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Geometry and number theory on clovers. (English) Zbl 1107.51007

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 ${r}^{2}=cos2\theta$ 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 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 ${r}^{m/2}=cos\left(m\theta /2\right)$. 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 ${p}_{i}\equiv 1\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}3\right)$. 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.

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