## 3 Fibrous Structures
### Father  
Feather is another keratinuous material like hair. The hierarchical structure that gives the feather its strength and flexibility is shown in Figure 7a [89]. In the barb of feathers, the \(\beta\)-keratins are arranged into a quasi-ordered, sponge-like structure. The barb's microstructure is composed of either sphere- or channel-type nanostructures. In contrast to hair, where pigments produce their color, light coherently scattered from feathers' nanostructure at specific wavelengths gives them their characteristic color [89; 90]. These optical properties can be explained in terms of the nanostructures in the feather that can be determined from its SAXS patterns [36; 40; 91; 90; 92; 93]. Knowledge of network connectivity, periodicity, size distribution, and order can lead to a better understanding of the structural origins of the optical properties for engineering this class of nanostructures.  
Figure 7b shows three types of SAXS diffraction patterns from (i) rudimentary structure, (ii) channel-type, and (iii) sphere-type structures. In these patterns, the ring-like diffraction pattern indicates an isotropic, short-range order in the quasi-ordered nanostructure [40; 91]. The central diffuse scattering in Figure 7b (i) indicates the typical absence of the organization of \(\beta\)-keratin molecules for rudimentary structures. The sharper rings in 7b (ii) and 7b (iii) reflect higher-order \(\beta\)-keratin in channel-type and sphere-type structures, respectively. The interconnected channels caused the broader rings observed in channel-type structures [91].  
The 2D diffraction patterns in Figure 7b were analyzed by plotting the azimuthally integrated intensities in the form of a 1D Kratky plot (I\(q^{2}\) vs. _q_) [90]. The peak position in this plot (Figure 7c) refers to the periodicity in the structure. The full-width at half maximum (FWHM) of the peak represents the nanostructure size distribution. The breadth  
Figure 6: (**a**) The observed intensity profile is compared with that calculated from that shown in the inset. The x-axis units of s = 2\(\frac{\sin\theta}{\Lambda}\); reprinted from [PERSON] et al. (2004) [57] with permission from Elsevier. (**b**) Close packing of the dimers made of twisted \(\alpha\)-helices of keratin; the dotted triangle represents the tetramer, a protofilament. (**c**) The same two helices in (**b**) packed in a ring structure; this entails only a small angular rotation of the protofilaments. (**d,e**) Hexagonal arrangement of the IFs that gives rise to the 90 Å equatorial peaks; Figure d uses the ring-core or the compact motif, whilst Figure e uses the ring motif of the IF; from [PERSON] et al. (2019) [38].  
of the peak is a good indicator of the size distribution; a narrow peak demonstrates a small deviation in size, whilst a broader peak demonstrates the broad size distribution. The third parameter, the peak height, represents the populations of ordered structures; a higher maximum peak height represents a more ordered arrangement [90]. SAXS has been used to analyze various bird species in a single taxon. By comparing and analyzing the diffraction patterns alongside a phylogenetic tree, evolutionary changes in the gyroid's structure are tracked, moving from quasi-ordered to ordered, as indicated by sharper higher-order peaks [36].