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!

Mathematical Idea Analysis

I was switched onto George Lakoff and Rafael Núñez' seminal work Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being this summer. Understanding it was quite difficult for someone like me who's not a cognitive scientist, but the writing is very clear and the ideas are fascinating. When we finally got all the way to the Euler equation in the fourth case study I felt like applauding!





I do have some issues with their interpretation of the consequences of the theory as a total refutation of the Platonic position (regarding the independent existence of mathematical objects), but that doesn't invalidate what they say regarding the cognitive structures behind mathematical thought.

However, I'm looking forward to sharing some of this thinking with this year's cohort of trainees. It's always good to look at the "nuts and bolts" of our subject, but here we'll be investigating how the nuts and bolts themselves are made!