I think I wrote you more details in a private email but it might be an interesting question for others: I have made several versions over the years and don't recall the exact paramers. But the rotation matrix is a 5x5 Block diagonal matrix which rotates in the first two coordinates and leaves the others. The 2x2 matrix is of the form [[ cos(f(t)) , -sin(f(t))]],[sin(f(t),cos(f(t))]] where f(t) is a periodic function in one case something like f(t) = cos(3t) + sin(2t + sin(2t)) which has the effect also that the speed changes. The actual rotation is then of the form S^T R(t) S where S is a fixed rotation in R^5. The scene in R^5 is the projected with a stereographic projection to space.
it depends on what one wants to do. In modern computer game engines like unreal engine 5, they say to be able to handle a billion triangles per frame (pretty impressive).When working in computational topology ,where one wants to do computations like computing curvatures or Euler characteristic (that is what I do), one gets to limits much faster. One of the difficulties in higher dimensions is to find cells. This is the clique problem and frustratingly hard (an NP complete problem). Computer graphics like in computer games deals with two dimensional cells (triangles) and can handle much larger networks. Google, which analyzes the connections between billions of nodes in the internet also has to be able to handle large networks.
yes the cross polytopes exist in all dimensions. In dimension n there are 2^n simplices. In dimension 2, we have the square with 4 edges in dimension 3 we have an octahedron with 8 triangles, in dimension 4, we have 16 tetrahedra and in dimension 5, we have 32 hypertetrahedra. There is always a bit of a confusion with respect to dimension as for example, the Platonic solids like the octahedron are objects in three space, but topologists see them two dimensional discrete spheres of Euler characteristic 2. In dimension d, the Euler characteristic is (1+(-1)^d). The object you ask about has again Euler characteristic 2. It has 10 vertices, 40 edges, 80 triangles, 80 tetrahedra and 32 hypertetrahedra so that 10-40+80-80+32=2. This object is the smallest 4 dimensional discrete sphere there is.
Love idea of the dancing perfects with the classical score ! GENIUS !
wow sweet! super cool work. highly relevent to everything! thanks you so much!
forbidden shapes
Beautiful!! And the music was perfect. :)))
pierdolisz
@@dzmg 😂
Exactly what I was thinking.
That beauty is slicing and dicing like a gear driven engine
That 120 cell is kind of a badass.
Almost as much as the 600-cell!
By sea and land and air: the unsinkable
Does anyone know how to notate Schlafli symbols for antiprisms and crossed antiprisms?
as far as I know, Schlaefli symbols are only used for regular polytopes.
@@OliverKnill There are regular antiprisms and crossed antiprisms.
On what axes are the 5d shapes rotating on?
I think I wrote you more details in a private email but it might be an interesting question for others: I have made several versions over the years and don't recall the exact paramers. But the rotation matrix is a 5x5 Block diagonal matrix which rotates in the first two coordinates and leaves the others. The 2x2 matrix is of the form [[ cos(f(t)) , -sin(f(t))]],[sin(f(t),cos(f(t))]] where f(t) is a periodic function in one case something like f(t) = cos(3t) + sin(2t + sin(2t)) which has the effect also that the speed changes. The actual rotation is then of the form S^T R(t) S where S is a fixed rotation in R^5. The scene in R^5 is the projected with a stereographic projection to space.
One wonders if there are models for cells with 8 billions cells? Is there computational power for those magnitude of forms?
it depends on what one wants to do. In modern computer game engines like unreal engine 5, they say to be able to handle a billion triangles per frame (pretty impressive).When working in computational topology ,where one wants to do computations like computing curvatures or Euler characteristic (that is what I do), one gets to limits much faster. One of the difficulties in higher dimensions is to find cells. This is the clique problem and frustratingly hard (an NP complete problem). Computer graphics like in computer games deals with two dimensional cells (triangles) and can handle much larger networks. Google, which analyzes the connections between billions of nodes in the internet also has to be able to handle large networks.
@@OliverKnill thank you!
1 or 2 or 3 for all but a grid, right? You can trace lively trees.
What is that music starting at 4:05? I think it is REALLY good...
Siegmund Mischinger It's "War Dogs" by juno reactor :)
What software did you use to generate this?
I have a list of tools mentioned at the end of the video. The graphics of the polytopes has been done with Mathematica.
This is beautiful
A kaleidoscope. Unparalleled!
I love using a 2d medium to convey a 4d concept.
Is the cross polytope polyteron made of 32 4-simplexes?
yes the cross polytopes exist in all dimensions. In dimension n there are 2^n simplices. In dimension 2, we have the square with 4 edges in dimension 3 we have an octahedron with 8 triangles, in dimension 4, we have 16 tetrahedra and in dimension 5, we have 32 hypertetrahedra. There is always a bit of a confusion with respect to dimension as for example, the Platonic solids like the octahedron are objects in three space, but topologists see them two dimensional discrete spheres of Euler characteristic 2. In dimension d, the Euler characteristic is (1+(-1)^d). The object you ask about has again Euler characteristic 2.
It has 10 vertices, 40 edges, 80 triangles, 80 tetrahedra and 32 hypertetrahedra so that 10-40+80-80+32=2. This object is the smallest 4 dimensional discrete sphere there is.
Music goes unnecessarily hard
A good badass ;)
ludwig is my great- grandpa lmao