More Polyhedra

1. Build the cuboctahedron from the four hexagons and twelve bobby pins as instructed.

2. How many tetrahedra does the cuboctahedron contain? How many half octahedra?

3. If you were to build a tight-fitting cubic box to mail it to a friend, what fraction of the box would the cuboctahedron occupy?

4. Will it fill space? If yes, convince your neighbour. If no, what other polyhedron is needed to fill the gaps?

5. Imagine a tessellation of squares, inhabited alternately by red and green people. At a given signal, all the red people begin to encroach at a uniform rate in all four directions on the territory of the neighbouring green squares. This continues until the green squares are completely taken over, which will happen at one instant. What is the new shape of each red territory after the incursion, knowing the original regions were square?

6. Imagine a tessellation of space by cubes, inhabited alternately by red and green people. At a given signal, all the red people begin to encroach at a uniform rate in all six directions on the territory of the neighbouring green cubes. This continues until the green cubes are completely taken over, which will happen at one instant. What is the new shape of each red territory after the incursion, knowing the original regions were cubic?

Philip Mallinson

PCMI July 2001