I shall be rather busy during the next couple of months, so there may be a noticeable drop in the frequency of posts on Complex Projective 4-Space. Nevertheless, I shall endeavour to maintain a steady stream of interesting articles, assuming I have time to do so.
In the meantime, here is a puzzle Dan Asimov discovered in an obscure journal: Find a set of n distinct points on the Euclidean plane, such that the perpendicular bisector of any pair of points passes through exactly two points in the set. I rediscovered the same solution as in the journal, and we believe that this is the unique solution up to similarity. I’ll not post any spoilers yet.
I’ll start by explaining the title of this section. The word ‘ruminations’ can refer to deep contemplations, and also to the process by which a ruminant re-digests its food.
In the world of golden rhombi, Christian Perfect constructed a couple of Wieringa roofs at the Newcastle MathsJam following my instructions. His first attempt ended up exhibiting too much positive curvature, but made a decent hat:
Tessellations of [non]-Euclidean space
Tim Hutton and I are experimenting with different honeycombs for the program Ready. He implemented the triakis truncated tetrahedral tiling recently, and created a couple of Wikipedia pages related to it. You may recognise it as the Voronoi diagram of atoms in a diamond.
We’re also trying out non-Euclidean tessellations. Analogous to the BCC lattice of truncated octahedra in Euclidean 3-space, there is a hyperbolic tiling of truncated icosahedra. There was some discussion about this fifteen years ago. One proof of its existence is its Coxeter-Dynkin diagram, relating it to a known hyperbolic tessellation.