Hex Haven

##Iterated Function System 3D Print

Joniel Augustine Jerome

September 16, 2024

George Mason University

Math 401: Mathematics Through 3D Printing

Introduction

Hex Haven is a hexagon-based iterated function system, represented visually. It is named for its appearance which resembles a kind of fortress. The iterative function repeats the hexagon pattern by splitting each into three hexagons on each iteration. These hexagons are translated to the center of each side of the pattern hexagon. The repetition over time results in a triangular shape near the top of the Hex Haven.

Mathematical Background

Generally, iterated function systems are ways of generating fractals, which are infinitely repeating patterns that can be endlessly zoomed into. Formally, they are a finite set of contraction mappings. Iterated function systems make use of affine transformations, which are different from linear transformations because they can make use of translations. Affine transformations are of the form Ax + b. In this, A represents the transformation matrix while b represents the translation vector. A is a combination of both the scaling and rotation transformations, but in the case of Hex Haven, only scaling was used.

For Hex Haven in particular, the transformation matrix A will be of the form: >[[1/2 0 0] [0 1/2 0] [0 0 1]]

This shows how the dimensions were decreased on each iteration by a factor of 1/2 in the x and y directions while the height of each layer was preserved. Each iteration was a layer on top of the previous iteration. No rotations were used for Hex Haven. But translations were used to get the desired shape. The function was iterated 7 times to achieve the final design of Hex Haven.

Coding

Coding this in OpenSCAD was done by creating a module for the base shape as well as for each iteration of the iterated function system. A module was also created to properly perform the recursion and achieve each iterated layer. These modules were used in conjunction with a for loop to achieve the desired shape. OpenSCAD's scale and translate functions proved very useful in completing this task.

Printing

Hex Haven was printed using a Bambu Lab P1P 3D printer at a 0.2 mm resolution with blue PLA filament. The printing took approximately 45 minutes. No rafts or supports were used in the printing. Slicing was done using UltiMaker Cura.

Sources

https://larryriddle.agnesscott.org/ifs/siertri/siertri.htm https://en.wikipedia.org/wiki/Iterated_function_system#:~:text=In%20mathematics%2C%20iterated%20function%20systems,They%20were%20introduced%20in%201981.