Topological Nets: Definition and History

1. Definition and Developments from Dürer

In classical geometry, the mapping of a three-dimensional shape onto a two-dimensional plane relies on the concept of a polyhedral net, a method popularized by Albrecht Dürer in 1525. A traditional Dürer net requires a polyhedron's faces to be unfolded along their edges into a single, flat, non-overlapping net that can be cleanly folded back into a 3D polyhedral volume.

By contrast, a topological net represents a fundamental departure from this constraint. While it remains a flat, two-dimensional blueprint designed to construct a planar or non-planar form, a topological net explicitly allows overlapping, linking, twisting, and intertwining while originating from a planar net. Instead of folding cleanly edge-to-edge to seal a solid volume, a topological net uses these planar interactions to establish the precise over-under crossing data that governs a continuous, self-interweaving knot or link structure.

2. History

The conceptual bridge between classical polyhedral nets and topological nets for polyknots and polylinks can be traced through specific artistic and structural milestones:

Continuous Nets: 

In his 2018 paper Weaving Double-Layered Polyhedra, sculptor and mathematician Rinus Roelofs illustrated a vital foundational step to this methodology. His work demonstrated how flat, repeating structural bands and interwoven rings could weave and fold to create single-surface structures (Figure 1). This method focuses on continuous, flexible bands rather than being composed of discrete, polygon sequences. 

Rinus Roelofs, from Weaving Double‑Layered Polyhedra (2018): Figure 4, demonstrating the evolution from Dürer’s cube net to the interwoven net of a double‑layered cube.

Discrete Nets: 

Building upon this foundation, the discrete topological net formalizes these overlapping pathways into a strict, uniform design system composed entirely of polygons as seen below.  One advantage of this method is that it creates opportunities for a variety of structures that would not otherwise present themselves. 

Additional nets for the Escher dodecahedral polyknot can be seen below . From these nets other nets for nonstandarized polyknots and polylinks  can be infered using the animations located elswhere on this site. 

While the nets featured here pertain to polyknots and polylinks, topological nets apply to a variety of other structures including knots, links, braids, and weaves as documented on the website: Topological Nets Albert P. Carpenter