644 JOURNAL OF COSMETIC SCIENCE development of new bio-based materials composed of keratin and other similar fibrous biomolecular assemblages (11,12). WHAT IS CURL? Curliness can be intrinsic or induced. Here our focus is on intrinsic curliness. Intrinsic curliness is the shape built into a hair shaft during its growth in the follicle, and it is this shape to which an untreated hair returns when relaxed in water and dried without mechanical constraint (13). Imposed curvature is any chemically or physically induced change to intrinsic curvature, for example, from drying a fiber in a constrained state, heating a fiber in a constrained state, or inducing disulfide exchange via a permanent set technology (4). For an individual’s hair, the intrinsic curliness is what induced changes must build upon, and it is this intrinsic scenario that we still do not fully understand. What shape is curliness? For a single hair shaft, curl is three-dimensional and varies along its length. Curl is therefore a coil, something like a spring, at any one point along its length. Describing three-dimensional curliness mathematically requires each point along a hair to be described in terms of curvature and torsion (14) (Figure 1A). Measuring unconstrained hairs in three dimensions is time intensive and expensive (Figure 1B). However, constraining fibers into two dimensions and measuring only curvature is quick and relatively inexpensive. Constrained curvature data have proven to be relatively comparable to the extent that the Figure 1. Measuring curliness. (A) Curvature and torsion are three-dimensional components of hair curl. Curvature is measured on a plane that is defined by three points. Here curvature is measured on the infinite plane defined by points 1, 2, and 3, and the plane defined by points 2, 3, and 4. Torsion at midpoint between 2 and 3 is the angle between the two infinite planes (1, 2, 3 and 2, 3, 4) divided by the distance between 2 and 3. Some examples, of varied curvature and torsion. (B) Example of a photogrammetric approach to three- dimensional hair shape using photogrammetry. (C) Example of a radial ruler against which constrained hairs are placed to measure curvature quickly.

645 WHAT CAUSES CURLY HAIR? data have become the basis of standard measurements in both the wool and hair industries. In the wool industry, automated microscopy techniques (notably the optical fiber diameter analyzer) rely on measuring curvature from small subsamples of single fibers from within multifiber “staples” (a tress) and generating curvature in units of degrees per millimeter (15). In hair research, the routine approach is to constrain single hairs under a transparent sheet and line up hairs against a circular ruler to give a radius or diameter in millimeters or centimeters (Figure 1C). Loussouarn et al. (16) developed a curl classification system based on hybrid measurements where curl diameter was augmented by other simple two- dimensional measurements that indicate elements of the three-dimensional shape of a hair, including along-hair variability. Combining measurements in a statistical model allowed origin-independent classification of hair curl into a system of type I (straightest) to type VIII (most three-dimensionally curly). DIFFERENT TYPES OF CAUSATION AND NONCAUSATIVE CORRELATION The question “what causes and influences hair curl” is more complicated than it first appears. Causation occurs in multiple contexts. Curvature and torsion in a mature, dead, hair shaft are caused by the chemistry and structural organization of the shaft. That shaft structure was caused by processes during hair growth in the follicle, including the spatiotemporally distributed processes of keratin maturation and cornification. Ultimately, germline genes and physiological factors defined the follicle’s program to grow a hair of specific curliness, with this cause being first implemented during embryogenesis and then reconfigured by successive cycles of telogen and anagen. What causes curliness at the level of inherited genes may be of little interest to on- the-head product developers, but the cause of curliness at the level of the hair shaft itself might inform new technology. Conversely, dermatologists, anthropologists, and forensic scientists will differ in what level of causality is relevant. Irrespective of whether we are investigating curliness at the level of the hair shaft, hair growth, or follicle cycle, causation occurs within a complex system. A hair shaft contains multiple layers (i.e., medulla, cortex, and cuticle) and sublayers (e.g., orthocortex, paracortex) that were derived from once-living cells (17). Within the cortex, for example, hair shaft structural organization can be described over a wide range of spatial scales, with each having potential contributions to fiber structure and mechanical properties. These include protein chemistry (e.g., the matrix and the keratin intermediate filaments it surrounds), the architecture of filaments into macrofibrils, and the overall fiber shape. Within an anagen follicle, the situation is more complex again because the follicle is a miniature organ that creates a tissue that then cornifies into a range of different hair morphologies. Mammalian hair diversity indicates that a wide range of functional fibers can be produced by a follicle. While investigating causes of curliness, we should therefore be mindful of the following: • Curliness may have multiple potential causes (e.g., fiber ellipticity, cortical organization). • The multiple causes may be nonexclusive therefore, a specific curliness is an emergent phenomenon (i.e., the sum of causes). • There may be features that correlate with curl, but do not cause curl, that are caused by a common third phenomenon.