Date started: Original work in 2015, new directions started September 2019

Leads: Laura Taalman

Abstract

In 2015, the author created an interactive 3D model called the Pentomizer, which can produce the 15 complete families of pentagonal tessellations and incorporate them into 3D designs and objects. This model was parametrically designed in OpenSCAD and made freely available on Thingiverse.

In this project we find a new use for the Pentomizer in 2D, to create surface designs and laser cut art. Pentagonal-based surface designs with translational repeats are possible because each of the 15 tessellation families has a multi-unit piece that tesselates using only translations, although there are some complexities involved in making those repeats actually function in software such as Adobe Illustrator.

We also modified the Pentomizer OpenSCAD code to create induced open- and closed-star patterns based on the pentagonal tessellation families. These star patterns can produce surprisingly rich patterns with a lot of interesting substructure. We include some preliminary examples here.

Future steps in this project include laser-cut models that can utilize depth and shadow to create additional interest and structure.

Media

Some of the many 3D models that were made possible with the Pentomizer OpenSCAD code

Pentomizer OpenSCAD output for a member of the Type 7 tessellation family

Closed-star tessellation induced by the Type 7 pentagonal tessellation above

Open-star "sliced" tessellation induced by the Type 7 pentagonal tessellation above

Simple stained glass surface design based on a Type 9 Rice tessellation of irregular pentagons, colored using a double 8-tile primitive unit pattern

Closed-star surface design induced by a Type 7 Kershner tessellation of irregular pentagons, with a four-coloring of stars based on the 8-tile primitive unit of this tessellation

Open-star "sliced" design induced by a Type 4 Reinhardt tessellation of irregular pentagons, colored to highlight the underlying square pattern

Laser cut open-tile design of a Type 9 pentagonal tessellation

Laser cut closed-tile design of a Type 9 pentagonal tessellation

References

• Taalman, Thingiverse 3D model, Pentomizer - Every tessellating convex polygon: https://www.thingiverse.com/thing:1008750
• Taalman, Hacktastic article, Print All the Pentagons: https://mathgrrl.com/hacktastic/2019/07/print-all-the-pentagons/