We have uploaded a 360° VR version of our video "Portals to Non-Euclidean Geometries". If you wanted to look upwards or downwards or some other direction while watching the video, now you can!
Many roguelikes have a choice of "class" for replay value. Could we have that in HyperRogue, with its focus on non-Euclidean geometry, and combat based on just positioning (no hitpoints etc.)?
Apparently yes! Here the Rogue switches his blade for a crossbow, and takes advantage of how the straight lines work in hyperbolic geometry in a different way!
As a consequence of hyperbolic geometry, the enemies naturally arrange themselves in a straight line, which this crossbow attack takes advantage of. This is work in progress that should be available soon (the classic HyperRogue combat is attack adjacent creature, which also takes advantage of this property, but in a different way). Of course there is more variety in HyperRogue from the choice of geometry, land structure, and change of genre from roguelike to action/FPS/racing/etc.
Still needs some improvements, but already available for some early access testing in HyperRogue 12.1x. HyperRogue 12.2 should be released soon!
Here the arrows reflect in mirrors, and also we have some mimics, who shoot mimic arrows. Last seconds show that the arrows indeed fly in straight lines, even after reflecting off mirrors. #screenshotsaturday#HyperRogue#HyperbolicGeometry
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!
Our games and videos are made with great free tools (which are not popular because nobody markets them). Safer that way, and true to the roguelike roots!
We know that parallel straight lines generate a nice tree-like structure (first video), but we get some new options! We can also use subdivide the hyperbolic plane into these "circles" which are actually infinite (second video), or we can subdivide like in the third video -- depending on which of the infinitely faraway points is the closest. (Suggested by Dylan, based on https://arxiv.org/abs/2303.16831).
@zenorogue Aesthetically, I’d want the stones to be a bit bigger, almost or even touching, to make it easier to see the groups. Also, even toroidal go is very different from standard go due to the lack of boundary. It would be interesting to see a game played on a hyperbolic board in the shape of a suitable right angled polygon.
You cannot tile a hollow sphere with hexagons. But you can pretend.
Some games and animations pretend they can tile a sphere from the outside [ https://twitter.com/ZenoRogue/status/1439246553877729286 ] but I have not seen this done with the inside. This is the WIP HyperRogue feature of embedding 2D geometries into 3D geometries; in this case, the Euclidean world map is embedded as a (hollow) horosphere in 3D hyperbolic space. Expect more weird visualizations based on this (:
Somehow we have not yet visualized this, so it was a bit surprising! That "stalactite" is infinitely long, and also infinitely wide -- we cannot go around it.
The floor is a hyperbolic plane embedded in Solv geometry (as the set of points with x=0), while the roof is equidistant to it.
And here is the (z=0) plane, which has Euclidean intrinsic geometry. We have had visualizations of this earlier (e.g., it appeared in https://youtube.com/watch?v=yqUv2JO2BCs). It looks more like a horotorus than like a horosphere because hyperbolic planes (x=0), (y=0) expand in opposite z directions.