# Kaleidocycles, Part 3

(All rights belong to their owners. Image reprinted here from the book for review purposes only.)

Ok, now we get into the Kaleidocycles. The basic principle of a cycle is a double wedge. Take two identical pieces of paper cut into triangles, and tape them together along one edge. Take two more identical triangles and tape them together along one edge as well. Open up the two wedges, rotate one 90 degrees off from the other, and then tape the matching free edges together to make the double wedge. Repeat this process to make 3 or 4 double wedges, and tape their long edges together to make a chain. Finish by taping the opposite ends of the chain together to make a ring. The more interesting feature of this ring is that you can turn it through itself (rotate the corners of the wedges toward the center of the ring, or away from the center). As long as the hinge points (where the long edges of the wedges are taped together) are moderately flexible, the ring will rotate easily, with no damage to the paper.

The authors created 3 kinds of cycle – hexagonal, square and twisted. I’ll start here with hexagonal. The name refers to the shape of the assembled ring, and consists of 3 double wedges in total. There’s very little difference in difficultly making one cycle over the other, it’s just a question of how big the glue tabs are, and how many tabs you have to glue. If you look at the top photo, the paper is creased and ready to start gluing. The large white triangles are the glue tabs. Apply glue to one tab (I started from the left end) and fold the paper over width-wise to get the opposite panel to fit over the tab. Hold the paper in place until the glue dries, then go to the next tab. When all 3 tabs are glued, then glue the ends of the ring together (I’ll show that step next time.)

The finished hexagonal cycle. Each double wedge has 4 faces. If a tile has only two colors, then the cycle repeats after just 2 turns. The authors messed with the color maps occasionally to have 4 colors, giving a 4-turn cycle. On one cycle, they fudged by adding a fourth blank face with the words “M.C. Escher”, to get a 3-color mapped 3-turn cycle and then the filler text.