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 polyhedron1 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-gons2 -- 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

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

  1. 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 []
  2. 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?' []

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