Movement in Play Space
Play space is cube N units in each dimension.
* Ordered triple
Current position: (x0, y0, z0)
Next position: (x1, y1, z1)
Play space boundary:
0 <= x < N AND 0 <= y < N AND 0 <= z < N
Valid moves:
|x1 - x0| <= 1 AND |y1 - y0 | <= 1 AND |z1 - z0 | <= 1
* Array
Index numbers assigned sequentially parallel to x axis for each y and
each z.
Play space boundary:
Position index p: 0 <= p < N^3
Transform functions:
x(p) = p % N
y(p) = p % N^2 / N
z(p) = p / N^2
Valid moves:
|x(p1) - x(p0) | <= 1
|y(p1) - y(p0) | <= 1
|z(p1) - z(p0) | <= 1
(Or enumerate)
* Statistics
** Cubic lattice interior node ratio
<interior nodes>/<total nodes>
interior nodes = (n - 2)^3 (n: n >= 3)
total nodes = n^3
interior ratio = (n - 2)^3 / n^3
| n | total | ratio | remarks |
|----+-------+---------+---------|
| 3 | 27 | 1/27 | |
| 4 | 64 | 1/8 | |
| 5 | 125 | 27/125 | |
| 6 | 216 | 8/27 | |
| 7 | 343 | 125/343 | |
| 8 | 512 | 27/64 | |
| 9 | 729 | 343/729 | |
| 10 | 1000 | 64/125 | > 1/2 |
| | | | |
