I have a question related to the following process.
Take a single square block and place it in the bottom left of a grid.
▄Then take whatever there is in the grid, shift it to the right by one and drop it on top of the existing blocks, essentially doubling the number of blocks in the grid. After each iteration, each column will the have in it the number of blocks already present in the previous iteration within the same column plus one column to the left. A few examples:
▄▄▄█▄▄▄▄██▄████▄███▄████▄██▄████▄████▄My question is regarding what would happen if we could keep applying this process indefinitely. Obviously we can't, but does it make sense to talk about what this shape approaches in the limit, as the number of steps tends towards infinity?
If we just look at the numbers, it seems like the whole grid would be filled, except for the very first column, which would contain only the very first block. All other columns diverge to infinity.
However, as we go towards the right side, later columns will start growing later on, but eventually end up growing quicker than any column to the left of it. So in the limit, can we say that every column is taller than any other column to the left of it?
And more interestingly, this is a grid, but as we zoom out, it seems to approximate some continuous function. Maybe a different function in every iteration? Maybe there is one function that it approximates better and better at each iteration, modulo some scaling?
Anyway, my main question is this: what if we modify the process, so that after every iteration, we scale down the resulting shape so that the height of the tallest column would stay the same. What shape does it approach then?
What if, in addition, we also scale it down horizontally, so that after every iteration, the width would also stay the same?
Given that the middle would always be growing much faster than the rest, it would seem like the shape approaches some highly exponential curve so that essentially the area below it would stay zero, i.e. the middle point (the tallest point) is infinitely taller than any other point. Is that a correct intuition, and if so, how would one go about describing it more rigorously?
To be fair I'm not looking for a complete answer to all of these questions, but some pointers to at least how to start would be much appreciated. Note that I came across this shape in a 3D setting, but simplified it to two dimensions for this question, for simplicity.
