It has become a “go to” line for me, when discussing “what is math research?” with my (utterly fantastic) relatives or (pretty fantastic) non-mathematical friends that “someone’s gotta keep inventing new numbers.” In an effort to mix a whole lot of influences for me, today’s post is inspired by three things: The first the following dinosaur comic:

From Ryan North's fantastic Dinosaur Comics.

The second influence is from Edward Tufte, who loved (and I assume loves) using data to give micro/macro readings, like bar charts where you use numeric data instead of bars- the “bars” give a macro reading of how big the number is, while the number itself gives detail.

The 10th row of Pascal's triangle.

Finally, I really liked the wikipedia article on the binomial coefficient, in particular the graphic on the rows of Pascal’s triangle. I’ve included in this post a printout of the 10th, 100th and 1000th row of the triangle. For interest’s sake, the Python code I used to print these is

def binom(n,k):

if k > n:

return 0

k = min(k, n-k)

c = 1

for j in range(1,k+1):

c = c*(n-(k-j))/j

return c

TOP = 1000

for k in range(TOP+1):

print binom(TOP,k)

100th row of Pascal's triangle. The numbers are written vertically, because captions look silly underneath thin figures.

I love that the simple act of writing down these numbers suggests that there is some pattern in the number of digits of the binomial coefficient.

The 1000th row. This actually took forever to make, since it is hard to take a picture of a text document. Turns out after all the compressing I did to get this, the "digits" are all just one or two pixels. Sigh...

For the curious, wikipedia reports that this is called “log concavity”, though a more satisfying answer (which I assume has already been found) might be to describe precisely and analytically the shape that the *n*th row of the triangle will make as *n* gets very large. Likely you would first rescale everything to go from, say, -1 to 1, and the height of the *k*th row would be the log of *nCk, *rescaled by the same factor. Anyways, how cool that 11 lines of code (or a dinosaur writing down 1000 integers) can raise such interesting questions!

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I think it would be more natural (but probably slightly more difficult) to have the number printed with the lowest digits at the bottom. In this case, the tail zeros of binomial coefficients would form a “secondary rainbow”, possibly with a different asymptotic shape.

The shape depends on the base (radix) in which you print out coefficients of the n-th row. A low base (eg. 2) makes for a steep one, a higher base (e.g. 16 or 26) makes for a flat one.

Apart from that, you can get some idea of the shape progression by just following the central coefficient (k = n / 2) for n getting large. The rescaled height of a peak = log(n C (n / 2)) / n. At n = 100, h = 0.29004. At n = 1000, h = 0.29943. At n = 9998, h = 0.30082, but here I don’t have enough precision at the moment.

The rest of the shape is ruled by its concavity.