







Second Construction --- 30 Cards



Above is a computer rendering of the construction. I chose this polyhedron because it has 60 identical faces and the faces have bilateral symmetry. But many other polyhedra can be adapted to this slide-together technique. That is left as an exercise for others.





















This card-construction activity appeared in the This card-construction activity appeared in the Math Monday column on the Make Magzine blog from the Museum of Mathematics

Here's a view looking directly at one of the twelve 5-fold stars.

Above is the dual to the rhombicosidodecahedron. It has 60 faces. You can derive it (approximately) by putting a dot in the center of each rhombicosidodecahedron face. The four dots that surround any one rhombicosidodecahedron vertex become the vertices of one kite-shaped face of this dual. So all the faces are equivalent. It is "uniform on its faces". This polyhedron is called either the "trapezoidal hexecontahedron" or the "deltoidal hexecontahedron" depending on which book you check. It is one of the Catalan polyhedra Our construction replaces each face with a card. The position of one card is sketched above, in white. It lies in the plane of one face, extending past the face at the corners of the card. Part of the card goes beyond the face at the 3-fold vertex, so when three cards overlap there, they lock. At the other end of the card, it does not quite reach the 5-fold vertex, so there are 5-fold openings in our construction. Also, the sides of the cards do not quite reach the 4-fold vertices, so there are 4-sided openings in our construction.This is an 8-inch diameter construction in which the 30 cards have more overlap, so it is trickier to make. All the joints are 3-fold locks, so it holds together very tightly. You can throw this around a room and it will not come apart, whereas the 60-card construction easily comes apart at its 5-fold stars. But because the locks are deeper, it is probably not a good construction to try first. The template to make your own is here . Both ends of the cards have equivalent cuts in this construction, so you don't have to worry about which end is which. If you already made the 60-card construction, this should be pretty straightforward once you figure out how to make one of these deeper 3-fold locks.I like the 5-fold stars which arise on the sides of the cards. The geometry of this construction is the same as the slide-togethers with squares , except that I've used a rectangle instead of squares. This requires that the relative size of the triangles and stars change. I've make one particular choice for that change here, based on my aesthetic preferences. In Francesco De Comité's version, shown here , he made a different choice. His has no 3-fold locks at all, as his cards are turned 90 degrees from my version. So his has triangular openings instead of locks, and the cards are less obscured in his version. This may make his easier to assemble, but also easier to fall apart. Take your pick or make both!