Pascal’s triangle, the Ramanujan principle, and what makes something look like a part of an ellipse or a part a parabola?

John Cook writes:

The nth row of Pascal’s triangle contains the binomial coefficients C(n, r) for r ranging from 0 to n. For large n, if you print out the numbers in the nth row vertically in binary you can see a circular arc.

He explains:

The length of the numerical representation of a number is roughly proportional to its logarithm. Changing the base only changes the proportionality constant. The examples above suggests that a plot of the logarithms of a row of Pascal’s triangle will be a portion of a circle, up to some scaling of one of the axes, so in general we have an ellipse.

Cook continues with an explanation of why the ellipse fits so well:

WoЇfgang pointed out that the curve should be a parabola rather than an ellipse because the binomial distribution is asymptotically normal. Makes perfect sense.

So I redid my plots with the parabola that interpolates log C(n, r) at 0, n/2, and n. This also gives a very good fit, but not as good!

But that’s not a fair comparison because it’s comparing the best (least squares) elliptical fit to a convenient parabolic fit.

So I redid my plots again with the least squares parabolic fit. The fit was better, but still not as good as the elliptical fit.

I think the reason the ellipse fits better than the parabola has to do with the limitations of the central limit theorem. First of all, it applies to CDFs, not PDFs. Second, it applies to absolute error, not relative error. In practice, the CLT gives a good approximation in the middle but not in the tails. With all the curves mentioned above, the maximum error is in the tails.

Beyond the issue of the tails, I think there’s a perceptual issue, which is that we learn about parabolas in their convex orientation, as here:

The other thing is that a circle or an ellipse is finite and a parabola keeps going forever. So, the very fact that this graph stops makes it look less parabola-like, as compared to the sort of graph you might make where you can visually follow the curve off the edge of the graph.

The Ramanujan principle

Also I wanted to connect Cook’s point, that a table of numbers expressed in positional notation is approximately a graph of their logarithms, to the Ramanujan principle:

Tables are commonly read as crude graphs: what you notice in a table of numbers is (a) the minus signs, and thus which values are positive and which are negative, and (b) the length of each number, that is, its order of magnitude.

The name of the principle comes from a famous story of the mathematician Srinivasa Ramanujan supposedly conjecturing the asymptotic form of the partition function based on a look at a table of the first several partition numbers: he was essentially looking at a graph on the logarithmic scale.