Hidden inside the proof of an octagon are all the necessary components of a similar proof of a hexagon, an equilateral triangle, and a square. While I did this set of proofs using a computer program, I find there’s an elegance and tranquility which comes from doing it with a straightedge and a compass. I’m looking forward to being able to find the similar proofs of the pentagon and heptagon within this same shape.
It’s one of the beauties of these forms that all of the polygons start from the same side length, and yet as they iterate, the outermost shape becomes larger. The length of the side doesn’t change, but the angles do — and so the shapes grow ever larger from the same seed. The pentagon will nest within the hexagon, and the heptagon within the octagon. I’m kind of Curtis to see if drawing the heptagon will reveal the pentagon, but I’m not holding my breath— both are notoriously tricky to get right.
As some readers may know, the seven core shapes — these four, plus the two named but not shown, plus the circle — are the basis of our Design Thinking schematic at school. Today, a student was panicking about his Spanish project. I showed him how to brainstorm, how to visualize his completed project, and how to name the problems with his design. We imagined his project a few times, a few different ways, settled on an overall design, and then we cut the parts for his project’s visual aid, assembled them in rough form. In the process, we discovered his project would need the words for "lime green" in Spanish, along with "orange" and "yellow" and "graph paper" and "photograph" and "map" and "gray". By dialing backward and forward around the design thinking schematic, through the diagram of the shapes shown below, my student friend gradually saw his project emerge from the chaos.
While he held the project tight, he could only imagine the linear process: first do this step, then this one, then this one, then this one, and so on. He was like the baseline of all the shapes, working through the infinite points of the line. By doing design thinking — by calling out the individual functions of his brain, like naming a problem or cutting away the impractical matters, —he leapt away from the original line, to build something much vaster and stronger than just stressing out and finishing his project at the last minute could have done. It’s going to take dozens more go-arounds, maybe, before he gets it completely. But there’s a hidden power in the design movement and in the Maker movement. It calls forth new powers of the mind in response to encountering problems, in much the same way that the Octagon — in my teaching, the shape that means "ask for help" — reveals the interdependent and elegant forms of three other polygons. Underlying all of it, the circle — "what shall I do next?" — brings us back again and again to the places where we began, more expansive and more fully encompassing the full measure of our powers than we were before.