# Rope Polyhedra Exploration

From EscherMath

**Objective:**
Understand the structure of the Platonic solids through dance.

**Materials:**
Six loops of rope, approximately 15 feet of rope for each, ideally two each of three colors. Ten dancers.

The Dr. Schaffer and Mr. Stern Dance Ensemble pioneered these techniques for creating polyhedra and other figures using loops of rope and cooperating dancers.

## Tetrahedron and Octahedron

- For two people: Practice making the tetrahedron as described in the first half of Polyhedra on a shoe string.
- For three people: Make the tetrahedron from part 1, then add a third person to make a 4-simplex, pentagram, and octahedron as in the video.

## Octaflex

- For four people, described in Octaflex: make an octahedron, come together to make a tetrahedron, and then a cube.

## String Quartet

- For four people, described in String quartet: From a triangle, make a tetrahedron, octahedron, six-pointed star, and then a cube.

## Platonic Ensemble

For ten people:

- Begin with six people forming an octahedron from six loops in pairs. Each pair of loops forms one of the three square 'equators' of the octahedron. Each person holds one vertex, so that each person is touching four loops.
- Each person now spreads their hands, taking two parallel loops apart and forming an icosahedron. Six of the edges of the icosahedron are doubled. These six edges form the short sides of a set of three golden rectangles that can fit inside the icosahedron, as shown here:
- There are now eight triangles whose edges have only single ropes. Each of these triangles needs to be pinched so that the midpoints of the three edges come together to make a new vertex. Four additional people can do this, each person pinching two of these triangles. This forms a dodecahedron with 20 vertices, and all twenty hands are needed to support it.
- The original six people release their holds, leaving four people supporting the vertices of a cube. Each loop will be one square face of the cube, each edge will be doubled.
- Each person releases a single hand, forming the tetrahedron.