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
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.
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! 🎨🔥
#HyperRogue #RogueViz #HyperbolicGeometry #NonEuclideanGeometry #NonEuclidean
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
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
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/
And here is another conformal projection. (Obtained from the last one using inversion.)
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
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
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!
Nerds are so freaking cool.
5-Sided #Square - Numberphile
https://youtube.com/watch?v=n7GYYerlQWs&si=hjIW1K30XBLPsAV3
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
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?
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.
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?)
