Euclidean space. It is denoted by E3. A geometric figure is said to be symmetric if there are one or more transformations which, when applied to the figure, leave it unchanged. Symmetry is thus a property of invariance to certain types of transformations. These transformations are called isometries.
Formally, an isometry of the Euclidean space E3 is a transformation T: E3 → E3 that preserves the Euclidean metric, that is, a transformation that preserves lengths (Coxeter 1969; Rees 2000):
for all x and y points belonging to E3.
The different isometries of E3 are obtained by combining rotation and translation (x ↦ α Rx + t, where R is an orthogonal matrix of order 3 and t is a vector of
For biologists wishing to explore symmetry in an organism, the correct identification of the symmetry group is important because it conditions the relevance of the morphometric analysis to come. The symmetry group of the organic form is always a subgroup of the isometries of E3. In particular, the translation has no exact equivalent in biology, since the physical extension of an organism is finite (the finite repetition of arthropod segments, for example). Translational symmetry is therefore approximate. The other isometries form a finite subgroup of Euclidean isometries, including the cyclic and dihedral groups, as well as the tetrahedral, octahedral and icosahedral groups of the Platonic solids that were dear to Thompson (1942, Chapter 9). It appears that, essentially, the symmetry patterns of biological shapes correspond to cyclic groups (rotational symmetry of order n alone [Cn], or combined with a plane of symmetry perpendicular to the axis of rotation [Cnh], or with n planes of symmetry passing through the axis of rotation [Cnv]). The bilateral symmetry of bilaterians corresponds, for example, to the group C1v, in other words, a rotation of 2π/1 (identity) combined with a reflection across a plane passing through the rotation axis.
1.4. Biological asymmetries
Another aspect of the imperfect nature of biological symmetry rests on the existence of deviations from the symmetric expectation (Ludwig 1932). These deviations manifest themselves to varying degrees and have distinct developmental causes. Let us consider the general case of a biological structure, whose symmetry emerges from the coherent spatial repetition of a finite number of units (e.g. the two wings of the drosophila, the five arms of the starfish). Different types of asymmetry are recognized (see also Graham et al. (1993) and Palmer (1996, 2004)):
– directional asymmetry corresponds to the case where one of the units tends to systematically differ from the others in terms of size or shape. A classic example is the narwhal, whose “horn” is in fact the enlarged canine tooth of the left maxilla, while the vestigial right canine tooth remains embedded in the gum;
– antisymmetry is comparable in magnitude to directional asymmetry, but the unit that differs from the others in size or shape is not the same from one individual to another. The claws of the fiddler crab show this type of asymmetry, the most developed claw being the right or the left, depending on the individual;
– fluctuating asymmetry is an asymmetry of very small magnitude and is therefore much more difficult to detect. It is the result of random inaccuracies in the developmental processes during the formation of the units that compose the biological structure. Fluctuating asymmetry is a priori always present, even if it is not always measurable. Its magnitude is considered a measure of developmental precision and has often been used (albeit sometimes controversially) as a marker of stress.
Geometrically, the morphological variation in a sample of biological shapes exhibiting a symmetric arrangement can thus be decomposed into symmetric and asymmetric variations. There is only one way to be perfectly symmetric with respect to the symmetry group of the considered structure (this is the case when all the isometries of the group are respected), but there are one less many ways to deviate from perfect symmetry as there are isometries in the group. Thus, the total variation always includes one symmetric component and at least one asymmetric component. Geometric morphometrics offers mathematical and statistical tools to quantify and explore this empirical variation.
1.5. Principles of geometric morphometrics
We are limiting ourselves here to the morphometric framework, in which the morphology of a biological structure is described by a configuration of p landmarks (Bookstein 1991; Rohlf and Marcus 1993; Adams et al. 2004). These landmarks, ideally defined on anatomical criteria, must be recognizable on all n specimens of the sample. Their digitization in k = 2 or 3 dimensions (depending on the geometry of the object considered) provides a description of each specimen as a series of pk coordinates. The comparison of the two or more objects thus characterized is done by superimposing their landmark configurations. The method most commonly used today is the Procrustes superimposition (Dryden and Mardia 1998). The underlying idea is that the shape of an object can be formally defined as the geometric information that persists once it is freed from differences in scale, position and orientation (Kendall 1984). These non-informative differences in relation to the shape are eliminated by scaling, translation and rotation, so as to minimize the sum of the squared distances between homologous landmarks (Figure 1.2). The residual variation provides a measure of the shape difference between the two objects, the Procrustes distance, which constitutes the metric of the shape space. In this space, each point corresponds to a shape, and two shapes are all the more similar, the smaller the Procrustes distance is between the points that represent them.
Figure 1.2. Principles of geometric morphometrics illustrated for the simple case of triangles (three landmarks in two dimensions). For a color version of this figure, see www.iste.co.uk/grandcolas/systematics.zip
COMMENT ON FIGURE 1.2. – a) Two objects described by homologous configurations of landmarks are subjected to three successive transformations eliminating the differences in position (I), scale (II) and orientation (III), in order to extract a measure of their shape difference: the Procrustes distance. b) The Procrustes distance is the metric of the shape space, a non-Euclidean space in which each point corresponds to a unique shape. In applied morphometrics, researchers work with a space related to the shape space called the Procrustes (hyper)hemisphere. c) The non-linearity of this space requires projecting data onto a tangent space before testing biological hypotheses by using traditional statistical methods.
Empirical applications of morphometrics are generally carried out in another space, the Procrustes (hyper)hemisphere, mathematically linked to the shape space, and efficiently estimating distances between shapes when the studied morphological variation is small compared to the possible theoretical variation (which is the case for biological applications). The shape space and the Procrustes hemisphere have non-Euclidean geometries