The Trisection of an Arbitrary Angle

A Classical Geometric Solution

  • Arthur Clair Rediske General Electric Company, Washington, USA
Keywords: Angle trisection; Geometric impossibility; An arbitrary angle; A angle; Compass; Straightedge; Classical geometry Irrational numbers

Abstract

This paper presents an elegant classical geometric solution to the ancient Greek's problem of angle trisection.  Its primary objective is to provide a provable construction for resolving the trisection of an arbitrary angle, based on the restrictions governing the problem.  The angle trisection problem is believed to be unsolvable for compass-straightedge construction.  As stated by Pierre Laurent Wantzel (1837), the solution of the angle trisection problem corresponds to an implicit solution of the cubic equation x cubed minus 3x minus 1 equals 0, which is algebraically irreducible, and so is the geometric solution of the angle trisection problem.  The goal of the presented solution is to show the possibility to solve the trisection of an arbitrary angle using the traditional Greek's tools of geometry (a classical compass and straightedge) by changing  the problem from the algebraic impossibility classification to a solvable plane geometrical problem.  Fundamentally, this novel work is based on the fact that algebraic irrationality is not a geometrical impossibility.  The exposed methods of proof have been reduced to the Euclidean postulates of classical geometry.

Author Biography

Arthur Clair Rediske, General Electric Company, Washington, USA

General Electric Company: Retired, Olympia, Washington, USA

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Published
2018-05-30