More paper-cutting patterns

Experiment with the constructions below. Put the best examples into your journal, along with comments that describe and explain what is going on. Be careful to make your examples large enough to illustrate clearly the symmetries that are present. Also make sure that your cuts are interesting enough so that extra symmetries do not creep in. Concentrate on creating a collection of examples that will get across clearly what is going on, and include enough written commentary to make a connected narrative.

1. Conical patterns. Many rotationally-symmetric designs, like the twin blades of a food processor, cannot be made by folding and cutting. However, they can be formed by wrapping paper into a conical shape.

   Fold a sheet of paper in half, and then unfold. Cut along the fold to the center of the paper. Now wrap the paper into a conical shape, so that the cut edge lines up with the uncut half of the fold. Continue wrapping, so that the two cut edges line up and the original sheet of paper wraps two full turns around a cone. Now cut out any pattern you like from the cone. Unwrap and lay it out flat. The resulting pattern should have two-fold rotational symmetry.

   Try other examples of this technique, and also try experimenting with rolling the paper more than twice around a cone.

2. Cylindrical patterns. Similarly, it is possible to make repeating designs on strips. If you roll a strip of paper into a cylindrical shape, cut it, and unroll it, you should get a repeating pattern on the edge. Try it.
3. Möbius patterns. A Möbius band is formed by taking a strip of paper, and joining one end to the other with a twist so that the left edge of the strip continues to the right.

   Make or round up a strip of paper which is long compared to its width (perhaps made from ribbon, computer paper, adding-machine rolls, or formed by joining several shorter strips together end-to-end). Coil it around several times around in a Möbius band pattern. Cut out a pattern along the edge of the Möbius band, and unroll.

4. Other patterns. Can you come up with any other creative ideas for forming symmetrical patterns?