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:
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.
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
if k > n:
k = min(k, n-k)
c = 1
for j in range(1,k+1):
c = c*(n-(k-j))/j
TOP = 1000
for k in range(TOP+1):
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.
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 nth 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 kth 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!