>>if you take a not-quite-so-narrow strip of paper, say one inch wide, and you tie it into an overhand knot, and then you gently tighten the knot and flatten it, you wind up with a pentagon

Doubters can refer to http://www.jimloy.com/geometry/pentagon.htm (see the fourth illustration), illustrating the regular pentagon. I'll spare you the geometric proof elsewhere on the web.

PS: Thanks to wofahulicohoc for telling me how to url-ize this.