As the description says, this is a buggy quotient space of stretched PSL(2,R). (So, no.)

At first it seemed Euclidean except for the twisting

This is the R'Lyeh theme from HyperRogue soundtrack.

This is the R'Lyeh theme from the HyperRogue soundtrack.

Not necessarily. A straight line (also called geodesic) is essentially the shortest curve between points. We are in non-Euclidean geometry, so the length of curves is measured differently. The visualization is based on the assumption that the light travels in straight lines. We can imagine our screen as a small rectangle close to the eye E. To render a point A, we draw a straight line connecting A to E, check where this line intersects the screen rectangle, and A is rendered there. (That is how the usual linear perspective works.) So now if you have a segment AC and a point B on it, which are projected to A', C', and B' on our screen. In Euclidean geometry, B' is on A'C'. This is also true in hyperbolic and spherical geometry, so in these geometries, the straight lines do not seem to curve. But in the hybrid geometries, such as the worlds connecting Euclidean and hyperbolic/spherical geometry, B' may end up to not be on A'C' -- this is caused basically by the geometry being different in 'x' and 'y' directions. (Probably would be easier to understand with a picture, but I hope this helps.)

@@ZenoRogue Yeah that makes sense actually, thanks! I guess it is similar to Light seeming to follow a non-straight path around massive objects, which curve spacetime in a way that's different to where we are observing it.

@@CorruptMem Yes, that is indeed quite similar. For more similarity, imagine a straight line behind a black hole (or something) -- it will no longer look straight to us, due to the "gravitational lensing". Or, try looking at some straight line through curved glass -- it will likely not look straight anymore.

The geometry itself does not say how gravity would work. With the gravity as suggested in this video and used in Nil Rider, they are possible. (But one could say that this kind of gravity does not make sense -- so it is not a problem with Nil, one could imagine weird gravity in Euclidean geometry too.)

Combining multiple space-bending ideas (portals, hyperbolic geometry, 4D, etc.) in one game is probably too mind-bending. And it is not clear if there is any coolness that could be gained from that rather than doing them separately.

They are horospheres. (More precisely, their external surfaces would be horospheres if they had no holes).

@@ZenoRogue Are horospheres identical to Euclidean planes or something else, like intrinsic flatness?

@@snacku7 Yes, they are intrinsically identical to Euclidean planes.

Can a horosphere be superimposed with a Euclidean plane or not?

@@snacku7 Not sure what you mean... there is an isometry between them (when the distance on the horosphere = the length of the shortest path on the horosphere, not through the hyperbolic space it is in).

This is twisted H2xR (Nil is twisted E2xR).

@@ZenoRogue Twisted? What does that mean?

@@snacku7 Basically the same construction on Nil but the NESW plane is hyperbolic instead of Euclidean. (We talk about Nil in "Nil Geometry Explained" and twisted in general in "Geometry with a Strange Name")

@@ZenoRogue Wait, isn’t E2xR just E3?

@@snacku7 Yes, it is. (But I think "twisted AxB" is better to understand as a single operation, not as AxB which is then twisted.)

We do have some VR videos on our channel. But not many people have the VR hardware so it seems better to concentrate on those who do not. (VR video seems better for videos which are mostly 3D visualizations, like "Portals to Non-Euclidean Geometries" which has a 360° VR version -- here we have lots of elements which seem better shown on flatscreen)

Which one? We googled tkmiz and we see no obvious similarity.

The authors of the background music are credited in the description.

Nvm it’s hyperrogue soundtrack

ADD: It's actually _Ascending and Descending;_ not _Waterfall._ My bad. Wondered why there was no water. And that's why the video is named as is! But would definitely like to see if _Waterfall_ proper is possible too (if not in this then in some more elaborated kind of geometry? As I note there are also the perspective-bending pillars going up to "support" the "canal", though I suspect we can just consider the whole thing abstractly to be a couple of Penrose triangles suitably joined [i.e. one side is the columns, the other two sides are the canal] so it _should_ still work particularly given Nil is homogeneous, even though not isotropic ...), and to see how a boat going over that rapid would behave.