The article introduces IMAGINARY, a project that started out as a single mathematics exhibition and has since grown into a full non-profit organisation over the last ten years. It tries to address questions such as: ‘How can we best communicate mathematics?’, with interactive and stimulating exhibitions, online resources, and exploring any imaginable way to communicate mathematical concepts. One of its defining features is that it also addresses current mathematics research topics.

This paper describes the weaving technique of a traditional Colombian hat called sombrero vueltiao and some of its patterns. The making of the hat–and weaving in general–involves interesting mathematical thoughts and concepts and offers many learning opportunities, for all of which we provide examples in this paper.