fivefold symmetry

I am fascinated by shapes with fivefold symmetry. Most of the models on this website have fivefold symmetry  this is what makes them so beautiful. I recently taught a series of lessons on these shapes to year 10 students. We started by creating some pentagons. In groups we made pentagons using zometool. Then the students put them together to create a dodecahedron and we explored the angles, edges, faces and vertices etc. We then discovered the golden triangles in the pentagon and investigated the relationship between the pentagon, pentagram and the golden ratio (see below). Following our investigation into golden triangles, we looked at tessellating shapes with fivefold symmetry, and then had a play with some Penrose tiles. 
paper pentagonsI find it amazing that if you tie a knot in a piece of paper you create a pentagon. Can you prove why this happens? If you like this fact, you will enjoy making this model on the right! 
pentagon proofs
Below are outline proofs of some pentagon facts, using as little words as possible. I am sure these could be proved without any words, in the style of the excellent book Proofs Without Words by Roger Nelson.