I was re-reading The Color of Magic (book #1 in Terry Pratchett’s “Discworld” collection) as one of this month’s reads in the GoodReads.com “SciFi and Fantasy eBook” group, and I encountered a passage that I remember wondering about before. It describes a temple in which the floor is paved with octagonal tiles. I was thinking about how if you take a bunch of octagons as we typically think of them (if you forgot what they look like, they’re 8-sided shapes just like a STOP sign), and try to arrange them on a flat surface by aligning them edge to edge, you’ll end up with a bunch of squares in between.
Go ahead: try it. I’ll wait.
See, I told you so.
However, while reading the novel this time, I was also referring to “The Annotated Pratchett File” — which, by the way, is a really useful sort of thing for the Kindle’s web browser. It actually mentioned this issue, and proposed a solution:
- [p. 98/87] “The floor was a continuous mosaic of eight-sided tiles, [...]”
It is physically impossible for convex octagons (the ones we usually think of when we hear the word ‘octagon’) to tile a plane. Unless, of course, space itself would somehow be strangely distorted (one of the hallmarks of Lovecraft’s Cthulhu mythos). It is possible, however, to tile a plane with non-convex octagons (and Terry nowhere says or implies he meant convex tiles). Proof is left as an exercise to the reader (I hate ASCII pictures).
I can go one better than ASCII graphics, and use “real” graphics. The secret, as alluded to above, is that in addition to some regular convex octagons…
…we also need some concave octagons, which look like four-pointed stars (count the sides: still eight, right?):
The final trick is to align the convex (stop sign) octagons point to point instead of side to side, and then there’s a nice space for our concave (star) octagons to fill the gaps. Now we no longer need four-sided square tiles (or lots of grout?) to fill the gaps:
So the only question left now is, was this what Sir Terry was thinking, or was he intentionally describing something that was impossible, or did he not even think about it at the time? I guess we’ll just have to wait until he reads this and then leaves a comment to let us know.