Slicing the Cube Apple

It's obvious how to slice a cube into squares, but less intuitive is how to find a slice in the shape of a regular hexagon. The easiest way to show that it's possible is to slice up a cube-shaped fruit. I chose an apple.

If you don't have cubular apples available at your local orchard, you can make your own starting with the normal kind. Because I wanted the hexagonal slice to be perpendicular to the rotational symmetry axis of the apple (I want to cut across the core), I started by marking the top and bottom of the apple, to show that they will be corners of the eventual cube. I marked three more vertices evenly spaced around the stem of the apple and about a third of the way down, and did the same around the base, so that I had six vertices zig-zagging around the equator of the apple, and then used these as a guide. For more detailed instructions with photos, see this page.

Then, it's a simple matter of slicing!

Starting from a vertex, the first few slices of the cube are triangles. Soon, however, you approach three more vertices.

At this point, the slices become first irregular hexagons, which soon even out to a single middle slice where there is a regular hexagon (at least, a single hexagon when slicing across this axis. There are four in all).

And voila! You have found the hexagon within the cube. If you continue to slice, the hexagons will become irregular again, until they are triangles, until there is no more apple left.

Bon appetit!