Non-Euclidean Portal (in Nil geometry)

This time we are going into a portal that is actually non-Euclidean!
This is a portal in the shape of a Penrose staircase, living in a space with Nil geometry.
Nil geometry is a kind of non-Euclidean geometry which makes "impossible figures" like Penrose staircases possible (in some sense). See • Playing with Impossibi... for an explanation.
If we do not take the portal into account, this is a closed manifold with Nil geometry (i.e, the space is wrapped like in Asteroids or in Manifold Garden).
Six worlds like described above are connected with portals. When we go through the inside of the Penrose staircase, we end up in another world. Each world has a 3x1 box of a different color. Each world also has fog of a different color.
Made with the HyperRogue engine aka RogueViz
Source code: github.com/zenorogue/hyperrog...
A playable Windows exe at roguetemple.com/z/sims/notknot... (works in VR too!)

[@ninjacat230]

oh no, youve found way to make nil geometry MORE confusing

[@CodeParade]

Is there any natural way to define gravity in Nil geometry? Or at least a plane? Since it's non-isotropic, I assume you also get different results depending on how you choose to orient the space relative to your reference.

[@AMan-xz7tx]

*sees comment
*looks at username
*remembers hyperbolica
"oh my god! it's happening!"

[@ZenoRogue]

Every point in Nil has three coordinates, let's call them x, y, and z as usual. Nil has a special direction, let's say it is 'z' (Nil has rotational symmetry around the 'z' axis). In this video, this is the direction orthogonal to the Penrose staircases. You can make 'z' the direction of gravity. The problem of this is that you get perpetuum mobiles everywhere that way (kind of obvious given that we have created Penrose staircases). If you are wondering about making a physics-based game in Nil, this issue seems to make it impossible.
While the set {(x,y,z) in Nil:z=c} is some weird thing, the sets {(x,y,z) in Nil:x=c} and {(x,y,z) in Nil:y=c} are Euclidean planes. (The side faces of "cubes" in this video are rectangles on these planes.) You can also have gravity orthogonal to these planes. This should work without any paradoxes.
See also here: ruclips.net/video/mxvUAcgN3go/видео.html

[@Ganondurk]

You can theoretically make use of the idea that gravity is dependent on one's proximity to large masses. By that logic, you could simulate gravity by simulating a massive object very far away, and put a gravitational force in the direction of the geodesic taking the camera's position to the mass.

[@CodeParade]

@@ZenoRogue Perpetual motion isn't really a deal-breaker. An air resistance term can always limit velocities to free-falling speed limits just like on earth.
Still, I like the idea of the x=c plane better, as it seems more "natural" to familiar gravity. Do you have any videos with visualizations of such planes? If your camera is inside the plane, then obviously it would look just like a flat line around you,. But it should look different with your head above the plane. And moving in the vertical direction would appear as a rotation along that axis right?
Still trying to wrap my head around Nil. There's very sparse information about it online, and I still can't understand what's going on in the 3D visualizations I've played with because I don't know what I'm looking at exactly and which direction the axes are.

[@ZenoRogue]

@@CodeParade
Maybe perpetual motion in itself is not a problem, but the actual problem is that there are (apparently) no (smooth) stable surfaces. This was a problem for the rough idea of a Nil-game I had. Maybe a bit of friction would indeed help, and maybe something like Marble Marcher in Nil would work.
This plane is a bit like a horosphere in hyperbolic plane -- it has intrinsic Euclidean geometry, but it appears (negatively) curved.
It seems I have no video focusing on these planes, but there is ruclips.net/video/p8TXlpETBOc/видео.html (planes in all geometries -- Nil is bottom center)
These planes are used in HyperRogue as Great Walls, so running HyperRogue with commandline parameters "-geo Nil -W Cros -ray-do" should show you them.
We need to make a better version of the explanation video I linked... :)

[@mjde9532]

This stuff is really trippy

[@columbus8myhw]

Ah yeah my old home was like that

[@lilytinax3140]

I'd always thought portals -as we know them- were already non-euclidean objects, this is very interesting!

[@ZenoRogue]

Yeah, at some point people started referring to portals and other weird geometric stuff as "non-Euclidean", this is not consistent with what it meant in mathematics (portals change the topology, not the geometry). I have written a post about this here: zenorogue.medium.com/non-euclidean-geometry-and-games-fb46989320d4

[@columbus8myhw]

ZenoTheRogue The reverse, surely? They change the topology but not the geometry EDIT: It's fixed

[@lilytinax3140]

@@ZenoRogue thank you so much, this helped me understand non-Euclidean geometry a lot better. Turns out, some sci-fi games I thought were non-Euclidean, don't even come close to it 😂

[@ZenoRogue]

@@columbus8myhw Yes, fixed.

[@vindi167]

0:25 reminds me of that dream i had where i looked to the side, and saw a perfectly non-warped penrose staircase, that worked, and it was so weird...

[@andrewfrankovic6821]

Speaking as a word person, not a math person, although I had an unusual affinity for high school geometry which sort of adds up to nothing anymore, I see a duality where space is a place for something to be, while at the same time space isn't there. The thing that is there, where space would supposedly be, is point of view, which could be emanating from somewhere else. I could fictionalize this further, but I've probably already gone too far into this jungle.

[@tubebrocoli]

oh this is super trippy @___@

[@Invalid571]

😵 😵 😵

[@renakunisaki]

This video smells like toast.

[@theespatier4456]

Damn that’s some 100 IQ shit right there!

[@The-Devils-Advocate]

100 IQ is defined to be average?

[@Daniel-uc9ll]

a