WOLFRAM|DEMONSTRATIONS PROJECT

Stepping through k=3, r=1 Cellular Automata Rule Space

​
steps
20
80
150
300
initial conditions
normal
no 2's
no 1's
no 0's
single
color
B&W
1's
27 neighbors
focus
runs
expand
pixel constrained
new initial conditions
↑ new ↑
↓ grab ↓
prior
store
…203
reset
nnnm
ECA 1•0
ECA 2•0
ECA 2•1
27 flips
identity
additive
main
nearest neighbors
entire subspace
2222
2212
2200
2122
2111
2101
2020
2011
2000
1222
1210
1202
1121
1111
1102
1021
1010
1000
0220
0212
0202
0121
0112
0101
0020
0010
0001
7009893509203
bit-flips = 9
L↔R
1↔0
2↔0
2↔1
(012)
7,625,597,484,987 rules in subspace
The 3-color, nearest-neighbor, one-dimensional (
k=3
,
r=1
) cellular automaton (CA) rule space has over seven trillion rules. In this Demonstration you can restrict your exploration to a subset of this rule space by using the checkboxes in the pink regions. Click the "↑new↑" button to choose a rule at random from your constructed subspace.
To understand how to use the checkboxes you first need to understand how a rule is specified: in a
k=3
,
r=1
rule space each pixel and its two neighbors must fall into one of 27 possible configurations known as "input neighborhoods". The rules specify the "output" associated with each of these input neighborhoods. So for example, if 2 represents black, 1 represents gray, and 0 represents white, the expression
2221
would represent the center black pixel of three neighboring black pixels transitioning to gray. The 27 neighborhood transitions of the current rule are shown in the blue regions.
To create a subset of rules you restrict the output of each neighborhood to a set of allowed values using the three checkboxes that appear below each
n
1
n
2
n
3
→k
expression. When checked, the left-most checkbox means this neighborhood can transition to 2. A checked center checkbox means this neighborhood can transition to 1, and when the right-most checkbox is checked this means the neighborhood can transition to 0. As you check and uncheck the checkboxes, the size of the constructed subspace is shown below the rule number.
The two tabs, "nearest neighbors" and "entire subspace", show rules closely related to the current rule. The "nearest neighbors" tab displays thumbnails of the 54 rules that differ from the current rule by a single digit in their base-three rule specification. The "entire subspace" tab displays thumbnails for all the rules in the constructed subspace in one grand overview. In both cases these thumbnails are buttons, so clicking a thumbnail displays that rule in the main tab.
See the Details section for a complete description of all the controls in the Demonstration.