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So I've started reading the book 'Measurement' by Paul Lockhart but have been stuck on this question for a while now. The question is "What are all the symmetrical polyhedra?". The author asks you to find a pattern related to adding up shapes and their angles around a sphere (e.g a triangle being 1/6th of a full turn, a square 1/4 and so on) so that the sum of the angles adds up to less than one full turn to prevent an intersection when the corners are folded up. I hope I've explained this enough, any help would be much appreciated!

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What do you mean by "symmetrical"?

If you mean regular polyhedra, then this includes a single type of regular polygonal faces as well as a single type of regular polygonal vertex figures. Additionallly asking for convexity and a positive angular defect, thon it is clear that you can have only 3 triangles, 4 triangles, 5 triangles, 3 squares, or 3 pentagons per vertex. This then leads to the 5 Platonic solids. https://en.wikipedia.org/wiki/Platonic_solid

You might want to release the convexity constraint, both for the faces and for the vertex figures. This will add the Kepler-Poinsot solids. https://en.wikipedia.org/wiki/Kepler–Poinsot_polyhedron

But when just asking for uniformity of polyhedra (vertex transitivity, any number of regular polygonal faces), then the according set is listed here: https://en.wikipedia.org/wiki/Uniform_polyhedron

If you would be interested in polyhedra with any number of regular polygonal faces and any number of vertex types of any shape, while restricting to convex solids only, then you will Resort for the set of Johnson solids: https://en.wikipedia.org/wiki/Johnson_solid

--- rk

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