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TRISECTION OF THE ANGLE USING ONLY LINES AND CIRCLES

Constantinos Challoumis National and Kapodistrian University of Athens (NKUA), Athens, Greece Philips University (PU), Nicosia, Cyprus Athens University of Economics and Business (AUEB), Athens, Greece University of West Attica (UniWA), Athens, Greece

Abstract

The history of the problem of trisection of the angle is rooted in the ancient Hellenic world, as a mathematical, physical, and philosophical problem. This paper shows proof of it. The structure of this research is divided into three parts: introduction, proof, and conclusion. Trisection of an angle is a fundamental concept in classical geometry, which has a long history in mathematics dating back to the days of ancient Greek mathematicians like Hippias of Elis and Archimedes [1, 3-6, 8]. The problem is to divide any given angle into three equal parts using only two instruments: a straightedge and a compass. Though angles can be trisected using classical methods, the problem has remained unsolved for more than two millennia [2, 7]. This problem is considered a turning point in the history of mathematics, changing the focus from geometric construction to arithmetic structure. It is one of the three universal problems, even before space-time, like the Delian problem and squaring the circle. In this context, the interpretation of classical geometric problems can be extended beyond their historical formulation to a deeper ontological level [9-12]. Mathematical structures may be understood as elements of a broader axiomatics of physics, where fundamental principles exist independently of empirical space-time constraints. From this perspective, problems such as the trisection of the angle, the duplication of the cube, and the squaring of the circle are not merely historical challenges but represent achronous constructs that precede physical reality. They reflect abstract entities that exist beyond space and time, indicating that mathematics serves as a formal language for expressing universal ideas that are inherently timeless. Thus, these classical problems may be understood as manifestations of pre-spatiotemporal logical structures existing in the space-time of ideas within the mathematical axiomatic physics [3].

Keywords

trisection of the angle,