Geometric constructions lie in the very heart and origin of geometry - "Geo-metry" is the measurement of land. For the practical task, one has some specific set of intruments and needs to be able to use them for building specific shapes or comparing land areas and so on. In the mathematical question, one can consider the following aspects:

• What are the permitted tools and operations?
• Which constructions one can perform using these tools?
• Which constructions are impossible for the tools by some deep reasons?
• Are there real world mechanical tools to perform these constructions?
• Is it possible to write a code so that the construction could be performed by a computer?
• Are there any constructions which we don't know how to perform but also cannot prove to be impossible?

One can consider the above questions in different geometries and based on different set of tools/operations. The classical example for the question is ruler and compass constructions in Euclidean geometry. For instance, it is easy to show that an angle can be bisected using ruler and compass, but angle trisection is impossible for a general angle. At the same time, there is a simple construction allowing to trisect a segment (or even cut it into n equal parts for any positive integer n).

Prerequisites: Algebra II. Corequisites: Geometry III.

Resources:

You can find some sources here.