Andrew Miller

29 January 1996

MA8

Four-dimensional Cube Conundrum

To determine the number of edges on a four-dimensional cube, I examined the construction and number of edges on a point, a square, and a cube. I could not visualize a four-dimensional "cube", but could calculate the number of its edges by creating a series of coordinates to describe each figure and extrapolating the information to "observe" the properties of a four-dimensional cube. If put on the set of coordinate axes (w,x,y,z), the figures would have the following points in space:

Point: (0,0)

Square: (0,0), (1,0), (0,1), (1,1)

Cube: (0,0,0), (0,0,1), (0,1,1), (1,1,1), (1,0,0), (1,0,1), (0,1,0), (1,1,0)

4Dcube: (0,0,0,0), (0,0,0,1), (0,0,1,1), (0,1,1,1), (1,1,1,1), (1,0,0,0), (1,1,1,0), (1,0,1,1), (1,0,0,1), (1,0,1,0), (0,1,0,1), (1,1,0,0), (1,1,0,1), (0,1,1,0), (0,1,0,0), (0,0,1,0)

With these 16 points for a theoretical 4-dimensional cube, one has to determine which points are connected to which. To do so, I looked at the pattern for a 2-D square and 3-D cube. The points that connected were only ones that varied by a total of one coordinate. In other words, the difference between the sum of the coordinates of each point has to equal one. For example, the coordinates (0,0) and (0,1) for a square connect, but not (0,0) and (1,1) or (0,1) and (1,0). The coordinates (0,1,0,0) and (0,1,1,0) connect, but (0,1,1,1) and (1,1,0,1) do not. If one can assume that the same rule holds for a 4-dimensional "cube", then one can simply list the pair of points that should connect, which I do, below:

(0,0,0,0) + (0,0,0,1)

(0,0,0,0) + (0,0,1,0)

(0,0,0,0) + (0,1,0,0)

(0,0,0,0) + (1,0,0,0)

(1,1,1,1) + (1,0,1,1)

(1,1,1,1) + (1,1,0,1)

(1,1,1,1) + (1,1,1,0)

(1,1,1,1) + (0,1,1,1)

(0,0,0,1) + (1,1,0,0)

(0,0,0,1) + (0,1,1,0)

(0,0,0,1) + (1,0,0,1)

(0,0,0,1) + (1,0,1,0)

(0,0,0,1) + (0,0,1,1)

(0,0,0,1) + (0,1,0,1)

(1,1,0,0) + (1,0,0,0)

(1,1,0,0) + (0,1,0,0)

(1,1,0,0) + (0,0,1,0)

(0,1,1,0) + (1,0,0,0)

(0,1,1,0) + (0,1,0,0)

(0,1,1,0) + (0,0,1,0)

(1,0,0,1) + (1,0,0,0)

(1,0,0,1) + (0,1,0,0)

(1,0,0,1) + (0,0,1,0)

This should show the number of edges on a 4-dimensional cube because it follows the definition of a "cube" independent of dimension. Each point connects to four others. Similarly, in three dimensions, each point connects to three others. For a square in two dimensions, each point connects to two others. This technique, although cumbersome, should work for a determining the number of edges on a "cube" in any dimension. There are a total of 32 pairs, and, hence, 32 edges on a 4-dimensional cube.