Pentagonal Tiling of a Plane (Type 5)

Joseph Kerns
M401
Assignment 1


Throughout history, artists and architects have attempted to make patterns fit to whatever surface they have available. Typically, there is beauty in patterns and repetition. Within those patterns, mathematicians have been able to work out some, but not all of the methods used to tile a plane. I’ll focus on just one such tiling, which has the following rules:


1. Only one shape can be used
2. No gaps between the shapes
3. The shape must be convex


With these rules in mind, let’s look at pentagons. In 1918 Reinhardt proved classifications for the first five types of pentagonal tilings of a plane. But, the pentagons didn’t stop there. 50 years later, some progress was made regarding the different tilings of the plane. All in all, there has been 15 types of pentagons that have been shown to tile a plane monohedrally. This may not be the complete list, but serves as the most complete that we have today.


For my project, I was tasked with creating a pentagonal tiling of Reinhardt’s Type 5 pentagons. The rules for this pentagon are pretty simple:


1. One angle must be 60 degrees
2. One angle must be 120 degrees
3. The line segments that create the aforementioned angles must be equal length.

 

This tiling can be oriented in two separate ways to cover an entire plane. Either the tiling has a unit composed of 6 or 18 pentagons, as shown below.
 

 

Utilizing OpenSCAD software, I modified the angles of the pentagon that deviates from the basic case by increasing the size of one of the angles. Below is the code and an image of the shape, along with a 3D printed version that is oriented in the tiling it creates.


 

 
 
 




Source: https://en.wikipedia.org/wiki/Pentagonal_tiling