Blooming caterpillars
When a polyhedron’s surface can be cut and flattened into a polygon, the resulting flattened shape is called a net, and the system of cuts is called an unfolding. A familiar example is the Latin cross of six squares, unfolded after cutting seven edges of a cube. This example is an edge unfolding: its cuts are on edges of the polyhedron, and the resulting net consists of faces of the polyhedron connected edge-to-edge. We don’t know whether all convex polyhedra have edge unfoldings (they do always have unfoldings of other types). We also don’t know whether every net has a blooming, a continuous motion from the cut surface of the polyhedron to its flat unfolded state, throughout which the moving surface avoids crossing itself and stays flat except at the uncut edges of the polyhedron. (Nets do always have continuous flattening motions if bending the faces is allowed; see my work on unfolding simply-connected developable surfaces). And when a net does have a blooming, we don’t know whether we can restrict the blooming to only increase dihedral angles monotonically.
Here’s a Wikipedia animation of a blooming for an edge unfolding of a regular dodecahedron, created by Chang Yu-Fan:
A partial answer to whether nets have bloomings was given in a 2011 paper by Demaine, Demaine, Hart, Iacono, Langerman, and O’Rourke, “Continuous blooming of convex polyhedra”. They showed that bloomings exist for every serpentine unfolding, an unfolding for which each uncut component of a polyhedron face is hinged along only two uncut polyhedron edges. The resulting net consists of a linear sequence of flat polygons connected by these hinges. The dodecahedron unfolding above is not serpentine, because the central pentagon touches five uncut edges instead of only two. Demaine et al. also construct a serpentine unfolding (but not an edge unfolding!) and its blooming for every convex polyhedron.
For serpentine edge unfoldings, it is very simple and intuitive to find a monotone blooming. Imagine that you have a solid physical model of your convex polyhedron, rolling face-to-edge-to-face across a flat tabletop, as you would do with countdown dice. Obviously, the tabletop will not obstruct this rolling motion. If you have a Hamiltonian path through the graph of adjacent faces of the polyhedron, as you would with well-constructed countdown dice and as you would with a serpentine edge unfolding, you can follow that path so that each face in turn is the one the polyhedron is resting on. Now, before rolling through the serpentine unfolding path in this way, wrap your polyhedron in its net, stuck very lightly onto the polyhedron (so it can peel off) with a much stronger adhesive on the outside surface, so that when the polyhedron rests on a face, that part of the net will become stuck to the table. Then if you roll through the face sequence of a serpentine edge unfolding, letting each face in turn stick to the table, the resulting motion is a monotone blooming! The parts of the net that are attached to the tabletop are attached in their final unfolded position in the net, and the parts of the net that are not yet attached to the tabletop are on the polyhedron, above the tabletop, so there is no self-intersection. At the end of the rolling sequence, all faces are stuck to the table in their flat unfolded positions.
This doesn’t always work as-is for serpentine unfoldings that are not edge unfoldings, causing complications for Demaine et al. The problem is that, at each step of the rolling process, a whole face of your polyhedron will lie flat on the tabletop, but only part of that face should detach from the polyhedron and stick down onto the tabletop. The rest of the face may need to stay attached to the polyhedron for now and detach later, and it might even have a two-dimensional region of overlap with the already-unfolded parts of the net. Some care with leaving the hinged edge angles slightly bent rather than completely flat can prevent this overlap. It is a bit messy, but these unfoldings do still always have monotone bloomings.
Instead, here, I want to describe a different class of edge unfoldings that always has monotone bloomings, constructed by an elaboration of the tabletop-rolling physical model. Serpenting unfoldings are unfoldings for which the graph of faces and uncut edges is a path; instead, I want to look at edge unfoldings for which the same graph is a caterpillar, a tree in which the internal vertices form a path.
To unfold a caterpillar edge unfolding, start in the same way with your net lightly attached to the polyhedron but glued on the outside so that it can stick more tightly to the tabletop. We are going to roll the polyhedron, but only along the path of faces that form internal vertices of the caterpillar. As we roll, the remaining faces that have not yet been stuck down to the tabletop in their final positions will still be attached to the polyhedron, above the tabletop, so there will be no self-intersections. But before we roll the polyhedron from one face to the next, find the leaves of the caterpillar adjacent to the face on which the polyhedron is resting. For each of these leaf faces in the net, one by one, peel it from the polyhedron and swing it down onto the tabletop. Each leaf swings through an empty wedge of space, between the polyhedron and the tabletop, with the hinged edge of the leaf forming the edge of the wedge. Because this wedge is empty, the leaf face is free to swing through it without self-intersection.
The Latin cross net of the cube can be an example. It is a caterpillar edge unfolding: its squares are connected in the pattern of a caterpillar, with a short path of two internal vertices. The following animation by “2DragonFreak” shows a different blooming for this net, where we also start with the cube sitting on one of its faces on a flat surface. Then, every other face moves everywhere all at once. Instead, our caterpillar unfolding consists of two kinds of steps, the steps where the polyhedron rolls onto a new face and the steps where adjacent faces swing down one at a time.
Instead, imagine starting with the whole cube sitting on the flat tabletop, with only its base square attached to the tabletop. Let three square walls of the cube (the three leaves of the caterpillar attached to the starting square) fall flat one at a time. This leaves the back right wall and roof still attached and in their original positions. Then, roll the parts that are still cubical (one remaining wall and the roof) onto the wall, keeping them at right angles as they roll together, and finally, let the square that started as the roof fall flat.
Unfortunately, it’s not obvious to me how to go beyond caterpillars in any general way. The connection pattern of the faces of any net always forms a tree, so the obvious approach is to find some way of recursively blooming some subtree of this tree while another part of the tree has already been flattened onto its final position and yet a third part is still in its original position on the polyhedron, waiting for the recursion to return. The difficult is doing this in a way that allows the motion in the recursive subtree to avoid both the flattened parts of the net and the remaining parts still on the polyhedron. In the caterpillar case, we did this by finding an empty wedge through which each face could swing, but in more complicated trees the free space has a more complicated shape and proving the existence of a continuous motion through it is correspondingly more complicated.
The caterpillar blooming has an extra property beyond monotonicity: the dihedral angles of the uncut edges change one at a time, once per edge, directly from the folded to the unfolded state. This seems special enough that maybe it is possible to find nets and unfoldings that do not have this kind of one-at-a-time blooming.