[Patterns from] bitwise functions

Bitwise functions typically yield nested patterns. (As discussed above, any cellular automaton rule can be represented as an appropriate combination of bitwise functions.) Note that BitOr[x, y] + BitAnd[x, y]  x + y

BitOr[x, y] + BitAnd[x, y]  x + y and BitOr[x, y] - BitAnd[x, y]  BitXor[x, y]

✖

BitOr[x, y] - BitAnd[x, y]  BitXor[x, y].

The patterns below show where BitXor[x, y]  t

BitXor[x, y]  t for successive t and correspond to steps in the "munching squares" program studied on the PDP-1 computer in 1962.

Nesting is also seen in curves obtained by applying bitwise functions to n and 2n

2n for successive n. Note that 2n

✖

2n has the same digits as n, but shifted one position to the left.