A cube of more than three dimensions. A single (2^0 = 1)
point (or "node") can be considered as a zero dimensional
cube, two (2^1) nodes joined by a line (or "edge") are a one
dimensional cube, four (2^2) nodes arranged in a square are a
two dimensional cube and eight (2^3) nodes are an ordinary
three dimensional cube. Continuing this geometric
progression, the first hypercube has 2^4 = 16 nodes and is a
four dimensional shape (a "four-cube") and an N dimensional
cube has 2^N nodes (an "N-cube"). To make an N+1 dimensional
cube, take two N dimensional cubes and join each node on one
cube to the corresponding node on the other. A four-cube can
be visualised as a three-cube with a smaller three-cube
centred inside it with edges radiating diagonally out (in the
fourth dimension) from each node on the inner cube to the
corresponding node on the outer cube.
Each node in an N dimensional cube is directly connected to N
other nodes. We can identify each node by a set of N
Cartesian coordinates where each coordinate is either zero
or one. Two node will be directly connected if they differ in
only one coordinate.
The simple, regular geometrical structure and the close
relationship between the coordinate system and binary numbers
make the hypercube an appropriate topology for a parallel
computer interconnection network. The fact that the number of
directly connected, "nearest neighbour", nodes increases with
the total size of the network is also highly desirable for a
parallel computer.