What Vi pointed out to me the other day was that an apple slice is exactly the opposite of blowing a bubble in the following way: When you start to blow a bubble, say through one of those little plastic bubble blower rings, the ring itself doesn’t change size, but the soap film gets bigger (because of all that air pressure you’re adding on one side).
What happens of course is that the soap film bulges out, and starts to form part of a sphere. The harder you blow, the bigger part of a sphere you get. So you see, when you increase the area of a soap film, but not the size of its perimeter, it starts to go from flat to spherical.
But what is going on with a dried apple slice? Vi pointed out to me that the area of the apple slice starts to shrink as it loses water. But the perimeter of the apple slice doesn’t go down by as much. So it’s the opposite of a soap bubble — instead of a surface with too much area for its perimeter, we have a surface with too little area for its perimeter.
So if a too-much-area-for-its-perimeter shape is a section of a sphere, what is a too-little-area-for-its-perimeter shape?
Vi gave a wonderful answer to this last week at the annual Joint Mathematics Meeting, which you can see in this video.
One thing I haven’t seen much mention of in these presentations is the topological variability of hyperbolic surfaces. Having six equilateral triangles meeting at each vertex you can make a flat surface, but you can also make a cylinder (this is the idea behind buckytubes as opposed to graphene.) With seven equilateral triangles meeting at a vertex you can make a wrinkly sheet like these crochet patterns or the apple slice you mentioned, but you can also make intricate surfaces filled with intersecting tubes and tunnels. There’s an infinite variety of different arrangements of these tunnels while still keeping the same number of triangles meeting at every vertex.
Another neat way to get a surface like this is to rip a plastic garbage bag. The edge near the rip stretches more than the rest of the bag, and so it forms these fractal wrinkles on wrinkles.