Math and Origami

Question

I am working on a project for class about the mathematics behind origami and write now I am looking into what is and is not constructible. I've gotten to the definition of origami constructible points which says the following:

The set of origami constructible points $P_0$ is defined as:
$P_0=\{P|(0,0),(0,1) \in P$ and $P$ is closed under origami constructions$\}$.

I am having a very difficult time understanding what this is saying and it is a key point in my project. If anyone could explain this to me I would greatly appreciate it.

Answer 1

It seems you should actually start with $\{(0,0),(0,1),(1,0),(1,1)\}$ as you have all four corners of the paper. Then you define what constructions are allowed. I imagine that folding on a line through any points you already have is one. Another would be bringing together two pairs of points that are the same distance apart. There may be more operations permitted. The points generated are from the intersections of fold lines and the edges of the paper. After you generate some points, you look what constructions are available using those points.

For an example, you can fold the paper in half horizontally, making a crease along $(0,\frac 12)$ to $(1,\frac 12)$ This generates the points $(0,\frac 12)$ and $(1,\frac 12)$ where the crease hits the edges. Now folding the bottom to that crease gets you $(0,\frac 14)$ and $(1,\frac 14)$ and it should be clear you can get all the dyadic fractions this way. Then you can fold $(1,0)$ to $(\frac 12,1)$ making a diagonal line and crossing many of the horizontal and vertical lines you already have. You imagine keeping this up as much as necessary until you can't get any new points. The set $P_0$ the set of points you can generate. It will be countably infinite, but every point in it can be reached with a finite number of operations.

Answer 2

I am guessing you are asking about Definition 3.3 on page 10 of The Mathematics of Origami by Sheri Lin, which cites a Math Monthly paper by Auckly and Cleveland, a version of which is available here. The Lin paper omits a symbol: It should read

$$P_0 = \cap\{P | (0,0),(0,1)\in P \text{ and $P$ is closed under origami constructions}\}$$

What's missing is the intersection symbol. Does this clarify matters?

Added later: The "$\cap$" in effect means that you're defining the smallest set containing two starting points that's closed under origami constructions. (A tacit lemma is that the intersection of two "closed" sets is again "closed.") You need two points to get a nontrivial smallest set. The intersection is well defined because at the very least you have the entire set of points in the plane as a set $P$ that is closed under origami constructions.

One might alternatively define the constructible points as the union of the finite sets that are produced by applying origami procedures to obtain "new" points out of "existing" ones, but to do so one would have to keep track of lines along the way as well. The paper(s) chose a cleaner approach in which they define a notion of origami pair (any set of points and lines satisfying their origami axioms), followed by a definition of a point set being closed under origami construction (namely, being part of an origami pair), followed by the definition of "the" set of constructible points.

I hope this helps.