Designing an optimal Norman window is a standard calculus exercise. How much more difficult (or interesting) is its generalization to deploying multiple semicircles along the head (or along head and sill, or head and jambs)? What if we use shapes beside semi-circles? As the number of copies of the shape increases and the optimal Norman windows…

This paper examines three historical geometric constructions for handcrafting stringed instruments. Using elementary geometry--in particular, triangles--these methods can provide quite good rational approximations to the irrationals that arise from tuning instruments in equal temperament. Interestingly, continued fractions help explain the…

This article presents some identities on the sum of the entries in the first half of a row in Pascal's triangle. The results were discovered while the author was working on a problem involving Laplace transforms, which are used in proving of the identities.