## Let’s Paint the Town Green: Mathematical Surfaces Behind Butterflies’ Colours

By Michelle Gardiner, Murdoch University

Let’s start with something that you can see: did you know that the green colour of some butterflies’ wings is caused by a nanostructure, without any pigmentation? No chemistry just physics! This nanostructure, called the Gyroid, is a double maze of two intergrown labyrinth-like channels. The tiny size is crucial to cause the coloured reflection through interference: the width of a human hair would fit about fifty of these (Wilts et al., 2019).

Figure 1. Schwarz Primitive surface, another bicontinuous nanostructure like the Gyroid

For a mathematician the Gyroid is an ordered bicontinuous triply-periodic saddle surface. Just like a soap bubble, these surfaces are the result of minimising surface area while maintaining volume. These structures are highly regular or ordered, forming a repetitive (or periodic) pattern in three dimensions, similar to a stack of bricks. ‘Bicontinuous’ means that the two sides of the surface each are a labyrinth-like maze which continue infinitely and never meet. In the butterfly nanostructure, one side is hollow and the other is filled with chitin (the butterfly’s equivalent of dead finger nail material, called keratin) (Wilts et al., 2019).

It seems like a miracle that nature can build such highly regular patterns at such a tiny scale. It may seem ironic to now start looking for similar structures with less regularity or order, but that is exactly what I set out to do in my AMSI Vacation Research Scholarship project. So why would you bother looking for disordered bicontinuous surfaces? Because there are some disordered nanostructures that cause colouration, for example in some parrot feathers. There are also plenty of other disordered bicontinuous structures in biology and chemistry, and having the mathematical tools to describe these will ultimately help us understand how they form and what they are good for.

So how could a third-year mathematics undergraduate student go about researching disordered bicontinuous saddle surfaces?  By learning some experimental mathematics and a good dose of computer visualisation, and some inspiration from arts and nature. ‘Nervous system’, a Boston-based arty 3D printing start-up company, describes an algorithm for modelling jewellery that looked just like a sea sponge. Their idea was to start with a tessellation of space into cells just like a soap foam, and then to remove half of the cells to create two intertwined labyrinths and then to evolve this to a smooth-looking surface. Bring in Lloyds algorithm which makes nice and even looking tilings, and Ken Brakke’s ‘Surface Evolver’ to blow numerical soap bubbles, and we can make it all happen on a computer and get something that looks close to a disordered bicontinuous surface.

Figure 2. Disordered bicontinuous surface created computationally during my research project

I have always known that computers have powerful graphical tools to help develop and visualise mathematical problems. I did not know however, doing this project myself would be such a valuable experience and encourage me to have a ‘can do’ attitude towards building my numerical skill set further. I started out with no idea what bicontinuous structures were; yet, since developing an admiration for the area I can see why academics, material engineers and artists alike are fascinated with these surfaces, just as is evident from Alan Schoen’s original models of the ordered minimal surfaces.

References:

Wilts, B., Clode, P., Patel, N. and Schröder-Turk, G. (2019). Nature’s functional nanomaterials: Growth or self-assembly?. MRS Bulletin, 44(2), pp. 106-112.

Michelle Gardiner was a recipient of a 2018/19 AMSI Vacation Research Scholarship.