# 3 Dimensional Deformations of Julia Sets

Date started: October 2019
Leads: Gabriel Dorfsman-Hopkins, Bernat Espigulé, Greg McShane

### Abstract

To any complex number one can associate a fractal called a Julia set, and if we vary the complex numbers continuously, the Julia sets vary continuously as well. Therefore, a path through the complex numbers leads to a continuous deformation of Julia sets, a concept which has led to many striking animations of deforming fractal patterns. We take this idea and extend it into the third dimension, using 3D printers to replace the time axis of the animations with the z-axis. Each layer of these 3dD prints is a filled Julia set, varying according to a path through the Mandlebrot set. Paths include straight lines along the real axis, as well as a traverse along the upper half of the main cardioid. As an interesting reformulation, we can imagine the total Mandlebrot/Julia set as a 4 (real) dimensional subspace of $\mathbb{C} x \mathbb{C}$, where the first coordinate is a point of the Mandlebrot set, and the second a point of the associated Julia set. Then our 3D prints are certain 3 dimensional cross sections of this larger 4 dimensional fractal space.

### Media

Exploration of the main cardioid:

Local explorations:

Printing the real axis:

Greg had some fun choosing paths and resizing to make various shapes of his choosing, including a chalice and a coke bottle:

The whole collection: