Mathematical Games Hosted by Ed Pegg Jr. [Episode 20: Icosahedra and Other Polyforms]

Hello, everyone. We're going to be starting very soon. Welcome to my talk. We're going to wait for the numbers to stabilize, and then we'll go ahead and start things up. And somewhere here I've got another window I need to look at. Okay? So welcome, everyone, to the 20th episode of Mathematical Games. And the icosahedron has 20 faces. And when I saw I was doing episode 20, I thought I would talk about polyhedra, since that was one item that Martin Gardner discussed extensively in his Mathematical Games column for Scientific American. I've been doing this series as sort of a homage to Martin Gardner and the Mathematical games movement he started because it was very influential for me as a mathematician. So, as I often mentioned, we've got many items and demonstrations.wolfram.com that relate to polyhedra. Almost a thousand demonstrations related to polyhedra. And. And one of Martin Gardner's first columns was about polyhedra. This is one from the 50s, and he talks about the regular Platonic solids. And he shows a way of folding an envelope into a tetrahedron, because the envelope is curiously related to the golden ratio. And it happens to work out well. But this item down here, a way of making a dodecahedron. If you. If you make a. A pentagon on cardboard and make five copies of it around it, you make this basic shape and you make two of those out of cardboard, and then you interlace them like this with a rubber band. As soon as you let go of it, it will spring into a dodecahedron. This was something I saw in his columns when I was really young, about eight or so. And when I saw this, I happened to get a piece of cardboard from a box and I made this shape. And that endlessly fascinated me. From that point, I was hooked on reading Martin Gardner's columns because I thought that was the neatest thing in the world. And this is very easy to make, by the way, so I highly recommend that to anybody. To make this particular construction of a dodecahedron where you take two pieces of cardboard and when you let them go, it springs into the correct shape. So, of course, we can do the Platonic solids easily. And I'll be showing a lot of code here in the Talk. So one of his other early columns discussed the Icosian game by Hamiltonian. And he basically had a board where you had to come up with a pass going through all the vertices that were connected. One thing Martin Gardner didn't have access to was a copy of the actual game. This is from the collection of James Dalgetty who has a big puzzle museum. But I believe he's recently given it to George Miller, who runs the Puzzle Palace. And he now has the biggest collection of puzzles in the world at his actual castle in Italy. And this is one of two known copies. Basically you had these numbered pegs and you need to make a connected root of them through this grid that connected all the vertices together. So of course, we can solve this pretty easily with Mathematica. Here's a possible answer where the path is highlighted in red. And we can do it in 3D2. Here's a path on a snub cube. I believe all of the major polyhedra are solvable, but there are examples of polyhedral graphs where there isn't a Hamiltonian cycle. I believe the smallest one has 30 something vertices. It's in MA. And if we want to make pretty pictures, we've also got plenty of functions for making various forms of, of them. One thing I was planning on doing is redoing a dual polyhedra function by George Beck using ribbon polyhedron as as a function rather than having the full code. But I didn't get around to it. Still, it's, it's, this is a nice function for making pretty polyhedra. So I started planning this talk about a month ago and something kind of dropped into my lap day before yesterday. So one related item to polyhedra is twisty puzzles such as the Rubik's cube and all of the major polyhedra have been turned into twisty puzzles at some point. So one question you might ask is what is the highest order possible for a Rubik's cube? You've probably seen the order 3 cube, order 4, order 5 cube. And you may have seen some world records by Oscar Van Deventer where he extended that up. And since then there's been other improvements with the mechanism. And then just a few days ago, we hit a level 49 cube, which is completely functional. Preston Alden finished it after working on it for three years. So that's, that's a big Rubik's cube there. 30 kilograms, 34 centimeters tall, 13,000 pieces and 14,000 stickers. Oh, four years to construct. And this came in just a few days ago. So one question you might ask is how many positions does this have if you go through all of them? So first, let's talk about a few numbers here. We've got a 24 double factorial where you skip even and odd. We've got 24 factorial which is all the combinations of 1 through 24 and that's this big number here. And we've got 24 factorial over 4 factorial to the 6th. And that's this number here. So these three numbers are used in this formula for the number of positions in a order in Rubik's Cube. And we can look at how these numbers grow. This, these numbers grow very quickly. So here's order five here. It's a huge number, but it's not so bad if you factor everything. So here's down here the number of positions and the order 49, Rubik's cube. And I believe it has 9300 digits. The number of particles in the universe is one to the hundred. So it's 93 orders of magnitude larger than the number of particles in the entire universe. So we'll never see all the positions of the 49 cube. So I mentioned permutations, order 24 permutations. Well, it turns out with signed permutations, you can represent all of the polyhedra, for example, for these three values, 0, 1 and the golden ratio. If you use cyclic permutations and you do all the possible signs of them, you get these 12 points, and those are the 12 vertices of an icosahedron. And it turns out that all of the Platonic and Archimedean solids can be easily represented with sign permutations. I say easily, but the snubdodecahedron, you might argue that that one's not easy. The, the values involved with the snub dodecahedron are somewhat nasty, but it's still, once you have the three values, it's not too difficult to actually make it. So here are pictures of all of them, including the snub Doe decahedron down here. And just in case you wanted these back into polyhedra, I came up with a little program to convert one of these mesh objects, which is a region, into a polyhedron again. So here's an example of the Ikase dodecahedron. Let's see. So I mentioned polyhedra. It's possible to make a polyhedron that doesn't have a Hamiltonian cycle. And you can find that in a study of polyhedral graphs. So here's an example of one here. It's just a graph, but we can convert that graph into an actual polyhedron. And we've got this function polyhedral. Polyhedral graph faces. Basically, you put in any polyhedral graph and it will make the polyhedron. So in order for graph to be polyhedra a polyhedral, it needs to be planar. Although if it's not planar. You won't be able to make a polyhedron out of it. Basically you need to be able to connect up all the edges so that they don't cross each other. And it also has to be three connected. So if you basically remove any three, you have to remove at least three edges in order to disconnect the graph. If it's still connected, no matter how you remove two edges, then it's three connected. So you can kind of see that every single edge here, every single vertex here is connected with at least three points. One thing that isn't three connected is say two graphs which are connected by one edge. Since you could basically cut that one edge and you would break the connection. That's an example of a one connected graph. So here in Poly list I have a collapse form of all the Polypedral graphs with 1 to 10 vertices and what's called the G6 format, which is nice and nice and neat. Let's see here. Here we grab all the vert all the G6 graphs with six vertices and we can we get faces out of them. And here's representations for all the polyhedra with six vertices. But I kind of say all the polyhedra with six vertices. But it's possible to move the vertices around and you get different shapes. So is there sort of a way of constraining the polyhedron so that it's. It's in one fixed form? Well, it turns out the answer is yes. And what's called the canonical form of a polyhedron, and we've got code for that. The canonical form basically puts the polyhedron around a unit sphere so that every edge is tangent to the unit sphere at the midpoint. And this is example. You can see the sphere inside here and inside each, each, each edge you can see where there's a tangency. That's the canonical form of a polyhedron. So since we have code for that, we can go through the seven polyhedra that have six vertices. And here's the canonical forms of each. We've got a prism here, the octahedron, a pyramid, a bipyramid of some sort. No, that's actually something stranger. And here we've got the weirdest of these. And it turns out the volume of that. This one here is curiously close to PI in its canonical form, the four decimal places. We can also look at the duals of all these graphs and we've got collapse code here and there. I've got about 40,000 graphs in here. These are all of the polyhedra that have a given number of faces. So here's all of the hexahedra, all of the polyhedra with six faces. We've got a bipyramid. We've got the anti wedge that has a volume of PI. Again, a pyramid. We've got the cube, which has six faces and a few other shapes. There are seven hexahedra, and these are their canonical forms. There's also 34 heptahedra, which are polyhedra with seven faces. And these are all of them. For example, this is the, the five prism here. And there's a lot of other different shapes. There's a 257 octahedra and 32, 300 decahedra. So these are all different ways of making polyhedral graphs. If, if you want to grab all of these graphs, they're at canonical polyhedra or the demo polyhedral kind of canonical polyhedra. This whole talk will be posted sometime soon at Community and you will find all the links there as well. We can also represent these graphs as Schlegel diagrams. And that's when you take one of the faces, make it regular on the outside, and then you build convex polygons on the inside to represent all of the faces. So, for example, down Here is the 5 prism. You've got a pentagon on the outside, a pentagon on the inside, and the rest of the faces are quadrilaterals. It's always possible to make a slagle diagram, which is also called a tut embedding. One classic formula with polyhedra is if you take the number of faces, add the number of vertices and subtract the number of edges, you always get 2, unless the graph or polyhedron happens to be a torus. If it's, if it's not a torus, then the Euler characteristic is two. And here's examples for all of the Archimedean solids. Faces plus vertices minus edges equals two in all cases. So I mentioned torus. Here's the smallest torus made from equilateral triangles. This is something Conway found. John Conway, he started with three octahedra and then added tetrahedra to make them start fitting together. And. But that's. These are still disconnected, I believe. Yes, that's. That's not a clean connection here. That's just a line. And then he added three more tetrahedra. And now it's a nice solid torus shape. So that's the fewest number of equilateral triangles you can use to Make a torus. You can do it with more. Here's a way of gluing eight octahedra together. And with the program polyhedron face reflect. You just reflect on an edge several times and you get the octahedra to fit nicely together. Oops, I wonder why that's not working. Let me try re evaluating this. It worked the first time. So if you keep gluing tetrahedra together, you get this spiral pattern and it actually is a helix. Here's what the helix looks like. And there's both a right helix for these, for this helix of tetrahedra and a left helix which is more compact. Like so. This is a nice demonstration by Sandor Kabai who has many polyhedra demonstrations in the demonstration site. We can also pack similar cuboids together in strange ways. This is what's known as Hoffman's packing puzzle. If you can get 27Amazon boxes of the same size and shape, you can measure the A plus A, A, B and C edges and construct a cube that has a edge of A plus B plus C. But it's very strange the way the boxes go together. It's. It's not at all intuitive how it works. And this is the classic Hoffman packing puzzle. Now there's a few cases of edges A, B and C where there's a different solution that exists. And you can squeeze in one more box. This was something that was found by Don Knuth. So if you want a hard puzzle, try to figure out the parameters of A, B and C which allows a 28 box solution for the Hoffman packing puzzle. And if you can't figure it out, you can look up Hoffman packing puzzle and newth and see his solution. Here's another torus which is known as the. Oh, I'm not actually sure how to pronounce his name. I've met him at several conferences. I really should have asked him how to pronounce his name. Siz Lossy. I'll just say every face of this has six edges. And all seven of these hexahedrons, which are irregular, all six of them touch each other. So this is a valid seven color Taurus. This is something else Martin Gardner talked about that I found fascinating. The this, this particular toroidal polyhedron. So one thing you can do with polyhedra is, is cut them into nets. I mentioned those that earlier with the dodecahedron where you have two pentagons with the notches, you put them together with rubber bands and it springs into a dodecahedron. Usually it's a bit More involved to, to make these from a net. For example, here's the net for the truncated octahedron. And we've got nets for all of these. But actually folding and gluing them together can be slightly tricky. Usually you, you want to add notches to help them fit together. If you have a polyhedron, you can get a whole bunch of nets just by running this little program. Random polyhedral net by Isidore Heafner, who also has many demonstrations over a thousand. So I've had this, this one shape here that I've shown several times. This is the hexicus icosahedron. And this is the most faces you can have for a fair dye, except for gluing two pyramids together. So this is sort of the largest possible fair die where all the faces are equivalent to each other. A isohedron. And you can actually make this shape. Well, I say rather easily. The code's not too long. It's somewhat involved though how quaternions work for this. But you basically start with, with these three quads, two, two and zeros, all ones and 011 over the golden ratio. And the golden ratio. And then you do the sine permutations, the even permutations of these quads. And then you can you get the vertices of the 600 cell which is a shape in the fourth dimension that has that, that defines 600 tetrahedra and the dual of that has a 120 dodecahedra. So the 600 cell made of tetrahedron and the 120 cell made out of dodecahedra are closely related. So with those 124 dimensional vertices you can make the icosians which are quaternions. And then you can take this one triangle here, I forget which triangle it is, it's in here somewhere. But all these triangles are equivalent. So they all have a form which is somewhat similar to this, but usually not with so many zeros. So you take this, this one and you multiply it by all of the quaternions using quaternion arithmetic and you get this shape and that's an equivalent for the icosahedral group. But it's somewhat faster and somewhat less involved because the quaternions are slightly easier than the 3 by 3 matrices. Instead of using nine items in the 3 by 3 matrix, you're only using four items. If, if you're able to consider that quaternions are easier to work with, and that's up for debate. But I, I find them fascinating. So here is the icosahevo group. And I'll show you what that looks like. It's all of these values. It's basically sign permutations of ABC and 001 where A is half the golden ratio, B is half the golden ratio one over half the golden ratio and C is one half. With all of these matrices you can pick one, one face, for example, one of these jagged faces here and you get the compound of 5 tetrahedra and all of these shapes. You basically give it one polygon multiplied by the 60 elements of the icosahedral group and you get the full shape. Here's the Mathematica spiky. And we can also remove faces. So here's just 15 of the 60 faces here. The accumulated.decahedran so when you're, when you have the icosahedral group, whether you're using the icosian version with quaternions or whether you're using the matrix version or whether you go to. I forget the name of the group, but it's one of our group datas. You can, you can represent a cosmohedral group and then multiply a given polygon by it and get a 3D shape. And here's one example of that. We've got these flat pieces here. When you make them correctly, you can multiply them by the icosahedral group and you get these, this nice star shape which is something that's sold at the Museum of Mathematics. This is also nice to 3D print. However, you can also make shapes from other groups. With the tetrahedral and octahedral groups you can also do some strange things. And here's a little known fact. You can take any triangle and make an, and make a octahedron out of it. And that's something that works for any triangle. Even, even obtuse triangles. With any acute triangle you can make a tetrahedron, but that, you can't make a tetrahedron out of obtuse triangles, but you can, you make an octahedron out of them. Here's another strange shape that uses the tetrahedral group, which is called the tetartoid. Every face is a pentagon and you can change the, you can change the values and you get different pentagons. You get close to a dodecahedron or rhombic. I find it very interesting how all of these, all of these work with that, this particular code. I, I saw the tet or toy DNA in a book. So I, I figured, well, there has to be code for making this somewhere. But I Wasn't able to find it, so I just derived the code on my own. And now it's kind of the standard. It's in Wikipedia and everywhere. This short piece of code for making the tet ortoid. And you can actually buy these at. If you look at dice art. Fun, I think it is. It's a site that sells mathematical dice. They really like this demonstration, so they asked if they could make dice out of it. And so you can get these skewed dice at. At their site. You can also do stellations of the polyhedra. For example, here's one of the stylations of the icosahedron here. There's 59 stallations, if I remember right. And the tricontahedron also has a lot of stellations. Here's the original tricontahedron, a polyhedron with 30 faces. There's 227 stellations. And we can grow these up like so to show, I think, 32 of the 227. Eventually I want to have code that will make stallations for any polyhedron. But that would be an involved project. Still something I want to do, though. And we can also go to distillation diagrams. Here's the faces that are filled in out of the whole plane of possible polygons that can be used. So here's the shape again, that was in the hexacous icosahedron. 120 different faces. And these are the 15 great circles of a sphere. And there's other sets of great circles that you can look at. There's a total of 31 great circles that were of interest to Buckminster fuller based on four different polyhedra. One thing that people@twistypuzzle.com have been wanting for 30 years or so is whether there's a mechanism that allows this shape to be scrambled. And so far, no one has been able to figure out a mechanism. They've been able to make a Mechanism for a order 49 Rubik's Cube. But this is an unsolved problem, and it's the biggest unsolved question in twisty puzzles, is how to make a mechanism that can do this. There's various puzzles you can do with polyhedra. For example, what is the largest cube you can fit through Another cube? And this is known as the Prince Rubert's cube problem. Well, it turns out that the red cube, the blue cube, is a unit cube with a volume of one. The red cube, which is passing through it, has a volume of 9, 8. So it has a volume 18 larger than the smaller cube. And if you shrink it down slightly, you'll be able to slide it through without the smaller cube breaking up. If you go for the full volume, full largest, it breaks the smaller cube up, but it's possible to do slightly smaller so that you have, you can basically slide a piece out of the small cube, slide the larger cube through it, and then put the smaller piece back in. And you're basically shown that you can fit a larger cube through a smaller cube. This concept of self penetration shows up quite frequently in puzzles. This object of golden ratio bricks, this is a common shape for burr puzzles. And if you look up burr puzzle, there's a nice piece of code for making burpuzzles that's available for free. So if you put down an irregular polyhedron, often it'll be a, it'll have a face which is not stable. So one question that Hepps asked was whether there was a tetrahedron that existed that would topple twice. It would fall onto one side and then fall into another side. And he was able to come up with a solution for that. This tetrahedron has two unstable faces and it's possible to have it do a double topple. And then there's a question is what is the smallest number of faces you can have on a polyhedron so that it has just one stable face? This was a unsolved question for a while. Conway and Richard K. Guy found a solution with 19 faces. But then later this guy Rashitov managed to get it down to 14 faces. And we've got a demonstration which shows it. Another thing, other than toppling polyhedra, you can also fit them together to fill space. And here's just a few examples. The truncated octahedron, the rhombic dodecahedron, hexangular prisms, cubes are all the common forms, but there's many others. There's actually thousands of space filling polyhedra. One thing I'd eventually like to do is have a nice program for showing them all. And here's one example. If you take a tetrahedron which is truncated and the small triangle here have edges of one and the longer edges here have lengths of two, you can make this space filling object, which is quite irregular. And I've got a link here. Let me show that site briefly. This site here, Moritz Schmidt has a list of all of the thousands of different space filling polyhedra, but they're not in a convenient form. Basically, I would like to get all these space filling polyhedra into a convenient form So I could do fun things with them. As kind of an aside from space filling polyhedra. This is a question asked by Regiomontanus. Where do you stand so that a statue appears as large as possible? And the answer is you want a circle that goes through the top and bottom of the statue so that that circle goes right through your plane of vision. That is where the statue will appear largest. If you go closer than that, the statue will be foreshortened, so it won't look quite as big. So you basically want to stand back a bit. And that's where the statue will look the largest. So one question is, why am I bringing up regiomontanus? He discovered one of the biggest errors in mathematics. And I'm wondering if anybody can guess who that, what that big error in mathematics was. We were talking about space filling polyhedra. This is the tightest packing for tetrahedra that is known. It was found in 2010, but for almost 2,000 years, people thought that the tetrahedron would fill space because Aristotle said it did. And they just kind of took it on faith. It wasn't until about 1470 that Regiomontanus said, no, this doesn't fill space. If you try to put five tetrahedra together, there's this gap. There's, there's no way you can make this object fill space. So that's one of the reasons he became famous, because he caught this, this long lasting mathematical error. And two different packings were, were found in short order. Let's see, this was. This became the tightest packing in 2006, but then it was beaten a few years later. And that's currently the record. So far, no one has beaten this packing for tetrahedra. Another thing you can do with polyhedra is bounce things around in them. So this is a. Oops. And it's not working. Let's see if I can make it work. So here's a, a loop which will go to each face of a tetrahedron. And all the bounces are legitimate. This is found by John Conway. And one question is what polyhedra have these loops where a bouncing ball will come back to where it was before. Here's a loop for the cube that was found by Steinhaus. But this is a unsolved problem question, especially for irregular polyhedra. Another question. With curves, it's easy to make a curve of constant width. And that basically means that if you put it down on a, on a line, the top point will always be one unit above the line. No matter how you roll the polygon. So these are different curves of constant width. And many of them will also make surfaces of constant width, but they don't quite work usually. And here's an arbitrary curve of constant width. You basically take the triangle and you make these three pairs of arcs and you get a curve of constant width. And there's a link to the demonstration. This will all be posted on Community sometime soon. And this is the. Hopefully I can get this working. Here's the Meister tetrahedra. It's somewhat involved in. In the making. If we add all these pieces and wedges, we get a stall at of constant width. It's more complicated than the reload tetrahedron, which doesn't quite work. And that basically just adds spherical segments to a tetrahedron. But it isn't a solid of constant width. But this Miesner solid is. And this is, this demonstration shows how to make it. And for a while this was the only known solid of constant width that actually worked. There was. There's two of them. But a few years ago. What year is that? Three years ago this paper came out and it gave a. A set of solutions for how to build more of these objects. And since then there's, I think there's, there's now five major papers about solids of constant width or, or surfaces of constant width with, with methods for making them based on this paper. So. But there's not many pictures out for these yet. It's just here's a method for fixing the problem. But there haven't been too many printouts of these new shapes. And one reason for that is it's difficult to find base shapes which will work with the method. They haven't been paired up yet. And so a question is what polyhedra will work with this method? Well, it turns out I made a list of the polyhedra about 10 years ago in this, this thing. Biggest little polyhedra, the one that everyone is using is the classic tetrahedron. But there's other shapes. This particular pyramid works. All of the vertices are within one of each other. And that's. This shape has been known for a long time. I should mention what a big little polyhedron is. And that is a polyhedron where the, where the farthest distance between vertices is one. Basically you want what is the largest polyhedron you can find so that none of the vertices are. Are greater than one apart. And one tricky part of that is you can't have vertices that are in. That are on opposite sides of zero, because that Wouldn't reach a maximum. For example, the icosahedron, all of the vertices have exact opposites. But you could make a larger polyhedron if you made something more like a caltropia. So here's a somewhat odd shape with six vertices, and we have red lines for all the vertices that have length one apart from each other. It's a somewhat strange shape, but it was discovered in 2003, and no one had managed to find any larger solutions until I did a search and started finding them. Here's the solution for seven vertices. For eight. For 9, they get progressively more complicated for 10, 11, 12. The 12 object. If you basically put on any face, there's one point sticking straight up. And that's kind of a feature of all of these shapes. There's always a flat shape on the bottom and a point at the top. So they're all basically caltrops. There's no matter how you throw them down, there's always a point sticking up. And that's a feature of these polyhedra where you maximize volume. But they're basically small. No two vertices are greater than one apart. So, 13, this shape with 16 points here, this is the largest one that I'm sure of the solution. And this one is very spiky. It's. It's not a tetrahedron. It's. It's not an icosahedron. It's stranger than that. So one question is, if you take these shapes and you put the Peabody's algorithm on it, what are the surfaces of constant width you get out of that, And I'm hoping that someone else can solve that problem. But I might tackle that myself because it'll be cool to have a whole bunch of new shapes of constant width that you can print out and play with. One other way of generating polyhedra is to just start with a lattice and start growing the lattice. You basically pick all of the points that are a certain distance away or less and ask what polyhedra grows out of that. And you can do that with different lattices. And you get all these different polyhedra. And these are known as the generalized Waterman polyhedra. One other question I wanted to look at is if you have a particular polygon, what are all the polyhedra you can make out of it if you use just that one shape? Well, it turns out you can make shapes that are not regular. Polyhedra basically have a prism and two pyramids. And this shape devolves into a hexagonal shape. More fun with the icosahedral group. This Is a nice demonstration by Sandor showing how these all fit together in the small rhombi icosahedron. Cayley graphs where you basically work with permutations and you find out how they can connect up. These often make regular polyhedra, but they also make four dimensional shapes as well. Let me see if I can find a polyhedron out of these. Here we go. A cube octahedron with K graphs. You can get these a wide variety of regular structures. I advertised a bit with the Engel 38 space filling polyhedron. So here's an animation of that. So this is a space filling polyhedron. It has 38 faces, it's a convex shape and it will fill space. And here it is surrounded by 38 copies of itself spinning around. This is the worst brick to use for building a house. This is highly complicated and this particular space group that's involved, group 214 of the 230 space groups I 4132 is the most complicated of the known crystal groups. And Steve Dutch has this description of it. This group looks chaotic, but visualizing it is easy. All you do is sit there until little beads of blood form on your forehead. In 1980, Peter Engel did a survey of all the different crystal groups and figure out a way of using what's known as Roynoi cells to predict the largest polyhedra that you could build with them. And more recently Morris Schmitz basically verified most of Engel's result, but he expanded them quite a bit. Engel with his computers from 1980 missed quite a few. But Morris Schmitz made a good survey of everything were possible in that, in that paper I linked to earlier. So it'd be neat to have all of these in convenience in convenient forms. It took, it took a while for well, 15 years for me to decipher the the code and everything to get this particular shape. So now I can actually look at it and play with it. But this was a. Once I got a clue from somebody and he had some Python code, I was able to take that and reverse engineer and. And after a lot of programming I was able to figure out how to make this shape and make it work. And I don't want to redo that for the other 10,000 space filling polyhedra. So it would be interesting to have somebody else do that. But I might get around to that sometime. It would be neat to have all of these space filling polyhedra available. So I usually end these talks with what are some unsolved questions. So one of them is can we take results that are known and apply the new methods to make new solids of constant width based on polyhedra? So I'd really like to see some 3D prints of new shapes space filling polyhedra. It would be nice to have all of them in a convenient form that people could play with rather than just annotations. In Schmidt's paper. What are bounce loops and arbitrary polyhedra? That's something that hasn't been solved. What is the largest small polyhedra? The the demonstration I showed earlier. I'm pretty sure I've solved 16 and everything up to 10 vertices, but I'm not sure of 11 vertices and anything over 16 vertices. So it'd be nice to have some way of nailing down solutions for some of those. However, with what's called the Thompson problem where you put vertices on a sphere as far apart as possible that also hasn't been solved past 12 vertices. It's believed they have the best solutions for 13 to 100, but they haven't managed to prove any of them. Can a mechanism be made for Big Chop that was the shape of the 120 faced object with the 15 great circles? What polyhedra can divide into similar copies of themselves? I did a few talks on reptiles, but this is still an unsolved question for three dimensions. Do all polyhedra have nets that don't self overlap? That's a long dumb solved question. It's possible to find some tetrahedra that have one particular net where you'll get self overlap. But in general there's so many polyhedral nets for any polyhedron that it always seems to be the case. There does exist 1. A net that doesn't have self overlap. I mentioned the Thompson problem. Can any of those be proven? For the hat problem, I did an early talk about the cat tiling. That was a newly discovered shape which fills shape in an irregular way. Are there any 3D equivalents of that there? There probably might be one if you use multiple copies of the 38 sided angle shape. I'm sure that gets really nasty, but there's probably some simpler shapes in three dimensions that will force unusual tilings. And also what shapes can completely surround themselves but not fill space? The that's well studied in two dimensions, but not in three dimensions. And that's what I had. I'd be glad to answer any questions if you post them in YouTube. Let's see, how might new polyhedra be defined in four dimensional space? For that I would start by looking at polychorons or polytopes at Wikipedia and then doing a Google search. There's a lot known about four dimensional Polyhedra, and there's a few nice programs for that somewhere. I saw a site that had all of the dice in four dimensions for 20 faces and there was like a hundred of them, and they're all very weird. It would be nice to have all of that put into nice code somewhere for people to play with. But there's a lot known about four dimensional Polyhedra. Anything else? All right, many thanks for attending my talk, and I'll be back in a month with another discussion on mathematical games. Many thanks.