I am teaching English in Japan and I have a student who speaks English well, and to keep up his level, in our weekly lessons would like to learn some subjects related to my degree in mathematics.

I would like to get across reasons why mathematics can be beautiful and interesting in a way that is accessible to a non-mathematician. Ideally this would help him understand some ideas in mathematics, avoiding the rigour of undergraduate courses. 

Are there any good resources, books, websites etc., which I could be pointed towards?
Or additionally does anyone have any good topics which I could look into?

One topic I have found is the development of cryptography, based on The Code Book by Simon Singh