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Symmetry

 

 

Notation

The indices for a cell in a d-dimensional hyper cube are represented as .

The value for the number in a cell of a d-dimensional hyper cube, whose order ("row size") is n, is represented by a d-tuple 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 Controller

A direction controller for for a d-dimensional hyper-cube is a d x d matrix whose elements are either 1, or -1.

1 1 1 1 1 1 1
1 -1 -1 -1 -1 -1 -1
1 -1 1 1 1 1 1
1 -1 1 -1 -1 -1 -1
1 -1 1 -1 1 1 1
1 -1 1 -1 1 -1 -1
1 -1 1 -1 1 -1 1

The matrix above is a direction controller for a 7-dimensional hyper cube.  Notice the arrangement of +1, and -1.

 

1 1 1
1 -1 -1
1 -1 1

This matrix is a direction controller for a 3-dimensional hyper cube, such as the "screen-shot" 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 0-dimensional axis.  The next to last row indicates the increment along the 1-dimensional axis.  The top row indicates the increment along the (d-1)-dimensional axis.

Using the hyper cube illustrated above, the example below illustrates how to proceed along the 1-dimensional axis from the value 64, to the value 83:

The next example illustrates how to proceed along the 2-dimensional 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.