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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 p r , where a0 and where the p i are distinct Fermat primes, i.e., odd primes of the form 2 n +1, n0. 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 p r , where a,b0 and where the p i are distinct Pierpont primes, i.e., primes > 3 having the form 2 n 3 m +1, n,m0. 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 =cos2θ 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 p r , where a0 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 p r , where a,b0 and where the p i are distinct Pierpont primes such that p i =7 or p i 1 (mod 4). It also puts these results in a more general context by investigating curves of the form r m/2 =cos(mθ/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 p r , where a,b0 and where the p i are distinct Pierpont primes such that p i =5, p i =17, or p i 1(mod3). 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:
51M15Geometric constructions
11G05Elliptic curves over global fields