# Origami Arithmetics

## Origami Arithmetics

Mathematics of Origami

Keywords: origami, galois theory, algorithms, group theory, graph theory

Prerequisites: 01018: Discrete Mathematics 2

Origami is the art of paper folding: starting with a square piece of paper, one can make wondrous shapes simply by folding the paper. Many intriguing mathematical questions naturally present themselves. Here are a few examples, but searches on Google reveal many more, and there are even conferences devoted to the topic.

• It is common to use only exact folding, i.e. where e.g. a fold is defined by a know point be sent on top of another known point. A point is “known” when it is the intersection of two exact folds that were made. A natural question in
this context is therefore:

What set of points can become known by a finite sequence of exact folds?

This question is similar to classical geometric questions such as “Using only compass and straight-edge, can you double the square or trisect an angle?”. To answer it, one delves into beautiful algebraic theory initiated by Galois.

• If one completely unfolds an origami shape, the folds left on the paper is called a crease pattern. A natural question is therefore, given a crease pattern, is it possible to fold along exactly those lines and end up with a flat figure? If so, what sequence of folds should one perform? These questions turn out to be computationally hard! In fact, it is often posed to Origami enthusiast as interesting puzzles. A necessary, known condition for flat-foldability, however, is e.g. Kawasaki’s Theorem. Inquiries such as these lead one into combinatorics, discrete algorithms and perhaps meta-heuristics.
• Modular Origami is the practice of using many simple units, each folded from one pieces of paper, together to form a large, complex shape, e.g. stars and bucky balls. A classical example is the Sonobe star made of 30 pieces of paper. Here, 20 small triangular pyramids are arranged on an icosahedron to make the star. Each pyramid is composed of three halves of units.Classical geometric–algebraic questions arise in new light in this setting: if I take 3 colours, and fold 10 units of each colour, is it possible to assemple the Sonobe star such that each pyramid has exactly those three colours? The answer is yes, but it is tricky to accomplish. One can therefore ask how many solutions there are, or perhaps which one is *best* (e.g. most symmetrical). These questions are group theoretic in nature.