The mathematics of how diffraction works to separate white light into component colors is a well-studied, predictable phenomenon. But how some iridescent butterfly scales keep showing brilliant iridescent colors when viewed from different angles has been a long-standing mystery. The answer to that question is revealed in a paper by Saranathan et al. (2010), who found that butterflies, through millions of years of selection, evolved a geometric nanostructure within their scale cells, previously known only to mathematicians. The first three-dimensional picture of the arrangement was computer generated by NASA materials scientist A. H. Schoen in 1970. He named the almost impossible-to-describe geometric shape a gyroid. Probably the best way to introduce students to the complexities of a gyroid is to let them climb through one – at least the educators at San Francisco’s Exploratorium think so. They built a 10-foot-high gyroid in the institute’s outdoor Geometry Playground, viewable at http://www.exploratorium.edu/geometryplayground/gyroid.html.

A gyroid is a labyrinth-like structure in which there are no straight lines or plane reflectional surfaces (Figure 1). The mindboggling shape is no doubt the key to the omnidirectional irradiance these gyroids, when nano sized, can emit. Saranathan reports that as a butterfly wing’s epidermal cell develops into a scale, its outside plasma membrane folds inward into the cytoplasm. The authors state that the plasma membrane, along with the endoplasmic reticulum, can self-assemble its bilaminar membrane into a gyroidal configuration. Chitin outside the cell is drawn in with the involuting membrane, thereby coating it. Maturing butterfly scale cells, like human hair cells, die and lose water. When dry, what remains is a scale packed with chains of chitin-coated nano-gyroids. The air–chitin interface is the diffracting medium, the thickness of which determines the iridescent color. It is the overall gyroid’s three-dimensional shape that causes the omnidirectional distribution of the diffracted light (Saranathan et al., 2010).

Figure 1.

Gyroid. Created by Bathsheba Grossman. A plastic rendition of a “gyroid” formed by a computer-guided 3D printer. To create the model, Grossman entered Schoen’s formula (cos x · sin y + cos y · sin z + cos z · sin x = 0) into Mathematica and exported a mesh-form from that into Surface Evolver (www.bathseba.com).

Figure 1.

Gyroid. Created by Bathsheba Grossman. A plastic rendition of a “gyroid” formed by a computer-guided 3D printer. To create the model, Grossman entered Schoen’s formula (cos x · sin y + cos y · sin z + cos z · sin x = 0) into Mathematica and exported a mesh-form from that into Surface Evolver (www.bathseba.com).

For the researchers, it was mathematical analysis that cracked the case. Rows of nano-gyroids filling a butterfly’s scale are not three-dimensionally visible using either light or electron microscopy. Saranathan’s team employed a technique famously used by Rosalind Franklin for generating the data demonstrating that DNA molecules form a double helix: X-ray diffraction. As Franklin’s famous “Photo 51” shows, X-ray diffraction produces a pattern of dots on film known as “Bragg’s scattering peaks.” Using trigonometry, along with deductive reasoning, a picture of the diffracting or reflecting molecules’ positions in a specimen can be worked out. Saranathan’s team used small-angle x-ray scattering to compile a pattern of Bragg’s peaks for several species of butterfly scales. The Bragg pattern they had to analyze was more complicated by several magnitudes than what is seen in Franklin’s Photo 51, but contemporary computer processing power can manage the task. A. H. Schoen’s never-before-seen geometric figure (save computer generated) has been discovered naturally existing in butterflies.

## Taking the Cover Photograph

To create the highly magnified yet sharp picture of arranged butterfly scales, computer processing was again employed as a vision-assisting tool. The butterfly scales on the microscope slide are not flat enough (some are curled, others tilted) to lie evenly within a microscope objective’s focal plane. Live viewing of a slide requires the continual adjustment of a microscope’s fine focus, but a photograph captures the view in a slice in time. Using a photographic macro lens with its diaphragm stopped down, instead of a microscope objective, would provide a greater depth of focus, but the tradeoff between depth of focus and resolution would be unacceptable. Light diffracted by the diaphragm’s edges would deviate from the image, forming an optically refracted path, and thus deteriorate picture quality below ABT’s cover standards. The solution to overcome the resolving ability of visible light was to use a computational computer program called “stacking.” When stacking, a computer selects sharp areas from each of many images and combines them into one sharp picture.

To compile a group of digital pictures, or “stack,” the microscope was focused on the butterfly scale’s highest point and a picture taken. After moving the fine focus 4 microns downward, another picture was taken. The process was repeated 30 times to reach the focal plane at the bottom of the arranged butterfly scales. The stack was fed into a computer running the program Zerene Stacker (30-day free trial available at http://www.zerenesystems.com/).

According to Rik Littlefield, founder of Zerene Systems, the program he developed is based on National Science Foundation–funded research back in the early 1980s (Burt & Adelson, 1983). In an e-mail, Littlefield explained that the computer processing used for the cover photo is commonly known as PMax. According to Littlefield, “The computer program constructs a ‘Laplacian pyramid’ of difference images and then uses gradient magnitudes and multiresolution decomposition to pick out well-focused areas. They are then fitted back together seamlessly” (R. Littlefield, pers. comm.).

The butterfly scale arrangement’s size exceeded the microscope’s field, so the subject was divided into a slightly overlapping array of four quadrates. A focus stack of each was processed and then, using Photoshop, the “layers” were assembled into a single image – a process known in the macrophotography community (http://www.photo macrography.net) as “stack and stitch.”

## Acknowledgments

The arranged slide titled “Butterfly Scales – 100 pieces” was created by K. D. Kemp, East Brent, England. The slide is in the personal collection of the author. Thanks to Rik Littlefield for reviewing the manuscript for technical accuracy. Bathsheba Goldman created the gyroid model and explained how it was made. It is in the author’s personal collection.

## References

Burt, P.J. & Adelson, E.H. (1983). A multiresolution spline with application to image mosaics. ACM Transactions on Graphics, 4, 217–236.
Saranathan, V., Osuji, C.O., Mochrie, S.G.J., Noh, H., Narayanan, S., Sandy, A. & others. (2010). Structure, function, and self-assembly of single network gyroid (I4132) photonic crystals in butterfly wing scales. Proceedings of the National Academy of Sciences USA, 107, 11676–11681.
Schoen, A.H. (1970). Infinite periodic minimal surfaces without self-intersection. NASA Tech Note D-5541.