In order to build the dodecahedron we need a "map" or graph telling us how the PHiZZ units (the edges) should fit together. This is straight-forward as long as we remember that each face is a pentagon and each vertex has exactly three edges.
| Begin with a pentagon. |
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| Add an edge to each vertex so that they all have three. |
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| Add vertices so that each new face will have five sides (edges). |
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| Add edges to the new vertices. |
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| Add the last edges to the outer vertices. |
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...and you're done! Notice that the region "outside the graph" corresponds to the twelfth pentagon. The five exterior edges compromise its "sides"!
An interesting color scheme arises if you want to make each vertex consist of exactly three different colors.
| Begin with the graph constructed above. |
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| Find a path through the graph that begins and ends at the same vertex, and passes through each vertex exactly once. |
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| Alternate edge colors along this path (e.g. blue-green-blue...). Color all edges not on this path a third color (e.g. yellow). |
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| The final product in real life! Notice that each vertex consists of a blue PHiZZ, a green PHiZZ, and a yellow PHiZZ. |
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