Geometric Constructions through Ordered Sets
arXiv:2509.19361
Abstract
This work develops an approach to geometric construction problems in which the given figure is regarded as an element of an ordered set generated by a construction process. Instead of seeking an isolated figure, the method constructs a chain in which each element follows logically from its predecessors. Three classical problems involving proportional segments, a Gothic detail, and proportional angles illustrate how the required solution emerges within such a structure. In the third problem, discrete links and a continuously varying radius form a two-parameter discrete-continuous structure. The examples also motivate a limited interpretation of information conservation within an ordered construction.
16 pages, 3 figures. Substantially revised version: the exposition and geometric constructions have been clarified; Chapter 3, the summaries, and the conclusion have been revised