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Hyperbolic tilings library

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New: I added an Instructable about how to use this to make cookies:
https://www.instructables.com/id/Hyperbolic-Cookies/.

This is an OpenSCAD library for creating customizable tilings of the hyperbolic plane (in the Poincaré disk model) by triangle groups.

  • A triangle group is defined by a triplet (p,q,r) where 1/p+1/q+1/r < 1, and invoked as triangle_group(p,q,r). The number 0 represents the special value ∞. Any infinite numbers must precede finite numbers (i.e. triangle_group(0,2,3) is legal, while triangle_group(2,0,3) is not — this would be the same group anyway).

  • Tessellation data is computed from the triangle group with tgroup_tessellation_data(group, depth), where the depth parameter is the number of consecutive reflections from the center tile.

  • The modules building the 3d object use this tessellation data; a few modules are provided as examples, such as tessellation_simple and tessellation_cookie (which builds a cookie cutter for hyperbolic cookies). It should be quite simple to build your own modules starting from these.

Note that the same tiling (same triangle group) can give quite different objects depending on how you mark edges (you can e.g. construct a tiling or its dual this way, for example either 7 triangles or 3 heptagons meeting at each vertex).

Warning

Oh, and given the recursive, exponential complexity of these tilings, generating STLs is long (about 15 minutes on my laptop at depth 3, and a crash at depth 4), so that I strongly advise you to use a very small depth parameter for tweaking (as in, 0 or 1), and only when you have a design you like to go up to depth 3 or 4. Since this is basically a stress test for CGAL, it might also decide to crash your computer or teleport you to some unpleasant place. I am not in any way responsible for this.

  • 3D file format: SCAD and STL
  • Publication date: 2020/02/05 at 08:09

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