Playing with Cos(z)

​We did a neat project with Newton’s method yesterday:

Exploring Newton’s Method with kids

That project plus the fact that my son is starting to learn a bit of trigonometry got me thinking about what the transformation z -> Cos(z) “looks like.”

During the day today I printed the real and imaginary parts of this transformation:

After printing those shapes I had a different idea – I’d look at how the function z -> Cos(z) mapped circles centered at the origin. To see the images of different circles, I put the circle with radius R at a height R in the map. That ended up being a bit too squished, though, so I changed the height in the image to 3R. Here’s what it looked like in Mathematica:

[ I’m having a little trouble with the videos below. Maybebecause I took them on my phone – not sure – but hopefully they at least show the shape and a few of the ideas my kids had.]

Here’s what the 3d print looked like:

and here’s how the boys described the shape. My younger son went first:


My older son went second