KoG, Vol. 26 No. 26, 2022.
Izvorni znanstveni članak
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
Sažetak
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.
Ključne riječi
angle trisection; angle n-section; equilateral hyperbola; cubic; epicycloid
Hrčak ID:
288263
URI
Datum izdavanja:
28.12.2022.
Posjeta: 445 *