# Ask Uncle Colin: a topological conundrum

*Ask Uncle Colin* is a chance to ask your burning, possibly embarrassing, maths questions – and to show off your skills at coming up with clever acronyms. Send your questions to colin@flyingcoloursmaths.co.uk and Uncle Colin will do what he can.

Dear Uncle Colin,

I’m having a topology crisis. I’ve been trying to understand why 3.9.12 isn’t a valid semi-regular polyhedron and I can’t make sense of it. I’m driving myself to distraction!

-- Vertices, Edges, Notation Newbie

Ah, vertex notation, my old nemesis, we meet again.

No need to worry, VENN, there is a perfectly logical explanation. Allow me to explain it – after I’ve explained a bit about the notation.

The idea of vertex notation for a semi-regular polyhedron ((meaning that every face is a regular polygon – as opposed to a regular polyhedron (or Platonic solid), in which every face is the same regular polygon)) is to tell you how the faces around each vertex are arranged.

For example, if you imagine looking at the corner of a cube, you can see that three square faces meet at each corner. The vertex notation for a cube is 4.4.4 – a list of the faces, in order. An old-fashioned football (a truncated icosahedron) is 5.6.6 – around each corner, you have a regular pentagon and two regular hexagons.

It’s not restricted to a three-item list – something like 3.3.3.3 would mean four equilateral triangles fit around each vertex (making a regular octahedron).

Now, to get back to your question, VENN, the numbers serve a dual purpose: they also tell you what’s going on as you go around a shape. Going back to 5.6.6, going around a pentagon, you ignore the 5 – the shapes you encounter on your path are alternating 6 and 6 – it’s hexagons all the way. Going around a hexagon, though, you ignore a 6 and find that you alternate hexagons and pentagons.

That’s the key to why 3.9.12 doesn’t work: going around the triangle, you would need to alternate 9-gons with 12-gons ((I know they have names. Nonagons and dodecagons. I prefer 9-gons and, especially, 12-gons so you don’t have to think ‘is that 12 or 20?’)) – but you hit a problem: you have an odd number of sides to go around, meaning you’d end up with two 9-gons next to each other (not allowed) or two adjacent 12-gons (also not allowed).

In short, VENN, the problem with 3.9.12 is that it builds inconsistent vertices: if you put a triangle, a 9-gon and a 12-gon around a point, there’s no shape you can put next to the remaining side of the triangle that makes 3.9.12 at both of its remaining vertices.

-- Uncle Colin

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