This summer is one I will remember. I taught both differential equations and calculus II for the first time. Seeing these subjects from the instructors side has really opened my eyes to all kinds of details I never noticed before. One of the most striking new insights is how much these two courses have in common. They both rely deeply on sequences and series. Sequences are like a hallway in mathematics, one that connects many many many rooms.
I am working on two math book projects. The first is a Japanese-styled art book on the topic of sine and cosine. It's inspired by many of the lessons I taught this summer.
I want to bring all of the different ways that sine and cosine are presented in elementary and undergraduate mathematics in to one (long) pictorial document. I start with the differential equation, then solved it (using the series method from differential equations) producing and
Next I wanted a pictorial way to relate these power series to the unit circle. I have found it in this spiral (the first image shows how it is constructed as an involution):
The vertical and horizontal components will form the power series for sine and cosine respectively. Take the series of vertical line segments: and so on, the segments repeatedly over and under-shoot the accutal value of sine. The full paper by Leo S. Gurin, "A problem", can be found here.
I'm going to incorporate Gurin's spiral in to my book. I want to show the power series literally flying out of it, like they have come to life. I wonder if I can make it like the famous drawing of the sine curve projecting out of the unit circle?
Naturally, I already have planned to put that diagram in my booklet.
The Japanese-style book is perfect for series and periodic functions It's one long continuous piece of paper:
Yet very compact:
I'm also working on a very silly book about hypercycloids (that's the "math" name for the shapes drawn by spirographs, did I mention I collect spirographs?):
I'm trying to make it like a children's book, fun, light, a little silly:
I can't wait to share the final product.