I've just noticed my new puppy's chew toy is a dodecahedron! Even better, each hole has five different colours around it. That's optimal! Sadly though, it isn't the same five colours round every hole: the toy is made of six wobbly rings, so by the pigeonhole principle they all have to be different colours. (I say this because I'm colourblind and can't tell by looking that they're all different)

Now I really want to find a five-coloured dodecahedral dog toy, so Poppy can do more mathematically interesting chewing.

Now I really want to find a five-coloured dodecahedral dog toy, so Poppy can do more mathematically interesting chewing.

- +Ravi Vakil gave a very interesting talk at some big math conference on this toy (it's called the Nobbly Wobbly, designed by +Dick Esterle, who often comes to the Bridges math/art conferences). It's somewhat surprising that there are six colours - you would expect a symmetric thing with dodecahedral symmetry to have fives in it, but sixes? Are all combinations of 6 choose 5 colours represented on the faces of the dodecahedron? Plenty of interesting questions!Jul 27, 2014
- Ooh, I had no idea! I'd like to see that talk!Jul 27, 2014
- Yeah, the permutations of the colours that correspond to symmetries of this toy (reflections included) are one of the special copies of S_5 in S_6 that exist due to the outer automorphism!

https://en.wikipedia.org/wiki/Automorphisms_of_the_symmetric_and_alternating_groups#Exotic_map_S5_.E2.86.92_S6

[edit] No, that's not quite true, I think it's an A_5.Jul 27, 2014 - I saw this in the shop and took a photo, meaning to post about it, but forgot. :-SJul 27, 2014
- Following up on +Henry Segerman's comment: Yes indeed --- this six-coloring is actually more interesting than the five-coloring, partially because it is less obvious. For example, the outer automorphism of S_6 is in there. As a start: How many ways can you do such a six-coloring?Jul 28, 2014
- I thought I would add this link from CoM for a pdf which shows the colorings for the nobbly wobbly mapped to a cube you can cut out and assemble. They're numbered and paired.

celebrationofmind.org - Nobbly Wobbly Workshop9w - Thanks, +Dick Esterle!9w