Check out this new web toy I made. Explore the tilings of the hyperbolic plan...
- 2014-09-13 00:22:48+0000 - Updated: 2014-09-13 20:42:36+0000
Check out this new web toy I made. Explore the tilings of the hyperbolic plane by moving your mouse around to smoothly change the curvature of space. Take the curvature positive to discover the polyhedra.﻿

### timhutton/hyperplay

- 2014-09-13 18:21:27+0000 - Updated: 2014-09-14 13:15:08+0000
Cool. I had experimented with cellular automata on hyperbolic tilings (on the some {5;4} tiling, in fact), and wrote a simple simulator for them, it's in Java:
https://github.com/dmishin/Pentagrid/releases
And I have even found some spaceships: http://dmishin.blogspot.ru/2011/10/hyperbolic-cellular-automation.html

Unfortunately, {5;4} tiling appeared boring: rules with spaceships are instable. I tried to write a simulator for the {4;5} tilings, but failed to develop a method of cell addressing.
- 2014-09-13 19:40:39+0000
Very nice .  I like the addative coloring of the overlapping edges and polygons.  I like how you can move among all the geometries too.  Well done!
- 2014-09-14 02:13:00+0000 - Updated: 2014-09-14 02:14:26+0000
You need to find someone who studies McKay correspondence and have them take a look at this. (http://math.ucr.edu/home/baez/week182.html has relevant information, tagging )
- 2014-09-14 04:13:21+0000
Also, I think it might be prudent to make a table of curvatures which correspond to the table in the above paper. It's sort of funny that the dodecahedron has curvature 0.61
- 2014-09-14 12:32:30+0000
Beautiful!
- 2014-09-14 14:50:41+0000
Cool. Maybe explain the mouse controls also on the index.html. Huh, did i just find an icosahedron among pentagons? Hah, not quite :)
- 2014-09-14 18:08:31+0000 - Updated: 2014-09-14 19:48:08+0000
Thanks everyone. I'm still learning all this stuff so let me know if I've got anything wrong. , I think the curvature makes sense. Its reciprocal is the radius of the sphere, so 0.61 is what you'd expect for a dodecahedron in our model, where each vertex is distance 1 from the face center.
- 2014-09-14 19:46:24+0000 - Updated: 2014-09-15 09:36:47+0000
Yes, I noticed that! It's at curvature 0.894 in the {5,q} family. After some exploration I think it must be the great dodecahedron: https://en.wikipedia.org/wiki/Great_dodecahedron (Better rendering would help here.) Now I'm looking for other things! I think the great icosahedron is there too, at 0.983 in the {3,q} family. https://en.wikipedia.org/wiki/Great_icosahedron
- 2014-09-14 20:07:00+0000
I bet you're right .

> I think the great icosahedron is there too, at 0.98 in the {3,q} family.

Of course it is! I would have missed that one, thanks!
- 2014-09-14 20:14:53+0000 - Updated: 2014-09-14 20:15:18+0000
I've just updated the app with constant 3D rotation, to help with understanding what we're seeing. But it looks now like I need proper 3D rendering to show things clearly.
- 2014-09-14 20:30:37+0000
Question: what is the weird spikiness at extreme negative curvatures? Is this an artefact of our circle inversion somehow or a genuine phenomenon?
- 2014-09-14 20:53:54+0000 - Updated: 2014-09-15 00:45:12+0000
very roughly:
http://en.wikipedia.org/wiki/Schlegel_diagram

-1.115 Schlegel diagram of tetrahedron;
-1.054 Schlegel diagram of icosahedron
0.61 icosahedron,
0.815 octahedron
0.944 tetrahedron
0.983 first stellation of the dodecahedron
- 2014-09-15 12:20:55+0000 - Updated: 2014-09-15 12:22:01+0000
I think it must be the great icosahedron at 0.983 in the {3,q} family, not the first stellation of the dodecahedron, because the base shape is a triangle not a pentagram. But now I realise a cool way to extend the app to include the stellated dodecahedron: if we had a slider for each dimension. The first would turn a line segment into various polygons, including star polygons because of the way the reflection works. Then the second slider would reflect that polygon in its edges, making all the tilings and polyhedra we see in the current version but also all the polyhedra made from star polygons. Then a third slider could turn those into space-filling tessellations and 4D polyhedra like the hypercube. We'd probably stop at three sliders to avoid hurting my brain even more...
- 2014-09-17 15:42:23+0000 - Updated: 2014-09-30 09:57:10+0000
I've now made a version with 3 sliders: http://timhutton.github.io/hyperplay/index_sliders.html
It allows you to share the link directly to shapes you find. e.g. {5,3,4} is here: http://timhutton.github.io/hyperplay/index_sliders.html?f0=1.176&f1=0.714&f2=-0.347 Here's {4,3,4}: http://timhutton.github.io/hyperplay/index_sliders.html?f0=1.4142&f1=1.1547&f2=0
The curvatures are now based off the edge length of 1. Previously it was based on the polygon circumcircle radius of 1, which wasn't really sensible in this new multi-dimensional setting.
Share the shapes you find!
- 2014-09-17 15:46:52+0000
Wow!
- 2014-09-17 15:52:44+0000
- 2014-09-17 15:54:43+0000 - Updated: 2014-09-17 16:26:44+0000
Oh heck, I've got a bug in the curvatures. No more sharing links until it's fixed please! Sorry! Hold the line.
- 2014-09-17 16:04:13+0000 - Updated: 2014-09-17 16:08:42+0000
your amusing hair one was something like this I think: http://timhutton.github.io/hyperplay/index_sliders.html?f0=0.413&f1=0.403&f2=-0.066
- 2014-09-17 16:18:47+0000 - Updated: 2014-09-30 09:57:43+0000
- 2014-09-17 16:22:21+0000 - Updated: 2014-09-30 09:58:45+0000
So now we can do star polygons and star polyhedra. Here's the pentagram: http://timhutton.github.io/hyperplay/index_sliders.html?f0=1.903
which you can then bend into the first stellation of the dodecahedron here: http://timhutton.github.io/hyperplay/index_sliders.html?f0=1.903&f1=1.699
- 2014-09-17 16:25:38+0000 - Updated: 2014-09-30 09:59:17+0000
- 2014-09-17 17:03:07+0000
- 2014-09-17 17:08:57+0000
- 2014-09-17 18:09:42+0000 - Updated: 2014-09-17 19:23:31+0000
: The one you mentioned here: http://timhutton.github.io/hyperplay/index_sliders.html?f0=1.732&f1=1.633&f2=0 is the tiling of space that would happen if regular tetrahedra worked. They don't, quite. Instead they leave little gaps. Famously, Aristotle thought they did fill space. http://www.nytimes.com/2010/01/05/science/05tetr.html?_r=0
- 2014-09-17 18:19:25+0000
Thanks  for that elucidation.
- 2014-09-18 08:43:09+0000 - Updated: 2014-09-18 10:12:40+0000
The third slider now allows positive curvature too, using stereographic projection from the 4th dimension.
- 2014-09-18 08:55:00+0000
I'm making a list of useful coordinates here:
https://github.com/timhutton/hyperplay/wiki
Anyone is welcome to add more. Just make a github account and edit directly or send them to me.
- 2014-09-18 10:46:33+0000 - Updated: 2014-09-30 10:00:34+0000
Following on from the Aristotle story, if you bend space with the third slider only a tiny bit you can get the tetrahedra to close those gaps and fill space. You end up with {3,3,5}: http://timhutton.github.io/hyperplay/index_sliders.html?f0=1.732&f1=1.633&f2=0.618 (although we only show part of the mesh)
It has positive curvature, so it closes back on itself and actually makes a polyhedron in 4D made of 600 tetrahedra joined face-to-face: http://en.wikipedia.org/wiki/600-cell
- 2015-10-26 16:21:17+0000
Is there a way to export the tiling patterns? I'd like to use those in combination with some pattern generating software like Taprats
- 2015-10-26 20:19:55+0000
There's no export feature at the moment. What format should it output as?
- 2015-10-26 23:31:11+0000
First off Tim, let me congratulate you on an amazing piece of work. It is really amazing the shapes and tiling patterns that can be created just by moving he mouse on the simpler version! (I love both versions :)

It would be great to be able to freeze the geometry and export -
If a all possible, Encapsulated Post Script so it can be vector and used in Illustrator.

Since it is also wrapping 3D maybe an export to .stl or .obj also?

THANKS again for this!

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