Archive for April, 2009
After the dodecahedron comes the cardboard associahedron : it represents the combinatorics of moving parentheses to calculate an associative product of five things step by step. The vertices of the associahedron are thus the 14 different binary trees with 5 leaves.
Given four factors, there are exactly five ways of multiplying them using a binary operation: by moving parentheses according to the associativity rule, you go through five different trees in a cyclic way. This is known as MacLane’s pentagon coherence rule, which states that in not too weak notions of monoid, checking coherence for pentagon diagrams ensures that the definition of the product is well-behaved.
I made a cardboard dodecahedron for the needs of a talk.
If you draw 5-coloured stars on all facets, by choosing smartly the colours, you can get five coloured cubes whose vertices are vertices of the dodecahedron. This trick can be used to show that the symmetry group of the dodecahedron is the alternate symmetric group : it replaces a star by a star with a different arrangement of colours.
Since there are 12 facets and 5 ways of rotating each of them, 60 colourings can be seen by rotating the dodecahedron by direct isometries. However, it is NOT true that you can see the 120 possible colourings by allowing also reflections (the full isometry group of the dodecahedron is the symmetric group on five colours). An easy reason for this is that the colouring is invariant under symmetry through the central point (which is a determinant -1 transformation). You can also argue that reflections act as double transpositions of the colours of a star.
People also talk about five tetrahedra in a isocahedron, which can also be obtained in the dodecahedron by choosing a tetrahedron in each cube in a consistent way. The tetrahedra have faithful action of the isometry group: there are two sets of five tetrahedron, which are exchanged under signature -1 transformations, and even permutations of the tetrahedra correspond to direct isometries.