Balls and Straws
Diagonals of Polyhedra

 Introduction. Given a convex polyhedron, it is interesting to connect interior edges (a.k.a. "diagonals"). Every model on this page was obtained in this way. Five Tetrahedra. This is a famous configuration of five tetrahedra inscribed in a regular dodecahedron. The dodecahedral edges are orange and each of the five inscribed tetrahedra has been assigned a unique color:

The Compound of Five Tetrahedra.

 Borromean Rings. This is a famous configuration of 3 rings, no two of which are linked, but nevertheless comprise a non-trivial link. The rings are represented by the colors blue, yellow, and red. If these rings are the boundaries of golden rectangles lying in mutually-perpendicular planes, then the 12 vertices of the 3 rectangles coincide with those of a regular icosahedron.

The Borromean Rings.

 Semiregular Polyhedra. These two miscellaneous models were obtained from the cuboctahedron and the icosidodecahedron, respectively. The first has 4 inscribed triangles and the second has 10 inscribed triangles.

Four Triangles in the Cuboctahedron.

Ten Triangles in the Icosidodecahedron.