**1. Introduction**

Most living organisms present bilateral symmetry, meaning that the left and right sides of the body represent an almost perfect reflection of one another about the medial plane. However, perfect symmetry is virtually absent in nature, and minor, localized deviations from perfect symmetry are common. Asymmetry can thus be defined as a deviation of the shape from a perfectly mirrored image of the counter-side of a bilateral object. The observation and quantification of asymmetry patterns in biological structures are keenly studied by evolutionary and developmental biologists, anthropologists, and paleontologists. There are three different types of asymmetries in living organisms: (i) fluctuating asymmetry,

**Citation:** Melchionna, M.; Profico, A.; Buzi, C.; Castiglione, S.; Mondanaro, A.; Del Bove, A.; Sansalone, G.; Piras, P.; Raia, P. A New Integrated Tool to Calculate and Map Bilateral Asymmetry on Three-Dimensional Digital Models. *Symmetry* **2021**, *13*, 1644. https://doi.org/10.3390/ sym13091644

Academic Editor: Antoine Balzeau

Received: 22 July 2021 Accepted: 4 September 2021 Published: 7 September 2021

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(ii) directional asymmetry, and (iii) antisymmetry. The term fluctuating asymmetry (FA) applies to small left–right differences produced by developmental noises in the form of environmental and/or genetic stress [1]. Several studies identified FA as a good proxy for developmental instability. However, this assertion is still questioned, especially when it comes to the effect of habitat fragmentation, urbanization, and pollution on FA [2–6]. In humans, FA is usually linked to childhood diseases and poor genetic quality [7,8]. Different studies report a possible relationship between a mate's facial attractiveness and symmetry and usually support the notion that FA is higher in males than in females ([9–12], but see [13]). However, FA linkage to developmental disorders in our species remains contentious [14]. As an example, in a study carried out in the early medieval society from the Mikulˇcice settlement (Czech Republic), the higher degree of FA in females is deemed to be linked to the large variety of the female population due to patrilocality, although environmental effects cannot be ruled out [15].

Directional asymmetry (DA) refers to a skewed distribution of asymmetry when comparing the left to the right side of the body. DA has been largely observed in both vertebrates and invertebrates (i.e., the direction of coiling in gastropod shells, the presence of grossly unequal claws in male fiddler crabs [16]). Major examples of DA in humans pertain to handiness and brain lateralization, which in turn relates to the functioning of Broca's area for speech production [17]. Several investigations of DA in humans focused on differences occurring between males and females and usually support the notion that DA is higher in males [18,19]. DA was also used as an indicator of biomechanical loading in humans [15].

Antisymmetry (AS) is commonly defined as the inversion of the regular pattern of asymmetry, and it is widespread in both animals and plants [20]. The analysis of traits with antisymmetry may present a bimodal distribution in the most extreme manifestation, as in the case of left and right claw size in fiddler crabs. An extreme example of antisymmetry in humans is the condition known as *situs inversus*, which refers to the congenital mirrored position of most of the internal organs [21].

Studying and understanding asymmetry patterns also hold a prominent role in paleontology. Taphonomic and diagenetic processes (i.e., the postburial deformation of the organic material) can heavily affect the physical preservation of biological remains and obliterate their natural symmetry. The majority of fossils thence present damages and missing parts, as well as severe, plastic deformations due to compressive and shear forces. Incorrect identification of the nature of taphonomic distortions may misguide the recognition of diagnostic features, producing taxonomic and evolutionary misinterpretations [22,23]. More than DA and FA, which are virtually impossible to determine in the vast majority of fossil species, paleontologists are interested in quantifying the loss of biological symmetry and in identifying patterns of compression and distortion on the remains to guide the restoration of their original shape and the correct interpretation of diagnostic features. In the last few decades, with the rise of virtual paleontology, several methods of digital restoration were developed. Mirroring procedures [24–27], retrodeformation (i.e., the restoration of specimen's symmetry [28,29]), and target deformation [30] are all examples of digital manipulation procedures aiming to produce the genuine shape the remains had before taphonomy impinged on them. Assessing the reliability of these techniques is therefore crucial for paleontologists and anthropologists interested in virtual restoration.

A number of methodological strategies have been proposed to compute and discriminate between FA, DA, and AS by using geometric morphometrics data [19,31–34]. However, these strategies are generally limited in terms of visual outputs, mostly offering a 2D visualization, and/or require multiple steps to prepare the data before the asymmetry analyses can be conducted. The low visual rendering makes these approaches suboptimal in terms of interpreting the topology and regional variation in the intensity of the patterns of asymmetry and is of little help when the goal is to produce a sensible virtual restoration of the features paleontologists are most interested in.

Herein, we present a new function written in R language, named *show.asymmetry*, that allows users to visualize and measure the left–right differences of bilateral biological objects, while mapping the extent of asymmetry on the object surface and calculating levels of FA and DA where appropriate. To test *show.asymmetry*, we applied the tool to (i) visualize and assess levels of asymmetry in male and female *Homo sapiens* skulls from contemporary populations, (ii) identify patterns of asymmetry in human fossil specimens and compare them to modern humans, and (iii) test the effect of retrodeformation techniques in restoring the original biological symmetry.

## **2. Materials and Methods**
