Mathematics and Origami

Question

I am reading through this paper about the math behind origami: http://www.math.washington.edu/~morrow/336_09/papers/Sheri.pdf
However, I am getting confused with definitions 3.3 and 3.4. I am not sure I understand what each one is saying. If someone could help I would really appreciate it.

Answer 1

What are you not understanding exactly?

Here is my interpretation of definition 3.3: What is the set of origami constructible points $\mathcal P_0$ as a subset of a set of points $\mathcal P$? This is a set $\mathcal P_0$ of points in $\Bbb R^2$ with the property, that

1. The points $(0,0)$ and $(0,1)$ lie in $\mathcal P_0$
2. All points in this set are closed under origami construction (this is defined in definition 3.2, 3.1 resp.)

Furthermore, definition 3.4 means the following to me? For an arbitrary set of points $\mathcal P$ in $\Bbb R^2$ you want to know all the distances between them. So you define the set of all possible distances between two points of your $\mathcal P$ as the set of origami numbers. This set contains each real number, for which they are two points in your $\mathcal P$, such that the distance between them equals exactly this real number.

Hope that helps you!