Progressions in Polygons

You know how it is. You start tinkering around with an idea and before you know it, it's a month later and you've written 40 pages' worth of stuff. That's what happened when I started messing around with patterns in the interior angles of polygons this summer. The picture above is of an octagram whose angles form an arithmetic progression.

I started out by looking at simple alternating angle patterns, then moved onto APs and GPs, and finally quadratic sequences. It's not an exhaustive piece of work by any means, but there are lots of interesting discoveries in there. For example, the AP relationships in a quadrilateral (illustrated below) force any such quadrilaterals either to be cyclic, or trapezia! I love it when this kind of unexpected simplicity pops out of what appears to be a complex investigation.

There are lots of places to go with this, too... specific sequences such as polygonal numbers;  into three dimensions with angular deficiency or skew polygons, etc. It's not difficult mathematics, but I suspect I may have poked round in the odd unexplored corner, even if it's only because nobody else could be bothered!