PAPER-FOLDING, POLYGONS, COMPLETE SYMBOLS, AND THE EULER TOTIENT FUNCTION: AN ONGOING SAGA CONNECTING GEOMETRY, ALGEBRA, AND NUMBER THEORY

The Greeks understood, around 350 B.C., how to construct, with Euclidean tools, regular N-gons for N = 2^cN₀, where N₀ = 1, 3, 5 or 15. Two thousand years later, Gauss proved that a Euclidean construction of a regular N-gon is possible if and only if N = 2^c × (product of distinct Fermat primes).

Here we are content to constuct arbitrarily close approximations to regular polygons. Our constructions lead to some interesting number theory involving the Euler totient function. For a given odd number b, and a given odd number , we construct a numerical array, called a symbol and describe an explicit procedure based on the symbol for constructing a regular star -gon.

We can combine the symbols for a given b to produce a complete symbol where each constituent symbol is called a coach. We present two theorems, the Quasi-Order Theorem and the Coach Theorem, which show how the numbers appearing in the symbol (and hence the steps in our constructions) are governed by the values of Φ(b) and the quasi-order of b mod 2. We then generalize the results to any arbitrary base.