Amazing video! Makes me wonder what patterns would be produced by projecting 6, 7, 8-dimensional cube lattices on a plane, etc. And what 3d tilings whould we get from projecting higher dimention lattices on 3d space
Yeah, these are great ideas. I can at least partially comment on them: Actually, this projection from five dimensions was discovered, while constructing the Penrose tiling with pentagrids. The details for this construction can be found in the paper by deBruijn (see video description for the references). The construction method can be easily generalized to hepta-grids and so forth. It should correspond to projections from 7-dimensional cubic lattices, but I haven't checked this. One such construction can be found here: community.wolfram.com/groups/-/m/t/2992328 Three dimensional tilings are tricky to visualize since they fill the entire space obviously. One would not project the faces but rather the three volumes. And they would turn into two kinds of parallelepipeds. But there is another neat thing that can be done. If the diagonal normal vector n = (1,1,1,1,1) is added to the projection plane, this plane becomes a three-dimensional space. When the faces of the hyper-cubes are projected into this space, all tiles do have the same size but they are tilted in two different ways such that a projection into two dimensions results in fat and thin rhombs. They generate a folded tiling, whose projection yields the Penrose tiling. This is referred to as Wieringa roof and a lot of resources are available.
This is really a superlative video! You have laid out each step in the projection of a five dimensional cube onto a plane with beautiful clarity. The animations are perfect and beautiful as well. The exposition is relatively easy to follow for such a complicated concept. I look forward to the next video on this fascinating topic.
Excellent video and explanation of the how the Penrose tiling is constructed as a projection from 5-dimensions! I especially enjoy seeing the variations that occur when the projection plane is rotated in 5D. Definitely looking forward to the follow-up video detailing the Blender files!
Hell ya bruva, that was awesome. Imma share this with all of the mathematical discord servers I hang out in, this video is criminally under-viewed. Hook it up with that blender file, or a video about that blender file. Looking forward to it.
I am actually fascinated by this idea of projection, I am curious about generalization of this procedure to Bravais Lattices or the crystallographic lattices of higher dimensions. I am really excited for the next video on this!
This is most remarkable and will take a bit of studying. The Penrose tiling is aperiodic, meaning the tiling is non-periodic, and this is a most remarkable property. Thank you so much for this video.
Marvellous! I was disappointed you skipped over how a projection from the 4d lattice looks, but I think you get a glimpse at 21:30 when you're in 5d, but adding the rotations one at a time?
I quickly looked into this as well. But I didn't find anything exciting. It's not completely obvious what plane to choose in four dimensions. If you follow exactly the same lines, you'll find a boring pattern of squares from different orientations, not even rhombs. I hope that I can cover a few more things in follow up videos. Thank you for sharing your thoughts.
Great video! If I understand correctly, this cut-and-project technique could also be used to create random triangle square tilings (see e.g. the article "Square-triangle tilings: an infinite playground for soft matter")...
Cool stuff! My humble self only generated penrose tilings by subdivision. Is there also a higher dimensional structure behind other penrose tilings (such as the kites and darts), or the newly found einstein hat aperiodic tiling.
There is a claim, that the hat tile also can be obtained from projections of higher dimensional lattices. "Direct Construction of Aperiodic Tilings with the Hat Monotile", Ulrich Reitebuch, 2023. But I haven't had time to study it yet. It's on my agenda. arxiv.org/abs/2306.06512
I think it would be super cool to see that tiling but instead of coloring by face, you give each 5D cube a different color, so we can see how the tiling is divided up
What about the common faces? I could try a color gradient through the dimensions. For three dimensions there would obviously be the RGB space. Any idea for the remaining two dimensions? Or, I just distribute the hue values across all the cubes of the sample lattice. I think about it.
@@Number_Cruncher I actually hadn’t considered that these are lower-dimensional faces being projected rather than 5D volumes. So maybe this isn’t such a good idea, I don’t know
@@jakobr_ It probably wouldn't look exciting. If the color was somehow "smoothly" distributed across the lattice, the coloring of the tiling would be a color gradient depending on the orientation of the plane.
The rendering is done in blender. The blender files are generated with python scripts that are in spirit similar to manim. You create objects, text, etc. You can make them appear, transform and move. Recently, I included the creation of complex geometry node graphs with scripts. I guess it would not be feasible to create them manually. For this animation I needed ten 5x5 matrices that rotate five basis vector. You can see the node groups at the end of the video. And some of the groups contain fifty or more nodes. I'm preparing a follow up with more details.
It looks to me that in the animation that generates the tiling, some of the rotations “flip over”, causing the projection plane to cross over an axis (or a plane spanned by axes, hard to tell). I find these flips the hardest parts to follow. The rest of it is easier, just another side getting exposed to our new point of view. But that flipping over is very disorientating. Maybe a different initial rotation state would remove these?
It's nice to see that you really try to imagine these rotations. Unfortunately, I'm not really sure what you mean by "flipping over". Let me add instead some of my observations hopefully addressing your concern. The rotations are smooth and represented by the squares on top. Nevertheless, the change in the tiling is not smooth. As long as the tiles are just changing size, everything is fine. But at some point, tiles get replaced by others from different dimensions. There is no way that this can be done smoothly. Maybe this is what you mean by flipping. This can also be seen in the three dimensional analogy. It's also nice to shift the lattice. Then you "flip" from one Penrose tiling to another. When you shift the unit cube you have to make sure that all five components of the shift vector add to zero to flip from one Penrose rolling to another one. This all can be explored in the blender files. Let me know when we talk about different things.
@@Number_Cruncher The thing I’m talking about is a little hard to describe. Imagine you’ve got a 3D cube, colored so that opposite faces match, and you’re looking at one edge, so you see two faces. Rotate the cube until one of those faces disappears, and keep going until the same color reappears on the other side.
This is very cool. I didn't realize these flips before. But now I can see them. And I agree, it's most likely that tiles get replaced by tiles with the same color from the opposite side of the cube. And the rotations 2, 3 and 7 go beyond 90 degrees, which seems necessary for the flips to occur. I'm not sure, whether it is possible to go from the parallely aligned plane to the slope for the Penrose tiling without flips. It was not completely trivial to find these primitive rotations in the first place. I wonder whether some people have special mental skills which allows them to understand these rotations and the related 5d geometry better than others.
@@Number_Cruncher If those people do exist, I’m definitely not one of them, haha. I guess I’m just more used to recognizing that sort of pattern, having dipped my toes into some 4D games. I don’t have the full picture of the geometry, but I’m occasionally able to see how some weird visual movements come about by extrapolating from my 3D experience.
@@diproton No, I don't think this is possible in blender. However, it should not be too difficult to implement the algorithm in Julia or other web based languages.
This year's summer of math exposition is a community competition. It is not conducted by 3b1b. There is a discord server with more information. discord.com/invite/FF3QuhM6
Indeed this is always an issue. However, I hope that the animation and continuous deformation from the tiling to the grid and vice versa can partially compensate this problem.
@@Number_Cruncher Well, describing planes using complex eigenvectors seems very hacky and unnatural. Instead, eigenbivectors provide the natural notion of an invariant plane of a rotation without the need to pull complex numbers out of nowhere (and in general, everything to do with planes and rotations benefits greatly from the use of bivectors, especially when it comes to higher dimensions). There's also something to be said about how GA makes projections more natural, but I wouldn't count that against this video since it only applies to a formula that is on screen for a few seconds. I understand that it might be a bit much to ask you to use a mathematical framework that most people will probaly not even have heard of, when that isn't even the topic of this video, but this is one of those "once you see it, you can't unsee it" type situations where I'm cursed with the knowledge of how it could be better.
Now I understand. I'm not familiar with geometric algebra. At first I didn't understand your comment. I thought that there was something wrong. So you would have preferred, seeing wedge products appear, right? Yes, I also was sloppy when I wrote down the 5D real and imaginary parts as eigenvectors, which they are only in their complex combination. How would one compute the eigenbivector from the rotation matrix?
@@Number_Cruncher As Chris Doran points out in a blogpost from 2020 titled "Complex Eigenvalues in Geometric Algebra", complex eigenvectors can be seen as a representation of eigenbivectors. A nicer, in my opinion, way would be to compute the second compound power of the matrix (see blargoner's video on the box product), which is the matrix representation of the extension of the linear map represented by the original matrix to the space of bivectors. With that, you can just do regular linear algebra to diagonalize it and thus find out its eigen"vectors" and corresponding eigenvalues, which are the eigenbivectors and corresponding eigenvalues of the original matrix/linear map. This method might be more expensive than "just" using complex numbers, but it's more natural because it stays in the context of real numbers.
Thank you for this amazing reference. I've never heard about the box product before. It looks like really cool stuff. I'll play with it a bit and maybe "improve" the algebra, when I explain the geometry nodes.
@@ongrys2000 this is the most remarkable result. I tried to highlight it in the one-dimensional setup. And by the way, the projection operation is non-linear and non-invertible.
Amazing video! Makes me wonder what patterns would be produced by projecting 6, 7, 8-dimensional cube lattices on a plane, etc. And what 3d tilings whould we get from projecting higher dimention lattices on 3d space
Yeah, these are great ideas.
I can at least partially comment on them:
Actually, this projection from five dimensions was discovered, while constructing the Penrose tiling with pentagrids. The details for this construction can be found in the paper by deBruijn (see video description for the references). The construction method can be easily generalized to hepta-grids and so forth. It should correspond to projections from 7-dimensional cubic lattices, but I haven't checked this. One such construction can be found here:
community.wolfram.com/groups/-/m/t/2992328
Three dimensional tilings are tricky to visualize since they fill the entire space obviously. One would not project the faces but rather the three volumes. And they would turn into two kinds of parallelepipeds.
But there is another neat thing that can be done. If the diagonal normal vector n = (1,1,1,1,1) is added to the projection plane, this plane becomes a three-dimensional space. When the faces of the hyper-cubes are projected into this space, all tiles do have the same size but they are tilted in two different ways such that a projection into two dimensions results in fat and thin rhombs. They generate a folded tiling, whose projection yields the Penrose tiling. This is referred to as Wieringa roof and a lot of resources are available.
I had no idea the Penrose tiling was just part of a 5D cubic lattice projected into 2D space. More people should see this video, this is very cool!
This is really a superlative video! You have laid out each step in the projection of a five dimensional cube onto a plane with beautiful clarity. The animations are perfect and beautiful as well. The exposition is relatively easy to follow for such a complicated concept. I look forward to the next video on this fascinating topic.
Excellent video and explanation of the how the Penrose tiling is constructed as a projection from 5-dimensions!
I especially enjoy seeing the variations that occur when the projection plane is rotated in 5D.
Definitely looking forward to the follow-up video detailing the Blender files!
I am...beyond excited that you did these in blender and geometry nodes.
Hell ya bruva, that was awesome. Imma share this with all of the mathematical discord servers I hang out in, this video is criminally under-viewed.
Hook it up with that blender file, or a video about that blender file. Looking forward to it.
Thank you for sharing. The blender file video is in preparation.
I am actually fascinated by this idea of projection, I am curious about generalization of this procedure to Bravais Lattices or the crystallographic lattices of higher dimensions. I am really excited for the next video on this!
This is most remarkable and will take a bit of studying. The Penrose tiling is aperiodic, meaning the tiling is non-periodic, and this is a most remarkable property. Thank you so much for this video.
It took me a while to really appreciate aperiodicity. But it's a rather deep and important issue, definitely worth persuing a bit further.
I support these efforts 🙏
thanks, that is awesome!
Marvellous! I was disappointed you skipped over how a projection from the 4d lattice looks, but I think you get a glimpse at 21:30 when you're in 5d, but adding the rotations one at a time?
I quickly looked into this as well. But I didn't find anything exciting. It's not completely obvious what plane to choose in four dimensions. If you follow exactly the same lines, you'll find a boring pattern of squares from different orientations, not even rhombs. I hope that I can cover a few more things in follow up videos. Thank you for sharing your thoughts.
Wow
Great video! If I understand correctly, this cut-and-project technique could also be used to create random triangle square tilings (see e.g. the article "Square-triangle tilings: an infinite playground for soft matter")...
Thank you for the reference.
Cool stuff! My humble self only generated penrose tilings by subdivision. Is there also a higher dimensional structure behind other penrose tilings (such as the kites and darts), or the newly found einstein hat aperiodic tiling.
There is a claim, that the hat tile also can be obtained from projections of higher dimensional lattices.
"Direct Construction of Aperiodic Tilings with the Hat Monotile", Ulrich Reitebuch, 2023. But I haven't had time to study it yet. It's on my agenda.
arxiv.org/abs/2306.06512
I think it would be super cool to see that tiling but instead of coloring by face, you give each 5D cube a different color, so we can see how the tiling is divided up
What about the common faces? I could try a color gradient through the dimensions. For three dimensions there would obviously be the RGB space. Any idea for the remaining two dimensions?
Or, I just distribute the hue values across all the cubes of the sample lattice. I think about it.
@@Number_Cruncher I actually hadn’t considered that these are lower-dimensional faces being projected rather than 5D volumes. So maybe this isn’t such a good idea, I don’t know
@@jakobr_ It probably wouldn't look exciting. If the color was somehow "smoothly" distributed across the lattice, the coloring of the tiling would be a color gradient depending on the orientation of the plane.
This is beautiful. I guess I have to learn Blender now.. 😐
What software did you use for the animations?
The rendering is done in blender. The blender files are generated with python scripts that are in spirit similar to manim. You create objects, text, etc. You can make them appear, transform and move. Recently, I included the creation of complex geometry node graphs with scripts. I guess it would not be feasible to create them manually. For this animation I needed ten 5x5 matrices that rotate five basis vector. You can see the node groups at the end of the video. And some of the groups contain fifty or more nodes. I'm preparing a follow up with more details.
It looks to me that in the animation that generates the tiling, some of the rotations “flip over”, causing the projection plane to cross over an axis (or a plane spanned by axes, hard to tell). I find these flips the hardest parts to follow. The rest of it is easier, just another side getting exposed to our new point of view. But that flipping over is very disorientating. Maybe a different initial rotation state would remove these?
It's nice to see that you really try to imagine these rotations. Unfortunately, I'm not really sure what you mean by "flipping over". Let me add instead some of my observations hopefully addressing your concern.
The rotations are smooth and represented by the squares on top. Nevertheless, the change in the tiling is not smooth. As long as the tiles are just changing size, everything is fine. But at some point, tiles get replaced by others from different dimensions. There is no way that this can be done smoothly. Maybe this is what you mean by flipping. This can also be seen in the three dimensional analogy.
It's also nice to shift the lattice. Then you "flip" from one Penrose tiling to another.
When you shift the unit cube you have to make sure that all five components of the shift vector add to zero to flip from one Penrose rolling to another one.
This all can be explored in the blender files.
Let me know when we talk about different things.
@@Number_Cruncher The thing I’m talking about is a little hard to describe. Imagine you’ve got a 3D cube, colored so that opposite faces match, and you’re looking at one edge, so you see two faces. Rotate the cube until one of those faces disappears, and keep going until the same color reappears on the other side.
It happens on rotations 2, 3, and 7
This is very cool. I didn't realize these flips before. But now I can see them. And I agree, it's most likely that tiles get replaced by tiles with the same color from the opposite side of the cube. And the rotations 2, 3 and 7 go beyond 90 degrees, which seems necessary for the flips to occur.
I'm not sure, whether it is possible to go from the parallely aligned plane to the slope for the Penrose tiling without flips. It was not completely trivial to find these primitive rotations in the first place.
I wonder whether some people have special mental skills which allows them to understand these rotations and the related 5d geometry better than others.
@@Number_Cruncher If those people do exist, I’m definitely not one of them, haha.
I guess I’m just more used to recognizing that sort of pattern, having dipped my toes into some 4D games. I don’t have the full picture of the geometry, but I’m occasionally able to see how some weird visual movements come about by extrapolating from my 3D experience.
❤
Is there a webgl interface for the section /Towards Penrose Tilings/?
@@diproton No, I don't think this is possible in blender. However, it should not be too difficult to implement the algorithm in Julia or other web based languages.
What is SoMEpi? I can't find information on it.
This year's summer of math exposition is a community competition. It is not conducted by 3b1b. There is a discord server with more information.
discord.com/invite/FF3QuhM6
"We color code this difference in green and orange" so the red-green color blind will have a hard time spotting it. :/
Indeed this is always an issue. However, I hope that the animation and continuous deformation from the tiling to the grid and vice versa can partially compensate this problem.
This video suffers from a lack of Geometric Algebra. :(
What exactly do you miss, or what is done incorrectly in your opinion?
@@Number_Cruncher Well, describing planes using complex eigenvectors seems very hacky and unnatural. Instead, eigenbivectors provide the natural notion of an invariant plane of a rotation without the need to pull complex numbers out of nowhere (and in general, everything to do with planes and rotations benefits greatly from the use of bivectors, especially when it comes to higher dimensions). There's also something to be said about how GA makes projections more natural, but I wouldn't count that against this video since it only applies to a formula that is on screen for a few seconds.
I understand that it might be a bit much to ask you to use a mathematical framework that most people will probaly not even have heard of, when that isn't even the topic of this video, but this is one of those "once you see it, you can't unsee it" type situations where I'm cursed with the knowledge of how it could be better.
Now I understand. I'm not familiar with geometric algebra. At first I didn't understand your comment. I thought that there was something wrong. So you would have preferred, seeing wedge products appear, right?
Yes, I also was sloppy when I wrote down the 5D real and imaginary parts as eigenvectors, which they are only in their complex combination.
How would one compute the eigenbivector from the rotation matrix?
@@Number_Cruncher As Chris Doran points out in a blogpost from 2020 titled "Complex Eigenvalues in Geometric Algebra", complex eigenvectors can be seen as a representation of eigenbivectors. A nicer, in my opinion, way would be to compute the second compound power of the matrix (see blargoner's video on the box product), which is the matrix representation of the extension of the linear map represented by the original matrix to the space of bivectors. With that, you can just do regular linear algebra to diagonalize it and thus find out its eigen"vectors" and corresponding eigenvalues, which are the eigenbivectors and corresponding eigenvalues of the original matrix/linear map. This method might be more expensive than "just" using complex numbers, but it's more natural because it stays in the context of real numbers.
Thank you for this amazing reference. I've never heard about the box product before. It looks like really cool stuff. I'll play with it a bit and maybe "improve" the algebra, when I explain the geometry nodes.
The Penrose tilings are nonperiodic. How can a nonperiodic structure emerge from the perfectly periodic grid under a linear transformation?
@@ongrys2000 this is the most remarkable result. I tried to highlight it in the one-dimensional setup. And by the way, the projection operation is non-linear and non-invertible.