"The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful." — Poincare


And so, Boland and I have, we believe, discovered the Boland-Wrymouth Generalized Formula for Polynomial Expansion. It is a lovely little poem of mathematics which takes Newton's Binomial Theorem and it's connection to Pascal's triangle, and generalizes the results from binomials and 2-dimensional triangles to polynomials of any length and n-dimensional tetrahedrons.

Illustrations will have to wait, but here's the basic draft:


"... gives the terms for any expansion of a polynomial (a1 + a2 + a3 + ... + ad)^nth power..."


The formula is so elegant and simple (trust me) that I immediately became depressed, for surely this has been done already. Later, I became a little less depressed, because something this simple and elegant would have surely been taught at the high school or college level, if it were well-known.

And then, I cheered up still further when I realized that, to a certain degree, it doesn't matter ifit has been done before. This high-school / middle-school / junior college teacher, and a high school junior, managed to build upon the work of Pascal and Newton, working in tandem.

If that will have to do, then that will have to do.

Congratulations, Boland. We did it!