Cubes, Cylinders and Triangles

If you don’t have the bricks available that I used as substitutes for a rhombic dodecahedron, you can still make simple models jut using cubes: Take an ordinary cube, and choose three edges, one in each coordinate direction, and so that they don’t share a vertex. There are, up to rotations, two ways of doing so. Let’s call them blue and red. Make a few dozen of the blue cubes.


Now comes the tricky part: You are only allowed to attach two cubes so that they share one of their blue edges. This is fairly easy in zero gravity, or in your favorite computer software, like Minecraft. The structure you get this way is yet another version of the Laves graph. This looks clumsy, but it is useful for prototyping things. It also gave me the idea of a further reduction that is even harder to hold together but much more elegant: Replace each marked cube by the equilateral triangle that has its vertices at the midpoints of the marked edges.


Now one even has plenty of room to show the two intertwining Laves graphs simultaneously. What one cannot see very well in the above ethereal image is that if one orthogonally pierces a cylinder through the midpoint of any triangle, the cylinder will periodically hit other triangles in the same way, without interfering with any other triangles or cylinders.


Out of the sudden, there is structure. And it gets better: Because the cylinders don’t interfere, we can make their radii so big that they reach the vertices of the triangles. This way the cylinders will touch precisely at the vertices of the triangles. This means that the cylinder packing that uses cylinders in all four directions of the diagonals of a cube can be used to construct the Laves graphs: Determine where the cylinders touch. Each of these points belongs to two equilateral triangles equitorially inscribed in the two touching cylinders. Use the triangles centers as vertices of the Laves graph, and connect them by an edge if the triangles meet at a vertex.



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