A characterization of regular n -gons whose pairs of diagonals are either congruent or incommensurable
- PDF / 730,052 Bytes
- 11 Pages / 439.37 x 666.142 pts Page_size
- 51 Downloads / 167 Views
Archiv der Mathematik
A characterization of regular n-gons whose pairs of diagonals are either congruent or incommensurable Giovanni Vincenzi
Abstract. It is well-known that the side length of a regular hexagon is half the length of its longest diagonals. From this property, one can easily see that for every positive integer m > 1, any regular 6m-gon contains two non-congruent diagonals that are commensurable. In this paper, we show that if n is not a multiple of 6, then all pairs of diagonals of different lengths of a regular n-gon are incommensurable. This yields a characterization of regular n-gons whose pairs of diagonals are either congruent or incommensurable. The main result gives positive answers to some questions on this topic. Mathematics Subject Classification. 11H99, 97F60. Keywords. Incommensurability, Sides of regular polygons, Diagonals of regular polygons.
1. Introduction. One of the most famous and ancient results in mathematics pertains to the incommensurability between the diagonal of a square and its side, which the Greeks (probably) solved via the method of infinite descent (see [3,5], and references therein). On the other hand, it is quite easy to establish examples of regular polygons in which two non-congruent diagonals are commensurable. In fact, for every positive integer m > 1, any regular 6m-gon has pairs of non-congruent diagonals that are commensurable (see Fig. 1). It is known that also for pentagons, the method of infinite descent can be applied to prove that diagonals are not commensurable with the side, and it is somewhat surprising that for n > 6, this method cannot be applied in general (see [3]). In Havil’s book (see [7, p. 28]), we find the following: . . . Quite remarkably, the argument fails thereafter; that is, trying to force the infinite regress for regular polygons of sides greater than
G. Vincenzi
Arch. Math.
Figure 1. The green hexagon is interior to a regular red 24gon. The diagonal l 4 (P24 ) is half of l 12 (P24 )
6 is doomed to failure, as shown in a beautiful but lengthy argument by E.J. Barbeau. This seems to suggest that in order to investigate the general problem of incommensurability among diagonals of regular polygons, we need more algebra or/and analytic methods. Note that we may consider any side of a regular polygon Pn as a particular diagonal: indeed, any segment connecting two vertices of Pn can be considered as a diagonal of Pn . Therefore, by the previous consideration, the first question about this topic becomes the following: which diagonals of a regular n-gon are commensurable with its side? The first step in this direction was made by Barbeau himself, who algebraically showed that the shortest (proper) diagonal of a polygon is always incommensurable with its side (see [3]). It is also known that all pairs of diagonals of a regular heptagon are either congruent or incommensurable, but there are few general results of this type for n-regular polygons (see [1,2,6,8,9,12–15,17]), and the following main question is still open: Which pairs of diagonals o
Data Loading...
