#noneuclideangeometry — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #noneuclideangeometry, aggregated by home.social.
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A new way to play HyperRogue. In the new mode, you cannot see beyond the walls, or whatever obstacles are there that block your lines of sight!
Almost all roguelikes work like this, but it seems surprisingly rare in non-roguelike 2D games. Probably one of the reasons why roguelikes are so incredibly immersive? (Please comment with any non-rogueish 2D games with line-of-sight you know!)
Due to the use of non-Euclidean geometry, HyperRogue does not normally need such a restriction. However, this mode adds it. It does not make the game visibly harder because a similar restriction is added to the enemies: if they cannot see you, they cannot track you! Of course, not counting the ice wolves which track you by heat (which makes them more powerful in this mode than usual), hyperbugs which track you by smell, and ghosts which track you spiritually.
An interesting gamedev trick, by the way: HyperRogue calculates line-of-sight only for you, not for the enemies. An enemy cannot track you through tiles YOU cannot see. This may sound funny -- technically, they should not track you through tiles THEY cannot see -- but if there is a tile on the path that YOU cannot see because it is behind an obstacle, there is also one that THEY cannot see!
#roguelike #NonEuclideanGeometry #nonEuclidean #HyperRogue #gamedev #gamedesign
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Our brain reconstructs the 3D scene based on the angles perceived by both eyes. So (under some natural assumptions) a human in hyperbolic space would perceive it as the Beltrami-Klein ellipsoid model -- their eyes would be at the foci of this ellipsoid!
Note that, in the picture in [2/3], for clarity, the distance between foci was much larger than the distance between the eyes of the Princess. Here is a picture that is more accurate to the size of HyperRogue characters. The cutting plane is more "flat" and the perceived shape of hyperbolic space is closer to a B-K ball than obvious ellipsoid.
(This was inspired by @theking's 2019 post https://www.reddit.com/r/Hyperrogue/comments/aq4i7k/euclidean_depth_perception_in_the_hyperbolic_plane/ abut what hyperbolic space looks like, and the "stretched B-K" result was originally proven by Magma in the HyperRogue discord -- however, Magma's proof only used continuity to show that the ellipsoid model is azimuthal in two points -- but not that they are actually foci.)
[3/3] #hyperrogue #vr #RogueViz #nonEuclidean #noneuclideanGeometry #mathart #geometry
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A minor, but still quite large, update to HyperRogue!
Suggest moves after 3 incorrect attempts, grow Ivy through narrow chasms, use compasses, rulers and non-Euclidean tools in the built-in drawing tool, more intuitive touch dragging, bugfixes and more!
https://github.com/zenorogue/hyperrogue/releases/tag/v13.1l #noneuclidean #roguelike #mathart #tessellation #HyperRogue #RogueViz #noneuclideanGeometry
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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
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A monster kills a monster, only to be killed by another monster, and somehow, there are enough monsters for every monster to kill forever.
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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
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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.
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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/ )
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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...)
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Developer's intention: A beautiful visualization feature!
What the users share: A chaotic art generator! BREAK THE LIMITS! 🎨🔥
#HyperRogue #RogueViz #HyperbolicGeometry #NonEuclideanGeometry #NonEuclidean
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Every tile in a hyperbolic tiling is randomly colored red or blue. How far should we move to find a specific pattern (a shape constructed out of tiles, with specific colors)?
The distance is usually proportional to \(n\), the number of tiles in the pattern -- this is because at a given location the pattern appears with probability \( 1/2^n \), and there are \(\Theta(c^d)\) possible locations in distance d.
So, for example, HyperRogue normally displays 582 tiles, and for every specific coloring of these 582 tiles, it should be somewhere in about 742 steps (and all of them in about 753 steps). If we used symbols (26 letters + space) instead of colors and were looking for a specific message of 1000 characters, it should appear in radius 6000.
Problem: however, it is not clear how to find such a pattern (and whether it would be still true for a given pseudorandom generator). What it the most elegant coloring algorithm with similar properties but where requested patterns can be found by following some procedure? (If you do not know how hyperbolic geometry works, might be easier to think about binary trees)
#NonEuclideanGeometry #NonEuclidean #HyperbolicGeometry #HyperRogue #roguelike #procgen
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The largest Christmas decoration in a videogame!
#mathart #noneuclidean #NoneuclideanGeometry #roguelike #HyperRogue -
We cannot tell how fast we are moving (for example, do not feel that Earth is moving very fast). This is related to how the objects move at constant speed in a straight line if no force is acting on them.
This is not the case in spherical or hyperbolic geometry, though (assuming a naive model of time*). In this visualization, every point in the yellowish "ghost" moves in a straight line at constant speed. The captain could tell how fast they are moving by measuring these distortions.
* not the case in (anti-)de Sitter spacetime, as in Relative Hell. https://zenorogue.itch.io/relative-hell
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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/
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And here is another conformal projection. (Obtained from the last one using inversion.)
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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#RogueViz #nilrider #NonEuclidean #NonEuclideanGeometry #mathart #gamedev
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Some social network analysis on Metroidvanias.
Data from https://www.reddit.com/r/metroidvania/comments/1dwqjp0/if_you_had_to_pick_5_of_the_best_metroidvanias/, two games are connected (with a weighted edge) when they appear in the same answer.
We arrange the games on the tiles, aiming to minimize the sum of (distance) * (weight). Hyperbolic geometry tends to be be good for this kind of visualization (closeness correlates to edge weight).
Tehora Rogue has run a community detection algorithm (using Gephi) on this network multiple times. This algorithm would find 4 to 6 communities in different runs, the 4 communities shown are based on aggregation of the results.
(There are "green" and "yellow" communities of sets of five games posted by "trolls" who have posted games not mentioned by anyone else, and the main genre splits into "blue" and "red" -- if you know these games, do you have any idea what the blue/red split could mean?In the second picture, the red / blue / green components (excluding trolls) are determined by how often the game appeared in the same community as Iconoclasts / Cookie Cutter / Castlevania: Harmony of Despair. These three games were chosen because they frequently belonged to different communities.
#metroidvania #mathart #socialnetworkanalysis #gamedesign #rogueviz #NonEuclideanGeometry #NonEuclidean #tessellation
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Our new video about the Thurston geometry we have not previously explained in our videos, "the universal cover of the 2x2 special linear group over reals". Why such a name? An exciting travel through spaces of motion, product, and twisted product geometries!
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Finding suitable embeddings for connectomes (spatially embedded complex networks that map neural connections in the brain) is crucial for analyzing and understanding cognitive processes. Recent studies have found two-dimensional hyperbolic embeddings superior to Euclidean embeddings in modeling connectomes across species, especially human connectomes. However, those studies had limitations: geometries other than Euclidean, hyperbolic, or spherical were not considered. Following William Thurston's suggestion that the networks of neurons in the brain could be successfully represented in Solv geometry, we study the goodness-of-fit of the embeddings for 21 connectome networks (8 species). To this end, we suggest an embedding algorithm based on Simulating Annealing that allows us to embed connectomes to Euclidean, Spherical, Hyperbolic, Solv, Nil, and product geometries. Our algorithm tends to find better embeddings than the state-of-the-art, even in the hyperbolic case. Our findings suggest that while three-dimensional hyperbolic embeddings yield the best results in many cases, Solv embeddings perform reasonably well.
Full video: https://www.youtube.com/watch?v=GQKaKF_yOL4 arXiv: http://arxiv.org/abs/2407.16077 with Tehora Rogue #NonEuclideanGeometry #connectome #RogueViz
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CW: high frequency sounds, hyperbolic geometry
Using hyperbolic geometry to create a weird musical instrument.
The drawing tool in HyperRogue lets us to draw on the hyperbolic plane; we combine this with the scrolling animation. This scrolling animation makes the pen move on the surface -- the further we are from the central horizontal line, the faster it moves (a property of hyperbolic geometry); it should of course make some scratching sound when it is drawing, and the frequency of this sound should be proportional to its speed. Also we draw on the Klein quartic to make the display more interesting.
#NonEuclidean #NonEuclideanGeometry #HyperRogue #RogueViz #mathart
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Not many #WorldTessellationDays on mathstodon yet, so here is a non-Euclidean honeycomb (=3D tessellation)!
Do you recognize this geometry, and how this honeycomb was constructed?
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Four new alternate land structures added recently to HyperRogue! As usual, the great walls are straight lines.
(a great walls following a periodic pattern
(b) lots of great walls crossing at 90° angles
(c) lots of great walls crossing at 60° angles, with some surprises waiting in the corners.
(d) no great walls, but using the "landscape method" to determine the boundaries between lands.
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The pattern is broken somewhere in this animation. Can you find the offender?
Probably easier to find the offender in the second one. (Also, can you tell how this was done?)
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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.
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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!
#NonEuclideanGeometry #rogueviz #youtube #mathart #HyperbolicGeometry #noneuclidean