ZenoRogue | Negatively curved cubes @ZenoRogue | Uploaded May 2020 | Updated October 2024, 28 minutes ago.
In youtu.be/XUIYga-AfLI we simulated spherical geometry using an Euclidean engine. Here we simulate Euclidean space using hyperbolic geometry!
We start with the cubic tiling of Euclidean space, and proceed by replacing them by more and more curved hyperbolic cubes.
As the hyperbolic cubes get larger and larger, more of them (k) fit around the edge.
At k=6 (0:10) the vertices are infinitely far away (ideal vertices). At k over 6 the vertices are even further (ultra-ideal) . The largest k equals 8. For k=8, the four original cubes around every edge are repeated twice, so everything agrees.
The second half of the video shows a more regular construction in Euclidean space.
In youtu.be/XUIYga-AfLI we simulated spherical geometry using an Euclidean engine. Here we simulate Euclidean space using hyperbolic geometry!
We start with the cubic tiling of Euclidean space, and proceed by replacing them by more and more curved hyperbolic cubes.
As the hyperbolic cubes get larger and larger, more of them (k) fit around the edge.
At k=6 (0:10) the vertices are infinitely far away (ideal vertices). At k over 6 the vertices are even further (ultra-ideal) . The largest k equals 8. For k=8, the four original cubes around every edge are repeated twice, so everything agrees.
The second half of the video shows a more regular construction in Euclidean space.
