Lines in positive genus: An introduction to flat surfaces

TL;DR

This paper introduces the study of lines on flat surfaces of positive genus, aiming to engage students with geometric concepts through visual explanations and exercises, highlighting the beauty and complexity of the subject.

Contribution

It provides an accessible introduction to the mathematical theory of flat surfaces and lines on them, emphasizing hands-on learning and exploration for undergraduates.

Findings

Exploration of line patterns on flat surfaces

Illustration of geometric motions and their effects

Encouragement of active student engagement in learning

Abstract

This text is aimed at undergraduates, or anyone else who enjoys thinking about shapes and numbers. The goal is to encourage the student to think deeply about seemingly simple things. The main objects of study are lines, squares, and the effects of simple geometric motions on them. Much of the beauty of this subject is explained through the text and the figures, and some of it is left for the student to discover in the exercises. We want readers to "get their hands dirty" by thinking about examples and working exercises, and to discover the elegance and richness of this area of mathematics.