Visualising Contact Structures in 3D Through Paintings

Mathematics and art, often seen as distinct realms, share a common pursuit of structure, abstraction, and beauty. In the setting of differential topology, contact structures on 3-manifolds present a particularly rich geometric landscape.

We begin with an introduction to two flavours of contact structures: tight and overtwisted—the latter being well understood, while the former remains more mysterious. Through the tools of surgery and convex surface theory, the classification of tight structures has been extended to Seifert fibered manifolds. Finally, we turn to the frontier: What new insights can 4-manifold techniques bring, and what questions still lie beyond our reach?