# Sculplexity: sculptures of complexity using 3D printing

I have just had an unusual paper published by European Physics Letters (doi: 10.1209/0295-5075/104/48001). 3D printing (or additive manufacturing technologies to give the field its more formal name) is very much in vogue: the first 3D printed gun, Obama’s 2013 state of the Union address, and so forth. I like to know about new technological advances, even better if I can play with it. So how can a theoretical physicist get involved with 3D printing?

One of the ways I learn about a new topic is to set it up as a project for final year undergraduate students or as a masters level project. We have some of the best students in the UK here so all I have to do is come up with an outline and the students usually rise to the challenge.

What I needed was a connection between my work in complexity and 3D printing. This link came from one of those serendipitous moments. Another advantage of being at Imperial is that it is part of a Victorian complex of arts and science institutions so we are surrounded by national museums. One is the V&A (Victoria and Albert museum) which is dedicated to all aspects of design, from all ages and all locations. It also has some amazing tea rooms, covered in tiles designed by William Morris, great when I have a visitor and a little more time. It was on one of those trips that I just happened to walk past something that caught my eye. At one level it is just a table. However I saw this table as a branching process. To the designers, it was inspired by the tree-like structures of nature. The designers had embraced technology to do this, using Computer Aided Design (CAD) and 3D printing. For on further investigation this was the Fractal.MGX table designed by WertelOberfell and Matthias Bär, the first 3D printed object the V&A had acquired.

Branching processes occur in many problems in complex systems and they have a long history of mathematical investigation. So here was the link I was looking for. The question I asked was what other physics models familiar to me could be turned into a 3D printed object? How would you actually do this conversion? Does the tool, the 3D printer, impose its own limitations on the type of mathematical model we can use? Are these new limitations interesting mathematically in their own right? Until now researchers had only seen these mathematical models visualised using two-dimensional representations, often not even using perspective to give the impression of a 3D object. Making a 3D printed object opens up uncharted territory. My project shows one can move from traditional visualisations to new “tactilisations”. So can we gain new insights by using touch rather than vision?

The approach might also be useful for outreach as well as research. The same things that got my students and I interested in the project might intrigue school children, a retired judge or whoever. These objects might be particularly useful when explaining science to those whose sense of touch is better than their sight. However we could also go back to where this project started and see if models of complexity can produce something of aesthetic value alone – hence Sculplexity: sculptures of complexity.

The basic idea is simple. A 3D printer builds up its object in layers. So the height of the object can be thought of as a time. Suppose I have a model which defines a flat (two dimensional) picture. Typically this will be a grid with some squares full and some empty. The model also has to describe how this picture evolves in time. For instance there is a famous example known as Conway’s Game of Life for which are there many 2D visualisations. What I do is use the model at each point in time to define what the printer should print at one height. The next time step in the model will then define what to print on top of the first layer, and so forth.

In fact while the basic idea is straightforward, the implementation turned out to be much much harder than I expected. It is to the real credit of the undergraduate students working with me on this project, Dominic Reiss and Joshua Price, that we pushed this idea through and actually got a final 3D printed object representing our modified forest fire model. OK so our final result is a bit of a lump of black plastic compared to the inspiration for this project, the Fractal.MGX table in the V&A. But this is just the start.

Now that we have shown that this can be done, there is so much more to explore. We are adding another dimension to representations of mathematical models but the possibilities for 3D printing are endless. All we have done is made the first step in terms of 3d printing and mathematical models. We have highlighted the key problems and given at least one way to fix them. I can already see how to extend the existing approach, new solutions to some of the problems, different ways to create an object from a wider variety of theoretical models. Imagination and ingenuity are all that are required.

#### Note added

I have since found some other work using 3D printing to visualise maths which is full of useful ideas. So look for the work of Aboufadel, Krawczyk and Sherman-Bennettand as well as that by Knill and Slavkovsky listed below.

#### References

- Reiss D.S., Price J.J. and Evans T.S., 2013.
*Sculplexity: Sculptures of Complexity using 3D printing*, European Physics Letters**104**(2013) 48001, doi 10.1209/0295-5075/104/48001.

Copy of*Sculplexity: Sculptures of Complexity using 3D printing*on personal web page.

Chosen to be part of IOPselect, a collection of papers chosen by the editors for their novelty, significance and potential impact on future research.

(altmetrics for paper). - Evans T.S., 2013. Images of 3D printing output for Sculptures of Complexity – Sculpexity. http://dx.doi.org/10.6084/m9.figshare.868866
- Reiss D.S. and Price J.J., 2013. Source-code for Complex Processes and 3D Printing, https://github.com/doreiss/3D_Print, doi 10.6084/m9.figshare.718155 .
- Reiss D.S., 2013. Complex Processes and 3D Printing, project report, http://dx.doi.org/10.6084/m9.figshare.718146.
- Price J.J., 2013. Complex Processes and 3D Printing, project report, http://dx.doi.org/10.6084/m9.figshare.718147.
- 3D printing used as a tool to explain theoretical physics by Laura Gallagher, Imperial College News and Events, 9th December 2013
- “3D-printed models of complex theoretical physics“ The IET Engineering and Technology Magazine, 9th December 2013.
- Aboufadel E., Krawczyk S.V. and Sherman-Bennett, M. “3D Printing for Math Professors and Their Students“, arXiv:1308.3420, 2013.
- Knill O. and Slavkovsky E., “Illustrating Mathematics using 3D Printers“, arXiv:1306.5599, 2013