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Looks like an infinite line of doors, but the things behind the doors look like strange, non-Euclidean shapes... because they are non-Euclidean. After the recent videos ("Non-Euclidean Geometry AND Portals") some of you have been asking about portals between different geometries... so there they are!
It is not clear how to make a portal between the Euclidean space 𝔼³ and the hyperbolic space ℍ³, because we would like the portal surface to be the same intrinsic shape on both sides, and also of same extrinsic curvature.
Thus, we could have a square portal in 𝔼³ and a square portal in ℍ³, but that does not work -- an Euclidean square has four angles 90° each, and a hyperbolic square has smaller angles, so they are not actually the same shape!
We could also try connecting this square portal with a "square" cut out of a horosphere (as in the video "Temple of Cthulhu in 3D") but then the horosphere is curved in ℍ³, so the effects would be more like a curved mirror in Euclidean space (i.e., caused by the portal itself, rather than by the space).
But this problem does not appear in 2D (between 𝔼² and ℍ²) since lines have no intrinsic curvature. We can move this solution to 3D by adding the third dimension in the Euclidean way -- thus, we get a portal between 𝔼²×ℝ=𝔼³ and ℍ²×ℝ. We can take any product tessellation in ℍ²×ℝ, and choose the tile height so that the portal will have a square shape on both sides.
We can also create a portal between ℍ²×ℝ and ℍ³. To do this, we will use the right-angled dodecahedral honeycomb (aka {5,3,4}). It is a tessellation of the hyperbolic space constructed out of dodecahedra, where all the faces are pentagons and all the dihedral angles are right. (See "Right-angled pentagon" for a cool visualization.) Just like in the Euclidean cubic honeycomb (tessellation by cubes as seen in Minecraft), faces of these dodecahedra are arranged in planes. So you get planes tessellated with right-angled penteagon (aka {5,4}). We can also create a ℍ²×ℝ based on this tessellation ℍ², and then their pentagons can be naturally connected with a portal.
We could do the same construction to make a portal from 𝔼³ to 𝕊²×ℝ to 𝕊³. Exactly the same approach would connect the 16-cell (aka {3,3,4}, 16 right-angled tetrahedra tessellating the sphere) to the product tessellation based on an octahedron. However, in this video we take a more interesting approach: the triangular face of the 24-cell ({3,4,3}, octahedra with dihedral angles 120°) has the same shape as the triangular face of the cubooctahedral tessellation of 𝕊².
Of course we can then also create a portal directly from the square vertical face of ℍ²×ℝ and the square vertical face of 𝕊²×ℝ. Since these are squares, we can do this in a more interesting way (rotate by 90° on the way).
This video does not use the cool smooth animation engine (used in most recent videos) because it is somewhat difficult to generalize to intra-geometric portals. It is manually controlled in real time. (This demo should be added to RogueViz later -- with features based on interest (playing HyperRogue in this does not seem feasible, but: fully featured map editor? portals between Sol and H2xR or H3? any ideas for new kinds of portals?)
The perspective is different in every geometry. We move with constant speed* -- sometimes when going through a portal the apparent speed changes, but that is due to perspective acting like fast zooming in H3 and acting very weird in S3. (You can learn how to recognize the geometry by perspective in the "Non-Euclidean Snowballs" video.)
That's all for now. Have fun watching and please comment! Play HyperRogue or join the HyperRogue discord to learn the cool math used here.
unless we crash into something... in S3 sometimes it appears you would crash into something but it is actually a faraway thing appearing close due to geometric lensing, and then you crash into something that was actually real
See also: • Portals to Non-Euclide... (and many other portal videos on this channel)
#NonEuclidean #RogueViz #HyperRogue