Abstract
In this article we continue the consideration of geometrical constructions of regular n-gons for odd n by rhombic bicompasses and ruler used in [1] for the construction of the regular heptagon (n=7). We discuss the possible factorization of the cyclotomic polynomial in polynomial factors which contain not higher than quadratic radicals in the coefficients whereas usually the factorization of the cyclotomic polynomials is considered in products of irreducible factors with integer coefficients. In considering the regular heptagon we find a modified variant of its construction by rhombic bicompasses and ruler. In detail, supported by figures, we investigate the case of the regular tridecagon (n=13) which in addition to n=7 is the only candidate with low n (the next to this is n=769 ) for which such a construction by rhombic bicompasses and ruler seems to be possible. Besides the coordinate origin we find here two points to fix for the possible application of two bicompasses (or even four with the addition of the complex conjugate points to be fixed). With only one bicompass one has in addition the problem of the trisection of an angle which can be solved by a neusis construction that, however, is not in the spirit of constructions by compass and ruler and is difficult to realize during the action of bicompasses. As discussed it seems that to finish the construction by bicompasses the correlated action of two rhombic bicompasses must be applied in this case which avoids the disadvantages of the neusis construction. Single rhombic bicompasses allow to draw at once two circles around two fixed points in such correlated way that the position of one of the rotating points on one circle determines the positions of all the other points on the second circle in unique way. The known case n=17 embedded in our method is discussed in detail.
Highlights
From ancient time on it was a problem of serious and of recreational mathematics which of the regular n-gons may be constructed by compass and ruler and all “simple” constructions were known without a proof of the completeness of their possibilities up to the appearance ofGauss on the scene at the very beginning of the 19-th century
In present article we investigate the factorizations of the cyclotomic polynomials for low odd n in polynomial factors of 3-cycles which contain not higher than quadratic radicals in the coefficients and explain how this can be obtained in explicit form
The technical design of rhombic bicompasses, in particular, if the arm length should be variable within certain limits, we do not consider as our problem to which we may contribute
Summary
From ancient time on it was a problem of serious and of recreational mathematics which of the regular n-gons may be constructed by compass and ruler (straightedge without marks) and all “simple” constructions were known without a proof of the completeness of their possibilities up to the appearance of. Gauss showed that the basic numbers n for such constructions are the prime Fermat numbers Fl =22l +1,(l =0,1, 2, ) with the long known cases corresponding to =n F=0 3 and =n F=1 5 and with the first unknown and surprising case at this time n= F2 = 222 +1= 17 (e.g., [2]-[13] and the more popular articles of Gardner [14] [15]). This results from the solution of the cyclotomic equations for these cases. We give in explicit form for low n the factorization with only quadratic radicals in the coefficients but with other than 3-
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