Semi-Euclidean Portals
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- Опубликовано: 13 сен 2021
- 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 
Наука
you connected between “normal” space with curved space with a portal. I don’t think other games do this, so you’re the only one.
Crazy what you've achieved here, I think this may be the first time anyone has ever stitched together different geometries like this.
I would like to see a multiplayer aircraft shooter in this ambient. Bullet trajectories would be fun.
imagine if the bullets tragetory depends on the "house" where u are
In hyperbolic space it would become *exponentially* harder to aim.
@@TheSummoner almost impossible, unless you shoot hyperbolic bullets, exponentially opening their range.
It would be very hard to chase someone lol
Doubtful, gravity less 3d shooters alone down work from motion sickness
Nice description, interesting.
Also, you can add timecodes to the description to better understand in which space we go to.
Thanks! Not sure if it is practical given that geometries change all the time :)
Here is a Twitter thread with more precise naming: twitter.com/ZenoRogue/status/1437786373235302410
I like the unsettling sound of the microphone and the clacks of your controller moving the camera as i find it grounding to be unsettled watching this
Holy… this is art…
You guys are doing great work for popularizing maths. Dziękuję!
Found myself thinking "Jeeze, tiny S3 is so rude"
Great visualization! How did you avoid getting lost while flying around?
practice, and also him having made the scene as well
Now I want to see what non-euclidean Descent or something looks like.
I think this might be the greatest video on RUclips. No exaggeration. It's something entirely new, and brilliant, and awe-inspiringly beautiful. An innovation unlike anything ever seen before.
Thanks! The new video should be great too: ruclips.net/video/yqUv2JO2BCs/видео.html
I have heard of a so called "semi-euclidean geometry" called the Dehn Plane, where lines diverge like in hyperbolic space, but the sum of the angles of a triangle are AT LEAST 180 degrees. en.wikipedia.org/wiki/Dehn_plane
Aaahh! The keyboard sounds! Click clack click clack!
4:40 Ok he’s gonna go into the wall… W A T
she's* also no im not being rude :P
but yeah thats cool
Now this is pretty epic.
7:21 there are two slight jerks as you cross between the three geometries. Are these inevitable or just a programming error?
You show how sometimes objects in S³ appear very close (if there is no fog) but it's just lensing. However, touching is via electromagnetic interaction, and EM obeys inverse-square laws in Euclidean space, and those would be lensed by S³ in the same way that light is, so it might actually be possible to "touch" things on the opposite side of the space after all.
I don't know how to such calculate forces in S³ properly as the force can reach round the space many times to reach you, always at similar strength, which seems like a non-convergent infinite series.
I think they are because of programming inaccuracies (not doing the portal transition exactly at the moment when we cross the portal; computing the derivatives numerically as (f(x+ε)-f(x))/ε instead of the correct values).
Actually there some bugs in the formulas to transform between portal coordinates and space coordinates. They are fixed in the new video: ruclips.net/video/yqUv2JO2BCs/видео.html
Is it possible to have a dynamic local geometry shift, say centered around a point moving through E3, having a volume of negative curvature and a "smoothing out" by a region of positive curvature to flatten the effect back out?
This makes me nauseous. 10/10.
i trying to think what would happen if u try to connect a triangle portal and a pentagonal portal of both hyperbolic and spherical geometry and idk, distort the portals in a way that makes the triangle fit and cover the area of the pentagon
i also trying to think why solv and nil geometry aren't here but i think it would not work because it would create the same effect as the horocycle portal u mentioned in the description
Solv has XZ and YZ planes which are hyperbolic, and XY planes which are "horo-tori" with Euclidean geometry. The horo-tori indeed have the same problem as horospheres, but XZ and YZ hyperbolic planes should work and could be connected with H2xR or H3.
I think the method used to connect product spaces should also work for twisted product spaces: twisted S2xR, twisted E2xR (=Nil) and twisted H2xR (aka SL(2,R)).
These H2xR/H3/Solv would use honeycombs based on the binary tiling and S3 would use honeycombs based on the Hopf fibration, while the current video uses regular/Archimedean/product ones, so I see no natural way to connect them in a single video, and I think it is too much content anyway. (The binary tiling does not yield closed manifolds -- so it would look different from the current video.)
This kinda feels scary, especially with the lack of light.
What's the geometry that you enter at 2:14?
S2xR (aka S2xE).
@@ZenoRogue Thanks for the reply!
Odd that there's no music.
Unintentional, but a bit funny I guess. I wanted to not record anything during filming and then add music to it, but somehow it did record sound from the microphone and music was not added (probably because audio was already there) and I did not notice before uploading.
Oh no. SEMI-Euclidean?
Dear god, this is where the HorrorTerrors are from!
This is our Furthest Ring.
What about a trangle and hexagon portal where the euclidean space unfolds like squaring a complex number grid
is the really high pitched noise intentional by the way
where is the download to this? or did you already put iin roguevis?
Where can I find this on Rogue please?!
how can you get 3d or portals?
what's with the microphone?
I wish can see this unsquished can see everything
Ok I figured it out the red 9ne is a triangle and everyother shape are a cube
3:54 big brainfцck here
It's that a sqare or a trangnale I still can't tell?3:10
great video (the noise is killing my ears)
zeno, im lagging a lot in rogueviz demos, how do i stop the lag?
You mean low framerate?
* Some demos are just slow (the videos are not rendered in real time)
* either for a good reason (generally, rendering non-Euclidean geometries is sometimes very hard computationally) or because they have not been optimized well
* if you have bad hardware (CPU/GPU), they might work better on good hardware
* it is often possible to reduce the quality -- which improves the framerate but has its costs (for example by reducing the rendering range by pressing '1' 'r' )
@@ZenoRogue also in the non euclidean portal collection its restricting me to the current geometry, how do i disable it?
Euclid's museum
What does R geometry mean?
ℝ means "real line".
Some sources use the following notation: ℍ²×𝔼
We prefer ℍ²×ℝ because we consider ℍ²×𝔼 misleading -- there is only one geometry in one dimension (one-dimensional hyperbolic ℍ¹ and one-dimensional Euclidean 𝔼¹ are the same, so we just call them ℝ). The reason why this dimension seems to act in a Euclidean way is because we use the Cartesian product ("x"), ℍ²×ℍ (or ℍ²×ℍ¹) would still be the same geometry, not ℍ³.
@@ZenoRogue Is there such a thing as S^1? Or would I start breaching into complex numbers?
@@LaserBread One-dimensional sphere S^1 is just a circle.
It is still the same intrinsic geometry as E^1 and H^1, but it is looped (thus, a different topology).
Is this Joe Rogan dimension?