“I've watched you now a full half-hour;
Self-poised upon that yellow flower
And, little Butterfly! Indeed
I know not if you sleep or feed.”
–William Wordsworth
For centuries, poets have hailed them as miracles of nature – few images match the simple beauty of a butterfly flitting between sun-dappled flowers. Yet, for all that time, this creature still perplexes science with its mysteries, much as it did Wordsworth in his day. One such conundrum is this: just why do butterflies have the colours and beauty they do?
It’s been known for some time that butterflies have “structural colour” – their intricate photonic crystal scale structures reflect light in certain ways so that the interference produces what our eyes interpret as green, blue, red. We can even see the structures behind this phenomenon, thanks to electron microscopes. What isn’t known is why these particular structures exist...
It is believed that the scale-structures initially form from certain proteins surrounded by the endoplasmic reticulum – a sort of gooey fluid central to the butterfly’s cell biology. The network is in fact the inverse of the endoplasm, but in the early developmental stage, a colloidal (i.e. on the boundary between two substances) template forms, and over time the scales solidify into a rigid periodic network made of chitin [1]. The current theory says that the resulting template is a “minimal surface”, because nature obeys an over-riding “Principle of Least Action”. Basically, if you’ve ever wondered why it’s easy to be lazy, this is the reason: nature abhors doing more work than it needs to. When applied to the case of the endoplasmic reticulum, this principle suggests that the scale template has a geometry that minimises the amount of surface tension it has to deal with.
Minimal Surfaces – What are They?
Minimal surfaces are everywhere – from the shape of a PringlesTM chip to the bubbles in champagne. Strictly speaking, they have “zero-mean curvature”… but an easier way to understand them is to say they have the smallest surface area for given boundary conditions [2]. PringlesTM have minimum surface area given certain conditions for packing efficiency, and bubbles have the smallest surface area possible to reduce surface tension (which could destroy the bubble).
In antiquity, when Queen Dido of Carthage was told she could have as much land as she could enclose with an ox’s hide, she cut it into one long strip and fenced off a circular arc… that’s because the circle is a 2D minimal surface: it has the smallest perimeter of all 2D shapes with the same volume, so the shrewd queen got the most possible land!
So What does the Butterfly Look Like, Anyway?
Thanks to the incredible power of electron microscopes, certain structures have been identified in the butterfly’s scales. No single type is found everywhere, but there are localised patches of particular arrangements. Some are easier to model than others, including the following:
(i) The P-type structure (Figure 1) is a six-fold connected network. Basically it’s like lots of nuclei in space, each connected to six other nuclei. This structure appears not only in butterflies, but also in many other organisms [3].
(ii) The D-type structure (see Figures 2 and 3) is a tetrahedral network, rather like that found in a diamond lattice. This arrangement is known to exist in at least four types of butterfly, observed with a scanning electron microscope [4].
(iii) The square mesh model (see Figure 4) resembles a mattress of sorts.
The Plan of Attack
If a suitable model can be found for the network structure (see “Mathematical Miscellany 2”), then the minimality of the butterfly’s scales can be studied precisely. If a theoretical minimal surface could be constructed from data for the periodicities of the butterfly (which have been measured using electron microscopes) and somehow compared with the butterfly, then the “closeness” of fit should somehow indicate the degree of minimality that the butterfly exhibits. This is the general vibe of the more complicated mathematics that is required.
However, true to nature’s style, for preliminary research the “principle of least action” comes into play – make the model as simple as possible (to make things easier). Usually the butterfly has unit cells that are non-cubic, but for a first model, we can assume cubic periodicity. The basic method is:
(i) Given the periods for the butterfly, can a condition for the Fourier coefficients be found so that the butterfly satisfies the minimal surface equation? (All the models studied feature trigonometric terms, and the coefficients of these trigonometric functions are the Fourier coefficients.)
(ii) Does the theoretical minimal surface constructed
in (i) compare well with the butterfly? More specifically, does the theoretical model have the same “tube” radius as the actual butterfly? If so, then there is a good chance the butterfly is minimal.
So, part (i) in principle seems the most difficult to do, and indeed it is! For a simple model, such as the P-type structure, the equations rapidly become ugly… indeed, one of them features a fraction whose numerator has 57 terms! Imagine your worst nightmare… it would surely cower before this mathematical monstrosity. For even our simplest model, then, all seems lost…
If at First You Don’t Succeed…
Or does it? Fortunately, some sensible methods of approximation can be used. The technical details are left for “Mathematical Miscellany 3”, but the general idea is this: if we enlarge or shrink our surface slightly so that its equations are in some sense “zero”, then perhaps we can find a condition such that the curvature equation (which must be zero for a minimal surface) looks like the surface equation. Since this was “zeroed” at the start, this would mean the curvature is “zeroed”, and we have a minimal surface!
The ultimate test of this method, of course, is the accuracy with which it responds to numerical tests. Fortunately, though, the method happens to work rather well, providing approximately zero coefficients for the Fourier terms in the curvature – the condition for a minimal surface.
And What’s the Butterfly’s Take?
Since an approximation proved essential, it is also important to check that it is valid. However, having assumed cubic periodicity, we must now also find perpendicular vectors corresponding to the parallelepiped vectors. The most obvious way is to use the same magnitudes. See “Mathematical Miscellany 4” for the data obtained this way.
A good, concrete example is necessary to illustrate this whole procedure. Take the P-type structure with the function . If we look at the zero-contour surface, that is, the surface , then the vectors calculated essentially give , , and . Using the minimal surface equation, we can find the Fourier terms that have the form cos(kx), cos(ly), and cos(mz) and set their coefficients equal to a constant multiple of a, b, and c respectively (the coefficients in F). This is our “minimality” condition. We can then calculate a, b, and c, to obtain a surface that we think is approximately minimal. Now, to check our approximation, we substitute the coefficients and periods back into the minimal surface equation (see “Mathematical Miscellany 1”… What is the result? We get an expression with all coefficients approximately zero (of the order 10–2 or 10–3). So, the curvature is approximately zero: the approximation works.
Now, we move to the second part of our method… We compare this theoretical minimal surface with the observed butterfly surface. First, we graph the theoretical surface (see Figure 4). Using some numerical techniques, and also by taking cross sections of this 3D graph, we can find an approximate value for the vertical and horizontal radii of the “tubes”. The mean result turns out to be about 70 to 75 nm, comparable to the 50 nm observed in the butterfly [5]! According to this data, the butterfly is indeed close to a minimal surface, exactly as the original hypothesis predicted.
Extending the Model
Essentially, so far we have found that to certain approximations the butterfly scales are in fact minimal surfaces. A natural step is therefore to relax some of our assumptions and approximations, namely the non-cubic periodicity. It seems credible that by deforming our scale units into a cubic cell, we alter the dimensions. Remarkably, however, the mathematics yields similar results!
In fact, when the non-cubic periodicity is taken into account (see “Mathematical Miscellany 5”), the final graph is simply a rotation and skewed version of the cubic case (see Figure 5). We find that the same mean radius is obtained, which provides stronger evidence again in favour of the minimal surface hypothesis. All of this suggests that the current theory may be correct.
Conclusions and Further Work
So what has this all taught us? Certainly, it looks like there is some promising new evidence to support minimal surface hypothesis. However, we have only done a detailed study on the P-type scales (with non-cubic periodicity); though they may certainly show very promising signs for being minimal, there is no guarantee that the other structures will… Butterflies may still go ahead and “break the rules”, as we thought. Fortunately, there is nothing about the mathematical technique employed that pertains specifically to the P-type structure, and we should be able to analyse in more detail the other networks.
Due to the discrepancy between the observed and calculated radii, however, it would seem that a better approximation technique is needed. If there is to be any conclusive evidence, rather than the somewhat approximate and speculative results here, we need more rigour and exactitude in the mathematics.
But then again, who knows? Nature may still have many more tricks up her sleeve, and butterflies may continue to confound scientists… that’s part of their beauty, part of their charm.
Acknowledgements
I should like to thank those who helped with the project, specifically Dr. Leon Poladian and Dr. Maryanne Large, my supervisors, and also to fellow students Ben Fulcher and Michelle Rigozzi. Thanks also go to Shelley Wickham, whose thesis in similar areas proved enlightening at times.
References
Cited resources include:
1. A. Argyros, S. Manos, M. C. J. Large, D. R. McKenzie, G. C. Cox, and D. M. Dwarte (2002); Electron tomography and computer visualization of a three-dimensional “photonic” crystal in a butterfly wing-scale (Micron 33, 483 – 487).
2. Many of the mathematical theorems here I either derived, or learned from Wolfram’s Mathworld website: www.wolfram.mathworld.com.
3. S. T. Hyde, S. Andersson, Z. Blum, S. Lidin, K. Larsson, T. Landh and B.W. Ninham (1997); The Language of Shape (Chapter 7).
4. Photos used with permission from the Optical Fibre Technology Centre at the University of Sydney; they appear in the Honours thesis by Wickham, Shelley (2004).
5. L. Poladian, M. C. Large, K. H. Lee, S. Wickham, W. E. Padden, and N. Beeton (yet to be published); Suppression of iridescence in low-contrast three dimensional photonic crystals.
Other sources include:
1. Eric A. Lord, and Alan L. Mackay (2003); Periodic minimal surfaces of cubic symmetry (Curr. Sci. 85, 346 – 362).
2. http://www.susqu.edu/brakke/evolver/examples/periodic.html.
3. S. T. Hyde, G. E. Schröder (2003); Novel surfactant mesostructural topologies: between lamellae and columnar (hexagonal) forms (Current Opinion in Colloid and Interface Science 8, 5 – 14).
Opening photo obtained from website http://www.twilightbridge.com/op/gallery/butterfly/nib004 (Optical Percept Gallery).
Closing photo courtesy of the Optical Fibre Technology Centre at the University of Sydney.