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Folding regular polygons and how it leads to a theorem about numbers. (English)
Parabola 34, No. 3, 3-12 (1998).
In our paper (1) (Hilton,P. and Pedersen,J.: How to construct regular 7-sided polygons - and much else besides! Parabola (1998) v. 34(1) p. 2-12. For a review see ZDM 1998(5) under G40 3531) we showed which convex polygons could be folded by a period-2 folding procedure ‒ these turned out to be those polygons whose number of sides, $s$, had the form $s= \frac{2^{m+n}-1}{2^n-1}$ and the procedure is then the $(m,n)$-folding procedure. However, $s$ is not, in general, an integer; indeed the condition for it to be an integer is precisely that $n\vert m$. What happens if $n\nmid m$? Then $s$ is some reduced fraction $\frac ba$ and the procedure described in (1) based on the $(m,n)$-folding procedure and the FAT algorithm produces what we call the regular star $\{\frac ba\}$-gon, that is, a connected sequence of edges that visits every a$^th$ vertex of a regular convex $b$-gon. We may then regard the regular $N$-gon as a regular star $\{\frac N1\}$-gon. It turns out that we can only fold a regular star $\{\frac ab\}$-gon by a period-2 procedure if we can fold a regular $b$-gon by a period-2 procedure ‒ this follows from the result quoted in (1) that the gcd of $t^A-1$ and $t^B-1$ is $t^D-1$ where $D= \text{gcd}(A,B)$. Thus the question remains of how to fold a regular star $\{\frac ba\}$-gon if $b$ is not a folding number. We will answer this question in Section 2 if $a$ is odd. If $a$ is even, there is an additional secondary procedure required. (Of course, if $\frac ba$ is the reduced form of $s$ then $b$ and $a$ are both odd.) But note that, in any case, we mayalways assume $a<\frac b2$, since a star $\{\frac {b}{b-a}\}$-gon is just a star $\{\frac ba\}$-gon described backwards! We will describe in Section 3 a beautiful theorem of number theory which emerges immediately from a description we give in Section 2 of the general folding procedure. Indeed, what is especially pleasing ‒ and highly unusual ‒ is that precisely the same data lead, on the one hand, to a set of explicit folding instructions for folding certain star polygons and, on the other hand, to an astonishing result in quite a different part of mathematics, namely, an algorithm for calculating what is called in number theory the quasi-order mod 2 of an arbitrary odd integer. (From the Introduction)
Classification: G40
Keywords: quasi-order mod 2