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John Cook is an applied mathematician working in Houston, Texas. His career has been a blend of research, software development, consulting, and management. John is a DZone MVB and is not an employee of DZone and has posted 172 posts at DZone. You can read more from them at their website. View Full User Profile

Dual Polyhedra for Kids

11.23.2012
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Here are a dodecahedron (left) and icosahedron (right) made from Zometool pieces.

dodecahedron and icosahedron

These figures are duals of each other:  If you put a vertex in the middle of each face of one of the shapes, and connect all the new vertices, you get the other shape. You could use these as a tangible way to introduce duality to kids.

There are lots of patterns that kids might discover for themselves. The dodecahedron has 12 faces and 20 vertices; the icosahedron has 20 faces and 12 vertices. At each vertex of the dodecahedron 3 five-sided faces come together; at each vertex of the icosahedron 5 three-sided faces come together.

The two polyhedra have the same number of edges. You can see this by taking one shape apart to make the other. A more sophisticated explanation is that Euler’s theorem says thatV + F = E + 2. When you swap the roles of V and FV+F doesn’t change, so E cannot change.

Here’s a hint on making an icosahedron with Zometool. Stick the red struts with the pentagonal ends into every pentagonal hole on one of the balls. Now if you connect each of the outer balls to each other, you have an icosahedron. You can leave the red pieces inside, or you can use a few of them as a temporary scaffolding to get started, then remove them.

If you do leave the red pieces inside, it’s hard to put the last few pieces in place because the shape is so rigid.

icosahedron with struts to its center

Published at DZone with permission of John Cook, author and DZone MVB. (source)

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