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.

The first documented use of topological nets can be found in the paper, Weaving Double-Layered Polyhedra published in 2018 by mathematical artist Rinus Roelofs. In that work, he clearly references Dürer’s treatise when applying nets to his own work on entangled polyhedra. 

On this site, that idea is generalized and extended further. By not only 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 Möbius 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.

Yet topological nets are more than representations of known topological forms. Just as polyhedral nets contributed to the development of polyhedral geometry, topological nets provide a foundation for the study of polygonally realized topological structures. Their significance lies not only in modeling existing objects but also in revealing new geometric relationships, classifications, and families of structures that emerge from regular polygonal construction.

In this way, topological nets support two complementary directions of inquiry. They extend geometry into the study of structures traditionally regarded as topological, including knots, links, braids, weaves, and surfaces. When realized through regular polygonal constructions, these objects acquire geometric properties such as symmetry, angle relationships, area, and perimeter. At the same time, topological nets extend topology into the study of structures traditionally regarded as geometric, including polygons, polyhedra, and tessellations. Through folding, twisting, linking, and entanglement, these geometric forms acquire topological properties such as knotting, one-sidedness, orientability, and connectivity.

By bringing these perspectives together within a common system of polygonal construction, topological nets reveal a reciprocal relationship between geometry and topology, opening new avenues of investigation in both topological geometry and geometric topology.

For the visitor: 

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

Structural Geometric Topology Albert P. Carpenter 

For readers interested in additional illustrations and inspiration related to this work, see the companion volume Topological Nets: Knots, Links, Braids and Beyond by Albert P. Carpenter available at: Topological Nets: Knots, Links, Braids and Beyond 1, Carpenter, Albert P. - Amazon.com