Original scientific paper

https://doi.org/10.31896/k.26.5

János Bolyai's Angle Trisection Revisited

Hans Dirnböck ; Klagenfurt-Woelfnitz, Austria
Gunter Weiss ; University of Technology Vienna, Vienna, Austria

Abstract

J. Bolyai proposed an elegant recipe for the angle trisection via the intersection of the arcs of the unit circle with
that of an equilateral hyperbola c. It seems worthwhile to investigate the geometric background of this recipe and use it as the basic idea for finding the n^th part of a given angle. In this paper, we shall apply this idea for the trivial case n = 4, and for 5. Following Bolyai in the case 5, one has to intersect the unit circle with cubic curve c. There, and in the cases n is greater or equal to 5, we find only numerical solutions, which shows the limitation of Bolyai's method. Therefore, we propose another construction based on epicycloids inscribed to the unit circle. By this method is even possible to construct the (n/m)^th part of a given angle.

Keywords

angle trisection; angle n-section; equilateral hyperbola; cubic; epicycloid

Hrčak ID:

288263

URI

https://hrcak.srce.hr/288263

Publication date:

28.12.2022.

Article data in other languages: croatian

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