Escape from flatland

In 1884, midway in time between the widespread acceptance of Darwin’s theory of natural selection and the rediscovery of Mendel, the English clergyman Edwin Abbott Abbott (writing under the apt pseudonym of A. Square) published his classic Flatland, A Romance of Many Dimensions (Abbott, 1884). The inhabitants of Flatland, a two dimensional space (essentially a sheet of paper) have distorted views of even two-dimensional objects (such as triangles), whose appearance changes as they change their orientation with respect to the viewer. Viewed from one perspective, they appear as a line, whereas seen from another angle they suggest a polygon. On rare occasions when visitors from spaceland (3-D space) appear, their appearance dynamically changes as well. A sphere passing through flatland starts as a point, becoming an ever-growing circle that reaches a maximal size and then shrinks again down to a point before disappearing. Flatland was a truly ground-breaking work (motivating later excursions into other spaces, such as Burger’s Sphereland, Dewdeny’s The Planiverse and Stewart’s Flatterland), suggesting the inherent distortions and ambiguities when viewing a geometrical object from a lower-dimensional perspective. On a cold winter night on the first of February in 1898, 14 years after Flatland appeared, Hermon Bumpus was collecting 136 house sparrows (Passer domesticus) stunned by a severe winter storm in Providence, Rhode Island. Only 64 of these survived, resulting in the famous Bumpus data set on multivariate morphological traits and survival (Bumpus, 1899), and leading in the future to many attempts to map a complex geometrical space onto lower dimensions. However, the geometrical complexities of this problem were largely unnoticed by biologists. It was not until 1917, when D’Arcy Wentworth Thompson published his classic On Growth and Form, that biologists started to think very seriously about geometry. Thompson’s ideas lead to the field of morphometrics, whose practitioners routinely think in tensor spaces and all sorts of other delicious aspects of geometry. Geometry has also played a key role in evolutionary theory. The year following Thompson’s book, R. A. Fisher’s classic 1918 paper appeared. Although widely recognized as starting the field of quantitative genetics, it is perhaps less appreciated that this paper was also heavily geometric, introducing the world to the orthogonal decomposition of components of variance (indeed, introducing the term variance itself). Fisher’s 1930 model of the probability of a new mutation being advantageous was also heavily geometric, and the notion of Sewall Wright’s genotypic fitness surfaces, which George Gaylord Simpson expressed as fitness given measures on phenotypic space, still dominates much of current evolutionary thinking. Fast-forwarding in time we pass Dickerson (1955) who was among the first to suggest that univariate heritability estimates can give a very misleading picture of response to selection on an index of traits. Finally, we reach Lande (1979) and Lande & Arnold (1983) who left us with the legacy of genetic constraints through the G matrix of genetic variances and covariances, and the matrix c of the quadratic partial regression coefficients as an estimate of the curvature of an individual fitness surface. Both of these matrices are compact representations of a complex geometric space. Thus organismal and evolutionary biology has a rich (but under-appreciated) history of geometry. However, biology also has a Flatland history, in trying to consider complex high-dimensional spaces in simpler (lowerdimensional) terms. Just as a triangle can appear as either a line or a polygon in flatland, different projections into lower dimensional spaces can result in very different perspectives of the same complex geometrical object. Against this background appears the current Target Review from Mark Blows (Blows, 2007), whose lab has produced a number of remarkably interesting (and important) papers over the past few years. Blows correctly stresses that viewing the complex geometries inherent in the G and c matrices from a Flatlandian element-by-element analysis gives a very distorted picture of their true nature. To more fully see this, let us explore a couple of simple cases involving just two traits. Consider selection response first, with two students working on different traits in the same study organism. Student one finds that trait one has an additive genetic variance of 10 and a directional selection gradient of 2, suggesting an expected response of R1 1⁄4 rAð1Þ bð1Þ 1⁄4 20. Likewise, the second student finds rAð2Þ 1⁄4 40 and b(2) 1⁄4 )1, suggesting a response of )40 for trait two. In reality, suppose the G matrix is: