#hyperbolicgeometry — Public Fediverse posts

Showing off how in
#HyperbolicGeometry there are exponentially many nearby cells! @zenorogue speaking at #RoguelikeCelebration

The Nielsen-Thurston Classification, discovered by Jakob Nielsen, is a classification of all people named Thurston. Nielsen proved that all Thurstons are isotopic to one of three kinds of Thurstons:

a) William
b) Dylan
c) Not a hyperbolic geometer

Note that not all geometer Thurstons are actually William or Dylan: most of them are just isotopic to William or Dylan.

#NielsenThurstonClassification #Thurston #WilliamThurston #DylanThurston #MappingClassGroups #HyperbolicGeometry

The Nielsen-Thurston Classification, discovered by Jakob Nielsen, is a classification of all people named Thurston. Nielsen proved that all Thurstons are isotopic to one of three kinds of Thurstons:

a) William
b) Dylan
c) Not a hyperbolic geometer

Note that not all geometer Thurstons are actually William or Dylan: most of them are just isotopic to William or Dylan.

#NielsenThurstonClassification #Thurston #WilliamThurston #DylanThurston #MappingClassGroups #HyperbolicGeometry

The Nielsen-Thurston Classification, discovered by Jakob Nielsen, is a classification of all people named Thurston. Nielsen proved that all Thurstons are isotopic to one of three kinds of Thurstons:

a) William
b) Dylan
c) Not a hyperbolic geometer

Note that not all geometer Thurstons are actually William or Dylan: most of them are just isotopic to William or Dylan.

#NielsenThurstonClassification #Thurston #WilliamThurston #DylanThurston #MappingClassGroups #HyperbolicGeometry

The Nielsen-Thurston Classification, discovered by Jakob Nielsen, is a classification of all people named Thurston. Nielsen proved that all Thurstons are isotopic to one of three kinds of Thurstons:

a) William
b) Dylan
c) Not a hyperbolic geometer

Note that not all geometer Thurstons are actually William or Dylan: most of them are just isotopic to William or Dylan.

#NielsenThurstonClassification #Thurston #WilliamThurston #DylanThurston #MappingClassGroups #HyperbolicGeometry

The Nielsen-Thurston Classification, discovered by Jakob Nielsen, is a classification of all people named Thurston. Nielsen proved that all Thurstons are isotopic to one of three kinds of Thurstons:

a) William
b) Dylan
c) Not a hyperbolic geometer

Note that not all geometer Thurstons are actually William or Dylan: most of them are just isotopic to William or Dylan.

#NielsenThurstonClassification #Thurston #WilliamThurston #DylanThurston #MappingClassGroups #HyperbolicGeometry

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

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

Developer's intention: A beautiful visualization feature!

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

#HyperRogue #RogueViz #HyperbolicGeometry #NonEuclideanGeometry #NonEuclidean

Developer's intention: A beautiful visualization feature!

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

#HyperRogue #RogueViz #HyperbolicGeometry #NonEuclideanGeometry #NonEuclidean

Developer's intention: A beautiful visualization feature!

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

#HyperRogue #RogueViz #HyperbolicGeometry #NonEuclideanGeometry #NonEuclidean

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

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

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

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

Revisiting an old hyperbolic interface idea for visually browsing thousands of images in my personal note taking & media management tool... Made with https://thi.ng/geom & https://thi.ng/webgl

Instancing is used already, next steps: add texture atlas to actually show images and move the circle inversion & LOD filtering to the vertex shader...

#ThingUmbrella #WebGL #HyperbolicGeometry #CircleInversion #UI #DataViz #TypeScript

@[email protected] This is an actual conversation I witnessed a few years ago.

Person A: I'm trying to remember, what was the name of the theorem that a hyperbolic metric on a surface is determined by 9g-9 curves?
Person B: The 9g-9 theorem.
Person C: Named after William 9g-9.

@divbyzero Quick mock-up of a hyperbolic version

#Math #Maths #Mathematics #HyperbolicGeometry #Hitomezashi

Published today by @philogb

Exploring Indra’s Pearls with WebGPU

https://medium.com/@philogb/exploring-indras-pearls-with-webgpu-e0f4a745c2f6

#Fractals #HyperbolicGeometry

Experimenting with adding some more ways to visualize how Kleinian group fractals map onto themselves, because I'm wondering if we can use image-based rendering to draw them.

The yellow lines show where the point maps to under the two Möbius transforms and their inverses.

Image-based rendering for IFS (Iterated Function Systems) was presented in a 2004 paper:
https://www.uni-konstanz.de/mmsp/pubsys/publishedFiles/WiSa04.pdf
https://www.win.tue.nl/~vanwijk/ibifs/

Did anyone try that for Kleinian groups yet?

https://timhutton.github.io/mobius-transforms/dfs_recipes.html?id=special&ta.x=1.870000&ta.y=0.080000&tb.x=1.870000&tb.y=-0.100000&tab.x=1.790000&tab.y=-1.948000

#Fractals #HyperbolicGeometry

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

#Mathober #Mathober2023

The prompt for day 10 is 'Nowhere-Neat'. In mathematics, a nonwhere-neat tiling is one where no two tiles share an edge (they do meet at their edges, but one edge is always a different size or offset, so they don't share the entire edge).

As it turns out, the tiling I made for day 9 was already nowhere-neat (https://mathstodon.xyz/@OscarCunningham/111229992806552483). But yesterday I screwed up the colouring in the image. I tried to use three colours so no adjacent tiles have the same colour. But in fact this is impossible. You can see at the bottom of the image two white rectangles are next to each other.

So I corrected the colouring to use four colours, worked out how to colour the parts with negative x coordinate, and mapped the whole thing to the disc model of the hyperbolic plane.

#Math #Maths #Mathematics #Hyperbolic #HyperbolicGeometry #HyperbolicTilings #Fibonacci #GoldenRatio

#Mathober #Mathober2023

The prompt for day 9 is 'Hierarchy'. This reminded me of the binary tiling (https://en.wikipedia.org/wiki/Binary_tiling), where every square is arranged in a hierachy with a manager and two subordinates.

So I decided to figure out how to do the same thing with base ϕ (https://en.wikipedia.org/wiki/Golden_ratio_base) in place of binary.

#Math #Maths #Mathematics #Hyperbolic #HyperbolicGeometry #HyperbolicTilings #Fibonacci #GoldenRatio

For discoverability, here are some topics I like, as hashtags:

#ArtificialChemistry
#ArtificialLife
#CellularAutomata
#ComputerVision
#Fractals
#HyperbolicGeometry
#ReactionDiffusion

This entire fractal is fully defined by these two transforms being animated.
https://timhutton.github.io/mobius-transforms/dfs_recipes.html?id=special&ta.x=1.870000&ta.y=0.080000&tb.x=1.870000&tb.y=-0.100000&tab.x=1.790000&tab.y=-1.948000

The swirliness of the transforms themselves (they are 'loxodromic') seem to make the fractal visually appealing. One day I will try using fluid dynamics motions to explore this idea.

#Fractals #HyperbolicGeometry

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...)

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...)

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...)

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...)