

NotationThe indices for a cell in a ddimensional hyper cube are represented as . The value for the number in a cell of a ddimensional hyper cube, whose order ("row size") is n, is represented by a dtuple of integers in base n . For example, the hyper cube illustrated below has dimension 3 and order 5. In this hyper cube, the integers in each 3 tuple are represented in base 5, since the order is equal to 5:
and
Direction ControllerA direction controller for for a ddimensional hypercube is a d x d matrix whose elements are either 1, or 1.
The matrix above is a direction controller for a 7dimensional hyper cube. Notice the arrangement of +1, and 1.
This matrix is a direction controller for a 3dimensional hyper cube, such as the "screenshot" illustrated above. A direction controller indicates the increment in the tuple values between adjacent cells. The last row in a direction controller matrix indicates the amount that is added to each digit in the tuple to calculate the value for the tuple in the adjacent cell along the 0dimensional axis. The next to last row indicates the increment along the 1dimensional axis. The top row indicates the increment along the (d1)dimensional axis. Using the hyper cube illustrated above, the example below illustrates how to proceed along the 1dimensional axis from the value 64, to the value 83: The next example illustrates how to proceed along the 2dimensional axis from 64 to 90: The addition used to increment each tuple is performed modulo the order ("row size") of the hypercube. There are no "carried digits"; this means that if the sum of two digits is greater than the base in which the numbers are represented, the sum does not affect the digits to its left. 
