Oops, forgot to pack this extra piece. There’s no way that’s gonna...wait WHAT?
A brief discussion of what's going on:
To start, let's do this trick in 2D first. We start with two squares, where one is the goal size and another slightly smaller square will be chopped into puzzle pieces.
To start, let's do this trick in 2D first. We start with two squares, where one is the goal size and another slightly smaller square will be chopped into puzzle pieces.
If we take four copies of the smaller square and position them at the corners of the goal square, we get this:
If we then rotate the blue squares about their centers until their sides touch, we get this:
By trimming this set of blue squares to the size of the original red goal square, we see the resulting puzzle pieces:
The four blue pieces can be rearranged back into a single blue-sized square, which is how the puzzle would start, with a small red piece sitting on the side.
In 3D, it is essentially the same trick. We start with a red goal cube, and define a slightly smaller blue cube that will get chopped up for puzzle pieces.
Then, we position eight copies of the blue cube at the corners of the red cube. Note that there is a small gap between them, just as there was in the 2D example.
Next we rotate the blue cubes about their centers until the sides touch. In 2D, this was a simple rotate about the center point. In 3D, this is a rotate about the (1,1,1) axis of each blue cube.
This is easier to see if we trim the blue cubes by the size of the red cube. These are the resulting puzzle pieces.
The additional piece (which was just a small square in 2D) is the portion of the red cube that remains after the blue cubes are removed. These two images show the remaining geometry before and after the rotations.
That's it. It should be noted that the simple 2D example produced four identical puzzle pieces. Perhaps surprisingly, the 3D cube subdivides into eight distinct pieces. Which makes it a bit of a challenge to assemble in either configuration.
As I’ve had more time to play with the pieces, I realize there only four distinct piece shapes. The (1,1,1) piece and the (-1,-1,-1) piece are unique, and (because of three-fold rotational symmetry) there are two other types of pieces which each occur three times.
