Mathematical Games Hosted by Ed Pegg Jr. [Episode 13: Surfaces]

Hello, everyone. I'm waiting for things to settle. Okay, there's my talk. They're just getting started. I'll wait a minute or so for things to. To. For people to come in and do. Feel free to post any questions or anything that you might like. I'm Ed Pegg Jr. And I run demonstrations.wolfram.com and a few other sites. So I'll be talking today about mathematical surfaces and some of the things related to them. While putting together this talk, I actually managed to learn quite a bit about services. Okay, so this is part of my mathematical game series. I met Peg Jr. And in. Starting about 50 years ago, Martin Gardner started the mathematical games column in Scientific American. And he ran that for 25 years. Every month there was a new thing that he talked about in mathematics. And that became a very influential topic for recreational mathematics in general. And it brought in a lot of people to mathematics, including me, where I, I saw how math could actually be interesting rather than just sort of the less interesting things taught in grade school. So I became a mathematician as a result. There's many different services in our demonstrations project demonstrations.wolfram.com and I will be showing several of these in the course of this talk. And if you'd like to get more code or anything like that for these, you can go into the demonstrations and find some of the code that I used for this talk before that. Welcome to 2024. So here's one way of representing 2024 is the sum of cubes, and 2024 is both a tetrahedral and a dodecahedral number. So here is a tetrahedron with, with 2024 spheres in it. And if you cut up on the green lines, you get the dodecahedral representation of 2024. There's a way of stretching these so that you get an actual dodecahedron shape. But I didn't do that animation yet. And there's a various ways you can reach 2024. For example, it's the sum of the first 22 triangular numbers. Here's how we did 2023, and that sort of works for 2024. If you go with the square root of 7, it's part of the 12th convergent. Another weird thing about 2024 is if you square the number, you get the composition of two square numbers as the squares of 64 and 24. There's a bit of math fun you can do with the year 2024. So back to services. What is a surface? It's a generalization of a plane. So here's A A plane. Here we have three points defining a plane. And here's the Hessian representation for the plane. There's the normal for the plane and there's the. There's the distance. And with the Hessian form, you can do various things with, with planes, the dot product is handy. So here's a representation for one simple surface, the plane. So since we're going to be dealing with surfaces, what are the different ways you can show a surface? So here's several different ways we can do a. In Mathematica, we can do a contour plot. And here's a surface that I just kind of made up. I'm not sure if this is a. A name surface. I just typed something and I got this and I thought, oh, that looks vaguely interesting. Here's a complex plot. It's another way of doing a surface. For this particular complex expression, we can do a parametric plot where we specify the X coordinate, Y coordinate, and Z coordinate in terms of two variables. So here's a parametric surface. We can also do spherical coordinates. So those are four fairly standard ways of representing a. A 3D mathematical surface, contour, complex parametric, spherical plot, 3D. So before we get too much into surfaces, we should discuss curves first, because surfaces are a 3D generalization of two dimensional curves. So one classic curve is the conic section. Five points will define a conic section. And if you just do five points randomly, you'll almost always get either hyperbolas with 95 over 144 chance or an ellipse with 49 over 144 chance. So here's a sample of a conic section with this equation. And that's what this conic looks like. With four points you define a hyperbola, something discovered by Isaac Newton. You actually define two, two parabolas. If I said hyperbola earlier, a four points defines two parabolas. So here's one parabola through those four points, and here's the other parabola through four points. And I'm using it for this code. I'm basically using Isaac Newton's code. It's actually right in universal arithmetic. So I just adapted that slightly to match mathematical code and, and put it in the function repository. Three points defines a circle. If you put in these three points, it's a circle with that center with this radius. So once you have a curve, you can generalize it to a surface and with circle to sphere. That's fairly easy. And four points will define a sphere. If you put in these four points, which are similar to these three points, we're basically adding the point 2, 2, 2. We get a different center and a bigger radius. So with curves slightly covered here, we can move on to two surfaces. With surfaces, you usually need nine points to define the simplest surfaces other than spheres. So for example, the quadric surfaces, which are the 3D equivalents of conic sections. Quadrics usually need nine points to define them. So here's nine semi random points which are based on the powers of this number. I'll show what this number is. If you look at, If you look at these digits, if you split up these digits into points, you get this surface. We can use the resource function quadric surface plot to show the same ellipsoid, but it gives more detail. So with, with this function you can get more information about a particular surface. So ellipsoids are one surface hyperboloid of two sheets, again defined by nine semi random points gives another surface. Hyperbolic paraboloid is another surface. And you can define it with, with nine points. But it's slightly more interesting to do it like this. You basically take a, a tilted frame and you connect up points with straight lines. And with these straight lines you get this curved surface which is a hyperbolic paraboloid. It's a double ruled surface. Through any point on the hyperbolic paraboloid you can draw two straight lines that are on the surface and there's two known double ruled surfaces, the hyperbolic paraboloid being one of them. And we can, we can change the weave and the stretch to get various changes in the, in this, in the surface. And this is a common thing with, with wireframe pictures where you basically put up pins on a framework which is warped in some way. And then you just connect everything up with, with wires or thread or fishing line and you get these nice patterns. So that's the hyperbolic paraboloid. We can also do a hyperboloid of one sheet. So here's again nine random points making a hyperbole of one sheet. And this is also a double rolled surface going here. For any point, you can draw two straight lines through it that stay on it. One interesting thing I like to do when I'm making spaghetti is to hold up the bundle of spaghetti and to twist it and you basically get a hyperbolic paraboloid. And that's very handy to do before you put it into the pot of boiling water. That helps spread out the spaghetti so it doesn't clump. And if you do it just right, it just falls into the pot perfectly. We can change the number of bars and the twist and things like that. So something very similar to this is the catenoid. So here's two rings connected by a film. And we can do this as a revolution plot. It's basically the hyperbolic cosine. We do a revolution plot based on that. You can also do it as a parametric plot. Hyperbolic cosine. Hyperbolic cosine. And just the, the value that we're taking the hyperbolic cosine of. And the reason we want to take a look at this is because it's a natural form. If you dip two rings and to soap, the soap film left behind is a catenoid. And this is based on a curve known as the catenary curve. If you take a chain and two points and, and dangle it, the curve that's formed by the chain is a catenary. It's, that's, that's the parabola there. Here's the catenary, here's the parabola. They're very similar to each other. In fact, they originally thought that this curve that the chain made was a parabola, but it turns out it's, it's the catenary curve instead. So this is a, this is a way of minimizing energy for a chain. And we also want to look at minimal distances. The, the, the. For a soap bubble, a sphere minimizes or maximizes the volume for the given amount of soap. For a number of points, we can take a look at the Fermat point, which is the point at a minimal distance to other connecting points. And when they were originally studying soap films, they thought, well, this is the point. If we just bang nails into a piece of wood and dip it in soap, all the soap films should converge on that point. But that's not what happens. They found out that you would get these 120 degree angles when you move the soap around. So if, if you want to play a little game, this Steiner trim demonstration basically challenges you to make this number down here as low as possible. And the way to do it is to find the places where all the angles are 120 and that's, that's where the minimum is. Let's see if I can get close to that. Somewhere around here. And it's easier if you have code to do that automatically. Finding the Steiner tree is a, is a non trivial problem. But that's something I think I'll try, I'll try to tackle sometime soon. So if you put these four, four points as nails into a board and dip them into soap the soap film will automatically go to the minimal place and the, the angles you'll see in the soap films will be 120 degrees. And the normal soap bubbles act in the same way. If you have three soap bubbles, this angle here where the soap bubbles converge will be 120 degree angle. That's where the minimum is found for intersecting soap bubbles. So getting back to the catenoid, this is also a soap film, but unlike the sphere, this is, there's a sphere has internal pressure working on it. So it's not kind of a purely mathematical object because there's this one extra thing working on the soap film from the inside, it's containing a volume, whereas here it's basically just a pure minimalization of, of soap between two rings. So mathematicians in the 18th century thought that the catenoid was more interesting than the sphere for something to study. And they eventually found out that it was the catenary curve that was defining this object between two rings. And they also knew of one other shape which was a minimal surface, and that is the, the helicoid. Here. You can, if you twist a, a wire and dip that in film, you'll, you can actually see a, a, a helicoid form in soap film. So they had these two shapes which were minimal surfaces. And they found out that the mean curvature, which I'll get into later, the mean curvature at every point on the surface was zero. It was basically flat at every point on the surface and it could bend one way in one direction and bend another way in another direction. But the mean of those, the, the balance between them turned out to be zero. So these surfaces that had a zero curvature in their mean curvature were thought to be very interesting. So they wondered if there was more. And they noticed that there's a transformation from a, a catenoid to a helicoid and back wire. Strauss came up with the parameterization that allows the changing from, from one to another. So they started looking for more of these minimal surfaces, wondering if there's some other ways you could dip a wire framework into soap and come out with a new mathematical object. So one that they found was the helicoid with a hole. Let's see if I can get that to work. It doesn't seem to want to work it. Oh, there we go. Oops, I wonder why that one doesn't want to work now. It worked earlier. Well, the helicoid with, with hole is one of them. Oh, this will be inconvenient if nothing shows up. Here we go. The circ minimal surface was one of the first that was found, it's this object here. Note that there's two holes this way and three holes the other way. So there's one minimal surface that was found early on in the 18th century by Heinrich Shirk. Here's the parameterization of it as a surface. And once he found this kind of the, the race was on to find more of these unknown mathematical surfaces. I, I believe this is one that you can actually create with soap films. And there's various forms of this in, within this demonstration. Shirk minimal surface. If you go there, you can see some of the different variations of this curve. The code tends to be somewhat slow. Sometimes it can take up to 30 seconds for these to show up. So I can try some of these and see if they change. Yeah, I think these ones are relatively fast. I think that's the same one. Here's the double leaf periodic form. If you build up a wire framework, this is one that you can, you can create. And some of these take longer to produce. So let's see here and here's another form of doing way of doing the, the second circ surface. Let's see if I can get that to work. I'll wait on that and then see if we can get this one to, to work. Let's see, I'm seeing some of these comments here. Video. The spaghetti twist. It's pretty easy just to do that. You just hold the bundle of spaghetti and twist it with your hands and you'll get this automatic result. Here's the second Shirk surface. That looks like a catenoid though. So I'm not, not sure I trust it. It might be just. Oh, there it is. There's the second shirt surface done with ellipsoids tangent to each other. So once, once they found some of these, the race was on to find new and interesting minimal surfaces. Here's another one here. Catalan was another that managed to find some of these surfaces. I'm not sure why these are not showing up immediately. So there's the Catalan surface. It's self intersecting and here's the parameterization for it. It's a bit more complicated than the others. And we can get the Riemann minimal surface by Bernhard Riemann. I believe this is another that you can actually build with soap. It would be interesting to have a catalog of actual soap films, have the function Mathematica generation and then a wireframe with soap film actually showing the curve is done. You can kind of see he's joining catenoids here to produce this object. An infinite number of catenoids, each overlapping with each other. Yes, these are constant mean curvature surfaces where the. Where the mean curvature is zero everywhere. To answer Dugan, Hammock. And I thought these pictures would all show immediately. Let's see if these other ones show up. There we go. Here's the inner per surface. We can change the radius. Another surface where the mean curvature is zero. So it's in the minimal surface and the chin. Gag stator surface, which is an inner surface with. With two handles. I think there's a pure chin here. Yes, here it is here. This is the chin gak statter surface. This is one of the newer surfaces that was found in 1982. I'm not sure how fast those are. And then we have two surfaces by Schwartz and his student Niovius 1865. This is the Neovius surface here with this implicit equation. And hopefully this will work. HH Swart did this surface here, the P surface. So that's the P surface, the D surface. If it'll come out, there's the. The D surface. And after he found those two, his student found the in surface after named after Niovius. You can see there's lots of holes going through this object. And if you build up a cube with the yellow wireframe, you can see this weird surface on the inside forming in scope. Theoretically. So this is a more complicated minimal surface. Let me see. We can say with one recursion. Let's see what happens when I change the range on this. And there's various other services you can play with here. This is the Sports minimal surface and Neovius minimal surface. But there's a whole bunch of other surfaces given within this particular demonstration. Some of these are used in biomaterials and other special compounds just. Just because once you sort of build the framework, you can get these surfaces to just appear as a minimalization out of soap or other materials. Plastics for example. The things like aerogels and things like that can be built with things like this. The Hindenburg minimal surface still back in the 1800s is another self intersecting curve and has this parameterization. And yes, all of these are constant mean curvature surfaces. I'm not sure what it means with when they're self intersecting. Obviously you couldn't get a soap film to self intersect. I'm pretty sure. Maybe, maybe not. I don't think so. But I think although all the ones that are non self intersecting can be done with soap films. But I could be wrong. So this was found by solving what was known as the Burlene problem. What is the Borlane problem. Well, find a minimal surface passing through a given curve. And for the cycloid Catalan found what's called the catalytic Catalan surface. For circle, the catenoid was already known for parabola. The Hindenburg surface was found by Hindenburg. And the Beurlen equation is just basically this. So with that you can get things like the Beerlein helix. So here's the. I'm probably saying his name wrong. Here's the beuroule and helix here, which is a highly complicated object. This is starting with the given curve, the. What is it? The helix, and finding a minimal curve Going through that helix with a given set of normals, you get this complex minimal surface known as the Beerlein delects. So they managed to prove that the Beerling problem always had solutions. Of course, many of them were very nasty. The, for example, the helix here we've got a third of a page of code solving that. And that was very challenging to find in the 1800s. For most curves, you're dealing with two, three, four or hundreds of pages of code to describe the surface. So those were not found in the 1800s. That's just too difficult for a human to go through and solve and let alone make pictures of. So most of those weren't really investigated until much later when we had computers. Here's another minimal surface. The board's minimal surface and the parameters can be changed. And the guy Roy is another minimal service found by my. My friend Alan Showin. I actually wrote a book with him. Here's his minimal surface called the Geroid, which turned out to be one of the more useful of the minimal surfaces. And it has a very simple implicit equation there it is there. Costa also found one basically taking a torus and removing three points, and you get this complex surface here. And here's another version of that by Enrique as Stuart Dixon did that first one, nice complex surface here. And that is the Costa minimal surface. Now here's. Here's one called a non surface. It has. It has zero curvature. And here's a picture of it not interactive because this one is very slow. However, this surface doesn't actually exist. They were able to deplot a lot of it, but they couldn't finish it up. And it turned out that due to this object, this conjecture, called the Hoffman Meeks conjecture, with a minimal surface of genus G, the number of ends must be G +2. I'm not sure if that conjecture has been proven or not, but when they tried to come up with this particular surface, it would have been a counterexample. However, the part in the center of this object doesn't actually work. This turns out to be a degenerate surface. So it actually cannot be completed. They can do a partial solution of it, but they can't complete it. So we can make a picture of the partial, but it's not possible to complete. So it's called the Horgan minimal non surface because it's it fix. It works as a minimal object, but it has too many holes in it to be considered equivalent to a plane. There's just too much where it's undefined. A Thompson minimal surface, this is one discovered in the 1980s and this has a not too bad parameterization. So there's, there's many surfaces that have been discovered, including this minimization of a Klein bottle with one puncture. So this is a complete non orientable minimal surface which is equivalent to a Klein bottle when it's. When there's a completion element given to it for how the, how the boundaries work. And I think I may have doubled this slide. So chin. Oh, this is a different demonstration actually. The chin cat stat surface related to these, the inner per surface and the others. Let's see here. So what is mean curvature? I've, I've mentioned that the mean curvature of all those surfaces was zero. So here's one way we can calculate the mean curvature. So we can do that with the Shirk minimal surface, which has a very simple parameterization. U V and log cosine U secant V. This parameterization gives the circ minimal surface. So we can dig in. The UU partial derivative of that is 00 minus sequent U squared. So I'm basically using the code here. Here's the first derivative, the first partial derivative in terms of U, the partial derivative in terms of V. And the first part of this big equation is the UU partial which is here, the DU partial which is here, and the partial V which is here. When we combine them, we get these three objects and then we take the determinant of that, multiply it by that and then the. And the dot product there. So this is the first part here. And if you continue doing that and then divide by this last part, this last part here, you basically get it all simplifies to zero. So calculating the curvature is non trivial. But once you've got the code done, you don't really have to do it again. You can just toss your parameterized curve into the function. And this is in the Wolfram function repository, by the way. Just search for Mean curvature. Basically it'll do all that work for you and you can calculate what the mean curvature is. Here's the catenoid. You toss in the parameterization. The mean curvature is 0. Hberg minimal surface. Toss in the parameterization. Mean curvature is zero in a per minimal surface parameterization. The Bohr's minimal surface has this parameterization. The Thompson minimal surface has this parameterization. And Catalan's minimal surface as that has this parameterization. Now you can see that the mean curvature of all of these minimal surfaces is zero. So that's for a, the parametric form of the mean curvature. For an implicit function. You need this formula and it's a little bit nastier to work with here. We're also again taking derivatives of the function in various ways and then we calculate the trace and do this calculation like so. So with the helicoid, when you do this, this, this series of steps here, you get zero. This mean curvature form is also in the function repository. Whether you have a, an implicit equation or a parametric equation, it will work for both. So it turns out that things aren't quite as, as clear cut when you're working with these functions. When you do the, the, the mean curvature for the Schwarz mineral surface, for example, which has this nice implicit equation, you get this back, which doesn't look like zero at all. However, we have this formula to work with. So we can use sine y plus sine z plus sine x equals zero. And take a look at this here. Well, sine x plus sin y is equal to negative sine Z. So basically I make the substitution here and put a minus in front of it. This next part here, that's equal to sine y minus sine y. This part here is equal to minus sine X. So if we do all those substitutions on the top, everything that is positive up here now becomes negative down here. We get this equation instead. Well, sine x plus sine y plus sine z is equal to 0 and sine 3x plus sine 3y plus sine 3z is equal to 0. So this top part here is equal to 0. I don't didn't calculate here, but I'm pretty sure that this is non zero. So due to that, this, this whole thing can simplify down to zero. But it's messy, it's not done automatically. For the shirk first minimal surface we have this implicit equation. But it, it only gets down to here and I don't know how to use the substitution here to get. Actually I do this part Here X, x, z cos Y which is here, that's equal to positive cosine. So if you make that substitution, you get negative cosine x plus cosine X which is equal to zero. So this part up here is zero. So the mean curvature of the shirt first minimal surface is zero. The gyroid. I don't know how to get this down to zero. There's probably some sort of substitution you could do by, by changing some of these equalities around, but I don't know how to do that. If anybody can can come up with that, that would probably be an interesting community post. If anybody can simplify the gyroid, the minute mean curvature of the gyroid down to zero, I'd like to see that. And for other implicit functions like the Neovius minimal surface and the inner per minimal surface, things just get nastier and nastier. So those are some things you can do with minimal surfaces. And here and the calculus involved with how to show that these are actually minimal surfaces where the mean curvature is zero. We can also move away from minimal surfaces and talk about other properties of some of these. For example, with a cubic surface, it's possible to put 27 lines on the surface and they intersect like this. This is known as the CLEPS surface and there's the equation for it. This is not a minimal surface. I don't know what the mean curvature is, but it's not a constant. It's possible to put up to 24 lines on a surface somewhat easily. And here's a, a picture showing the, the 24 lines on this particular surface somewhere here, There we can get places where the, where the curve doubles up. These are called double points here. And another question is what is the maximum number of double points you can get on a particular surface? These, these places where everything bunches together. So that's a cubic surface. 20, 27 lines is maximal. They started looking for other maximal lines and they found the sure quartic had the maximum width 24 or 64 lines for a order 4 equation. But if you're dealing with real space, the maximum is actually 56 lines. And there is the, there is the surface that has 56 lines. I have not yet finished putting all the lines on the surface, but I plan to do that in a future community post. The maximal number of lines on a quintic surface and above is unknown. That's an unsolved of some unsolved problem. So if anybody wants to, to play around with that, there's really very little known about Higher order surfaces for the term in terms of number of lines. So I mentioned double points. Here are a whole bunch of double points on an Arctic surface. That's the Barth sextic. Here's the Cayley cubic, which we just saw with four double points. There's a little tetrahedron at the center there. For a quartic, the maximum number of double points is 16. Wonder if there's a place for the. So, okay, These take a while to render, unfortunately. So double points on a surface has, has a whole bunch of different, best known surfaces for double points. And I found these, these rather interesting surfaces. And we can also look at topological spaces and one of the ones that is well known as the Boy surface. So similar to the Klein bottle and the projective plane, the cross cap, there's this object called the Boy surface, which is a non orientable surface, but you're able to make these pictures of it. So let's see. I've talked about quite a few surfaces. It turns out we have a database called surface Data that has many different surfaces. We have all sorts of different properties that are possible to look at, such as the number of punctures and the object. For example, I showed a Klein bottle with three punctures. Some of them are minimal surfaces. Somewhere here it says the word minimal. And here's a list of all the various surfaces that we have in this database. Surface data. Oh, it's. And minimal surfaces is one of the classes. We can take a look at the minimal surface class and it'll list all these surfaces with that property. We can also look at all the cortex surfaces or quadratic surfaces or periodic surfaces, for example. And for example, here are the minimal surfaces listed in Surface Data. I've gone over more than is listed here, so I think there's about 10 surfaces I touched on that aren't actually listed here. And here is a grid showing that the mean curvature of these minimal surfaces is zero for most of these, except for the double sphere, which is a degenerate surface. The double sphere looks like a sphere, but it's. You basically take the equation of a sphere, the implicit equation of a sphere and square it and it still looks like a sphere, but it has strange properties. I'm not sure how to describe the double sphere in a way other than it just looks like a sphere. So that's what I had prepared. And I can start looking at all these questions here. Let's see here. What is the qualitative difference between the orthocenter and centroids of the surface? I'M not sure. For example, with a triangle, just a triangle. I can tell you a lot about the orthocenter and the centroids. With a tetrahedron, the orthocenter is undefined. So I'm not sure that surfaces in general have orthocenters just. Just because in three dimensions the orthocenter isn't always defined. Many thanks Ultron for your, for your comment there. Let's see. Surfaces minimize surface based on boundaries. Much more complicated. Yeah, I can, I can believe that the intersecting surfaces are not as simple as I mentioned. Soap films that self intersect I don't think are possible. I would also like to see videos of more soap bubbles in terms of mathematical surfaces. That would be an interesting thing to put together. Let's see if anybody else have, have any. Let's see, is limit of number of lines on a boundary. Well, we saw with the hyperbolic paraboloid and the hyperbola of one sheet that you could have an infinite number of lines on a surface. So I'm not sure if there is a limit to the number of lines, but those are the only two known double ruled surfaces. I didn't cop, I didn't cover the, the ruled surfaces, but we have, we have a surface data class. We can take a look at that. Let's see surface data, Ruled surfaces. These are the ruled surfaces we have information on. Oh, I did not know that these, these were. Oh, it just took a while to come up. Oh, there's quite a few ruled surfaces. Many of these I don't know anything about. The oiloid is rule surface. Oh, I'll have to take a look at those minimal surface between two closed boundaries. What defines. Okay, what is the relationship between double points and the centroid? Okay, what is must all allowable lines pass through a given double point? Not that I know of. One thing I noticed while doing this is the first surfaces I looked at were were bubbles. And then looking further, I got into doubles and I was reminded of the line from Macbeth, double, double, boil and trouble or something like that. And I think that can close things out. Many thanks for attending my talk and I'll be here next month with another edition of Mathematical Games, a homage to some of the things that Martin Gardner talked about in scientific America. As many thanks for attending my talk and if you'd like to write to me, here's my address here. I'll be glad to answer any questions that people send to me. If you have any good demonstration ideas, feel free to send those in and you can also post on community. Wolfram.com so many thanks, everyone.