Jason Cawley
Wolfram Science Group
Phoenix, AZ USA
Registered: Aug 2003
Posts: 712 |
The key bit of code to make 3D pictures of CAs is shown in the implimentation note on page 927, at the very end of the note. For clarity, you can use a function like -
displayCA3D[array_] := Show[Graphics3D[Map[Cuboid[-Reverse[#]] &, Position[array, 1]]]]
- to display data from a 2 color, 2D CA in three dimensions. That will just show a block ("cuboid") at the position of every "1" in the pattern. You might make that an initialization cell in a notebook to investigate these, for example.
Then any of the 2D CA evolutions we've already talked about - provided it is 2 color - can be stored in a variable as "data", and passed to the above to display it. (For more than 2 colors, you need to modify the function above, because it just stacks cubes where it sees 1s in the underlying data, ignoring everything else).
The picture on page 180 comes from outer totalistic, 9 neighbor CA number 174826. The picture on page 172 comes from outer totalistic, 5 neighbor rule 942. So, using what we discussed in the previous thread, you can get the underlying data from -
data = CellularAutomaton[{174826,
{2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}},
ReplacePart[Table[0, {100}, {100}],
1, Table[{50, 44 + i}, {i, 1, 11}]],
30];
or
data = CellularAutomaton[{942,
{2, {{0, 2, 0}, {2,1, 2}, {0,2, 0}}}, {1, 1}},
ReplacePart[Table[0, {80}, {80}],
1, {40, 40}],
30];
Then show either one with
displayCA3D[data]
I'll explain the lines in the "data =" parts again. The first bit is just the call to CellularAutomaton and the rule number. The next bit specifies 2 color, outer totalistic and either 9 or 5 neighbors, using the "kernel" matrix for cell weights (and range 1 in each of 2 dimensions, the {1,1} at the end of that line). The next bit sets up an empty array, 100x100 or 80x80, and says to insert the part that comes next, into that blank array, as an initial condition. The next bit says what black cells to start with, where - a row 11 cells long in the first case, a single black cell in the second, in either case centered in the blank array. The last bit just says to evolve the CA for 30 steps.
I recommend only doing one of these 3D graphics in a single notebook, because you will tax your memory when actually rendering the 3D graphic. The data is a snap. But your CPU will have to work to render the 3D graphs of stacked cubes, and your memory will probably start paging - and thus slow down considerably - if you try to do two of them at once. Also, keep the number of steps low and the size of the background array managable. If you've pushed your machine too far, try aborting the evaluation and then retry with a smaller number of steps. Once the evaluation has finished, grab the corner of the graphic and drag it to expand the picture. You will see a lot more of the pattern, clearly, when you have a modest number of steps expanded to fill half the screen.
I hope this helps. Fine questions, by the way.
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12-16-2004 10:07 AM
