 All right, well it's it's one o'clock mountain time so let's go ahead and get started. I'm really happy to welcome our first keynote speaker Daniel Pecker. Daniel is involved in the design and realization of complex forms and architecture through both research and practice. After studying architecture at the architectural association in London. He's worked at a rupes advanced geometry unit and is currently part of the specialist modeling group at foster and partners. He's also the developer of the software kangaroo, which enables a wide range of interactive physical simulation, form finding and optimization within the rhino 3D CAD software. I want to personally add I've been a follower of Daniel's work for for a long time. It's really he's does amazing things is a, he's posting a lot of things recently they're really interesting on his Twitter feed. I encourage you all to follow that and, and he's contributed to several forums and blogs that I've been following his own website it's always always interesting to see what he's been thinking about. Thanks, David, and thank you to the organizers for inviting me up to confess it's a little intimidating talking to an audience of mathematicians it's not something I do often and not a mathematician myself by training I originally did architecture, as they said. And I feel I'm sometimes better at putting ideas into pictures than I am at putting them into words, but I'm going to give it go to try and do a mix of both today. But please forgive me if I make any terrible mathematical mistakes in my descriptions and correct me at the end. I want to start this journey with an experience I had. I think it was about 15 years old and I went to a sculpture exhibition by the Brazilian artist subject come on go. And this piece of marble, a still image doesn't really do justice I tried to replicate. It was actually the sculpture, as I remember it. And it was the first time for me that I felt like a piece of stone had told me a joke. It was like, it's one thing and you walk around and it was like the geometry was playing. It was humorous it was joyful and it really made an impression on me that a piece of sculpture without being figurative without representing anything else purely the geometry could have that sort of impact. And that's something that stayed with me. Another kind of key point in my journey is was as a young architecture student I was making a lot of models paper as I'm sure many of you origami and kind of things. And I was making these modular octahedra slotting together six squares of paper from the, the notepad by the fire. And then I started assembling these into larger forms and making this sort of, I guess, I think, I guess so they go ahead and surrounded by his octahedra and I was making it sound. It didn't quite match up but I was at first assuming that my model must not be accurate enough I haven't, you know, is the, the thickness of the paper or whatever. And this was around the time when it first started using computer modeling and my tutor at the AA, the architectural association at the time they introduced us to Rhino. And I started trying to model the same thing as I've been doing paper. And so, yeah, join the one octahedra two three four five. And there's a gap and I checked and yeah that's that gap is really there that's a geometric feature and the computer was able to reveal that to me, something that I could miss through physical modeling and that was one of the first things that convinced me, like, yeah, this is really something useful is you can engage with it in a kind of similar way that you do with physical models but at the same time, you have copy paste and 15 digit accuracy and all these things that let you discover a different type of things than you might be able to purely making things by hand. But illustrating mathematics. I was one of the things that came to mind when when I first saw the title of this this event. There's a quote from Einstein that's always stayed with me. He was interviewed by Hadamard being asked about how mathematicians think and what what role words and images play in their thought process and he asked a bunch of physicists and mathematicians and the thing from Einstein's answer that really struck me was, he says words don't come into it early on, and this combinatorial play, the elements are mainly visual and muscular type. And I found that a really curious phrase and I mean visual yes okay thinking in pictures but muscular I think it's not that he's flexing and what now I think it's, it speaks to something about our sense of physics yet we talk about grasping something and it's about physical intuition we can use our experience of interacting with real world objects to to engage with geometry and mathematical ideas in a different way. Inspirations along, along this way as well as an architecture student there are a few books that really made an impression on me. The first two that more good towards architects but really just wonderful explorations of. Geometric ideas and everything from polyhedra to minimal services and. And yeah, a few other books I did I see that Caroline series is here in the event it's I thank you it's a wonderful book. The tiling and patterns from badmanship that when I worked at the advanced geometry unit that was we call it a Bible because it was something we refer to so often to these these great examples to aspire to terms of visual communication of mathematics in my view. And as time went on in my sector studies I think I'm very interested in this idea of form finding and in architecture for a few pioneers. Gaudi was one early user upon finding and particularly prior to using ideas like the hanging chain models where you can take something hanging on the gravity inverted and get a form which works as a compression structure. And I love this idea that you can kind of use physics to teach you how best to resist physical forces and find these forms in that way. Um, so hanging hanging chain curves that we want to model a continuing one way we could do is to roll a parabola traces focus and the curve we get them is exactly. I continue to find things up. Or if we are interested in bending of wires or in strips of metal. The elastic curves. And again, there are the geometric constructions I learned about this one from the as a great thesis by Ralph Levion, who wrote about splines and their applications to fonts. But this year, if you, you roll along a hyperbole and trace trace the vertex then you get exactly the elastic again. It's not going to be fun, but if we want to actually design with it we need something obviously much more general purpose. So we might want to also do the kind of form finding for how to do it with so film to find minimal services that we could then use as tensile structures. And that was what led me to develop kangaroo, which is a form finding tool embedded in the cat application. And of course there were form finding tools long before this there was. They were mainly used by engineers, and they were things where you have to set up your inputs, wait for it to calculate and then adjust. And I really wanted to bring in something a bit like what I saw in game engines, where you can have a really rapid dynamic feedback and use that to inform your design process so this is the application that some of you are starting to learn it these for grasshopper and rhino these days. And it lets you plug together in this visual programming environment of grasshopper, many different types of goals I call them which are essentially energies to be optimized for. So we've all started using it and they've been making initially lots of billions little experimental structures as compression only things. Some tensile structures. As time went on I would try and add more different things for it to be able to optimize for in this case, bending and looking at different. energy formulations that I could incorporate in this case. This is nice simple model from cigarette address and Mike Barnes for the elastic up and great lines up it matches physical experiments. And we use this for some things like the bridge shells which are these structures which you can assemble them so they start out flat as a completely regular grid. So you can assemble them up for which you by forcing in the sides. And you need to be able to know what shape it's going to form hence the computation, and then fix the diagonals to shape as a structure, or inflatable structures, jumping a huge order of magnitude in scale but a few years later I'm working at foster projects and trying to apply some of these same principles to much larger buildings. Or even in completely different domains, it sort of extended, it started out very much as simulating physical form finding, and then grew to incorporate more kinds of geometrical constraints and this case, it was about making identical radius cylindrical rods and optimizing so that that distance between the central lines of the rods was the same so that it will be touching. Or another very recent application of it I just saw the other day. And this is a great example of where you might need to use form finding because where as most application is areas where the geometry and the forces have to be intimately tied together. So you could cut a piece of steel into an arbitrary shape and within some limits is going to stay that shape. But if you design a fabric structure, it needs to be negatively challenging. And if you design a technology structure you need to do it in a way that the forces will be balanced. And that's where the form finding process comes in. So what it uses is something called dynamic realization first described in 1965 by Alistair Day. And the basic idea is very simple it's just taking a load of out of balance forces, tracing the displacements with a pseudo dynamic approach with a bit of and eventually you converge to a situation where those energies are minimized and originally was talked about the use in static analysis but over time it's been shown that you can use a very huge range of things. So, somewhat of a mentor in this has been Christopher Williams from University of Bath. And he's particularly famous and known for his application of it to British Museum Great Court Ruth. What you see here on the left was the initial grid or mesh that they started from after many iterations they arrived at this particular triangulation, and then through relaxation, finding a smooth transition between the rectangular and the circular boundaries. I think I've always loved about this project which I think is, you wouldn't always notice is how nicely the top of the portico on the right lines up with the triangulation. So it's not just not just the simplest triangulation, you might think of when you connect triangle connect rectangle and circle but this is an element of design in there and there's an aesthetic balancing balance. In terms of applications that in my work at Foster's more recent stuff. This is a cross rail terminal. And again, it was a similar thing in this space and much more reduced to the constraints where we had triangular panels which had to stay within limits for what was possible with the ETFB pillows and the themes. But the middle section of the building is completely regular and completed, but the angle of the end being is is inclined quite a bit more than the rest of the grid. And hopefully it all looks quite smooth and you don't see it see a jump or a difference. And that's that's because it's been relaxed to to optimize for the these changes in angle and while keeping within the limits. Fabrication oriented things that you're using it for, like in this case, trying to make planar panels so a common thing in architectures we have a roof or a facade and we, we need to make it flat and it's going to be glass then glass is a lot more expensive than flat glass, but you can't make flat for the panels, unless they meet certain geometric constraints. So I've been trying to develop tools that make it easier to design within those constraints. So that's an example of that in action very recently. So that's a rapid overview of what kangaroo is and what it does but I'm going to talk now a bit more about two particular geometric topics that I found really fun to explore and some examples of the kind of role that visualization and illustration play in my process. So this is a fairly common requirement in say product design where you might want to distribute a set of perforations in the speaker for example. And you might start out with a regular regular arrangement. It turns out to be a surprisingly complex and rich problem even if you want to have all the circles the same size if you've got even just a circular boundary if you have a set number of circles how do you fit largest possible circles within that boundary. And I found it quite shocking that something from some simple conditions then the best known solutions are actually often quite asymmetrical and surprising. So other approaches to circle packing common one. You'll see, and I say circle packing and I realize that mathematicians often use that in a much more specific sense designers tend to say circle packing for all sorts of distributions of circles. So in this case, you might start by randomly placing some and then placing another one checking if it's inside an existing circle. And then projecting it if it if it is sizing it sequentially that kind of thing but say I was interested in fitting equal sized circles in boundaries or on surfaces. So I'm starting to use kangaroo in this case for the geometric constraint but no two points are closer together than a certain distance. So it checks the distance between all the points and you have to do a bit of sorting and optimization in the code if you don't want that to be too slow. And then you can push them apart and you can get pretty nice distributions whether circles are all touching some of their neighbors. But those of you familiar with circle packing will know that there's some some not so nice features in here there's a lot of gaps which are not three sided, which are unavoidable if you're sticking to circles of exactly the same size. And actually, I mentioned dynamic relaxation but the specific form of dynamic relaxation used in kangaroo to the current version. It's, it's actually very similar to an approach given in this wonderfully titled paper divide and concur. And the basic idea here, and it was actually, I only discovered this paper much later I learned of a similar technique through other routes but I think this is one of the earliest examples of it. The basic idea is essentially these alternating projections, you satisfy one set of goals. And you satisfy another set of goals independently. And then you recombine them into geometry here and it's a result and repeat that and keep repeating that and you'll converge to to the intersection of your goals if it exists. And so actually on the left here you see sequential projections, which was actually quite popular in game physics. A few years ago, and it is good. It converges very quickly but for the kind of things I was interested in. I wanted to also explore goals which may be conflicted where there was no overlap and I wanted to find a balanced solution, which the average projections that's where as the sequential one you'll always keep jumping back and forth between two conflicting goals. And circle packing is I just keep getting within myself where I keep getting people asking for new new versions of so I think I'm amazed by how many different questions that can be on fixed radii fixed families of radio different boundaries tangent to boundaries points. In the boundary on the boundary. So if you haven't found the physical experiments, this is the Brazil nut effect. So, as I said, if you, your box of muesli get shaken around in the truck, then people notice that all the Brazils tend to rise to the top. We can simulate that even from very simple granular physics. And then indeed, we end up with the big ones at the top, not just through. Not just because of the light of it because of the size of the circles. But if we want real circle packings. I say real in the center compact circle packings where the gaps between them are three sided. There's some, obviously, the great work of Ken Stevenson, but I learned to this approach through this paper from the potlums group, which, and they came up with a very nice simple geometrical condition on the, the four lens of to adjacent triangle circle. So if they have two triangles that share an edge, and you the sum of pairs of opposite angles has to be equal, and that guarantees that the serve the in circles of those triangles will be tangent to the common point on the line. If you optimize for that, then you can do things like generating these structures where the beams because they said the normals of the circles you can use as the nodes of the beams so each of these yellow rectangles doesn't have any twist so you can make it out of flat sheets because the normals of the circles are co-planar because they're touching the same point on the common edge. And this can also give you proper circle patterns, proper compact circle patterns if you take circles through these points of tangency. You can apply this to to three dimensional models on on a curved surface, they won't be, unless it's a sphere they won't be perfectly tangent because it's actually tangent spheres but if it's a smooth enough surface and your circles are relatively you can get pretty close. You can also apply things like periodic constraints in kangaroo. And so this is an example of taking different triangulations and finding was actually a unique circle packing. So, given a triangulation, there is one only one. There are two circles. And that's something that I keep coming back to this idea of finding the, and using an energy to minimize and get to a unique representative of some topological form. So you might. So you want to get the, you might start from some arbitrary geometry like this ugly triangulation in the lower right, and you can use optimization to find a better representation of that. So, another, another question I got asked about circle packings or point distributions was someone asked me how to model a golf ball, and my initial thoughts seems yeah quite simple I'm sure it's like the duty to go and take my face either and subdivide it, project it, you know, I thought that was, it was going to be very simple. But the more I looked into it, the more confused I got because I was checking the dimple counts on some of the, the popular designs for golf balls, and I couldn't find any level of subdivision. And I think it's Adrian, how about how you, even with the twist, there was no way to get that number of dimples, and it's, it's quite hard to see in just from visual inspection, but actually quite a lot of golf balls use, not as a withdrawal, but tetrahedral symmetry. So here's an example, again with a symmetry constraint so we're doing collisions but also colliding them with their neighbors across the symmetry constraint, cubic geometry, or for tetrahedral geometry. And, yeah, this the symmetry gets quite disguised by the time you, you reach the end result. So that's actually the arrangement on, on one of the more popular golf balls. Next time you have a golf ball in your hands, see if you can spot the, the four, the four corners of the tetrahedron. Yeah, I couldn't resist taking one of these and trying to optimize it for proper compact circle backing. I noticed that in the literature the, their examples seem quite far off being compacts. I wonder if there's any improvement possible there with the blinding on mathematics to that. So that covers like given a thick, you can either start from lots of points and distribute them through collision and get a distribution that way or you can start from a given mesh and optimize that for circle tangency. So you want both you want adaptive numbers of circles and you also want support and you'd see, and that's where we mentioned comes in, and that's something I'll come back to a little bit later when I talk about services. So this is an example of how you can take a surface and through a series of small topological optimizations alternates being alternating between the topology optimization, optimization and the geometry optimization, you can get in a way the best of both. So adaptive numbers of points and also, well in this case equal radio but the beginning circles. So this is the kind of applications have in mind for this. Sometimes in jewelry, they do these have a designs where they want to cover services or complex within complex boundaries have huge numbers of time gems placement space. So that's the kind of applications of it to more pavilions. Different type of circle packing here this is a beautiful paper by Stephen Settlement and others. And they describe an energy which I was able to implement in Kangaroo which like the tangent in circles energy this gives you for triangles this is a gave a set of angle conditions which allows tangent in circles, between adjacent not all what's happening circles that guarantees they have in circles and also the tangent where they have an edge. And so you can then optimize for this in kangaroo and use this to explore these types of measures. And there's actually some really nice links between these measures and discrete minimal surfaces, actually if you take. If you enforce this condition about the tangent in circles of quads, and also that the edges of tangent to a sphere, you can do this duality operation where you swap the directions of the diagonals, and you can switch. If that if it's on a sphere then you get a discrete minimal surface, or for any of these surfaces like you can take any discrete isophanic surface and switch it into a different one. Mobius transformations is something I keep coming back to and I will talk a bit more about it, but later on. So here, just seeing a bit more clearly the remeshing how. If you fix the points along the boundary then you've actually, as I mentioned before, you've got a unique circle packing for that apology. But here it's allowing it to change by adapting the apologies as you try this flipping some of the edges. I'm not going to go too long into other other variations of this, because I want to get on to, well, a little bit on conformal mapping. So you can even packing of circles in another three boundary, and I think of circles with the same topology in another boundary you can generate a conformal map between those two. But which made a big impact on me is this, this one by Tristan medium. If you don't know it's really a wonderful book. I had a strange encounter with it I was an architect or student I didn't. I mean I studied some mathematics in school but I started to get interested in some, some geometrical ideas and I hadn't quite figured out how I was going to be able to model them and I was browsing through the bookshop. And this book, like I knew it was all way above my head for the mathematics in it but there was just something about it that spoke to me. And it takes a really visual approach to explaining all these ideas and there's a nice bit in the preface of the book where he says he compares the state of mathematics as he sees it to a society where you can only re-sheet music and you can talk about music and can analyze music but you can never actually perform it. And to him, that's the way it is with geometry sometimes and he certainly makes great use of it in the book. So yeah, it's something I keep coming back to and I just love the possibilities with combining sources and syncs and streamlines and that so the project that it actually came from was this, what I called the atomic surfaces so I've been making these physical models of helicoids, I've found a way of folding paper into these helicoids and then so that you could join them together and into larger surfaces. I was trying to find a way to do that in a smooth way and that was what some of the ideas in the complex analysis but eventually led to this, this way of generating surfaces that if you take horizontal sections through these surfaces then you get the adequate potentials of your sources and syncs. The main focus now I want to go into is spanning surfaces of knots and links and approaching it from a few different angles and I think it's a rich topic and it has a lot of mathematical and software history. Three pieces of software, particularly now very old software that's very influential, and I've tried to in a way take elements of these, these tools and combine them into one coherent thing. So that you can actually use these features together so surface of old but can practice. You can get compact and build minimal surfaces, not blocked or relaxing knots and then see the view for finding see the surface of knots and links. Then a few artists that inspired me, but cheaper grace man was one of the early pioneers of three printed sculpture, making a lot of mathematical ideas and I know some really beautiful work. So perhaps my is a little bit out of place in that to ever held from my understanding she doesn't actually work through equations or computers, but it's actually all modeled very much by hand to me to which I find fascinating that I mean they, many of the pieces are very close to minimal surfaces. Some of them look like constant but it just says, I think it's, it just shows that our physical intuitions about geometry can be quite powerful. And it makes sense to try and use that. It's just a beautiful piece, more of these minimal or close to minimal services. And, of course, color sequence. He's been a big figure not just in creating sculptures but also writing about them and about the software and the techniques to generate them. So I think quite a key moment from my understanding of the way that mathematics of minimal services develop this, the cost of minimal services, the computer played a very interesting role in actually verify that it was embedded. I think it's quite nice that they, they describe how they actually were able to miss very early days I think early 80s it's not before to actually use the computer to check things and get some, get some feedback that wasn't always obvious just through through looking at the equations. I've been trying for many years to model minimal services or close to minimal services on various knots and links. Initially a lot of it I was starting with manually modeled quadrilateral meshes. So here I would have started from, from a simple base mesh and subdivided it, and then relax that. So that can be quite challenging actually designing that quarter mesh becomes quite a challenge in itself. So that led me to techniques for triangulated services. But there, if you want to relax triangles and you treat the edges just to springs, and the density of your triangulation ends up having a huge effect on the result that you get. So this in, in geometry processing graphics does very widely used coat handle of fashion which solves this problem. So you can actually relax triangle, and it can even simplify further, so the bottom right you can actually just relax triangle. By pulling the points towards the opposite edge with a force proportional to the length of the edge. And that can give you proper minimal services up to a point so you still say you can start from a good match and you can relax it and get something like a catnod in the middle and that's great but then so you want to change the boundary. Then at some point you're trying those, even with the way to you still get too distorted to actually get a nice result when you relax it. So that's where we mentioned comes back in. So it's this idea that given a triangulated mesh, you want to keep the triangles as nice as possible in various and what what you count as as nice as possible for the various different things. In this case, if you want to to collapse an edge. So if you have an edge, which is too long, then you might want to split it into two triangles. You want to want to split it want to get an edge which is too short you might want to collapse it to remove those triangles. And you can do these things iteratively and alternating with the relaxation is there are many, many papers written on on this topic that these are a couple of the ones I found useful and I think. And what that gives you, you can start from some horrible mesh like on the left so you can, and that's from doing various solid geometry operations in cash you often end up with these quite ugly meshes. But then you can repeatedly apply these these remeshing steps and up with with triangles which are actually very close to equilateral and mostly they than six inches. In the last recent years. So, so that's this is something I've developed more for right now actually added a feature into right now for these high quality triangular meshes. And a lot of the work that has gone to clean into future preservation so how do you remesh to get even triangles and also keep track of sharp features. So put all that together and you have meshes which could behave like safe films you can have large changes in the boundary conditions large changes to the geometry and making just the edges of springs using yet for triangles you can get good minimal services, and you can, you can have quite dramatic changes in in the way the boundaries interact and you can even change whether they intersect or, you know, make make topological changes to your services. You can also add pressure into the mix so this is minimal success with volume constraint say it and volume and sculpting with it, we can start to to come up with forms with constant mean curvature. For me, a lot of these studies, I don't often distinguish between illustration and exploration for me it's often the same thing, and I need to make make geometrical models often to understand ideas, and it's, it's a way for me. to take things out, and then hopefully sometimes turning those into tools, useful for other people or useful for explaining things. But, but the prime, primarily, I'm often just trying to understand the idea better myself. It's. This is replicating some of the experiments by Friday after I mentioned that started taking a safe form with a thread floating into it and using this design. fabric membrane shapes. Just through the process of playing with it I was quite surprised when I started, I tried to model certain classical minimal services and didn't always get the results expected because it's. I learned that actually this, this, this makes sense on some of these things, although they are minimal services they're not necessarily stable and in the initial configuration without other symmetry constraints applied. So then coming back to applying these to not some links you can take, take a not curve. In this case pulling it so it's now an unnot but keeping the surface attached to it, and we can make these big changes to the geometry while keeping our mesh attached and minimal. But not if we want to, to find from me from a given topology we want to start from all these different knots and get to the same thing. Then this is somewhere where relaxation and coming is for. I think that I've been doing that in essentially you take an interaction between you discretize it and you take all the points of the not and have them in this case repel all of the other points, usually much higher density subdivision and this. And then, when you apply this you can, you can then start relaxing these knots to find. It's, it's quite pleasing. I find sometimes to start from something very sketchily drawn and then end up reveal some symmetry that you weren't aware was that just from the initial play then not trying to try to make these into intuitive tools. That people can interact with and in all of kangaroo a lot of it is about aiding the development of intuition and I think play is a really essential part of that. So, doing things and seeing the results quickly lets you get a feel for how things behave and I think can help with how you think about. That together, so the not repulsion and the minimal services together with the remeshing and starting from what is actually a triple not relaxing and remeshing and we can quickly jump to something like this. So briefly about conformal mapping before and complex analysis. I'm just going to mention a couple of other work in great examples of, of illustrating mathematics that that have really inspired me some of these quite old but I think still hold up brilliantly. So in this case, talking about the links between various transformations and stereographic projection. One, one website in particular was Thomas banjox work. He has this wonderful page on the flat tourists in the three sphere and that really got me fascinated with what the three still was and trying to get to grips with that. For example, I won't go into too much detail about these but just pointing to the swimming for examples and of course, sort of liner and every segment work. They've done some wonderful, many great examples and illustrations of stereographic projection, vibrations of knots which is something I want to get on to next. So two ideas of inversion, geographic projection, one of the first bits of code I write for rhino was a tool to invert arbitrary measures. And I felt what what I wanted to to explore I've seen a lot of examples of mathematical transformations like inversion or formation rotation, but I struggled to grasp them because they were with unfamiliar objects. So, as it was in rhino and we can have three models from all sorts of great sources. I felt like if you can put recognizable objects to these transformations then it becomes a little bit more intuitive like, while not everyone's going to recognize an inverted tetrahedra that will recognize an inverted room. And I think the these conformal conformal transformations of free space, I just find, not just great fun to play with but I think it is, it's one of those things that's just, it feels like it's just on the, the border of what's possible to get physical intuition for. Obviously there's huge areas of mathematics which are don't have any obvious geometrical realization, and then there's, and there's three dimensional geometry where there's a direct realization but there's three spaces it feels like it's on that fascinating cast between the two where you can, and the computer can really help it's starting to get a feel for some of those things. So, great example of a minimal surface sculpture, minimal surface in the three spare this, I'm sure you all know sculpture boys surface. And I got interested in trying to model something similar study from, so this is the minimal surface with three planar ends that it gives us and I was playing and trying to develop an intuition for it through this. This is the projection and for the rotation. This is the form of it which you see in the sculpture. But if you turn it a bit further all the way around. I felt like this is the interesting that this is the part of the surface where where it all happens like it goes off to infinity and so closing up this is the triple point is what you really want to see. You're able to turn this into into a surface and so a lot of the time I'm trying to take geometrical objects and make them easier to grasp often very complex forms. That's not necessarily complicated like difficult to grasp but consisting of few elements and trying to present them in a way which makes it easier to to get a feel for them to sort of have it as something you can turn around in your head. I think self intersections often often conceal that and make it very hard to grasp what the surface is really doing. Another great inspiration here, of course, is George Francis who this book is full of amazing hand drawn illustrations and lots of great ways of explaining topological concepts. I'm also pioneer in computer graphics as well so later on he also made some some wonderful renderings of these types of services and particularly the other ones I once I really like are these minimal services. This is described by Blaine Lawson in this case the Klein bottle. But again the self intersection makes it quite hard to follow. So I took that and I tried to cut it in a way where you could see it in the same way that they did for the real project play. I'm just aware that with the time I want to skip a little bit of this to get on to some of the other stuff. In grasshopper what I've done has taken a little bit of code for the to describe the four dimensional rotations, but then in grasshopper you can turn these into components some of your experiencing these days, why them up and very quickly play with them. And the mix of going back and forth between the normal 3D environment where you can pan and rotate and all that stuff and equations and also planning stuff together in a way you can just link one thing and in this case, starting with one point transforming it giving it a list of what things transform those or using those lists to make surfaces. It all becomes very quick and fun and easy to play with them into a surface, you can get your, maybe a span with a circular edge. And when this case the Klein bottle. So it's got four holes, so you don't see that it's closed, but this is. This is more than Klein bottle of minimal surface in the free spirit with some holes cut out to reveal the areas where itself intersects. Three examples of different topological services or less minimal services in the free spirit. Another, another type of spanning services or not some links that I had fun playing with this. So you've got a tourist not where you can sweep that tourist not by taking a four dimensional rotation, stereographically projected and varying the two rates of rotation. And if you then make a spanning surface so you can relax the surface on that curve, you can then do the same four dimensional rotation to that surface, and the points along the boundary will follow the original curves. And that way, you can get the services which sweep through space is what I understand they're called open book vibrations. But the last, the last area of this that I want to talk about something that I'm quite excited about at the moment. So that's that previous one showed a way technique can use when they are tourist not. I want to do same time spanning services and get see for services and vibrations for arbitrary not I can process great paper from some physicists about solid angle, and this is so the solid angle. So we can visualize it as we take a spear around the point, move that point around the curve and then we can check how much of the area of that sphere is cut out when we extrude the curve to the point. And it's assigned area so you notice that it switches color as it passes the outside of the boundary. So we can evaluate that function for every point in space, and then we can get a field so in this case this is the color is showing the value of that solid angle field, as we move the plane through that through that not. Or we can write it and can see that it has it wraps around its periodic so we choosing a coloring so you don't see that jump. But then you can take a level set of that surface and then that can give you another sparring surface on on your not. So in this case, a secret surface on a trip or not. But you can do it on arbitrary. So you don't have that intention as before for the trip or not. So I've been having a lot of fun playing with these ideas that I think actually it was 10 second when I was starting to discuss some of these ideas he suggested to look at the figure eight not as nice example. I think that could be interesting to see. Also tried it with Borromean rings. They're in this case seeing one page of the open book turning around. But we can also, if we take two opposite pages, we can get a surface which passes through the links and get something like this. So we're looking up what other not survive, but in this case, pair of trefoils and yes it seems seems to work to generate that. Or in this case, two components so taking the white head link showing to play with that. I'm thinking that that was quite surprisingly confusing at first when it does this. The geometry is has to rotational symmetries but the surface doesn't. But of course, it's when you evaluate the solid angle which which way round your game matters so the orientation of those curves makes a difference in the surface you get. So that's my suggestion so. First, a lot of this stuff. online and had some really enjoyable engagements with all sorts of physicists and mathematicians. And so someone suggested. I think the components actually one ring the side angle is doubled and some to this the single solid angle at the other, and that was something very close to the. Maybe a span with a circular edge that I talked about for that. This was the end I got mentioned that can do this for the army and rings are here waiting one of the curses so the blue one has. So the red one by three, this old angle for the red one by two and the green one, just one. And you still get these. These nice surfaces which don't intersect and not not all not worth this so you take a non fighter not and you get singularities as you start to. I'm sort of exploring far beyond the depth of my, my true mathematical knowledge but I hopefully doesn't seem completely ridiculous to play with these things in a geometrical way. And it's something that I like to do to try and get a better understanding of these things. And then from the learning about different fight the not so found this, this wonderful paper which shows a sequence. So you can add in extra. You can add same bands, the repetitions of the same band and gets the figure eight not the bar me and rings and an infinite family of these, these five knots. The one after the bombing rings and tried relaxing that so using the same repulsive not relaxation I talked about before, and pushing around it finds its way to something that is symmetrical. You can take that, and find. We can't play that. I'll put the video somewhere else later that's so was a rotation of the same solid angle surface on that link. I'm getting close to time so I'm actually going to start wrapping up now. Hopefully ties together in some way but some of the key points I really want to try and get across from as a, as a non mathematician and my experience of this is like, it's really easy now to like you, you look at some of those examples I talked about like George Francis talking about how they've made these visualizations over in the paper about the cost service and they have they really went to a lot of work to make these visualizations but then they still found it worth it even though taking days and you know very laborious. Now you can do the same thing in seconds in free and cheap tools anyone can do it and interact with it real time. So, encourage people to try it and I know that's exactly why you're here so so I think that's good. I think that illustration will roll it can do it can make connections between different domains and with those from other disciplines. A lot of the stuff for the mathematical ideas I've ended up engaging with I never would have started learning about them if I haven't seen some of these examples of this motivation that I mentioned. I think they can be more than just to talk communication. I think that's the first piece that's illustrating can actually be a way of thinking and making illustrations can. It puts it into a different you can use your physical intuition you can think of things as physical objects in a way that is much harder for you to visualize them. Finally, just to conclude, these are all reasons I try and justify but ultimately, it's just fun, it's just beautiful and there's so many things to discover and you play and you keep finding new things and keep playing. So let's all thank Daniel. Really amazing. We've got our show and asked in 15 minutes but but I think probably lots of you have lots of questions for Daniel so let's let's take a couple minutes to see if there are any questions. Can you either just turn your mic on or put it in the chat. Hi. Yeah, I was wondering if any of the software you showed us, it would be possible to like generate a data set based on the figures you're looking at like maybe I just like care about. I don't know the angles of some curves like just just some variables like is there a way to keep track of that. Yeah, I'm not sure I caught the whole question so it's in kangaroo is the tool in grasshopper and you can you can set as many or as few goals as you like so you can, you can take a square and say these two edges are parallel and no other goals. So yeah, you can, the idea is that you can mix and match and select. Does that answer the question. Kind of I'm like thinking about like say I had a bunch of different variations on knots and I wanted to generate a data set. Like, and if I was playing with it visually and like as I went like can I maybe like mark things to like like figures that I want information about I'm not maybe this doesn't quite make sense what I'm thinking of. I'm just I guess like the question is about like generating a data set like if I can I can have like my visual like a set of figures and then is there like an automated way that you know of that I could kind of track information. So the, I mean, not necessarily. So, there's like a whole ecosystem of tools within right now in grasshopper and the relaxation is one part of it, but you can, you can write scripts and use link with other things so you can. Yeah, there are many ways you could be data you can have data coming in from Excel and going out to rabbit, you know you can you can link all the different softwares. That Wesley is next. So, I have a kind of different and more general and maybe less well posed question right so you spoke about nowadays right we have a lot of tools we have a lot of opportunities for producing these nice helpful visualizations that you know, many years weren't possible or maybe were very difficult. And so, maybe my question is more kind of looking forward are there maybe emerging technologies emerging tools merging ideas that you see as maybe having the potential for really, you know, pushing things forward or being that next kind of big step that makes sense. That's a good question. What I think I virtual reality is, it's obviously one of the things that comes to mind is having a resurgence and the geometry visualization was, you know, a lot of the very early pioneers were actually, you know, some of the earliest uses of the art were in. And the interactivity is another one I think is very exciting I think now. And it's also very easy to make a make something and put it on the web in the way that people, you know, anyone can click the link on their phone can start seeing what happens if you change this parameter and see other systems just on their favor. I mean, I think generally geometry, geometry processing ways of generating geometry is still a very active field there's still many questions. Around meshing. Level set surfaces and science distance fields, I think are very, lots of exciting developments going on that too. We're really limited on time but I know Aaron, Saul and Elliot had their hands up so let's let's get to you I think we're just have time for those three and then we'll have to have to call it a day. Yeah, hi thanks a lot Daniel for a beautiful presentation. Speaking of interactivity which you just mentioned I just was wondering, you know, you showed some amazing animations and some things that that you know playing with soap films and and see for surfaces and changing the knot and things like that. I wonder if any of those tools are available for people to play with and interact with. Do those exist online somehow. Yeah. So, what I show is mainly happening in Rhino, which runs locally there are ways, and you can actually access right now running on a site. I can show links like I've sometimes put. You can have a web page which links to grasshopper definition which is running on a computer somewhere else. So, I mean, I think there are there are also many dedicated online platforms which are also make that even easier and faster. I'm always working in my name grass because of the other. I mean, for the other applications and my, my day job about the United Design stuff for architecture so I tend to stay my least mostly in this platform but I think there are certainly technologies which can make these things easier to access and interact with. Daniel the the soap film surfaces that you that you showed that we're capable of changing topology was that your mesh machine component. So, it's a, it's a development that is from the same line of development. I've now got a few tools. There's, there is a remesh. So, like a feature for the thing we measure in Rhino now which is something I thought. And there's like if you want to get good triangulations for doing analysis on it can be useful. There's also as part of kangaroo that's what I call the life soap component, which does. It's a safe film which can about quickly change its topology, which was quite challenging to getting because keeping track of the indexing of the particles and when you're when you're changing it dynamically was to go. So, I just wanted to start by saying thank you very much that was a really beautiful talk. I also want to ask a question or perhaps make a suggestion. Have you thought at all about the spherey version problem. I would. I was very much like to tackle that definitely. Yeah, and I think this. I know. George Francis as well was. I'd like to look at more. Yeah, I'd be happy to talk more and actually just generally to everyone like, if you have ideas of things which you think would be interesting to visualize I'm very much open to suggestions or I maybe, if I could follow up me, could I ask will you be having an office hours where we could mob you. Yeah, I'll be online for the rest of the rest of the event. So, I mean, maybe not with UK times and it's probably in the earlier part of the day. I'll be more likely to be around. Yeah. Thank you again. Oh yeah. Thank you for the talk. So I was wondering, all of the energy functions you talked about in this were physically inspired, like they were simulating, I don't know, like a bubble film or something like that. I was wondering if you tried any, like more esoteric or abstract energy functionals and if you got any that gave like particularly cool results. Yeah. There are many. Yeah, so it started out with physically based ones but what actually to be a quality goals, which is not, I can't think of any real physical analogy but taking the power of angles and send those angles have to say quote but then doing that to all of the angles of all the triangles on a mesh or something. So this, yeah, there's a whole load now of non non physical energies in there. Yeah. Happy to talk more, give more examples. Yeah, there's a lot. It started out as, you know, digital version of physical form fighting and it's eventually morphed into something much broader about general geometric energies. I just remember I saw something at some point where they gave an energy that optimized that things ended up having like, like right angles and like gave this nice like cubic structure to things. I was wondering if there's stuff like that that you came across. Yeah, absolutely. I know that you mean, actually that's, I heard that that was the one what they originally trying to do it was different, different optimization that went wrong but they like the result of the wrong version that I turned it into paper. Yeah, there's, like, isn't it a very interesting one that developability energy, which takes a mission tries to make it piecewise developable based on some work by getting prayed and others. And that's, it's completely non physical and the tail behaves like that. But yeah, and from optimizing the certain geometric constraints you can get all sorts of surprising and interesting forms. Okay, thank you. Let's all thank Daniel one more time I was really excited by your talk to Daniel so just.