Steinhaus Dodecahedron

A dodec­a­he­dron (from Greek δώδεκα — twelve and εδρον — face) — is one of five regular regular poly­hedra. It has 12 faces, which are regular pentagons, 30 edges, 20 vertices.

The idea of a model is proposed in Hugo Stein­haus' book ”Math­e­mat­ical snap­shots” (first edition in Russian: Moscow-Leningrad, 1949).

A model can be manu­fac­tured of thick card­board. Corru­gated card­board, for example, that boxes for office paper are made of, would suffice. A rubber band — like the one a roll of money is binded with — is also needed. In this case the dodec­a­he­dron's edge should be around 5 centime­ters long.

The layout of the dodec­a­he­dron half can be printed on a sheet of regular paper to mark the shape for cutting from card­board.

If you now release the model, e.g. by tossing it, the rubber band will pull the halves together, turning the flat struc­ture into a regular dodec­a­he­dron.

The final stable posi­tion of the rubber band is the shortest path (geodesic) on the dodec­a­he­dron. This posi­tion is char­ac­ter­ized by the fact that for any pair of adja­cent edges the angles between the rubber band and the edge are equal, and it does not ”want” to shift rela­tively to the edge.