Take a look at these beautiful designs!
Are you aware of how they are made? The artist starts by selecting a set of points: these could be on straight lines (as in the first diagram), on a circle (as in the second) or a more complex curve (as in the third). He then proceeds to nail pins into a board at these points and connect pairs of pins with threads. The threads are straight but the eye sees curves in them. These is plenty of scope for variations, as the artist can choose (1) the set of points and (2) which points are to be joined to each other.
What is the mathematics behind these curves? Some of the questions that can be raised are: What is the relationship between the nail positions, the threads and the curve? Is it possible to select nail positions for a given curve? Can such investigations be carried out in 3D?
The downloadable document offers suggestions for exploration along these lines, together with useful links.
There are some files you can download for free here, which will help you experiment more with such patterns!