ZenoRogue | Too many shapes to see all of them @ZenoRogue | Uploaded August 2021 | Updated October 2024, 3 hours ago.
0:00 a confusing maze
It appears that we have a triangle of solid walls in front of us. Let's try to walk around it.
Surprisingly, this "triangle" has seven sides!
What's going on?
0:15 hyperbolic geometry
This maze is actually based on hyperbolic geometry. The hyperbolic plane can be tessellated with triangles, so that 7 triangles meet in every vertex. We place solid walls on some of these triangles, forming heptagonal structures. We try to do it so that each vertex of the tessellation has two triangles filled with walls. Then, we look at the scene obtained from the first person perspective.
0:25 change the curvature
However, rather than using the hyperbolic perspective, we construct this scene out of Euclidean triangular prisms rather than hyperbolic ones. We cannot do this in Euclidean space, but mathematically it makes sense, we just glue fragments of Euclidean space together. Some gamers call such gluing construction "non-Euclidean geometry" -- that's not really correct, such surgery generally changes the topology, not the geometry (except the cone lines). At 0:25-0:30 you can see the curvature changing gradually, from hyperbolic triangular "prisms" where the space itself is curved, to Euclidean triangular prism.
0:30 just one triangle
In the previous scenes, we have filled two triangles to make the scene look normal (until we move around). What if we filled just one triangle with wall at every vertex? In the Euclidean approximation, these triangles look like very narrow walls (but they still have three sides).
0:40 one square out of five
We can actually tile the hyperbolic plane in other ways. Generally, do an Euclidean tessellation, but add more shapes (7 triangles, 5 squares...). Here is what we get if the hyperbolic scene has 5 squares at every vertex.
0:50 one pentagon out of four
We can also use pentagons in hyperbolic geometry. Four pentagons meet in a vertex, but if we create an Euclidean approximation, only 3? of them fit normally. So if one of four has a wall, that wall will appear to be ⅓ wide (have just 36 degrees).
Can you imagine what these maps look like viewed from above in hyperbolic geometry? Or, in the triangular version, what would happen if if no triangles, or three triangles, were filled at every vertex? If there were no walls, would we see all the triangles in the world? See some answers here: twitter.com/ZenoRogue/status/1423663341944377345
Play HyperRogue and join the HyperRogue Discord for more non-Euclidean fun! Thanks to jpburelle and Rocco for some ideas.
0:00 a confusing maze
It appears that we have a triangle of solid walls in front of us. Let's try to walk around it.
Surprisingly, this "triangle" has seven sides!
What's going on?
0:15 hyperbolic geometry
This maze is actually based on hyperbolic geometry. The hyperbolic plane can be tessellated with triangles, so that 7 triangles meet in every vertex. We place solid walls on some of these triangles, forming heptagonal structures. We try to do it so that each vertex of the tessellation has two triangles filled with walls. Then, we look at the scene obtained from the first person perspective.
0:25 change the curvature
However, rather than using the hyperbolic perspective, we construct this scene out of Euclidean triangular prisms rather than hyperbolic ones. We cannot do this in Euclidean space, but mathematically it makes sense, we just glue fragments of Euclidean space together. Some gamers call such gluing construction "non-Euclidean geometry" -- that's not really correct, such surgery generally changes the topology, not the geometry (except the cone lines). At 0:25-0:30 you can see the curvature changing gradually, from hyperbolic triangular "prisms" where the space itself is curved, to Euclidean triangular prism.
0:30 just one triangle
In the previous scenes, we have filled two triangles to make the scene look normal (until we move around). What if we filled just one triangle with wall at every vertex? In the Euclidean approximation, these triangles look like very narrow walls (but they still have three sides).
0:40 one square out of five
We can actually tile the hyperbolic plane in other ways. Generally, do an Euclidean tessellation, but add more shapes (7 triangles, 5 squares...). Here is what we get if the hyperbolic scene has 5 squares at every vertex.
0:50 one pentagon out of four
We can also use pentagons in hyperbolic geometry. Four pentagons meet in a vertex, but if we create an Euclidean approximation, only 3? of them fit normally. So if one of four has a wall, that wall will appear to be ⅓ wide (have just 36 degrees).
Can you imagine what these maps look like viewed from above in hyperbolic geometry? Or, in the triangular version, what would happen if if no triangles, or three triangles, were filled at every vertex? If there were no walls, would we see all the triangles in the world? See some answers here: twitter.com/ZenoRogue/status/1423663341944377345
Play HyperRogue and join the HyperRogue Discord for more non-Euclidean fun! Thanks to jpburelle and Rocco for some ideas.
