Construction of Regular Heptagon by  Rhombic Bicompasses and Ruler

A. Wünsche

doi:10.4236/am.2014.515229

Home > Journals > Physics & Mathematics > AM

AM> Vol.5 No.15, August 2014

Share This Article:

Open Access

Construction of Regular Heptagon by Rhombic Bicompasses and Ruler

Abstract Full-Text HTML

Download as PDF (Size:3601KB) PP. 2370-2380

DOI: 10.4236/am.2014.515229 2,687 Downloads 3,158 Views Citations

Author(s) Leave a comment

A. Wünsche

Affiliation(s)

Humboldt-Universit?t Berlin, Institut für Physik, Berlin, Germany.

ABSTRACT

We discuss a new possible construction of the regular heptagon by rhombic bicompasses explained in the text as a new geometric mean of constructions in the spirit of classical constructions in connection with an unmarked ruler (straightedge). It avoids the disadvantages of the neusis construction which requires the trisection of an angle and which is not possible in classical way by compasses and ruler. The 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 (arms) on one circle determines the position of the points on the other circle. This means that the positions of all points (arms) on both circles are determined in unique way.

KEYWORDS

Plane Geometry, Algebra, Geometric Constructions, Heptagon, Cyclotomic Equations, Neusis

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Wünsche, A. (2014) Construction of Regular Heptagon by Rhombic Bicompasses and Ruler. Applied Mathematics, 5, 2370-2380. doi: 10.4236/am.2014.515229.

References

[1]	Courant, R. and Robbins, H. (1996) What Is Mathematics? Oxford University Press, Oxford.

[2]	Stewart, I. (2004) Galois Theory. 3rd Edition, Chapman & Hall/CRC, Boca Raton.

[3]	Convay, J.H. and Guy, R.K. (1996) The Book of Numbers. Springer, New York. http://dx.doi.org/10.1007/978-1-4612-4072-3

[4]	Edwards, H. (1984) Galois Theory. Springer, New York.

[5]	Postnikov, M.M. (1963) Teorija Galoa (in Russian), Fizmatgiz, Moskva. (English translation: Postnikov, M.M. (2004) Foundations of Galois Theory. Dover Publications, New York).

[6]	Shkolnik, A.G. (1961) The Problem of Circle Division (in Russian). Uchpedgiz, Moscow.

[7]	Bold, B. (1969) Famous Problems of Geometry and How to Solve Them. Dover, New York.

[8]	Weisstein, E.W. (2013) Heptagon, from MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/Heptagon.html

[9]	van der Waerden, B.L. (1964) Algebra, 1. Teil. 6th Edition, Springer, Berlin.

[10]	Klein, F. (1884) Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade. Leipzig.