4 ms·

Here's a static penrose tiling. https://gist.github.com/vjeranc/265db912d4004c7c0b0f16ae5fdad327 https://gist.github.com/vjeranc/265db912d4004c7c0b0f16ae5fda..

Here's a static penrose tiling. https://gist.github.com/vjeranc/265db912d4004c7c0b0f16ae5fdad327 https://gist.github.com/vjeranc/265db912d4004c7c0b0f16ae5fda... Interestingly, sphere is not infinite so having this aperiodic tiling is pointless (still looks nice). Although, I never found out if there's a similar set of tiles (or a monotile) that can tile an infinite cone.

dllthomas 3y ago
> cone For physical tiles, clearly not as the curvature would be wrong. I'm sure there remain interesting questions, but it's not sufficiently obvious to me what projection makes sense that I'm able to get further into the problem.
- deleted 3y ago
  [deleted]
- empath-nirvana 3y ago
  A cone has the same curvature as the plane.
  - dllthomas 3y ago
    Not for any definition of curvature that springs to mind. There may well be definitions for which that's true (let me know, I'm curious!), but they're not what I meant. As I said, I was considering covering a cone with physical tiles. My point was that the same tile cannot be placed at different heights along the cone, because a slice along the tile will have to conform to ever larger circles.
    - petters 3y ago
      The Gaussian curvature is the same. That is the curvature that can be measured "from within" the surface. What you wrote is of course right as well
      - dllthomas 3y ago
        Ah hah, neat!
  - bollu 3y ago
    How do you define the curvature for a cone when it has a singularity at the apex? From the definion that I know (angle defect), the tip of the cone will have curvature
- vjerancrnjak 3y ago
  I now realized I wanted to say cylinder, not cone haha.
  - dllthomas 3y ago
    That's also interesting! Assuming it's possible at all, I think scale winds up mattering in a way that it doesn't for a plane, because you have to accommodate "meeting yourself" again some fixed distance away.
DesiLurker 3y ago
might be interesting to slowly 'move' the sphere through infinite aperiodic tiled plane.
- vjerancrnjak 3y ago
  vec2 st = (vUv.xy * h + time/100. + 1.); This will move it with time. The tiling is not infinite, it's inside a big triangle, so with enough time you will travel outside of it, but I guess with proper bounds you can move around or make the triangle bigger. vec2 st = (vUv.xy * h/(1.+0.5*cos(time)) + time/100. + 1.); Periodic zooming. Looks like the sphere gets dressed in a cape.

dllthomas 3y ago

> cone For physical tiles, clearly not as the curvature would be wrong. I'm sure there remain interesting questions, but it's not sufficiently obvious to me what projection makes sense that I'm able to get further into the problem.

deleted 3y ago
[deleted]
empath-nirvana 3y ago
A cone has the same curvature as the plane.
- dllthomas 3y ago
  Not for any definition of curvature that springs to mind. There may well be definitions for which that's true (let me know, I'm curious!), but they're not what I meant. As I said, I was considering covering a cone with physical tiles. My point was that the same tile cannot be placed at different heights along the cone, because a slice along the tile will have to conform to ever larger circles.
  - petters 3y ago
    The Gaussian curvature is the same. That is the curvature that can be measured "from within" the surface. What you wrote is of course right as well
    - dllthomas 3y ago
      Ah hah, neat!
- bollu 3y ago
  How do you define the curvature for a cone when it has a singularity at the apex? From the definion that I know (angle defect), the tip of the cone will have curvature
vjerancrnjak 3y ago
I now realized I wanted to say cylinder, not cone haha.
- dllthomas 3y ago
  That's also interesting! Assuming it's possible at all, I think scale winds up mattering in a way that it doesn't for a plane, because you have to accommodate "meeting yourself" again some fixed distance away.

DesiLurker 3y ago

might be interesting to slowly 'move' the sphere through infinite aperiodic tiled plane.

vjerancrnjak 3y ago
vec2 st = (vUv.xy * h + time/100. + 1.); This will move it with time. The tiling is not infinite, it's inside a big triangle, so with enough time you will travel outside of it, but I guess with proper bounds you can move around or make the triangle bigger. vec2 st = (vUv.xy * h/(1.+0.5*cos(time)) + time/100. + 1.); Periodic zooming. Looks like the sphere gets dressed in a cape.