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Zbl 1107.51007
Cox, David A.; Shurman, Jerry
Geometry and number theory on clovers. (English)
[J] Am. Math. Mon. 112, No. 8, 682-704 (2005). ISSN 0002-9890

This is a well-written paper on the possibility of dividing the circle, lemniscate, and other curves into $n$ equal arcs using straightedge and compass and using origami. It is a nice blend of geometry, number theory, abstract algebra, and the theory of functions. In his Elements, Euclid showed that a regular $n$-gon can be constructed by straightedge and compass for $n=3, 4, 5$, and $6$. In 1796, C. F. Gauss showed that a regular $n$-gon is constructible by straightedge and compass if $n$ is of the form $2^a p_1 \cdots p_r$, where $a \ge 0$ and where the $p_i$ are distinct Fermat primes, i.e., odd primes of the form $2^n+1$, $n \ge 0$. In 1837, P. Wantzel proved that the converse is also true. In 1895, Pierpont proved that a regular $n$-gon is constructible using origami, i.e., paper-folding, if and only if $n$ is of the form $2^a 3^b p_1 \cdots p_r$, where $a, b \ge 0$ and where the $p_i$ are distinct Pierpont primes, i.e., primes $>$ 3 having the form $2^n 3^m +1$, $n, m \ge 0$. The appearance of the number 3 reflects the fact that angles can be trisected by origami. These theorems can be rephrased using the fact that a regular $n$-gon is constructible if and only if the circle can be divided into $n$ equal arcs. In view of this, Abel considered the lemniscate $r^2 = \cos 2\theta$ and proved that it can be divided into $n$ equal arcs using straightedge and compass if and only if the circle can be so divided, i.e., $n = 2^a p_1 \cdots p_r$, where $a \ge 0$ and where the $p_i$ are distinct Fermat primes; see {\it M. Rosen}'s paper in [Am. Math. Mon. 88, 387--395 (1981; Zbl 0491.14023)]. The paper under review complements these results. Among other things, it considers division of the lemniscate by origami and proves that the lemniscate can be divided into $n$ equal arcs using origami if and only if $n = 2^a 3^b p_1 \cdots p_r$, where $a, b \ge 0$ and where the $p_i$ are distinct Pierpont primes such that $p_i = 7$ or $p_i \equiv 1$ (mod 4). It also puts these results in a more general context by investigating curves of the form $r^{m/2} = \cos (m\theta / 2)$. For $m =1, 2,$ and $4$, these are the cardioid, the circle, and the lemniscate, and for $m=3$, the curve is referred to as the clover. The paper under review proves that the cardioid can be divided into $n$ equal arcs, for all $n$, by straightedge and compass (and hence by origami since origami subsumes straightedge and compass). It also proves that the clover can be divided into $n$ equal arcs by origami if and only if $n = 2^a 3^b p_1 \cdots p_r$, where $a, b \ge 0$ and where the $p_i$ are distinct Pierpont primes such that $p_i = 5$, $p_i = 17$, or $p_i \equiv 1\pmod 3$. The problem of finding conditions on $n$ under which a clover can be divided into $n$ equal arcs by straightedge and compass is left open.
[Mowaffaq Hajja (Irbid)]

MSC 2000:: *51M15 Geometric constructions
11G05 Elliptic curves over global fields

Keywords: cardioid; clover; constructible; cyclotomy; elliptic function; elliptic integral; Fermat prime; Galois theory; lemniscate; origami; paper folding; Pierpont prime; regular polygon; Weierstrass $\cal{P}$-function

Citations: Zbl 0491.14023

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