Too many shapes to see all of them
Ғылым және технология
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: / 1423663341944377345
Play HyperRogue and join the HyperRogue Discord for more non-Euclidean fun! Thanks to jpburelle and Rocco for some ideas.
Пікірлер: 18
This is messing with my head and I love it
For the "stitched" non-Euclidean geometry, on a large scale the space is non-Euclidean, though mostly not on the small scale as all the curvature is concentrated into small volumes, right?
the nerds: its a hyperbolic surface with perspective tricks me (dumb): antichamber effects go brrrr
> 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). so would "non-euclidean topology" be a more accurate descriptor of such constructions?
@ZenoRogue
2 жыл бұрын
It does make sense, especially that there is a math term "Euclidean topology", so "non-Euclidean topology" would just be different topology. The problem is that it feels quite confusing: it appears that lots of people did have heard about "non-Euclidean geometry" (as a celebrated discovery in mathematics, something strange, or whatever), so when they heard "non-Euclidean" they hear "non-Euclidean geometry". So I would rather keep non-Euclidean for geometry. Also note that the topology changes if we do surgery in some weird way (i.e. what so-called non-Euclidean games are doing) -- but in this particular video, we actually obtain Euclidean topology.
0:35 everyone gangsta until notice that the pink wall at the backround transforms into a hole
Could you make one of your cool videos about the action of the modular group on the hyperbolic plane in its different projections please? I would really love to see it on the Beltrami-Klein disk in particular.
Im the person who gave the pentagon idea! And i feel like the pentagon one could strangly be good for mechanical stuff but im not so sure why
@columbus8myhw
2 жыл бұрын
That's a trippy one. It feels like the P1 Penrose tiling, which is made out of pentagons with the gaps shaped like thin rhombuses ("diamonds"), stars, and partial stars ("boats"). But these gaps all have wide angles as well as small ones, and in the video we only ever find small ones. A shape with only 36 degree angles works out to a self-intersecting pentagram, I guess. So it feels like the gaps are being filled with pentagrams somehow, but we can only see part of the pentagram at a time. Trippy.
horo-line-segment
Hey, watch that transparency. Remember the forward rendering pipeline does not treat transparency well with unsorted geometry. Needs sorted geometry or order-independent transparency hack, otherwise blending will be off. This will make it harder to grasp non-euclidean geometry.
@ZenoRogue
2 жыл бұрын
At what time does the problem happen? Except the "round" scene, this video is rendered using a ray-based algorithm (portals are challenging to render otherwise) and there is no transparency (again might be a bit difficult to define with portals). The walls are slightly reflective, I think you are confusing that for transparency (and take the difference between the video and your expectation to be transparency bugs).
@jviper2004
2 жыл бұрын
@@ZenoRogue Oh yeah, I was referring to the quasi-triangular section in the middle, which your rotating around, first 14 seconds of the video. I didn't know the scene was not quite standard forward pipeline rendered. Yeah, those reflections did look like translucency to me. The scene seems to resemble glass panes. It's hard for me to un-see that now. Plus with symmetry in non-euclidean geometry, it could get tricky to pin down.
this is what dmt is like
Nice!
what happens of there wasnt a triangle? would the floor bug out?
@ZenoRogue
2 жыл бұрын
See the Twitter thread link in the description, I think it answers your question (in 8/10).
@peculiarjack617
2 жыл бұрын
@@ZenoRogue i thought it would morph the floor in a weirder way, still good insight though