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Todd Rowland
Wolfram Research
Maryland

Registered: Oct 2003
Posts: 113

Walls in Rule 73

Rule 73 has {0,1,1,0} as a static structure, regardless of the neighboring cells, which serve as walls (A). These walls effectively divide up space into separate regions, preventing information transfer between them, as discussed on p.699. The text suggests that if any run of consecutive blacks cells is odd then no walls will form. The conditions are a bit more complicated than that though, as demonstrated by {1,0,0,0,0,1} (C).

Also, the notes on p.885 and p.954 suggest that any even-length run of black cells will lead to a wall, or a structure which prevents information transfer, but it is also actually more complicated. For instance, {1,1,1,1} forms a growing cyclic pattern (B), which is a fairly common pattern for rule 73, and an example of information transfer can be seen by comparing graphics (E) and (F).

What is true is that if the runs of black cells and the runs of white cells are all of odd length, then no walls will form. The basic reason we can talk about rule 73 in this way is that it acts on runs of same color cells in a local way, without interference from the neighboring runs. After one step, a sequence of n black cells, delimited by white cells, becomes a black cell followed by n-2 white cells followed by a black cell. After one step, a sequence of n white cells, delimited by black cells, becomes a white cell followed by n-2 black cells followed by a white cell. When n==1, the cell becomes a white cell in either case. So if all the runs of white and black are odd, then on the next step all of the runs of black are odd. To see that the runs of white cells are odd, it suffices to check all possible cases. There is a unique degenerate case where an even run of whites is produced. It has to have a wrap-around boundary, and a sequence of alternating white and blacks (so the total length has to be even also). But in that case, the evolution reduces to alternating all white and all black. No structures {0,1,1,0} form in that degenerate case (D).

From the above analysis, it is clear that any run of black cells of length 4n+2 or any length of white cells of length 4n will produce a wall in its center after 2n steps, as can be checked by running those initial conditions.

The text on p.699 suggests that wall formation might play a role in universal behavior of rule 73. It seems like wall formation is an irreducible property in general, which is a good sign for universality. For instance, note the wall formation in graphic (F), which depends on the information transferred, compared with graphic (E) which has no walls.

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Old Post 04-05-2004 04:35 PM
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Jon Awbrey


Registered: Feb 2004
Posts: 558

Cactus Form Of Rule 73

Todd,

Just by way of incidental kibitzing,
I notice that Rule 73 has the form of
a "genus and species" or "pie-chart"
proposition, where q is the genus
and p and r are the species.

The cactus expression and
cactus graph are as follows:

o-------------------o
| ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` |
| ` ` ` p ` r ` ` ` |
| ` ` ` o ` o ` ` ` |
| ` ` ` | q | ` ` ` |
| ` ` ` o-o-o ` ` ` |
| ` ` ` `\ /` ` ` ` |
| ` ` ` ` @ ` ` ` ` |
o-------------------o
| ` ((p), q ,(r)) ` |
o-------------------o
| ` ` ` q_73` ` ` ` |
o-------------------o

See the discussion in and
around Cactus Rules Note 5.

http://forum.wolframscience.com/sho...tid=830#post830

Jon Awbrey

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Old Post 04-05-2004 06:28 PM
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inhaesio zha


Registered: Oct 2005
Posts: 403

some other partitioned systems

http://inhesion.com/zha/ca/snapshot...detail.cgi?a=23
http://inhesion.com/zha/ca/snapshot...etail.cgi?a=128
http://inhesion.com/zha/ca/snapshot...etail.cgi?a=186
http://inhesion.com/zha/ca/snapshot...etail.cgi?a=203
http://inhesion.com/zha/ca/snapshot...etail.cgi?a=215
http://inhesion.com/zha/ca/snapshot...etail.cgi?a=466
http://inhesion.com/zha/ca/snapshot...etail.cgi?a=594
http://inhesion.com/zha/ca/snapshot...tail.cgi?a=1221
http://inhesion.com/zha/ca/snapshot...tail.cgi?a=1356
http://inhesion.com/zha/ca/snapshot...detail.cgi?a=41
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=28050
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=28051
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=28114
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=28123
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=28176
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=28177
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=28305
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=28369
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=28627
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=29576
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=29580
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=29697
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=29698
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=29699
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=29705
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=29827
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=29835
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=30721
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=30729
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=30857
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=31361
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=31724
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=31875
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=32096
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=32347
http://inhesion.com/zha/ca/snapshot...ail.cgi?a=32351

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