An aperiodic tessellation of the hyperbolic plane, found by Toimine in the #tessellations channel in the HyperRogue discord.

These tiles are equilateral, with edge length of 0.56358, which is less than the 0.56626 that was achieved by the regular {7,3} tiling (which was apparently the previous record for a tiling with equilateral convex tiles).

It is also possible to the connect two pentagons by their "bases" to obtain a funky variant of the {8,3} tiling. Then, we can play with the angles -- for "narrow" tiles, the edge length becomes even smaller, about 0.5436.

The animated visualization also shows some new cool features of the RogueViz expression parser: it can now automatically solve for the edge length which makes the tiling work for the given angle \(\alpha\)!

https://github.com/zenorogue/hyperrogue/blob/master/tessellations/sample/aperiodic_pentagons.tes

#mathart #rogueviz #noneuclideanGeometry

A yet another visualization of the distortions of the Mercator projection.

An ASCII map using the Mercator projection is projected back on a sphere. In the map, all the characters are of the same size.

#mathart #rogueviz #roguelike

Our new video!

In most of our videos, we have been playing with the non-Euclidean spaces. However, we have not played with the time dimension in them so far. In this video, we will combine spherical and hyperbolic geometry with relativistic effects.

#RogueViz #RelativeHell #mathart #NonEuclideanGeometry #RelativitySpace

https://youtu.be/PxnoSsjMrck

Do you want the hexes on a sphere to be regular?

Or do you want them to be of the same area?

Or do you want straight lines be actually straight? (in one of two ways)

And the same idea with squares.

#Rogueviz #NonEuclideanGeometry

A procedurally generated, non-human building. What creatures could live in buildings of this kind?

(There is a correct answer, but other answers are interesting too.)

#Rogueviz #mathart #procgen

The animation on the left shows clearly that the shortest path is the straight line. Unbelievably, there are possible worlds where this would NOT be the case!

For example, it is not true in Discworld from Terry Pratchett's novels, which is famously a flat world. In a flat world, the shortest path is NOT a straight line!

(Based on the map of Discworld from https://www.reddit.com/r/discworld/comments/odrog0/wip_discworld_high_res_map/ )

#NonEuclideanGeometry #RogueViz

Ported Seuphorica (Scrabble solitaire deckbuilder) to RogueViz for a more intuitive and powerful interface!

(1) Infinite square grid, with standard Seuphorica special powers. Letters E and R (inverse colors close to the gigantic EE top left) are "reversing", hence "RE" is accepted and "EE" is accepted multiple times. Note the word GEESE which uses a portal to get a multiplier for gigantic 'E' two times in a single word. "PETER" uses a mirror.

But since this is RogueViz, let us make the board geometry abstract, to have even more fun with geometry and topology!

(2) Usually, words can only go "right" and "down". In hyperbolic geometry, we have holonomy, so "right" and "down" are not globally defined. So we have to accept both directions. (Or, optionally, only accept words if they are valid both ways.)

(3) In this one, "right" and "down" are not globally defined either, but "horizontal" and "vertical" are (in a somewhat twisted way), so Seuphorica "horizontal" and "vertical" multiplier powers can work.

Although the hexagons somehow turn a horizontal word into a vertical one...

#RogueViz #NonEuclideanGeometry #mathart #noneuclidean #HyperbolicGeometry #Seuphorica #scrabble

(to be continued...)

Developer's intention: A beautiful visualization feature!

What the users share: A chaotic art generator! BREAK THE LIMITS! 🎨🔥

#HyperRogue #RogueViz #HyperbolicGeometry #NonEuclideanGeometry #NonEuclidean

Added some enemy turrets to Relative Hell! (A game in anti-de Sitter spacetime)

These turrets are as accurate as they could possibly be -- they see our ship, and compute the shooting angle so that the ship would be hit if it did not accelerate in the meantime. This can be seen in the last part of the video where the ship does not accelerate.

Note "see" -- of course, the speed of light is limited in this game, so their information is rather outdated... The bullets move at speed close to the speed of light.

The visuals shown here are not what the player would see but rather a slice of the spacetime at t=0 relative to the ship (the turrets are deterministic so let us assume the ship's AI renders the current state). The "wobbling" of turrets seems to be caused by the Lorentz transformations as the spaceship accelerates.

Also new color scheme inspired by pop spacetime explanations, and using a different projection -- this is essentially Beltrami-Klein but it looks conformal because of the map's rotation causing length contraction. Less straightforward but looks better and less cells need to be rendered, which improves the performance.

RogueViz collection: https://store.steampowered.com/app/2271110/RogueViz_Collection/

#NonEuclideanGeometry #NonEuclidean #RogueViz