Vi Hart — Mathematical Food
The Hyperbolic Nature of Dried Fruit Slices:
It's Actually Pretty Weird that Stuff Curls Up When It Dries
I used to take it for granted that stuff curls up when it dries. I was so used to this being the way the world worked that I didn't even notice how strange it was that drying something, instead of simply shrinking it, changed the entire geometry of the object. There's a simple way to describe the nature of this shrivelled shape: it's hyperbolic!
The hyperbolic plane can take many forms. In hyperbolic space it would be nice and flat, but it doesn't fit so well in our Euclidean space. Even if you take a small piece of it, it curls up in various ways, the same ways that a slice of dried apple might curl up. Most people, even mathematicians who have studied hyperbolic geometry in depth, have seen at most one or two visualizations of how bits of hyperbolic plane might actually look when curled into three dimensions, and even those are usually very simple and symmetric. This would explain why human kind went so long without noticing that dried fruit slices are hyperbolic planes.
But why do they do this? How could I model, mathematically, the transformation that turns a flat Euclidean slice into a hyperbolic one? To try to gain insight, I started doing a few experiments. One theory was that it was the difference in shrinkage across the slice due to the skin. If the skin couldn't shrink, but the meat of the apple tried to shrink away from it, it would result in the observed curling. The meat of the apple would try to form a sort of minimal surface across the comparatively rigid skin. This was easy to test: I simply tried drying some naked slices.
As you can see, removing the skin made it curlier, if anything. The above slice was also taken far from the center of the apple, where I hoped any potential effects from the core would be minimized. I also wanted to test the effects of gravity. Drying them flat made them want to curl up, despite that gravity would have them stay flat, but maybe suspending them would allow them to take a more natural form.
I tried two with skin and two without skin, rigged up on a wire stand while I dried them in a toaster oven at low heat for a few hours. All came out clearly hyperbolic for the most part, with one exception being the center point where the wire poked through:
I suspect this middle point is due to the heat of the wire combined with gravity making the slice want to slump over around it. I'll have to do another test where I suspend the apple from a different point.
Another test was to cut the apple from top to bottom instead of across the core.
The slices may not be as pretty and symmetric, but they definitely approximate hyperbolicness.
From these experiments, I've concluded that stuff curls up when it dries not just because of inconsistencies in the medium (such as peel, core, etc) but because of something fundamental to the drying process of something cellular. This is why we see the same sort of curly shapes in other dried food such as potato chips.
Many thanks to the incredible Patrick Desjardins, who started me on this path by musing that the above hyperbolic figure study looks like a withered apple slice.
Besides drying apple slices or drawing the hyperbolic plane, I also have pages on how to make it out of beads, bottles, and balloons. Or, if you crochet, I recommend this book: Crocheting Adventures With Hyperbolic Planes.
For more cool mathfood, see the Mathematical Food Index.