Up: Math 53 Selected Course Notes

# A Cube Passes Through Flatland:

 This movie shows the cube passing through Flatland face first. This is our original method of undertanding the cube using only two-dimensional methods. Here, we used time as the third dimension, and we thought of a cube as "a square existing for a while". This movie shows the cube passing through Flatland edge first. Here, the slice begins as an edge, then becomes a rectangle; the rectangle grows, becomes a square for a moment, and then gets wider than it is tall. At its widest, it is as wide as the diagonal of one of the square faces of the cube. The rectangle then shrinks back to an edge at the top of the cube. The most interesting slicing sequence for the cube is when it passes though flatland corner first. In this case, the initial contact is a point, which then becomes a small equilateral triangle. This triangle grows until it touches three of the corners of the cube (the three edges are sweeping out three of the faces of the cube, and are now half-way through each of these faces). At this point, the corners of the triangles begin to be cut off by the other three faces of the cube. The slice then becomes hexagonal, and at the half-way point, the slice is a regular hexagon, with the slice cutting each of the six faces of the cube in exactly the same way. As the cube progresses through Flatland, the slice turns again into a cut-off triangle (but inverted with respect to the original one) and finally becomes an equilateral triangle once more as three more vertices pass through Flatland. This triangle shrinks down to a point and disappears.

Up: Math 53 Selected Course Notes