Up: To the Fourth Dimension and Beyond!

# Cube and the Hypercube Shadows:

 222K
This movie shows a square in a plane moving into the third dimension to form a cube. The cube's shadow in the plane is added, and the square moves again so that you can see how its shadow moves in the plane. The cube is then removed and the viewpoint changes so that we look straight down on the plane and see the movement again. The action now takes place entirely in the plane, and you (or A. Square) must "imagine" the three-dimensional object that has this shadow.

This gives a depiction of the cube that is entirely two-dimensional, so it can be understood by the Flatlanders. It is imperfect, however, in that the parts of the cube seem to lie ontop of itself in two dimensions (for example, the red and blue squares overlap for a while) while we know this is not what is happening in three dimensions.

 329K
This movie is analogous to the previous one in that it shows the three-dimensional "shadow" of a cube moving through a fourth dimension to form a hypercube. This is the same as the tail end of the other movie, where all the action took place in the plane where the Flatlanders could see it. We now must think of ourselves as the "Flatlanders" in a flat 3D world at the bottom of a 4D space. The picture we see here appears to take place entirely in our three space, but is really just a shadow of the movement in the fourth dimension.

The movie first shows the red and blue cubes in the same location (it looks kind of purple). Then the red cube begins to move away from the blue cube into the fourth dimension and what we see is its shadow overlapping the blue cube and moving aside in three dimensions. Its corners trace out the edges of the hypercube as it goes. Then two of the sides of the cubes are removed so we can see inside them easier and the movement is repeated. The shadow of the hypercube is then rotated in 3-space so we can see it from another angle and the red cube moves back down until it overlaps the blue one again.

Up: To the Fourth Dimension and Beyond!