Weaving patterns inspired by the pentagon snub subdivision scheme

TL;DR

This paper introduces a pentagon-based subdivision scheme for creating fractal-like, weaving patterns in 2D domains, with applications demonstrated through paper models and puzzles that make the mathematical concepts tangible.

Contribution

It presents a novel pentagon subdivision scheme that generates unique weaving patterns and extends these patterns to other schemes, including tactile models for better understanding.

Findings

The subdivision scheme produces fractal-like structures.

The weaving patterns can be generalized to other subdivision schemes.

Paper models and puzzles effectively illustrate the patterns and process.

Abstract

Various computer simulations regarding, e.g., the weather or structural mechanics, solve complex problems on a two-dimensional domain. They mostly do so by splitting the input domain into a finite set of smaller and simpler elements on which the simulation can be run fast and efficiently. This process of splitting can be automatized by using subdivision schemes. Given the wide range of simulation problems to be tackled, an equally wide range of subdivision schemes is available. They create subdivisions that are (mainly) comprised of triangles, quadrilaterals, or hexagons. Furthermore, they ensure that (almost) all vertices have the same number of neighboring vertices. This paper illustrates a subdivision scheme that splits the input domain into pentagons. Repeated application of the scheme gives rise to fractal-like structures. Furthermore, the resulting subdivided domain admits to certain weaving patterns. These patterns are subsequently generalized to several other subdivision schemes. As a final contribution, we provide paper models illustrating the weaving patterns induced by the pentagonal subdivision scheme. Furthermore, we present a jigsaw puzzle illustrating both the subdivision process and the induced weaving pattern. These transform the visual and abstract mathematical algorithms into tactile objects that offer exploration possibilities aside from the visual.