## 5. Constructing the Perfect Tetrahedron, Octahedron and Icosahedron

 Recall how much pleasure we took in the fact that the diagonal cube and the golden dodecahedron retained all their inherent symmetry. Generally speaking, braided models lose some of the symmetry of the underlying geometric figure; indeed, our braided tetrahedron, octahedron and icosahedron all lost some of the underlying geometric symmetry. Thus it is natural to ask "Is it possible to braid the tetrahedron, octahedron and icosahedron in such a way as to retain all the symmetry of the original polyhedron?" We have recently discovered a way to do this. The problem was to design strips so that three strips cross over each other to form each (triangular face) in a symmetric way. Figure 18(b) shows a typical straight strip of 5 equilateral triangles with a slit in each triangle from the top (or bottom) edge to (just past) the center.8 The perfect tetrahedron is constructed out of Figure 18(a) where you will see how the 3 strips are interlaced initially. We leave the completion of the model as a challenge to you. Figure 19(a) shows the layout of 3 strips for the beginning of the construction of the perfect octahedron. We'll give you one more hint. When you use Figure 19(a) remember that the strip shown below it in Figure 19(b) has to be braided into the figure above it. Figure 18 Figure 19 A perfect icosahedron may be constructed from 6 strips of this type having 11 triangles on each strip. Over to you! But take heart - these models take several hours to construct. Just to prove that they really do exist we show the photo of them in Figure 20. Figure 20 8 Theoretically the slit could go just to the center, but the model is then impossible to assemble. You need to have some leeway for the pieces to be free to move during the process of construction - although they will finally land in a symmetric position so that it looks as though the slit need not have gone past the center.