little cocoons at home and different objects in order to build a big installation that we call Mathemalchemy, the alchemy of mathematics. I mean how in mathematics ideas come up in one context and come again in different contexts and how it's all beautiful and fun. And so let me show here to the, so you can find the Mathemalchemy.org site. So I think I'm not sharing yet. So let me first in the chat give you that link. Mathemalchemy.org. Okay. And on that website. So now I'm going to share. Yes. So you can find a 3D view of what is really a tabletop market. So this was a tool that was built last year by a the artistic director Dominique Herman of this project of this big installation, which in the last few weeks, we have been in full reality in at Duke, and it will stand there for those who are in the neighborhood in the next few months until November after which it will be packed up and travel to Washington DC where it will have its first official exhibit. And we visualize mathematics in so many different ways. For instance, you have here these these ball arches. So, in one of them, the diameters decreased geometrically, and so that photo length of the arch is finite of course because that's a converging series. In the other one, they decay like a power that's less than one. So I think it's like n to the minus a quarter. So the result is that the diameters are smaller and smaller and smaller but because the sum of these is divergent. This arch in principle will go on forever. I mean, it in our installation it delves into the ocean after which you can't see it anymore but in our imagination it goes on past it crosses the earth on the other side it goes past after it leaves the galaxy and so on. And we'll describe it that way of course, but they have these beautiful colors. But some some of us extra decoration. And if you look at which ones is third and the fifth and seventh. And not it's not the odd numbers because nine is nothing but 11 and 13 or decorates again you say 15 not yet that fits 17 and 19 again. Not 21 23 turns out to be on the door and as well. So it's not the primes. So something else is going on. So 27 is not of course 29 and 31 hour. And so we let people discover which one so every single component in the whole thing has we discussed it a lot with that's one visualizes so much mathematics in this one installation. And I actually have just yesterday, because we just finished the whole thing on Tuesday. I mean two days ago. Yesterday, I walked around and made some movies of the finished installation. And I'm actually I just, I haven't even watched myself this is the first time I watched them. I made with my phone. And so you see these adorned balls the balls themselves, everything is also exquisitely crafted are made in a Japanese embroidery system that's called tamari. If lots of mathematical ideas swirling around so that was the end of that short movie, another one, because I was changing points of view all the time. Here you have little squirrels actually trying out the civil or stuff in this and bringing different screens to screen out for multiples of seven and multiples of 11 and so on. On the other tiles on the floor actually illustrate primes in a different number field. You have a whole pile of books on which a little girl who's discovering mathematics is sitting, and it has the elements of Euclid but it also has books in Arabic and Sanskrit and Chinese and, and then it has books which are on weaving. I mean, leaving has a lot of math in it. But this book was not written by mathematician. This is another one. Have we seen this one yet. Oh this one. Yes, here from the books we go on. Here's the mountain, and you have a vortex sheet that comes out of the trumpet. You have. These are the balls. The balls and the embroidery so they're wrapped in yarn and then embroidered with, as I said the tamari technique. And here you see I think 17 and 19. No, that was already and 2931. And you see the balls going down in the ocean. And this is exquisite. This is the quilt. It illustrates many, many different things of cryptography. I mean, and there's some hidden messages in there. And it alludes to the whole mission itself and it alludes to many other things. It's beautifully made quilt. Dominique Herman is really a quilter, and it's, it's her technique is just beckable. It's wonderful. Here I think is the last of the little movies I made. Ah, this is the bakery close to my heart because I helped design it. Oh, look on the wheel here. The wheel of the of the display cart has a superposition of a gasket and. Okay, so that's, I think the end of the movies I had some more but I haven't transferred to my computer yet. And I should leave you some time for Q&A. Henry Sigerman has a beautiful little video that he made of the of the market and and and the whole intent of mathematical me and he's making a much more beautiful video which isn't ready yet. Okay, Q&A. Thank you very much and good for a beautiful talk. So in the chat. This is the video that I made about the project as it existed before we came together at Duke University a few weeks ago, and I have lots of footage which have yet to find the time to put together of what was happening. Well about a week ago. Let's open the floor to questions. You can either type in the channel. I'll say it or just turn on your microphone. If I can hear somebody typing or if that's just, that's in group. Is there a question for in group. And I see in the chat that people found that it was print primes. Yes. Oh, well that was me. There's somebody guessed it. Somebody guessed it. Yeah, somebody guesses. That's why I said somebody guessed it and then you confirm. So somebody did guess good. And some knitting. Yes, of course, actually in the knits have a message in them. Knits was in there in cryptography because knits were used in the Second World War to transmit messages. So people would send knits to from from to to to front lines and so on or in packages disguised as things that were for soldiers but they really have messages in them. So these knits do have messages we researched all that and question in the chat. I'll just read out the questions so that they go on the recording. Can you elaborate more on how you went from visualizing the averages to constructing the special way for Okay, so the visualization played not so much a role in constructing the special wavelet as in in understanding. And in fact, I have in that cascade wavelet that cascade animation a little bit more actually in the animation I will show how we did that with with, sorry, with the this is a scaling function for the four tap wavelet. That was, if you had good choices for the coefficients but suppose you change the choices slightly, and you repeat the construction in beginning it doesn't look so different. As you iterate, you actually get a very, very different than much less smooth looking object. And that's because it turns out that you don't visualization illustrates it, but you mathematically you analyze by looking at the limiting process and proving things about it. That you need to the coefficients to satisfy certain conditions. So in order to get orthonormal basis, the sum of the squares of all these coefficients have to have a certain property that have to add up to one or a half I forget in this normalization. You also want the sum of the even coefficients to be equal to the sum of the odd coefficients and I don't actually remember whether that's the case in the example I give here. It isn't so, and then you want the, you want the further property and if these are not satisfied, the visualization immediately tells you that things are off. So, so in that sense, the recent help illustrate that. But to me, the visualization in the conception was important in realizing that I was basically building the same thing in combinations. When I saw these blocks and each of these blocks raining down and then combining to give us one that the link between the equation which was non local, and the local algorithm was something that that I got from that visualization that imaginary visualization in my head and that's why I like showing it when I talk about another question here in the chat. In putting together math mouth me did you find it easier or harder to represent certain areas or aspects of math than you did others. Well, we didn't want to represent all of math. And in fact, we could fill many, many more projects with math that we completely omitted. The way we wanted it to be playful and beautiful and fun and to integrate with many different into woven stories and that would come up and emerge in different places. And we, we were mean many of us, not me because I'm an amateur in these, but many of us are were superb artists and craftspeople who had experience in conveying mathematics through their art and craft. But typically they would take one concept or and make a beautiful object. Now we were thinking of a whole whole world. I mean a wonderland and all these different techniques also. So it was very important to Dominican me who proposed the project that everybody would feel that their way of looking at it was represented fully, or as fully as possible in the communal work. But and so we had many, many, many discussions of what to put in and how to put it in and so on. And those discussions really determined. I mean, in little video that Henry posted, he said, whatever concept we could think of and that we could squeeze in it went. I mean, he didn't say it maybe in those words, but that was generally the spirit. I mean, and. But of course it was a finite object, a finite installation, and more of us were algebraist or typologists than analysts. And so there's quite a bit more discrete mathematics and algebra and topology than there is analysis, for instance. But that's just the way of, I mean, everything is delightful. And I don't think there's a single person among us who knew everything that is represented we were all explaining things to each other. I think actually, just in general analysis seems to me to be harder to illustrate than algebraic or typological. Oh, I don't think so at all. I mean, I, well, analysis is, well, I'm an analyst. I mean, analysis, the difference between analysis and algebra for my again, I have a visual metaphor for this which which I mean, analysis is the mathematics of the I mean, what if I stuff a bit more in here at what point will it start bulging and how much well algebra is is the mathematics of the of the the the thinker toys. I mean, you build beautiful things. And if you don't tap it with a hammer. If you're lucky nothing happens. But if you're not lucky, it just is gone. The whole structure is gone. And that's to me the difference. And both of me, I think, can be very, maybe very visual. Maybe one last question. We've got a bunch of more questions. Okay, maybe a couple more. So, so we're 10 years illustrations of Alice in Wonderland and explicit inspiration. We very quickly saw that we were creating a wonderland. And some of us resisted that a bit and others thought it was wonderful. But, but, so we also haven't really named the whole construction yet. I mean, I had come up with this idea of mathematical me as an alchemy of different mathematics and so on. And it's the project, the mathematical me project, the installation itself still will name it may end up with that name or not. It's something we still have to discuss in our next group meeting, but some of the names were come play with us in our wonderland or things like that but that was a bit long but yes the wonderland idea occurred to us to another question and I don't have a chat in front of you. I'll just, maybe we've got time for maybe one more. You mentioned that there were these planes in RGB space. I was wondering what a curve subspace would correspond to. Well, I, I, well what that would correspond to is mixtures of colors that you don't achieve by simply taking one and mixing others or you could have something in which pigments interact and start chemically and start mixing something else than mixing. I don't know whether painters would like that maybe Sabetta can answer there. I don't think that, well, I don't know I mean I guess there's a lot of fun things with sort of optics now where you have like, like, with dichroic things where the gradient of the surface gives you different colors so that's another thing. With dichroic things we wouldn't be able to capture with RGB. Actually, that's an interesting point for these these, these, these late medieval things, because a burnished gold background, which was then marked with punches which would reflect the light very strongly were important. That is something we could not model just with flat pictures. So the final modeling was to take the flat picture and put it on a 3D object, where you had, we modeled the reflection with with rendering. And if you if you look at the videos at the, the, the, the GISSI project, then you will see actually in the in the exhibit. Oh, I didn't say that but what happened in the exhibit was that they put all the panels together with a high resolution print out of our virtually aged panel, which was good enough that you had to look closely to see it was printed paper, then they had a new panel on one wall together with an explanation of all the pigments and techniques, then you had a third wall with a video of Chagod the Gospels, the artists who made them making this and which was very interesting and on the fourth wall you had a huge panel on which you saw the, because once you can age it you can rejuvenate. And so we rejuvenated all the panels and we show that in 3D and we had little iPads with interviews of the undergrads it was all done with undergrads because most of the techniques we use were not research techniques but were things that you had to work a lot in order to make it work for this application explained about how they had done things and how they had experienced it. And the exhibit from being something that they expected would only get a few experts and so on became something hugely popular the sense loved it they all incorporated it in the tours if they were given the choice of what to show and it had a tremendous public so that was a wonderful thing. Well, it's, it's past 2pm mountain time now so we should, we should thank Ingrid once again for a beautiful time. Thank you.