# Summer courses almost over, "children's" books in the works.

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.

Japanese-style book about sine and cosine

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, $y''+y=0$ then solved it (using the series method from differential equations) producing $\sin x = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1}$ and $\cos x = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n}$

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):

Constructing The Involute Pinwheel

a sequence of involutes: the vertical and horizontal components will form the power series for sine and cosine respectively.

The vertical and horizontal components will form the power series for sine and cosine respectively. Take the series of vertical line segments: $\sin x = A_1A_2 - A_3A_4 + A_5A_6 - \cdots$ 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.

Work in progress

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.

• John Robinhūdas

Nice photos you have here,you are good writer :),and the  in the last picture "come for a ride"   Make me laugh

• Anonymous

Wow this is definitely cool!  I definitely would like to see what the final product is going to be like.  I like how you are adding a little bit more fun to math.  Seems like whenever anyone sees it just on paper they might get turned off, but if its done in an artistic way - people can't seem to get enough of it.

I've seen a few books before actually cut up with a print of something on the side, which gives it a 3D impression.  I can't seem to find the images, but its basically like finding a big book, like an old dictionary thats nice and thick and something is cut into the pages.  Mind you it wrecks the book, but if you never open it up, it would be amazing to stick on the side of the shelf.

• Honey F.

The Japanese style notebook is beautiful, quite interesting. I love that it goes on continuously and not be distracted with page flipping. That is very useful for someone like me who is a prolific note taker.

• Jessica Barming

Your Children’s book sounds fun and interesting! When is your children’s book going to be out? I have an eight- year old son, who is into shapes drawings, I am sure he will find your book fascinating!

• http://mandyallen.com Mandy Allen

It's great how sometimes we find something that's suits the job just perfectly.  The notebook seems like just what you were looking for.  Learning is so much easier when things are displayed properly and set out in an easy to read fashion.  Your books sound fascinating.