Friday, August 22, 2008

The Balloon Icosahedron

The icosahedron is another platonic solid that has all of its vertices with odd degree. What makes it a little trickier than the cube is that it has twelve degree 5 vertices and 30 edges (instead of 8 vertices of degree 3 and 12 edges of the cube). To ease in complexity of remembering the balloon design, symmetry in the use of balloons is preferred. Essential this means that each balloon could rotated into the position of an other balloon. Symmetry dictates the number of balloons should divide the number of edges so 3 balloons is a good choice. This would have ten segments per balloon.

With three balloons, we have six ends that can take care of six of the twelve degree 5 vertices. The other six vertices with odd degree can be turned into degree 6 vertices by joining them to an interior point. The video below demonstrates the construction but perhaps the psuedo-gauss code is more explicit.

From the graph of the balloon, you can see that symmetry of the balloons since you can rotate the graph 120 degrees and the shading of the lines (which represent the colour of the balloon) all become swapped. A further symmetry is that it can be inverted from start to end. This describes a D3 symmetry group.

Here is the psuedo-gauss code:

1z 3 9 10 5a 6 9c 8d 2 6 11 12z

2z 1 5a 11 7b 8d 0 5 4e 3 8 12 10z

3z 2 7b 12 9c 4e 0 7 6 1 4 10 11z

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