Does this block puzzle approximate a continuous function?

Question

I have a question related to the following process.

Take a single square block and place it in the bottom left of a grid.

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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:

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  █
 ███
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  ██
  ██
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 ████
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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.

Answer 1

It seems you made a mistake in the last iteration. From what you describe, consistent with the results of all the other iterations, these should be the binomial coefficients. The way you’re generating them is akin to how they’re generated in Pascal’s triangle. The key relationship that allows this is

$$ \binom nk=\binom{n-1}k+\binom{n-1}{k-1}\;. $$

To answer your questions about them:

So in the limit, can we say that every column is taller than any other column to the left of it?

We would first have to define what precisely is meant by “in the limit” here. What we can say is that for each pair of columns, the process reaches a point after which the right-hand column will always be larger than the left-hand column. In terms of binomial coefficients, for all $k\in\mathbb N$, we have

$$ \binom nk\gt\binom n{k-1} $$

for all $n\ge 2k$.

And more interestingly, this is a grid, but as we zoom out, it seems to approximate some continuous function. Maybe there is one function that it approximates better and better at each iteration, modulo some scaling?

Indeed, it does. You can read in the Wikipedia article linked to above about various continuous approximations for the binomial coefficients. For instance, for many values they can be approximated by a Gaussian:

$$ \binom{n}{k} \sim \binom{n}{\frac{n}{2}} e^{-d^2/(2n)} \sim \frac{2^n}{\sqrt{\frac{1}{2}n \pi }} e^{-d^2/(2n)}\;. $$

To make this approximation more rigorous, note that the binomial coefficients appear in the binomial distribution, and in fact if we choose $p=\frac12$, that distribution is simply proportional to the binomial coefficients. The central limit theorem gives a precise meaning to the impression that this distribution tends to a Gaussian distribution as $n$ increases.

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?

Scaling only the height wouldn’t lead to a limit shape, since the width would keep increasing, but ...

What if, in addition, we also scale it down horizontally, so that after every iteration, the width would also stay the same?

That’s indeed exactly what the central limit theorem does. It scales the width and height, and the resulting shape tends to a Gaussian.

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?

No, that’s actually not what happens. For large $n$, the values around the peak at $k=\frac n2$ are all of similar magnitude; the width of the peak is of the order of $\sqrt n$, as you can see from the scaling in the central limit theorem.

Answer 2

As it is drawn, non liquet. However, just saw a video on binomial distribution and per the author and I can confirm, the binomial distribution approximates a Bell curve/normal distribution. Bereshit, due to $n$ being small in $(x + y)^n$, the likeness is only with respect to shape; the binomial distribution looks like a bell, but pixelated (like a low resolution pic). Note that this seems to be because of the binomial/choice theorem's symmetry.

As $n \to \infty$, the binomial distribution begins to acquire the continuousness of a normal distribution. Now the binomial distribution is similar to a bell curve, not just shape-wise, but also with regard to being nondiscrete.

That is to say, for the discrete to resemble the continuous, you have to alter the size of individual building blocks (that's the heart of calculus, presumably). However, there are many classic, instantly recognizable, continuous functions like parabolas, exponential growth, logistic growth, etc. and we may be able to use discrete entities to replicate their shape.