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.)

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/ )

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!

A conformal mapping from ℍ²×𝕊¹ to 𝔼³. This is obtained by using the half-plane model for ℍ², let's say {(x,y): y>0}, and then adding the third dimension by rotating it in the 'yz' plane around the 'x' axis.
#NonEuclideanGeometry #NonEuclidean #RogueViz #mathart

Some improvements in Nil Rider before the Steam release of RogueViz [1]! Previously the game simply ended when you rode off the surface. Now, you can ride off to reach the layer below -- and because of how straight lines work in this geometry, you can drop directly below!

As explained in our video [2] without gravity objects follow helices, with radius smaller when the slope is closer to vertical. With gravity, the slope changes in time, so the projection of a freely falling unicycle is quite a funny curve (pink).

[1] https://store.steampowered.com/app/2271110/RogueViz_Collection/
[2] https://www.youtube.com/watch?v=FNX1rZotjjI

Two-point equidistant projection of the hyperbolic plane, but one point is in the center, and the other point is in the infinity, and changes its direction during this animation. The frame where horocycles are mapped to straight lines is insighftul. (Basically, a circle of radius 𝑟 around the center of ℍ² is mapped to a cirlce of radius 𝑟 around the center of 𝔼², and concentric horocycles are similarly mapped to straight lines; these two conditions determine where every point is mapped.) Based on an idea by bengineer8u.

By the way, our video "Portals to Non-Euclidean Geometries" https://youtu.be/yqUv2JO2BCs has just passed 1M views!

A game of Go on a hyperbolic manifold, played in the HyperRogue discord server, ended like this.
#noneuclidean #noneuclideangeometry #rogueviz #bringsurface #baduk

Our new video! We take you on a journey through a small game world and showcase the non-Euclidean transformations of its third dimension.

Full video on YouTube: https://youtu.be/Rhjv_PazzZE #noneuclidean #mathart #mathviz #NoneuclideanGeometry #rogueviz