Decomposing Regular Polyhedra

Geometry Level 4

For which regular polyhedra can you partition the vertices of the Polyhedra into some sets of 4 vertices such that each for each set of the 4 vertices, if we join every vertex to every other, we get a regular tetrahedron.

(A) : Icosahedron

(B) : Regular Tetrahedron

(C) : Cube

(D) : Dodecahedron

(E) : Octahedron

A, B, C A, B, C , D, E B, C, D A, B, C , D A, C, D D, E

Roberto Nicolaides by Roberto Nicolaides , 26, United Kingdom

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1 solution

Anand Chitrao
Aug 14, 2016

Suppose a polyhedron can be partitioned into sets of 4 vertices say S1, S2,..., Sk. Then for each set Si, joining each vertex in Si to every other vertices in Si is creating a complete graph on 4 vertices which is precisely a tetrahedron. So we cont the regular polyhedra which have number of vertices, a multiple of 4. Those are B, C and D.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Hi Anand,

Nice observation :)

I realise I did not put regular tetrahedron into the question ( I have edited this in now )! Can you see what the answer is if we add this new restraint?

And note that the icosahedron has 12 vertices.

Roberto Nicolaides - 4 years, 10 months ago

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