Uploaded August 2020 | Updated September 2026, 1 week ago
The three-dimensional sphere 𝕊³ is a Lie group. After defining the metric tensor g at one point, we can use the Lie group to define a homogeneous geometry in the whole sphere. Of course we could obtain the usual metric in 𝕊³ that way, but we could also obtain e.g. the Berger sphere by stretching one coordinate (this was done in youtu.be/KBYPQaoBgz0 ), or something more weird.
Here, we are using g(v,w) = ⟨Av,Aw⟩ where A(x,y,z) = (x,y+x/2,z), and ⟨⟩ is the usual inner product. The dodecahedra are stacked in the 'z' coordinate (they are regular in the original 𝕊³, but of course no longer regular in this weird geometry; they are a line of 10 of 120 dodecahedra in the 120-cell). The first scene shows a rather nice view, the second and third scene are more random directions. The last scene is using a different, much more random matrix for A.
Every dodecahedron in the loop has its antipodal dodecahedron also in the loop. In the usual spherical geometry, a light ray hitting an object X then hits it antipodal object X', then X again, and so on. However, these images will not be visible, as they would be visible in exactly the same spot. Since the geometry is slightly changed here, the antipodal object and afterimages are visible in a slightly different location. We also get weird geometric lensing effects, where objects seem to appear out of nowhere, kind of like fata morganas.
Mostly to see what it would look like -- it seems difficult to understand what is going on here (other than what was said in the last paragraph), or to find any practical uses.
The three-dimensional sphere 𝕊³ is a Lie group. After defining the metric tensor g at one point, we can use the Lie group to define a homogeneous geometry in the whole sphere. Of course we could obtain the usual metric in 𝕊³ that way, but we could also obtain e.g. the Berger sphere by stretching one coordinate (this was done in youtu.be/KBYPQaoBgz0 ), or something more weird.
Here, we are using g(v,w) = ⟨Av,Aw⟩ where A(x,y,z) = (x,y+x/2,z), and ⟨⟩ is the usual inner product. The dodecahedra are stacked in the 'z' coordinate (they are regular in the original 𝕊³, but of course no longer regular in this weird geometry; they are a line of 10 of 120 dodecahedra in the 120-cell). The first scene shows a rather nice view, the second and third scene are more random directions. The last scene is using a different, much more random matrix for A.
Every dodecahedron in the loop has its antipodal dodecahedron also in the loop. In the usual spherical geometry, a light ray hitting an object X then hits it antipodal object X', then X again, and so on. However, these images will not be visible, as they would be visible in exactly the same spot. Since the geometry is slightly changed here, the antipodal object and afterimages are visible in a slightly different location. We also get weird geometric lensing effects, where objects seem to appear out of nowhere, kind of like fata morganas.
Mostly to see what it would look like -- it seems difficult to understand what is going on here (other than what was said in the last paragraph), or to find any practical uses.
