You might remember James Grime from a couple posts about logarithms many years ago. Or from, you know, Numberphile, or dozens of other places online or in person. He’s a very nice chap, and I think the only person I’ve ever seen recognised in public and asked for a selfie.

Anyway. James, in putting together an exhibit for MathsWorld UK about Truchet tiles, found (1) my Chalkdust article and (2) an error in it - which, of course, I’d like to put right.

I made a hasty ((read: incorrect)) assertion about Truchet cubes that consist of three predominantly black tiles and three predominantly white ones:

With three white faces, there are again two possibilities: either the three faces meet at a vertex, or they wrap around like a tennis ball.

There are indeed only two ways to arrange coloured tiles in this way. But there are four ways to arrange Truchet tiles.

Mixed corner

If you think about the diagonals of three same-coloured tiles that meet at a common corner, there are two possibilities: either they meet at the vertex, or they form a ring around it, as pictured.

Two possibilities

Tennis ball

It took me a bit longer to see that there are also two different tennis ball states, but there are:

  • Whichever cube we build can be positioned with the middle of the three white faces pointing towards the viewer.
  • You can build one cube with the diagonal of this face pointing from southeast to northwest, and a second cube with the diagonal pointing from southwest to northeast.
  • You cannot rotate either cube to match the other (turning it upside-down is the only rotation that leaves the face colours unchanged, and this doesn’t flip the diagonal).

The conclusion

This sort of wrecks the conclusion of the article, since it’s no longer the case that the cube arrangements have the same structure as the decagons (there are now four self-inverse cubes instead of two). That’s a shame, but I suppose it explains why I couldn’t find a reason for them to be the same.

On science

I wanted to finish up by again holding up James’s behaviour here as an example of how to do good maths. He:

  • Found his answer disagreed with someone else’s
  • Looked into it carefully
  • Got in touch respectfully and clearly giving his reasoning
  • Allowed me the chance to go over my work again.

That’s how it ought to be. Mistakes happen. Disagreements happen. We can point them out – and respond to having them pointed out – with diplomacy and grace. Thanks!