These pages walk you through the analogs of the cube in lower and higher dimensions, developing the sequence: point, line, square, cube, hypercube. It begins the investigation of the hypercube by counting some of its parts, and by locating the cubes that form the faces of the hypercube.
Spheres Sliced in 2D and 3D
Flatlanders can understand a sphere as a sequence of circles changing over time. The flatlanders see time as a third dimension, but we see the third dimension as a physical one. Similarly, we can understand a hypersphere from the fourth dimension as a sequence of spheres changing over time. We use time as a means of representing a fourth physical dimension.
Folding Cubes and Hypercubes
Here we look at how unfolding the square faces of a cube can help us to explain a cube to Flatlanders. The shadows can be seen by the people in Flatland, and they can try to use these shadows to interpret the folding that we are doing in three-space. Similarly, we can unfold a hypercube into three-space, and watch its shadow as it folds together in four-space.