Topological Nets

Welcome to Topological Nets

Topological nets begin with a familiar idea: a net is a sequence of polygons in the plane that folds up without overlapping to form a polyhedron. This classical notion goes back to Albrecht Dürer’s 1543 treatise on measurement and has long served as a foundation for geometric exploration.

On this site, that idea is generalized and extended. By relaxing the traditional requirement that polygons must fold without crossing, nets are allowed to overlap, intertwine, and link. This simple shift opens the door to a wide range of entangled structures, including knots, links, braids, and weaves.

Once the definition is broadened, further possibilities emerge. Some nets twist and fold like the Mobius strip. Others form substructures that can be assembled into larger, more complex models such as handlehedra, Klein surfaces, polyknots, and polylinks. In every case, the constructions remain grounded in a single guiding principle: each model is built from regular polygons.

Topological nets continue the long tradition of using nets as mathematical manipulatives. Just as polyhedral nets have supported geometric learning for centuries, these innovative nets offer hands‑on access to advanced ideas in topology. Because they can be explored through both paper models and graphic designs, they make sophisticated concepts newly accessible in and out of the classroom.

Use the category guide below to explore the galleries. For a deeper look at how topological nets fit within a broader framework of structural geometric topology, see the conceptual overview on my main site:

Structural Geometric Topology Albert P. Carpenter