**4. Discussion**

There are several strategies available to evaluate asymmetry from landmark-based datasets [31,34]. Despite the presence of different methods, none of these offer a fully integrated tool to calculate and especially to map asymmetry from and to mesh-based models. Furthermore, the new function *show.asymmetry* is able to evaluate asymmetry in both multiple datasets (array) and single specimens (matrix), returning colored meshes showing the pattern of asymmetry in two different ways: the 3D map of Euclidean distances and the 3D map of local variations of area.

For example, in the R package 'geomorph' [45], the function *bilat.asymmetry* provides a 3D scatterplot of the distortion of landmarks or the 3D colorless meshes warped according to the detected pattern of asymmetry. Furthermore, landmark clouds can be useful or easy to read when dealing with a small number of points or with relatively simple 3D structures. However, when using complex 3D geometries (such as crania) and/or large numbers of landmarks, other graphical outputs, such as heatmaps, are a more welcome option [34]. Nonetheless, even built-in functions such as *bilat.symmetry* require multiple steps to be performed by the users in order to be implemented. Specifically, the two sets of coordinates defining the two sides are mirror images, and hence they must be reflected for landmark alignment (multiplication of the raw data matrix by −1 is required). On the same page, the approach described by Neubauer et al. [34] requires performing the singular value decomposition (SVD) of the raw data matrix of asymmetric vectors rather than performing a standard PCA of the mean-centered data. These steps must be performed before testing for the presence of any asymmetry pattern, increasing the chances of misuse and lengthening the time to perform the entire set of analyses.

When dealing with fossil specimens, it must be considered that taphonomic processes may sensibly alter the original shape of fossil remains. Whereas cracks and missing parts are undisputable accidents of the preservation process, the compressive and shear stresses acting upon the remains over prolonged periods of time may bring about plastic deformations that could be misinterpreted as 'natural'. This, in turn, may have important consequences on the correct recognition of the phylogenetic position and taxonomy of the remains [22,23]. For instance, the 'roofed' appearance of the neurocranium in the Steinheim skull was interpreted as evidence of its plesiomorphic condition [46] but may be better indicative of taphonomic alteration [40] (Buzi et al., this volume). A similar misinterpretation might have complicated the interpretation of Ceprano *Homo heidelbergensis* calvarium [47]. Digital restorations help in driving the restoration of the original shapes ye<sup>t</sup> obliterate true object symmetry (sensu [31]) and are uninformative as to where and to what extent asymmetry applies in the first place. The algorithm of *show.symmetry* provides exactly this piece of information and therefore helps to understand the processes behind the taphonomic distortions and their total amount. As demonstrated in the first case study, male *Homo sapiens* skulls are on average more asymmetric than female skulls. Whereas this result does not generalize and was not thought to provide an answer to a complicated question as to whether females, as compared to males, really tend to have a lower level of cranial asymmetry [15], it stills shows that *show.asymmety* retrieves even small differences between closely knit individuals. Similarly, *show.asymmetry* confirms that retrodeformation procedures actually reduce cranial symmetry below the natural level, even applied to a highly deformed skull such as Steinheim. Importantly, the tool successfully estimates and maps levels and direction of asymmetric deformations directly on the fossil remains, which may provide critical information when the recognition of the processes behind the deformation and the proper fossil restoration are at the stake.
