A while ago, I mentioned the concept of holyhedra: polyhedra, all of whose faces are multiply-connected. The first known example had 78 585 627 faces, and was later superseded by a highly symmetric example with as few as 492 faces. This is far from optimal, though, as yesterday I received an e-mail from Nathan Ho describing a 12-faced holyhedron!
This leaves open the question of the minimum number of faces a holyhedron can possess. We know that for the deceptively similarly named golyhedra, the answer is 11 as demonstrated by Alexey Nigin’s wonderfully minimal example:
The corresponding lower bound of 11 can be derived from the following simple derivation:
- By numerical considerations, the number of faces can only be 0 or 3 (modulo 4).
- If an orthogonal polyhedron has 8 or fewer faces, then one of the three axes has at most two faces orthogonal to it. Clearly, they must point in opposite directions and have equal area, so the polyhedron cannot be a golyhedron.
In other news, Hans Havermann has attacked the ménage prime problem with a computer search, exhausting all ménage numbers with fewer than 110 000 digits. Consequently, the expected number of ménage primes with fewer than 1000000 digits is about 0.195.
Also, I mentioned a while ago about the project to create a 31c/240 spaceship in Conway’s Game of Life. Dave Greene finally implemented my shield bug design, although changing one of the details resulted in the completed spaceship looking nothing like a shield bug whatsoever!
It’s an immense 934856 by 290486 pixels in size, making it (by area) much larger than the 17c/45 caterpillar that began the tradition of naming these large cumbersome track-laying spaceships after insects. The (23,5)c/79 and (13,1)c/31 projects appear to have been abandoned.
On an unrelated topic, a discussion with Tim Hutton and Simon Rawles in a pub in Cambridge has culminated in them investigating various embeddings of surfaces by crocheting them. Most recently Tim has produced a Chen-Gackstatter surface in this manner:
Another interesting blog by a computer scientist is that of Susan Stepney. One of her recent posts featured a video of a rendition of a particular music video in the Alexandroff Corridor of Newnham College, Cambridge, which Trinity has recently reproduced in an attempt to encourage the artist to perform at the forthcoming (149th) May Ball: