2x2 2-symbol L-system average patterns

The simplest 2x2 block replacement system patterns, averaged across generations.

Thumnail chart
Tiny thumbnail chart

About average patterns

Notes

2x2 2-symbol L-system average pattern thumbail chart

lists of all systems pattern chart average pattern chart (tiny) self-symmetry chart Fourier transform chart

Period-doubling diagonal system, link to

Sierpinski right triangle swap system, link to

Corner mirror inverse swap system, link to

Double period-doubling line system, link to

Diagonal corner cross swap system, link to

Sierpinski right triangle system, link to

lists of all systems pattern chart average pattern chart (tiny) self-symmetry chart Fourier transform chart

2x2 2-symbol L-system pattern tiny thumbail chart

lists of all systems pattern chart average pattern chart (larger) self-symmetry chart Fourier transform chart

About average patterns

[To Do]
What do the averages across generations add to understanding the systems?
The systems can be considered three dimensional discrete systems, with the generation number providing a third coordinate. The average across generations is a projection along this third dimension that is easier to display as an image, and shows something about the evolution of the system by generation. [To Do: Provide an example of a volume rendering.]

Noteson

    There are 256 different ways of filling a pair of 2x2 rules with one of two symbols ( 256 = 2^2x2x2 ). There are 49 patterns on this chart, but one pair (Corner periodic) is a mutual inverse and one pair (On and Blinker) is only different in luminance.

[To Do: Examples of the following.]

    A first rule that is all black, such that the second rule is never reached, is not included as it is not a two symbol system.

    One pattern is included twice, one the inverse (symbol interchange) of the other. While I don't count this as a unique pattern, the system occurs is properly located in two spots.

    The 0 -> [0,0,0,0], 1 -> [1,1,1,1] system never returns to symbol zero. The pattern is equivalent to its fixed point after the first generation. But it is a proper unique system if the evolution is considered.

------------------------------------
There are no restrictions on use of the images on this page. Claiming to be the originator of the material, explicitly or implicitly, is bad karma. A link (if appropriate), a note to dow[at]uoregon.edu, and credit are appreciated but not required.

Comments are welcome (dow[at]uoregon.edu).