Is Dodecahedron tesselation somehow possible?

Question

In this video (at 3:25) there is an animation of planets inside a dodecahedron matrix (or any data-structure that best fit this 3d mosaic). I tried reproducing it with 12 sided dices, or in Blender, but it looked impossible, because there is a growing gap inbetween faces.

The article what-are-the-conditions-for-a-polygon-to-be-tessellated helped me understand which polyhedron are and which ones are not possible to tesselate. So for regular matrixes, we are limited to some irregular polyhedron, like the Rhombic dodecahedron.

My questions on this:

• What is the technique used to represent a dodecahedron matrix in this video
• Is it only a 3D voronoi tiling, and were lucky it has 12 pentagonal sides in a region?
• Is it an optical illusion? Maybe the underliying structure is just a 3d Hyperbolic tiling, and it only looks like a regular dodecahedron matrix from our perspective?
  • If so, are there more of these "distorted representations of space" used to tesselate more polygons/polyhedrons
• Are there online resources for writing geometry shaders with this technique?

Answer 1

In fact, this is exactly the opposite of a hyperbolic tiling; it's a spherical one!

If you look closely, you can see that there are four regular dodecahedra meeting at each vertex. Since four dodecahedra packed around a single point have a little bit of wiggle room (as you've confirmed for yourself with d$12$s), this means that rather than needing 'more' room in your space to fit in all the dodecahedra, you actually need less; you need space to curl up on itself in order to pack the solids together tightly. (By contrast, the hyperbolic honeycomb mentioned in the comments fits eight dodecahedra around each vertex, in the same way that a cubic tesselation fits eight cubes around each vertex - so it needs a lot of extra space to fit all of those dodecahedra into.)

For an analogy, consider trying to pack three pentagons around a single point in the plane; there's a little extra room left over, and if you curl things up to fit exactly three pentagons around your first vertex, then continue fitting exactly three pentagons around each of those pentagons' vertices, etc., you'll soon find that things close up on themselves cleanly and you've built yourself a dodecahedron.

If you perform the same construction using four dodecahedra 'in space', you'll find that things close up cleanly on themselves in much the same fashion; and if you continue, then — just as in folding pentagons you establish a 'tiling' of the surface of a sphere by pentagons and thus get the dodecahedron — in four dimensions you get a regular tiling of the surface of a 3-sphere by dodecahedra. This is a well-known four-dimensional polytope, the so-called 120-cell.

Answer 2

As User7530 pointed out, there is - in the video - a heavy distortion of the earths near the left side of the screen and it must therefore be a hyperbolic tiling.

Also, L'Univers Chiffoné is a book that explains the possibility of living in a hyperbolic (a non-euclidian) space. its written by the french astrophysicist Jean-Pierre Luminet.