Blocky Knot Portal v1

See also the *new version*! • Blocky Knot Portal v2
A knot portal made of blocks.
You get back to the same place when you make three (small) loops around the portal frame. As a consequence, there are actually 24 rooms, and 4 knots.
The idea of a knot portal is not new -- you can play the beatiful KnotPortal by Moritz Sümmermann, based on the idea of Bill Thurston:
imaginary.org/program/knotportal
although the algorithms we have used here seem to be significantly different from KnotPortal.
Twitter thread: / 1362426573517955074
Source code: github.com/zenorogue/hyperrog...
A playable Windows exe at roguetemple.com/z/sims/notknot.zip (a bit buggy, don't go into the walls; works in VR too!)

[@Nerdule]

Oof. Trying to understand how the knotted portal divides up all these spaces really hurts my head. It's an incredible effect, though.

[@DShiznate]

These videos just don't feel the same without the temple of cthulhu track ;)

[@evilsorosfundedgovernments433]

if you have the music open on a separate tab you can kinda simulate the experience

[@ZenoRogue]

The Temple of Cthulu soundtrack is more for non-Euclidean geometry / infinite descent themes.

[@OCTAGRAM]

Old video "Not Knot" referred to a knot made out of 3 non-intersecting circles. Increasing the order lead to Lobachevskian geometry and dodecahedra fundamental domain with knot borders becoming some sides of dodecahedra.
That knot was not a trefoil one.

[@smtsjhr]

looks cool

[@elietheprof5678]

When you do LSD at the YMCA changeroom.

[@columbus8myhw]

(Hi! I'm Akiva.)
To provide some more explanation: This is the order-3 cover of the knot. This means, traveling around any piece of the knot three times brings you back to where you started (and, furthermore, for this knot, this is the "largest" space with such a property). It turns out that this creates 24 worlds.
If the portal were a normal flat circle, the order-3 cover would have only three worlds. Any loop that can be created from deforming a circle is called an "unknot". The number of worlds generated by a portal does not change when the portal is deformed, so the order-3 cover of any unknot also has only three worlds. Since 24>3, this video provides a proof that this knot (called the "trefoil") is not the unknot, meaning it cannot be deformed to a circle.
(The order-2 cover of a trefoil has six worlds, as demonstrated in the fourth video of this playlist: ruclips.net/p/PLMcvPzZcXt2R0YiGp_8XAmaR_jCALP7nc. This also provides a separate, easier, proof that the trefoil is not an unknot. For more, read this paper: link.springer.com/article/10.1007/s00283-020-10028-8)
EDIT: Now that this question's been answered, I _have_ to share it. math.stackexchange.com/q/4030689/166353

[@columbus8myhw]

(I'm not talking to Zeno here, I'm talking to everyone else reading these comments)

[@columbus8myhw]

Question: Do portals have any interesting topology?
Answer: Yes, but not the way the video game _Portal_ does them! This is for two reasons. One: Portals in _Portal_ are always against a wall, rather than free-floating in the middle of a room. Two: Portals in _Portal_ are always flat. Once you move away from these...
(I still maintain that the topology in the game _Portal_ isn't interesting)

[@pretzelboi64]

Quite impressive. Did you achieve this with regular portals and multiple passes?

[@ZenoRogue]

Thanks! No, this is raytraced (the space is subdivided into cubes connected in a weird way, and a ray is traced through all the cubes until it hits the wall).
As far as I understand, KnotPortal by Moritz Sümmermann uses the multiple passes method (not sure whether they are "regular portals", depends on the definition I guess).

[@alan2here]

Very nice :) thank you. Will it make it into a game?

[@ZenoRogue]

For now this is a rather specific example (with no interesting gameplay), so I do not think it should be included in HyperRogue.
It will be probably included with RogueViz. (It already is if you compile it yourself)
If someone has any ideas to make a game out of it (whether using RogueViz or own implementation), that would be great I think!

[@toimine8930]

Is the space itself also curved or it's just the fov

[@paulfoss5385]

I would guess FOV because of the {4,4} tilings.

[@ZenoRogue]

No, it is Euclidean geometry. It was quite difficult to make a good video because of the FOV limitations, so I have used the Panini projection to obtain 150 degree FOV.

[@ninjacat230]

@@ZenoRogue panini projection is when you only do it on one axis, doing it with both axis is called something else

[@ZenoRogue]

@@ninjacat230 We use the stereographic Panini projection, as defined in: tksharpless.net/vedutismo/Pannini/panini.pdf (no vertical compression)
Not sure how to do it in both axes, unless you mean the plain stereographic projection. Anyway, it is not done in this video.

[@ninjacat230]

@@ZenoRogue yeah, i was thinking of stereographic

[@TomtheMagician21]

Wow

[@roccoleader279]

How is this done?

[@ZenoRogue]

The algorithm is decribed here: github.com/zenorogue/hyperrogue/blob/master/rogueviz/notknot.cpp

[@ansonx10]

Wait, so is it a portal or is it knot?

[@theespatier4456]

No

[@igorjosue8957]

knortal

[@trymbruset3868]

These Minecraft mods are getting out of hand

[@ZenoRogue]

Not a Minecraft mod, although definitely inspired by these Minecraft mod videos :)

[@trymbruset3868]

@@ZenoRogue hehe, it was meant purely as satire in good humour. I really enjoy your content, and really think that with advances in computer science, and the implementations thereof within the gaming industry, we're likely to see more locally non euclidean level designs in 3d games. The visualisations and renderings you make surely helps a lot of develops understand the otherwise abstract world of geometry

[@igorjosue8957]

@@ZenoRogue imagine if someone actually do this in a mod

[@ZenoRogue]

@@igorjosue8957 Yeah, that would be cool -- lots of cool things have been done in these mods, so I think it should be possible (not sure whether their rendering algorithm is powerful enough in this special case).