I enjoy experimenting with freeform and hyperbolic crochet techniques. For more information on crocheting models of hyperbolic geometry check out this link: Math & Fiber. Here is an example:
This is a crocheted hyperbolic plane made from one skein of Red Heart Super Saver yarn in Peruvian. The edge is a metallic copper-colored yarn I had in my scrap stash. I don’t usually like to work with acrylic yarn, but in making crochet sculptures it does create a stiffer form than natural yarns. Part of the trick is to use a smaller crochet hook than you would usually use with a worsted weight yarn.
To make a hyperbolic plane in crochet, start with making some chains. I began this piece with 24 chains. Then determine how often you will increase your stitches. For a perfect model, use the same rate of increase throughout the whole piece. In this example I increased in every 4th stitch. Each row is made using single crochet stitches (American terminology). If you increase in every stitch it will get very curvy very fast. If you increase less often (try every 8th stitch) it will be curve more gently with larger waves. At first it may seem flat or only a slight bit wavy. With every row it gets wavier. When done you can arrange it many different ways (see pictures below).
More examples of can be found on my Pinterest board here: Hyperbolic Crochet.