And for example sequences that consist of random successions of specific blocks can yield any of the types of spectra shown below—and can sound variously like hisses, growls or gurgles. … But so far as I can Frequency spectra for long sequences obtained by concatenating blocks in random orders.

For all one ever need do is to work out the remainder from dividing the position of a particular square by the size of the basic repeating block, and this then immediately tells one how to look up the color one wants. … And in the case shown on the next page the rules for this system are such that they replace each square at each step by a 2×2 block of new squares.

And then, depending on the colors of these elements, one of several possible blocks is tagged onto the end of the sequence. … Examples of tag systems in which a single element is removed from the beginning of the sequence at each step, and a new block of elements is added to the end of the sequence according to the rules shown.

The constraint that no triple of identical blocks appear together turns out to be satisfied by the Thue–Morse nested sequence from page 83 —as already noted by Axel Thue in 1906. … For any given k , many combinations of blocks will inevitably occur in sufficiently long sequences (compare page 1068 ). … But some patterns of blocks can be avoided.

In the expanded tag system evolution, successive colors of elements are encoded by having a black cell at successive positions inside a fixed block of white cells.

Generalizations [of cyclic tag systems] The implementation above immediately allows cyclic tag systems which cycle through a list of more than two blocks. (With just one block the behavior is always repetitive.)

[Converting from CAs with] more colors Given a rule that involves three colors and nearest neighbors, the following converts each case of the rule to a collection of cases for a rule with two colors: CA3ToCA2[{a_, b_, c_}  d_] := Union[Flatten[Table[Thread[ Partition[Flatten[{l, a, b, c, r} /. coding], 11, 1] 〚 {2, 3, 4} 〛  (d /. coding)], {l, 0, 2}, {r, 0, 2}], 2]] coding = {0  {0, 0, 0}, 1  {0, 0, 1}, 2  {0, 1, 1}} The problem of encoding cells with several colors by blocks of black and white cells is related to standard problems in coding theory (see page 560 ). One approach is to use {1, 1} to indicate the boundary of each block, and then within each block to use all possible digit sequences which do not contain {1, 1} , as in the Fibonacci number system discussed on page 892 .

Image averaging Walsh functions yield significantly better compression than simple successive averaging of 2×2 blocks of cells, as shown below.

Block rules [examples] These pictures show the behavior of rule (c) starting from some special initial conditions. … Starting with a block of q black cells, the period can get close to this.

This specification gives a list of three blocks {b 1 , b 2 , b 3 } and the final initial conditions consist of an infinite repetition of b 1 blocks, followed by b 2 , followed by an infinite repetition of b 3 blocks. The b 1 blocks act like "clock pulses", b 2 encodes the initial conditions for the tag system and the b 3 blocks encode the rules for the tag system. … The core of the right-hand block grows approximately like 500 (Length[Flatten[rules]] + Length[rules]) , but to make a block that can just be repeated without shifts, between 1 and 30 repeats of this core can be needed.