Question: I was working on a problem for my job that involved transitioning from a binary vector of length 7 to another binary vector of length 7. The only rule in the transition is that no "1" element in the "from vector" can become a "0" in the "to vector". If I create all possible states and check for valid transitions (like shown here:[http://imgur.com/gcJELC1](https://imgur.com/gcJELC1)) I end up with this (when scaled): [http://imgur.com/BqhqIOj](https://imgur.com/BqhqIOj).

Once I listed all possible binary states I but a 1 if it was an allowed transition, and a 0 if it was not allowed (i colored the 0's red). Here are some examples with smaller vectors:
(0,0,0)->(0,0,1) is allowed
(0,0,0)->(0,0,0) is allowed
(1,0,0)->(1,0,1) is allowed
(1,0,0)->(0,0,1) is not allowed. The first 1 went to a 0, breaking the only rule, so the whole transition is listed in the matrix as a 0 because it is an invalid transition.

Answer: 
I think the emergence of the 2D fractal is a property of the ordering of the coordinates, which themselves have a fractal nature in binary notation. See the edges of this figure:

[![counting fractal](https://i.stack.imgur.com/vIYLU.png)](https://i.stack.imgur.com/vIYLU.png)

Source code for the figure, a visualisation of $2017$ in the context of [OEIS A134169](https://oeis.org/A134169), is on my [blog post](https://mathr.co.uk/blog/2017-01-08_rollover_2017.html).
It's a cellular automata modelizing a form of Sierpinski triangle.

See rule 60 in (<http://mathworld.wolfram.com/SierpinskiSieve.html>).

See this paper considering it as a biological model (<http://journals.plos.org/plosbiology/article?id=10.1371/journal.pbio.0020424>)