Hypercube definition


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From The Collaborative International Dictionary of English v.0.48 [gcide]:

  hypercube \hypercube\ n.
     A mathematical object existing in more than three dimensions,
     analogous to the cube in that each two-dimensional facet of
     the surface is a square; a generalization of a cube in more
     than three dimensions.

From The Free On-line Dictionary of Computing (27 SEP 03) [foldoc]:

          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}.

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