# Notes

## Section 8: Structures in Class 4 Systems

Structures [in rule 110]

The persistent structures shown can be obtained from the following {n, w} by inserting the sequences IntegerDigits[n, 2, w] between repetitions of the background block b:

{{152, 8}, {183, 8}, {18472955, 25}, {732, 10}, {129643, 18}, {0, 5}, {152, 13}, {39672, 21}, {619, 15}, {44, 7}, {334900605644, 39}, {8440, 15}, {248, 9}, {760, 11}, {38, 6}}

The repetition periods and distances moved in each period for the structures are respectively

{{4, -2}, {12, -6}, {12, -6}, {42, -14}, {42, -14}, {15, -4}, {15, -4}, {15, -4}, {15, -4}, {30, -8}, {92, -18}, {36, -4}, {7, 0}, {10, 2}, {3, 2}}

Note that the periodicity of the background forces all rule 110 structures to have periods and distances given by {4, -2} r + {3, 2} s where r and s are non-negative integers. Extended versions of structures (d)–(i) can be obtained by collisions with (a). Extended versions of (b) and (c) can be obtained from

Flatten[{IntegerDigits[1468, 2], Table[IntegerDigits[102524348, 2], {n}], IntegerDigits[v, 2]}]

where n is a non-negative integer and v is one of

{1784, 801016, 410097400, 13304, 6406392, 3280778648}

Note that in most cases multiple copies of the same structure can travel next to each other, as seen on page 290.

From Stephen Wolfram: A New Kind of Science [citation]