Golden ratio in venation patterns of dragonfly wings | Scientific Reports – Nature.com

We have developed an empirical Golden-Ratio Partition model to interpret these preferred intervein angles in the dragonfly wings venation patterns. Our model is inspired by a method proposed by Takuya Okabe for the interpretation of phyllotaxis patterns in leaf growth in a variety of plants28. In Okabes work, a new adaptive mechanism was proposed based on the principle that optimization of the divergent angles between plant leave stems leads to minimization of the energy cost of the phyllotaxis transition. This model can explain not only the presence of the golden angle but also the occurrence of other angles such as the Fibonacci number ratios observed in nature.

Residuals from the Gaussian-peak fitting for the hindwing (a) and forewing (b). The angle positions of the strong residuals are indicated in the figures, and they appear to be identical for both forewing and hindwing. (c) Relative occurrence frequencies of the predicted preferred angles given by golden-rule partitions of the intervals between regular angle pairs (Eq.1) in perfectly shaped quadrilateral, pentagon, and hexagon venation cells (see inset). (d) Polygon shapes in the venation pattern of a dragonfly wing. It shows that hexagons, pentagons, and quadrilaterals are the most popular shapes in the pattern.

In our dragonfly wing case, the purpose of the venation pattern optimization is to use the least amount of support veins that support the very-thin membrane to minimize the weight of the wing, while still providing the biomechanical and aerodynamic functions that are required for a dragonfly to fly. This leads to the following considerations in our model.

First, for a given area in unconstrained space, the minimum line boundary length is a circle. However, it is impossible to pack circles in space efficiently. Thus, a circular shape can be approximated by polygons such as hexagons for honeycombs in beehives. The formation of the venation pattern on a dragonfly wing is very different, as their formation is constrained by the overall wing boundary that is shaped to be aerodynamically efficient. Therefore, the venation veins must be developed within this constraint, which means it would be impossible to have regular polygon patterns as in the case of the honeycombs.

Second, we hypothesize that it is this boundary constraint that forces the formation of irregular-shaped or distorted polygons in the venation vein patterns in the dragonfly wings. The distorted polygons follow the golden-ratio rule ((phi ) = 1.618) and low-order Fibonacci number series as rational approximates (1/2, 2/3, 3/5, , which approaches its limit (1/phi = 0.618)) to partition the angle intervals defined by ([alpha _i, alpha _j]), where ([alpha _i, alpha _j]) are two regular angles either within the same polygon group (e.g. 72(^{circ }) and 108(^{circ }) for pentagons) or from two different polygon groups (e.g. 72(^{circ }) and 120(^{circ }) between a pentagon and a hexagon). These regular angles are illustrated in Fig.4c inset.

Finally, a closer look at the venation patterns in dragonfly wings indicates that the most popular shapes by far are hexagons and pentagons, followed by quadrilaterals, as shown in Fig.4d. We argue that heptagons and higher-order polygons would mostly contribute to the broad peaks centered at 111.25(^{circ }) and 137.5(^{circ }) as they will contribute many angular intervals that eventually give rise to the broad distribution of the intervein angles discussed in the previous section. Therefore, in our analysis of the outliers, we will only consider the regular angles associated with the hexagon, pentagon, and quadrilateral shapes.

With the model outlined above, we can now estimate a set of preferred intervein angles (alpha ) in the venation pattern from the regular polygon angle intervals ([alpha _i, alpha _j]) and the partition ratio p/q, which equals the golden ratio (1/phi ) and the low-order Fibonacci rational approximates 1/2, 2/3, 3/5:

$$begin{aligned} alpha =[alpha _i, alpha _j] p/q = alpha _i + (alpha _j - alpha _i) p/q , end{aligned}$$

(1)

where (p/q = 1/2, 2/3, 3/5), , (1/phi ), and ([alpha _i, alpha _j]) equals to any two regular angles in perfect rectangles (0(^{circ }), 90(^{circ }), 180(^{circ })), pentagons (72(^{circ }), 108(^{circ }), 144(^{circ })), and hexagons (60(^{circ }), 120(^{circ })). Using this method, in Fig.4c we plot the calculated preferred angle locations from Eq.1, where each occurrence is labeled by a specific ([alpha _i, alpha _j] p/q) that produced that angle. In the calculation, we include two primary second-level partition ratios with the lowest orders ((1/phi ^2, 1/2phi )). Since our angle distribution histogram is sampled with an angular interval of (0.9^{circ }), we use the same set of angle values to position our calculated preferred angles. In addition, if an angle (alpha ) calculated by Eq.1 is within (1.4^{circ }) of the location of a residual peak in the measured histogram, then the angle (alpha ) is counted in the angle interval of that residual peak. This is because the histogram angular interval (0.9^{circ }) and the statistical uncertainty (pm 1^{circ }) in the least-squares regression fits add quadratically as the total statistical error on the angular positions.

As also shown in Fig.4c, our approach not only predicts the angular positions of all observed strong residuals in the intervein angle histograms, but also provides relative frequencies of their occurrences thus their peak heights. This is because the preferred angles defined by Eq.1 for different angle pairs may overlap for certain angles, making these angles more likely to occur than others with greater probability. As an example, (alpha {[90, 120]/phi ^2} = 90 +(120-90)/1.618^2 = 101.5^{circ }), and (alpha {[72, 120]/phi } = 72 +(120-72)/1.618 = 101.7^{circ }), both contribute to the residual peak at (101.7^{circ }), making it twice likely to occur compared to a residual to which only one (alpha {[alpha _i, alpha _j] p/q}) contributes. Summing up all possible occurrences, the peak height for each (alpha ) is shown in Fig.4c. As one can see, Fig.4c agrees reasonably well with the residuals plots in Fig.4a,b, in both the locations and the heights of the residual peaks, suggesting that our model is a good way to describe the key features in the venation patterns of dragonfly wings, and how these venation patterns should be formed in constrained space.

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Golden ratio in venation patterns of dragonfly wings | Scientific Reports - Nature.com