[Math] Four Dimensional Origami Axioms

Question

What are the axioms of four dimensional Origami.

If standard Origami is considered three dimensional, it has points, lines, surfaces and folds to create a three dimensional form from the folded surface. This standard origami has seven axioms which have been proved complete.

The question I have is whether these same axioms apply to four dimensional origami which has lines, surfaces, shapes, and produces shapes folded along surfaces in the fourth dimension. Or are there more axioms in four dimensional Origami.

I cannot find this question approached in the literature. Four Dimensional Origami does not seem to be a field of study yet.

Answer 1

In the $n$-dimensional origami question, you start with a generic set of hyperplanes and their intersections, which can then be some collection of $k$-dimensional planes. An "axiom" is a set of incidence constraints that determines a unique reflection hyperplane, or conceivably a reflection hyperplane that is an isolated solution even if it is not unique. The space of available hyperplanes is also $n$-dimensional. Each incidence constraint has a codimension. A set of independent constraints makes an axiom when their codimensions add to $n$.

It is easy to write down a simple incidence constraint on a reflection $R$ and compute its codimension. If $n=2$, then the simple constraints are as follows:

1. $R(L) = L$ for a line $L$,
2. $R(x) = x$ for a point $x$,
3. $R(x) \in L$,
4. $R(L_1) = L_2$, and
5. $R(x_1) = x_2$

The first three constraints have codimension 1 and the last two have codimension 2. Formally there are 8 ways that to combine these constraints to make axioms. However, 1 cannot be used twice in Euclidean geometry, so that leaves 7 others. These are the seven axioms listed on Robert Lang's page. As it happens, Huzita missed combining 1 and 3. If you could have hyperbolic origami, you would have 8 axioms.

You can go through the same reasoning in $n=3$ dimensions. The simple constraints can again be written down:

1. $R(x) = x$ for a point $x$ (1)
2. $R$ fixes a line $L$ pointwise (2)
3. $R(L) = L$ by reflecting it (2)
4. $R(P) = P$ for a plane $P$ (1)
5. $R(x) \in L$ (2)
6. $R(x) \in P$ (1)
7. $R(L) \subset P$ (2)
8. $R(x_1) = x_2$ (3)
9. $R(L_1) \cap L_2 \ne \emptyset$ (1)
10. $R(P_1) = P_2$ (3)

I've written the codimensions of these constraints in parentheses. As before, you can combine these constraints. Certain pairs, such as 3 and 4, can't combine. You also can't use 4 three times. If you go through all of this properly, it is not all that hard to make a list of axioms that resembles the ones in 2 dimensions. (Possibly I made a mistake in this list or missed something, but it is not hard to go through this properly.)

However, there is a possible subtlety that I don't know how to address. Namely, suppose that you do make some complicated configuration using combinations of these constraints, at first. Can you create a configuration with the property that the codimensions don't simply add? For instance, ordinarily you can't use condition 7 twice to define the reflection $R$, because the total codimension is 4, which is too large. However, if the lines and planes in this condition are related to each other, is the true codimension sometimes 3? I would guess that you can make this happen. If so, then potentially you'd have to add "unstable" construction axioms to the list. But then it is not clear whether an unstable construction axiom is actually needed, or whether an unstable axiom can always be replaced by a sequence of stable axioms.