Vi Hart — Mathematical Food
Apple Platonic Solids
Apples are one of the most delicious and easy-to-carve materials readily available, along with other fruits, and conveniently come in spheroids that are perfect for slicing into convex polyhedra! Besides the obvious aesthetic benefits of having a set of platonic applehedra of your own, they are also good for slicing and truncating. I used Granny Smiths because they have firm flesh and don't brown as fast as some types of apples (of course one could also use other fruits... see bottom of page!). Making these was an interesting exercize for me mathematically, because though I knew intellectually how to slice a sphere to have the slice intersect a set of points, it felt a bit unintuitive to actually do.
Perhaps the easiest platonic solid to carve is the cube, not because it is the simplest (that honor goes to the tetrahedron) but because it is the most familiar. It's also great for slicing into hexagons. The cube has eight vertices (corners), and I start by marking one on the stem and one at the base. Three more go symmetrically around the stem, about a third of the way down the apple, and the last three go around the base, about a third of the way up, so that the six points around the circumference make a zigzag.
You can see how the dots can be connected into quadrilaterals that approach squareness. I started by slicing just a little off each square face, and as I sliced more it became easier to see the cube within the apple. Once the apple is a little cubeish, it's easier to ignore the dots and cut how you need to to get squares and right angles.
The tetrahedron has four vertices and four triangular sides. I make one the stem, and the other three are arranged around the base of the apple. The first slice takes off the bottom of the apple, and three more perfect slices would finish the tetrahedron, one from the stem to each pair of vertices. I find this shape to be a bit unintuitive to carve because, unlike the other platonic solids, when it is sitting flat on a face it has a vertex on top rather than an opposite face.
It's easy to mark the vertices of the octahedron because it is the dual of the cube. The six vertices go where the middle of the faces of a cube would be: top and bottom, front and back, side and side.
I make four conservative slices meeting at the top, do the same to the bottom, and then work on evening it out. Good knowledge of the octahedron helps here, but a good guide is to lie it flat on each side and see if the opposite side is parallel.
The icosahedron is by far the most difficult, in my opinion. Its twenty sides are difficult to make distinct and triangular. Its twelve vertices are not too tough to roughly place, though: one each on top and bottom, and then five around the top and five around the bottom to create a zigzag around the equator.
The rest is just very very careful slicing, until hopefully you get something like this:
The dodecahedron is a little easier. It has twenty vertices, which sounds like a large number, so I started by placing the vertices the same as the icosahedron and then, because the dodecahedron is the dual of the icosahedron, I make those points the center of the faces instead, and draw a sort of voronoi diagram around them. This should give me twelve pentagons.
Slice slice slice and, hopefully:
These techniques can be applied to other fruits as well, such as melons, which also often start off much more spherical than apples which makes it easier to carve them into regular polyhedra:
Fruits often have cores or seeds to deal with, but in the above melon it created a nice opportunity to do it Leonardo style. This style also works well with pumpkins:
For more cool mathfood, see the Mathematical Food Index.