A CA-horus Line
Discover your inner Bob Fosse with the help of A New Kind of Science. The sequence of dance steps of your CA-horus line is determined by the rules of a 2, 3, or 4 color/nearest neighbor cellular automaton (CA). Design the dance steps using the larger stick figures in the middle of the Demonstration. Set the CA rule by clicking on the grid underneath the stick figures.
Snapshot 1: The rule specification grid at the bottom has one cell for each bit of the CA rule. For two-color rules there are eight bits/cells arranged in two rows. The top row has a red background indicating that this is the row which controls the behavior of the red pose; that is, when red is the center element of a three-element neighborhood. Since each neighbor can be one of two colors, there are four possible neighbor configurations represented by the small, colored square boxes, â —â . You determine the center (red) element's transitions to green or red by clicking on the single, small square box that appears below the â —â neighborhood. You will discover that it is not easy to program your dancers to do what you want them to do—especially when you use three or four poses.
Snapshot 2: The "reset" button returns the dancers to a nominal starting position. It will also set the bits of the CA rule in such a way that there is no change from generation to generation. For this three-color rule all the top row neighborhoods (as shown in the rule specification grid) transition to blue. But since blue is the color of the center element of all these neighborhoods there is essentially no change. Similarly, the middle row shows red as a center element remaining red. The bottom row shows green remaining green.
Snapshot 3: Hitting the "can-can" button creates a rule that repeatedly cycles through all three colors. Notice that like the "identity" rule above in snapshot 2, each row in the rule specification grid has just one color. But here the top/blue row transitions to red for all the neighborhood configurations. The middle/red row transitions to green. The bottom/green row transitions back to blue, completing the cycle. This choreography seems reminiscent of the line dancing in the Folies Bergère.
Snapshot 4: The "rockettes" button creates a rule that has every dancer doing what their right neighbor has just done on the prior step. Notice the transition states are colored in columns, not rows as they were in the other snapshots. Here the transition color is determined solely by the center element's right neighbor. This rule creates characteristic diagonal lines in the CA plot below the dancers.
The organization of a 1D CA rule's bits by their center element's value is an alternative to the canonical ordering of CAs developed by Stephen Wolfram.