# Platonic solids

Exhibit
Platonic solids On display at Ramanujan Math Park
Type Hands-on giant sculpture
Topics Polyhedra

The five platonic solids are present in almost every math museum. Here we feature the immersive presentation in Ramanujan Math Park.

## Description

The five platonic solids are built as painted steel frames in big size, and anchored to the ground in concrete pedestals. The sculptures are big enough as to allow children to get inside. There are some eye bolts attached to the edges of the polyhedron, that allow a rope to be anchored to specific points of the edge (middle points or other points with geometric properties).

## Activities and user interaction

Many activities can be done exploring the polyhedra, from counting faces, edges, and vertices, to verify Euler’s formula, to measure angles (face and dihedral) or lengths of segments, to drawing the polyhedra to get familiarity, or to see the projection by the Sun on the floor. A special activity is conducted with a rope. The rope can be attached and passed through the eye bolts thus creating new edges of new polyhedra, inscribed on the steel polyhedron. Different degrees of complexity are possible. For instance, a first activity would be joining the middle points of each edge. Another would be to to attach a piece of a different string to the previous rope and make nested polyhedra.

## Mathematical background

If we join the midpoints of the edges of a tetrahedron, we obtain an octahedron, which is easy to see and identify. If we join the midpoints of edges of an octahedron or a cube, we obtain a cuboctahedron, and if we do the same process with an icosahedron or a dodecahedron, we obtain an icosidodecahedron. The names or even the shapes are not so interesting as the fact that we obtain the same result for two pairs. This comes from the fact that these are pairs of dual polyhedra (having one the vertices at the center of the faces of the other), but they have the same number of edges and at the same positions, just rotated 90º. This can be seen in the giant models or with a smaller one.

Other constructions less obvious but very pleasant are for instance that if we join the edges of an octahedron not by the midpoints, but by the ratio 1/φ , where φ = (1+√5)/2 is the golden ratio, then we obtain a perfectly regular icosahedron.

These and many other properties can be explored while developing a spatial vision, getting a familiarity with the objects, and intuition on their relations.

## History and museology

The exhibits were conceived by V.S. Sastry, a Math Communicator, and were executed along with Sujatha Ramdorai, a Mathematics Researcher.

## Resources

• A video of the Ramanujan Math Park.
• A couple of books about polyhedra from a visual perspective:
• Pugh, A., Polyhedra: a visual approach, University of California Press, 1976.
• Wenninger, M. J., Polyhedron models, Cambridge University Press, 1974.