Went to the New York City Convention, last week, and it was fun as it always is, but as always, I feel a sense of not having explained myself sufficiently. I taught a couple of classes, both on twist folds, and it is too easy to forget that the language used to describe objects with radial symmetry is specialized and manifestly ill-suited to get ideas across. If you don’t fold from different polygons on a regular basis, there seems to be a bewildering multiplicity of landmarks and none of them seem to be the right ones. That goes double for regular polygons with an odd number of sides.
My second class was on cutting a nonagon from a square, using angle trisection. (I will recommend Robert Lang’s discussion of the subject as background — it begins on page 34.) I drew up some diagrams for teaching the method I used to get a one-cut nonagon, but they leave out the whys and wherefores. A nonagon has nine sides and interior angles of 140°, which means that we want to fold our square into nine equal slices of 40° each. Because 40° × 9 = 360°, right?
Getting to 60° is easy — everyone knows this trick, I should think.
Getting to 40° is not quite so easy, but can be done if you bring two points to two lines at the same time. Like this.
What you’re doing here is constructing three imaginary, congruent triangles, each one having an angle one-third of the original. And as Mister Coudert taught us all those years ago, “Crass Protestants Can’t Teach Catechism,” which was his allegorical way of expressing the theorem, “Corresponding Parts of Congruent Triangles are Congruent.” It is not a business-like way of speaking, certainly, nor a worldly way, but it does live in the memory better so. (And while we’re on the subject of Euclid, let me recommend an exquisitely beautiful edition from 1847 that I came across this week.)
Now you have all that, I will offer you the diagrams for the Nine-Sided Twist Star above. Enjoy. Oh, wait, you’ll need a little folding music while you work.