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You will write a simulation of frost.
The frost simulation takes place in an M-by-N square grid of cells. N
and M are both even. The top and bottom edges of the grid are
connected, as are the left and right edges, so that the grid actually
represents a torus.
Each cell may contain vacuum, a vapor particle, or an ice particle.
Initially, there is one ice particle. in the middle of the grid, and
the rest of the cells are filled with a random distribution of vacuum
and vapor.
Time is discrete, measured in 'ticks', by a counter. At each time
step, the grid is divided into "Margolus neighborhoods", which are 2x2
blocks of adjacent cells. If a neighborhood contains an ice particle
at time T, then all the vapor cells in the same neighborhood change to
ice at time T+1. If a neighborhood contains only vapor and vacuum at
time T, then its contents rotate a quarter turn clockwise or
counterclockwise, at random, at time T+1.
For example, using ' ' to indicate vacuum, '.' to indicate vapor, and
'*' to indicate ice:
+--+ +--+
|.*| ==> |**|
| .| | *|
+--+ +--+
+--+ +--+
| .| ==> | *|
|* | |* |
+--+ +--+
+--+ +--+ +--+
| .| ==> |..| or | | (50% chance either way)
| .| | | |..|
+--+ +--+ +--+
+--+ +--+ +--+
|. | ==> | .| or | | (50% chance either way)
| | | | |. |
+--+ +--+ +--+
+--+ +--+
|**| ==> |**|
| | | |
+--+ +--+
+--+ +--+
| | ==> | |
| | | |
+--+ +--+
The division of the grid into neighborhoods changes from tick to tick.
(If the neighborhoods were always the same, the vapor particles would
be confined to one neighborhood for their entire lifetime, which is
not realistic.) Suppose for concreteness that the grid is 4x6. On
even-numbered ticks, cell (0,0) will be in the upper-left corner of
its neighborhood, which implies that the grid is divided into
neighborhoods like this:
+--+--+--+
| | | |
| | | |
+--+--+--+
| | | |
| | | |
+--+--+--+
On odd-numbered ticks, cell (0,0) will be in the lower right corner of
its neighborhood, which divides the grid into neighborhoods like this:
| | |
-+--+--+-
| | |
| | |
-+--+--+-
| | |
(Remember that the edges of the grid wrap around, so there are still
six neighborhoods here:
1|22|33|1
-+--+--+-
4|55|66|4
4|55|66|4
-+--+--+-
1|22|33|1
)
Consider what might happen to a single vapor particle:
T=0:
+--+--+--+
|. | | |
| | | |
+--+--+--+
| | | |
| | | |
+--+--+--+
Let's say that the neighborhood around the particle happens to rotate
clockwise, yielding:
T=1:
|. | |
-+--+--+-
| | |
| | |
-+--+--+-
| | |
Suppose the new neighborhood turns clockwise:
T=2:
+--+--+--+
| | | |
| | | |
+--+--+--+
| | | |
| .| | |
+--+--+--+
Suppose the new neighborhood turns counterclockwise:
T=3:
| | |
-+--+--+-
| | |
|. | |
-+--+--+-
| | |
Suppose the new neighborhood turns counterclockwise:
T=4:
+--+--+--+
| | | |
| | | |
+--+--+--+
| |. | |
| | | |
+--+--+--+
As you can see, using these rules, vapor particles can wander freely,
but the total number of vapor particles is always conserved. If a
grid starts out with a high vapor density in one place and a low
density elsewhere, the densities will soon equalize.
+--+--+--+ +--+--+--+
| | | | |. |. | .|
| | | | |. |. |. |
+--+--+--+ ==> +--+--+--+
|..|..|..| |. | |. |
|..|..|..| |..|. | |
+--+--+--+ +--+--+--+
Your program should have some method for graphic display of the grid
contents at each step, some method for specifying the initial vapor
density, and whatever else you find useful.