Lectures on piecewise distance preserving maps

TL;DR

This paper provides accessible lectures on piecewise distance preserving maps from 2D polyhedral spaces into the plane, emphasizing clarity, exercises, and potential generalizations to higher dimensions.

Contribution

It offers a comprehensive introduction to the theory of piecewise distance preserving maps with detailed explanations and exercises suitable for advanced undergraduates and early graduate students.

Findings

Maps can fold polyhedral surfaces flat without distortion

Results extend to higher dimensions with appropriate modifications

Educational resource with exercises and solutions

Abstract

These lectures were a part of the geometry course held during the Fall 2011 Mathematics Advanced Study Semesters (MASS) Program at Penn State (\url{http://www.math.psu.edu/mass/}). The lectures are meant to be accessible to advanced undergraduate and early graduate students in mathematics. We have placed a great emphasis on clarity and exposition, and we have included many exercises. Hints and solutions for most of the exercises are provided in the end. The lectures discuss piecewise distance preserving maps from a 2-dimensional polyhedral space into the plane. Roughly speaking, a polyhedral space is a space that is glued together out of triangles, for example the surface of a polyhedron. If one imagines such a polyhedral space as a paper model, then a piecewise distance preserving map into the plane is essentially a way to fold the model so that it lays flat on a table. We only consider the 2-dimensional case to keep things easy to visualize. However, most of the results admit generalizations to higher dimensions. These results are discussed in the Final Remarks, where proper credit and references are given.