While I was making the first rick-rack dodecahedron I had an idea for a slight variation made by joining the rick racks together point-to-point instead of edge to edge. Since this would require a join between three edges, I first thought that I could use a warped hexagon instead of a warped square. However, I was completely wrong about that! The angle of the warped hexagon was far too small. After a bit of trial and error I found that three warped squares joined into a pyramid on two of their sides resulted in a triangular shape with the right surface angle. I then spent the better part of a day trying out different colour combinations and patterns to end up at the conclusion that they looked best just all in the dark blue colour. It’s a bit of extra work making the three squares for each join but they look great – the small pyramids create this extra spikey structure around the rick racks that I really like. Here’s the finished piece:
If you compare it to the original you can see the differences – in the new shape two rick-rack points meet at each join where in the original three points meet at each join:
You can see the difference in how the two are constructed in the diagram below – on the left is the variation and on the right is the original:
The two polyhedra – the original and the variation – correspond to a dodecadodecahedron (no that’s not a typo!) and a ditrigonal dodecadodecahedron. (Although the triangular faces on the new shape are convex not concave, but close enough!). The original shape is also similar to a hyperbolic dodecahedron (which I believe is a dodecahedron that tessellates in hyperbolic space, rather than is entirely hyperbolic itself).
I was also asked if it was possible to make smaller shapes such as tetrahedra or cubes out of rick-racks and warped squares. The crucial factor here is the angle of the warped squares – their maximum angle (the angle between the ‘V’ shape each one forms) is about the same as the angle needed to join the rick-racks together on a dodecahedron. They’ll squash down enough to join together rick-racks at a smaller angle, which should make shapes with higher dihedral angles (i.e. less pointy edges) possible, such as truncated icosahedra, but not those with small dihedral angles such as tetrahedra. Fortunately there’s an easy solution! You can make something something very similar to a warped square by joining two triangles together one on side, as shown on the below on the left, compared to the original warped square join on the right:
Unlike the square however, the triangles will bend to any angle along this join, which should make any regular or semi-regular polyhedra possible! To test the idea I made a tetrahedron using this method:
The rick-racks here have three peaks instead of five, but otherwise it’s made in exactly the same way as the original dodecahedron, just joining each one together with the pairs of triangles instead of with a warped square. The resultant shape is very pointy (and hard to photograph!) but much quicker to make!
I’ve now made a set of podcasts with various different numbers of sides so I can make a whole set of these polyhedra!