Algorithms and Geometric Constructions

TL;DR

This paper explores the formalization of geometric constructions as algorithms, analyzing classical problems like angle trisection and discussing the history and definitions of geometric algorithms.

Contribution

It provides a clear, formal definition of geometric constructions as algorithms, clarifying the conceptual foundations and historical context.

Findings

Classical problems are unsolvable with compass and straightedge.

A formal definition of geometric algorithms is proposed.

Historical analysis of geometric construction concepts.

Abstract

It is well known that several classical geometry problems (e.g., angle trisection) are unsolvable by compass and straightedge constructions. But what kind of object is proven to be non-existing by usual arguments? These arguments refer to an intuitive idea of a geometric construction as a special kind of an `algorithm' using restricted means (straightedge and/or compass). However, the formalization is not obvious, and different descriptions existing in the literature are far from being complete and clear. We discuss the history of this notion and a possible definition in terms of a simple game