The golden ratio has been important in art and architecture at least since Greek times. Many of its appearances in nature are exaggerated but never-the-less it is important in aesthetic design.
The golden ratio from wiki. As you can see it connects to many things in geometry and can be obtained in many ways mathematically.
Continued fractions are interesting in general, and it's amazing that its kind of the simplest continues fraction which is the golden ratio.
Understand the connection to Fibonacci numbers and the Fibonacci spiral (see the images on the wiki page).
Check out also this fun video from the amazing Vi Hart (by the way her father George Hart is a mathematician and artists/sculptor. He was involved in a course at Aalto University and produced some pieces which are on display)
There are many thing here to expand on and understand, both mathematically and in terms of history, art and architecture.
And don't forget that the Fibonacci numbers also can be extracted from Pascal's triangle.