Parasols are great fun. I've only just begun to explore them (Jan '08). I've named them "parasols" because they remind me of the way the little paper parasols I had as a kid used to fold and unfold.

To construct them, you basically make a central "spoke" and then start progressively reflecting this spoke in its faces, and also through the point at the top of the spoke. Brief details of how to make sense of a file are contained in this Word document, or click here for a Flash demo (related file moving_Claude1.cg3) giving further details.

Basic spoke. Spokes can be made from any regular polygon (and quite possibly other polygons - I haven't yet experimented!).This one is based on a regular pentagon. When you've downloaded the file, drag either of the big red points.

Beginning of a parasol. A spoke with some reflections in faces.
Further on a parasol. A spoke with reflections in some faces and also the top of the spoke.
Pentagonal parasol. This one has lots more reflections (there's no limit to the number you can do). It's also formatted to look pretty.
Triangular parasol. Formatted to look pretty.