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.

It took me a while to really appreciate aperiodicity. But it's a rather deep and important issue, definitely worth persuing a bit further.

Thank you for sharing. The blender file video is in preparation.

Thank you for the reference.

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 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.

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.

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.

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.