# Platonic Solids

*Material Required**: Drinking straws preferably the harder variety, rounded paper clips, needle and thread.*

A platonic solid is a solid whose faces are regular polygons. All its faces are congruent, that is all its faces have the same shape and size. Also all its edges have the same length. Platonic solids are regular tetrahedron.

The most common platonic solid is the cube. It has six faces and each face is a square. Another platonic solid is the regular tetrahedron which has four faces and each face is an equilateral triangle.

How many such Platonic solids are possible? It is an interesting fact of geometry that there are only five possible platonic solids. Apart from the cube and the tetrahedron, there is the octahedron, which has eight faces, all equilateral triangles, the dodecahedron, which has 12 faces, all pentagons and the icosahedron which has 20 faces, all equilateral triangles. This was known to the Greek philosopher Plato, who lived over 2000 years ago, and after whom the solids are named. Plato thought that the Platonic solids held the key to the structure of the heavens, something which the astronomer and physicist Kepler, who lived in the 17th century also believed for a while.

To make the Platonic solids, take straws of equal lengths and join them in the shape of the Platonic solid. The joints are made from rounded paper clips which can be bent to any desired angle at the junction. The two āUā shaped arms of the clip are pushed into the opening of straws. Open up these arms a little for a better fit.

All these Platonic solids, if hollow, can be fitted exactly or nested one within the other. Some of the nesting combinations are interesting since the corners of the inner Platonic solid touch the vertices or the edges of the outer solid. One such sequence of combinations, with all the five platonic solids nested one within the other is interesting.

In this sequence the outer most platonic solid is the Dodecahedron. A cube can be nested within the dodecahedron. The twelve faces of the Dodecahedron are regular pentagons. On one of the faces join two opposite corners. This forms one edge of a cube which can be nested inside dodecahedron. The edge-length of the dodecahedron is 0.618 times the edge-length of the cube. These two nested solids can be made again with straws and paper clips. By joining one diagonal on each face of the cube, one obtains a tetrahedron. The length of the edge of the tetrahedron is then times the length of the edge of the cube. This nested combination can also be got from straws and paper clips.

Joining the mid-points of all the edges of the tetrahedron gives an octahedron. The edge of the octahedron is then half the edge of the tetrahedron in length. To make this combination, first make the tetrahedron and mark the midpoints on all the straws. Take pieces of straw which are half the length of the straws on the tetrahedron (a little less than half actually). Now pass a needle and thread through the midpoint of a side of the tetrahedron, slip in the smaller piece through the thread and pass the needle and thread through the midpoint of the adjacent edge of the tetrahedron. Making the thread taut, we find that one edge of the octahedron is in place. Similarly join all the edges of the octahedron with needle and thread.

Inside the octahedron the remaining platonic solid, the icosahedron can be fitted. Divide each edge of the octahedron in the golden ratio 1:1.618 and mark the points. Join the points on adjacent sides in cyclic fashion and you get an Icosahedron. To make this combination with straws, first make the octahedron with straws and paper clip joints. Now take pieces of straw which are ** times the length of the straw forming one edge of the octahedron. (Actually the pieces need to be somewhat smaller so that they fit in nicely.) Mark the point on all the edges of the octahedron which divide the edges in the ratio 1:1.618. On adjacent sides these points must not be equidistant from the vertex, but must alternate in the ratio 1 and 1.618. Pass a needle and thread through these points, thread in the smaller pieces and make the icosahedron. This will require some patience, but the resulting figure will be worth the effort.