How similar can the nets of distinct polyhedra be?

Question

My school, and most math books do not cover 3-d geometry well, especially the topics of polyhedron nets. However, I see quite a few questions here are being answered about them. I was wondering about these questions:

a) Can two distinct polyhedra have the same net?
b) Can we cut the nets of two different polyhedra by their edge in two pieces so that the two pieces of each polyhedron net are congruent?

To explain (b) further: If we take the net of polyhedron P and the net of polyhedron Q, can we cut the net of polyhedron P (along an edge) into pieces A and B and cut the net of polyhedron Q (along an edge) into pieces C and D such that A is congruent to C and B is congruent to D?

Answer 1

The answer to (a) depends on what you exactly called a net of a polyhedron.

If a net is only a finite collection of edge-joined polygons, then two exact same nets can be folded into two different polyhedra. For instance, an octahedron net can also be folded into a boat.

Now, you can also ask for the pattern for gluing the net edges and only one polyhedron can be folded from it, it's Alexandrov's uniqueness theorem.

Answer 2

Very long past the question, but it turns out that the answer to (a), and therefore trivially (b) is yes. In fact, two distinct cuboids can have the same net. See this youtube video for an in depth discussion, including examples of one net which can be folded into three distinct cuboids.

Here is an example of a net which can be folded into either a $1$x$1$x$5$ cuboid, or a $1$x$2$x$3$ cuboid (the net is the purple bit).