Binary Replicator CA

Binary Numbers
and the "Replicator" Ruleset
2d Cellular Automata

1. First of all, some of them are fractal, with versions of the same general shape appearing at many different levels. This is not unusual for cellular automata.
   
   
2. Second, these shapes are additive, which means that two overlapping shapes in this ruleset will evolve in the exact same way as if the shapes were entirely independent (with the exception that two overlapping "on" pixels will appear as an "off" pixel). This is an uncommon, but not unheard-of property for CAs.
   
   
3. The most unusual and unique property of these shapes is that they have a clear and striking structural one-to-one relationship with binary numbers.
   
   If you take the first pixel and number it "0" and the next shape as "1" then each successive interation of the CA will correspond to a binary number.
   
   Once that has been done, correspondences can be seen between the numbers and the shapes. For instance, every number composed of a single "1" followed by a string of zeros (that is to say, a integer power of the number "two" as expressed in binary) will consist of a ring of nine pixels arranged around the perimeter of a square. The distance between the pixels will be proportional to the number of zeros.
   
   Similarly, any shape corresponding to a given number will reappear at a more spread-out scale (but composed of the same number of "on" pixels) if you add zeros to the end of its binary representation (i.e. multiply it by a power of two).
   
   Nor do the correspondences end there. If a large shape has a binary number that ends in a sequence corresponding to a smaller shape, then the larger shape will be composed of copies of the small shape.
   
   In general, the shape matching a number can be described at the largest level of scale by starting at the left end of the binary representation, and at the smallest level of scale by starting at the right end of the binary representation.
   
   For example, the number 10101 describes a large version of shape "1" made up of medium sized copies of shape "1" which in turn are made of small copies of shape "1".
   
   

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   The shape 111010 can be seen as shape "111" made up of small copies of "10".
   
   
   Image size reduced

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   The shape "101011" can be viewed as "a big shape '10' made of little '1011's" or "a big '1010' made of little '11's" with equal validity. It can also be seen as "a big '10' made of medium '10's made of small '11's"
   
   

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   Because of this property, the binary carpet shapes serve as a great example of a phenomena that can be interpreted at many different levels simultaneously.
   
   

NOTES:

30121202011202010122010121201202010122010121200121202012010121200121202011202 010121202010122010121200121202012010121200121202011202010120121202011202010122 010121202010121200121202011202010120121202011202010122010121201202010122010121 20012120201

Addendum:

Addendum 2:

Binary Carpet Fractal Addendum 3

1

1   1   1
1       1
1   1   1

1   2   3   2   1
2   2   4   2   2
3   4   8   4   3
2   2   4   2   2
1   2   3   2   1

1   3   6   7   6   3   1
3   6   12  12  12  6   3
6   12  27  27  27  12  6
7   12  27  24  27  12  7
6   12  27  27  27  12  6
3   6   12  12  12  6   3
1   3   6   7   6   3   1

1    4    10   16   19   16   10   4    1
4    12   28   40   48   40   28   12   4
10   28   70   100  124  100  70   28   10
16   40   100  132  168  132  100  40   16
19   48   124  168  216  168  124  48   19
16   40   100  132  168  132  100  40   16
10   28   70   100  124  100  70   28   10
4    12   28   40   48   40   28   12   4
1    4    10   16   19   16   10   4    1

1. A grid that is empty except for a square of eight state-1 cells for iterations corresponding to 1mod-n, 10mod-n, and all other powers of 10mod-n (state-zero-mod-n cells are considered empty).
   
2. A shape for iteration x that corresponds, at larger levels of scale, to the shapes for iterations of the form x * 10 mod - n.
   
3. Shapes that can be described at increasing levels of detail by reading their iteration number from the left, and at increasing levels of scale by reading their iteration number from the right.

Proof:

Assume that x = kpq, where p, q are different primes (not 1), and k is an integer.

The number x in base p is not of the form 100... (one followed by only zeros), which is another way of saying that x is not an integer power of p.

Since p is factor of x
(y mod x) mod p = y mod p
for all y.


Which means that (x mod x) mod p = x mod p

That means that x mod x cannot show the so-called "expanded square" (eight state-one cells arranged in a square of some size) because if it did, it would reduce to an expanded square in mod p, and we already know it doesn't, because only numbers of the form 100... show the expanded square pattern in mod p (at least according to our supposition).