Conference rusure::math

931.0. "Neighbors in Matrix?" by LARVAE::TURNER (Mark Turner) Mon Sep 19 1988 12:56

How many neighbors does a cell in a 4-D matrix have?  

My notion of "neighbor" may be a bit peculiar, so here it is:

	1-D: two neighbors
                
		1#2
	
	2-D: eight neighbors:

		123
		4#5
		678

	3-D: 26 neighbors:

             Thinking of cells as cubes:

		 6 (one for each face)
	      + 12 (one for each edge)
	      +  8 (one for each corner)
	      ----
                26

	4-D: ??

    I can't visualize this in 4-D; does the notion even apply there?
    All help appreciated.


					Mark

    In n-dimensional space, the position of a cell is given by:

	(x , x , ..., x )
	  1   2        n

    Its neighbors (including itself) are the cells:

	(x +d , x +d , ..., x +d )
	  1  1   2  2        n  n

    where the d 's may independently equal -1, 0, or +1.
               i
                                             n
    Thus, in n-dimensional space, there are 3 - 1 proper neighbors of a cell.

    1-D:	2	0+2(3**0)
    2-D:	8	2+2(3**1)
    3-D:	26	8+2(3**2)
    4-D:	80	26+2(3**3)
    5-D:	242	80+2(3**4)

    Thanks!  
    
        -- Mark