**Abstract**

This article explores the applications of geometry to the carapace of different organisms. The initial focus is on the three-dimensional shape of the turtle carapace to introduce the Gomboc shape which provides the self-righting feature to the turtle based on its unique property of equilibrium points. Then, the patterns of the two-dimensional shape on the turtle’s carapace are discussed, for which tessellations serve as a blueprint for the pattern and also contribute to its hardness. Some scute abnormalities can occur during a turtle’s growth. The Turing pattern describes the pattern formation and the reaction-diffusion equation is the mathematics behind it. The Voronoi pattern appears on the carapace and is ubiquitous in nature. The Voronoi pattern can be expanded to the scutoid shape, which is the newly discovered three-dimensional object that models epithelial cell shape. Other mathematical models, such as geometric morphometrics, and their application in studying the morphology of crabs are reviewed. While these analytical tools assist in taxonomic identifications and determining sexual dimorphism, they also indicate ecological change. The statistical analysis of the symmetry of the carapace, along with the effect of geometric patterns on impact resistance will be studied. Lastly, the application of mathematics to the carapace is discussed throughout the article.

**Introduction**

The carapace is a biological structure that is a part of various animals’ exoskeletons and is mainly known for its protective functions. In fact, its multi-layered structure is able to shield animals from external threats and forces (Moustakas-Verho & Cherepanov, 2015). However, it is not only the physical composition of this dorsal structure that contributes to its protective function, but also its geometry.

Mathematical models and concepts are particularly relevant to understanding the natural world. This is specifically useful when analyzing geometric structures such as the carapace. For example, with the use of equilibrium properties, one can understand the turtle carapace’s self-righting properties – what allows the turtle to flip from its back onto its front (Varkonyi & Domokos, 2006). One may also be able to comprehend how the turtle carapace’s pattern contributes to increasing its stiffness (Jayasankar et al., 2017) or how one may study crabs’ morphology using a mathematical tool that studies the geometry of objects, also known as geometric morphometrics (Responte et al., 2015)

Moreover, mathematical models additionally allow one to make connections between different aspects of nature, such as with the Voronoi pattern that is found on the turtle’s carapace. Therefore, the mathematical analysis of a carapace emphasizes the importance of its various geometric structures and how they contribute to its function.

**Gomboc Shape – Self-Righting in Turtles**

**Fig. 1**. A “comeback kid” children’s toy, which is an example of a mono-monostatic body with only one stable equilibrium point and one unstable equilibrium point (Adapted from Varkonyi & Domokos, 2006).

The turtle carapace has one obvious disadvantage. This is clear when observing a turtle that has been flipped onto its back because the shape of the carapace makes it challenging for the turtle to flip back onto its feet (Domokos & Várkonyi, 2007). Turtle carapaces have evolved to adopt a self-righting shape, thereby reducing or eliminating the issue of a turtle getting stuck on its back. These were discovered by mathematicians P. L. Varkonyi and G. Domokos. Consider first a short introduction to the geometry ideas at the base of their research.

In mathematics, shapes can be classified by the number of stable equilibrium points they possess (Varkonyi & Domokos, 2006). A stable equilibrium point is a point on the surface of a shape such that when the shape is positioned with its center of mass directly above it, the shape remains stable indefinitely. Two dimensional shapes or bodies have at least two stable equilibrium points whereas, three dimensional rigid bodies have at least one stable equilibrium point (monostatic). A common example of this is the children’s toy, “comeback kid” (Figure 1). Furthermore, this children’s toy is actually a mono-monostatic body because it has precisely one stable equilibrium point and one unstable equilibrium point (Figure 2). An unstable equilibrium point is a point on which a shape can be balanced but the slightest perturbation will disturb the stability and cause the shape to move (Summers, 2009). In two dimensions, this is a trivial definition, and all monostatic two-dimensional bodies are also mono-monostatic. While V. I. Arnold theorized that they do exist in three dimensions (Domokos & Várkonyi, 2007), the mathematicians P. L. Varkonyi and G. Domokos initially began the search for a monostatic homogeneous convex body (Domokos & Várkonyi, 2007). They designed the body from a slightly flattened sphere, adding flattened faces and some sharp edges. This shape was named the Gomboc. A convex body defines a body such that there are no ‘dents’ in the outline or on the surface of the body. A line drawn from any arbitrary point to another different arbitrary point on the surface or the outline of the body will remain completely inside the given body (Summers, 2009).

Applying these ideas to turtle carapaces, the ideal shape of a carapace would then be a mono-monostatic body as it is both self-righting and the most optimized carapace shape (Domokos & Várkonyi, 2007). Practically, a turtle will likely encounter many instances where it is stuck upside down with its carapace facing the ground and its plastron or belly facing upwards. As a terrestrial animal with a rigid shell, this is an incredibly vulnerable position for the turtle. Therefore, a self-righting carapace shape is essential due to the significant decrease in energy expenditure it provides.

As Varkonyi and Domokos wanted to know if the shape was already found in nature, they examined hundreds of pebbles, tortoises and turtles. Finally, they succeeded in finding the Gomboc shape in the carapaces of tortoises, turtles who have adapted to living on land (Domokos & Várkonyi, 2007).

**Fig. 2**. Stable equilibrium points shown as solid dots and unstable equilibrium points shown as grey dots in the leftmost column. R represents the relative distance between the carapace and the plastron. Consequently, a larger R refers to a taller carapace and a smaller R refers to a flatter carapace. The rightmost column represents turtle species carapaces with arrows indicating their own R value. The *G. elegans* refers to the turtle commonly known as the Indian Star Tortoise whose carapace is virtually monostatic, closely resembling the Gomboc shape (Adapted from Domokos & Várkonyi, 2007).

The Gomboc shape is nature’s solution to the challenge of self-righting land turtles. One application involving principles of self-righting is the Gomboc Pill (Figure 3) (Domokos, 2019). Due to its monostatic self-righting nature, it will always position itself upright in the stomach, hence is used to deliver insulin or other injectables via pills instead of traditional injection (Domokos, 2019).

**Fig. 3**. A diagram to demonstrate an application of the Gomboc shape as a Gomboc Pill (Adapted from Domokos, 2019).

**Pholidosis of Turtle Carapace**

The turtle’s carapace is composed of multiple biomaterial layers that interlock and are fused together to contribute to the overall structure resistance and stability of the carapace. This, therefore, leads to the protection of the turtle. However, what if the structural importance of the famous pattern that spans the carapace’s surface was considered (Figure 4)?

**Fig. 4.** Top view of a Juvenile Flatback Sea Turtle (Adapted from Pngkey, n.d).

The pattern formed on turtle carapaces, which is called pholidosis, is made of scutes–bony plates–that are bounded from each other by skin depressions. These scutes are arranged symmetrically across the central series, which is composed of five scutes (C^{1} to C^{5}) lying along the turtle’s vertebrae (Figure 5). On each side of the central series, there is a column of 4 scutes (p^{1} to p^{4}), which is known as the pleural series (Figure 5). Lastly, along the periphery of the carapace, there are 12 pairs of smaller scutes (M^{1} to M^{12}), due to the symmetry of the carapace, which are known as the marginal scutes (Figure 5) (Cherepanov, 2013).

**Fig. 5.** Structural pattern of turtle carapace where C^{1} to C^{5} is the central series, p^{1} to p^{4} the pleural series and M^{1} to M^{12} is the marginal series (Adapted from Cherepanov, 2013).

**Tessellation Patterns**

The carapacial scutes, specifically the ones forming the central series, can be observed to have a hexagon-like shape (Fig. 5). The repetition of these shapes enables researchers to classify the pholidosis of the turtle carapace as a hexagonal tessellation pattern (Ascarrunz & Sánchez-Villagra, 2022). Tessellations, by definition, are repeated patterns composed of polygonal shapes that are tightly fitted together without gaps or overlapping (Merriam-Webster, 2022).

One may question the mechanical significance of the hexagonal-shaped scutes on the turtle carapace. In comparison to triangle and square tiles, hexagon tessellations allow for the optimization of material stiffness, which is a material’s ability to resist a force, throughout the carapace (Jayasankar et al., 2017). This means that these tessellations minimize the material stiffness differences in the carapace, and therefore serve to reinforce the carapace’s protective function.

Unlike tilling, which consists of identical polygons, tessellated polygons can vary in shape and size. In fact, this variation is frequently observed in turtle pholidosis. However, the individual variability of scute patterning in turtles is not only with respect to shape and size, but also with respect to the number of scutes (Cherepanov, 2013).

**Scute abnormalities **

As Cherepanov (2013) explains, there are three main types of scute abnormalities that can occur. The first is the atypical shape or size of scutes, the second is the presence of an additional scute, and the third is the absence of one of the regular scutes. While the three types can be expressed separately in an individual turtle, there is also the possibility of more than one type being expressed in a single individual. The most common type to be expressed is the presence of additional scutes, and the most common region in which these abnormalities occur is near the bottom of the carapace (Figure 6, letters A, B, C) (Moustakas-Verho & Cherepanov, 2015). A recurrent example of a mixed scute abnormality type is one of a “zigzag” pattern (Figure 6, letters C, D, E). This pattern is when the two pleural series are not symmetrical and some central scutes are “split,” therefore forming a triangular-like shape. This also typically occurs when there is an additional scute in the central or pleural series (Cherepanov, 2013).

**Fig. 6.** Five examples of scute abnormalities in turtle carapaces. (A) The carapace of *Chelonia mydas* with an asterisk shows the presence of an additional central scute. (B) The carapace of *Testudo graeca *with asterisks shows the presence of an additional central and pleural scute. (C) The carapace of *Testudo graeca *with an asterisk shows the presence of an additional central scute and the absence of one pleural scute, with white arrows and red dotted lines showing the “zigzag” pattern where central scutes are split. (D) The carapace of* Geochelone carbonaria* and (E) The carapace of *Trachemys scripta* both have an asterisk shows the presence of an additional pleural scute, and white arrows and red dotted lines showing the “zigzag” pattern where central scutes are split (Adapted from Moustakas-Verho & Cherepanov, 2015; Cherepanov, 2013).

**Turing Patterns**

The biological process of scute pattern formation on turtle’s carapaces is a phenomenon that is still being researched today. However, many researchers have hypothesized that the formation of pholidosis in turtle carapaces is due to the same mechanisms that form Turing patterns (Moustakas-Verho & Cherepanov, 2015). Turing patterns (Figure 7) are mathematical patterns that can occur in nature from chemical reactions when in an equilibrium state (Maini et al., 2012). This follows Alan Turing’s reaction-diffusion model (Turing, 1952; Moustakas-Verho & Cherepanov, 2015).

**Fig 7.** Turing pattern (Adapted from Vittadello et al., 2021).

The reaction-diffusion model demonstrates how the interaction between an activator and inhibitor, which are also known as morphogens, can result in creating different patterns. The idea is that a random fluctuation will destabilize the equilibrium state and result in the activator activating neighboring activators, as well as some inhibitors. The inhibitors, however, will prevent and therefore slow down the activation of activators. That being said, there is a third factor that affects pattern formation, and that is the diffusion rate of the morphogens. This corresponds to the activators and inhibitors’ abilities to interact with nearby cells (Maini et al., 2012). The following relationships can be expressed in the expressions:

\begin{equation} \frac{du}{dt}=F(u,v)+D_u\nabla^2u \end{equation}

\begin{equation} \frac{dv}{dt}=G(u,v)+D_v\nabla^2v \end{equation}

where *u* is the concentration of the activator, *v *is the concentration of the inhibitor, *F* and *G *are functions expressing the interactions, as previously described, between the activators and inhibitors, and *D _{u}*∇

^{2}

*u*and

*D*∇

_{u}^{2}

*v*are the diffusion terms (Elsevier, 2001).

It has been found that to form a pattern, the diffusion rate of the inhibitor must be faster than that of the activators. When activator concentration increases, it will trigger a cascade reaction, thereby stimulating activators and inhibitors. Since the speed of diffusion of the inhibitor must be faster than that of the activator, as the inhibitor diffuses to other cells inhibiting their activators, there will be a higher concentration of activators in that first cell. This will result in a spot pattern (Figure 8). The formation of different patterns depends on the size and shape of tissue as well as on the diffusion rate of the activator and inhibitor (Maini et al., 2012).

**Fig. 8.** Computer simulation of Turing spot pattern where the red spots depict a higher concentration of activators while the blue corresponds to high concentration of inhibitors (Adapted from Maini et al., 2012).

As Moustakas-Verho et al. (2014) hypothesized, this mathematical model can be applied to the pholidosis formation of the turtle carapace. They explored a model that contained two reaction-diffusion systems. The first one is for the position of the scutes on the carapace, and the second system is for the growth of the scutes paired with the scutes interlocking with one another. Turing patterns can be generated by computers using the Gray-Scott model (Fig. 9) One may notice its similarities with turtle pholidosis, specifically its hexagon-like shapes that are closely fitted together.

**Fig. 9. **Turing pattern generated from the Gray-Scott model where the pink is the location of high activator concentrations and the blue/green the location of low inhibitor concentration. The white corresponds to activator areas increasing in concentration (Adapted from Munafo, 2015).

**Carapace and Voronoi Patterns**

The turtle’s carapace is characterized by its hexagonal shapes called scutes. These scutes follow a mathematical pattern called the Voronoi pattern. In the below image, it can be shown that the computer-generated Voronoi pattern is similar to the turtle carapace pattern. The Voronoi pattern contains mostly hexagons and pentagons, as well as the turtle carapace (Figure 10).

**Fig. 10**. Left: A green sea turtle (*Chelonia mydas mydas*) (All About Sea Turtles – Scientific Classification | SeaWorld Parks & Entertainment, n.d.). Right: A Voronoi pattern generated by the author imitating the green sea turtle carapace, using Frederik Brasz’s Voronoi pattern generator (Voronoi Diagram Generator).

**Voronoi Pattern**

The Voronoi diagram was first discussed as early as 1644 by René Descartes and was applied by Dirichlet (1850) in the investigation of positive quadratic forms. In the 20^{th} century, George Voronoi (1907) expanded the study of Voronoi diagrams to higher dimensions. The Voronoi diagram has a lot of applications in computer graphics, epidemiology, geophysics, and meteorology.

The Voronoi pattern is a partition of a plane into polygons, called cells. Each cell contains only one generating point, called a site (it can also be called a seed or a generator). Every point on the cell is closer to its site than any other site on the plane. Thus, the Voronoi pattern represents the best way to group all the points of a plane into cells with the shortest distance to the site. Accordingly, the points on the boundary of two adjacent cells are equidistant to both sites (Figure 11).

**Fig. 11**. Part of a Voronoi pattern. The two double arrows show that the boundary of two adjacent cells is equidistant to the sites. The Voronoi pattern is generated using Steven J. Fortune’s Algorithm, annotated by the author (Javascript Implementation of Steven J. Fortune’s Algorithm to Compute Voronoi Diagrams, n.d.).

**Why is the Voronoi pattern present everywhere in nature?**

The Voronoi pattern coincides with nature’s tendency to favor efficiency. Bees build their hive in the nearest neighborhood, birds look for the shortest path, and cells grow with the tightest fit with each other. The Voronoi pattern is present on the giraffe’s fur, dragonfly’s wing, honeycomb, and epithelium on most of the organs (Figure 12).

**Fig. 12**. Left top: photograph of the honeycomb (File: Bee on His alvear.jpg – Wikimedia Commons, 2012); left bottom: photograph of a dragonfly’s wing; right top: photograph of giraffe’s fur (Girardeau, n.d.); right bottom: surface view of a vertebrate peritoneum (lining of the abdominal cavity). The epithelial cells show the Voronoi pattern (McMillan & Harris, 2018).

The epithelial cells are the cells that compose the epithelium, which is a thin and continuous protective layer on the surface of organs and blood vessels. Epithelial cells are usually packed together. The patterns on the animal’s skin originate from the pack of epithelial cells. Epithelial cells are arranged in the Voronoi pattern, as they seek for the tightest fit with the other epithelial cells to compete for space.

**From 2D to 3D – Voronoi to Scutoid**

In 2018, a piece of news created a buzz in the world: “scientists found a new shape – scutoid!” The paper written by Gómez-Gálvez was published in Nature Communication and he described a three-dimensional solid, a scutoid, that is usually bounded by a hexagon and a pentagon on both sides (Fig. 13). Scutoid describes the shape of the epithelial cells when being packed. The reason can be explained through the Voronoi pattern.

**Fig. 13**. Two identical 5-6 scutoids. For the left scutoid, the hexagon side is on the top and the pentagon side is on the bottom. For the right scutoid, the hexagon side is on the bottom and the pentagon side is on the top (Gómez-Gálvez et al., 2018).

As explained previously, epithelial cells tend to follow the Voronoi pattern on a two-dimensional plane. Applying this idea to three-dimensional space, it makes sense that epithelial cells are packed in a prism shape with the same Voronoi pattern on the top and on the bottom (Figure 14).

**Fig. 14**. Right: a prism, which is a cylindrical object of a polygon. The prism is different from a scutoid as it is bounded by two identical polygons while a scutoid is usually bounded by a hexagon and a pentagon. Left: a pack of prisms, which shows the way that epithelial cells are stacked together (Gómez-Gálvez et al., 2018).

The prism shape can work for a flat layer of cells like the case above since the outer surface and the inner surface have the same area. However, when the layer is curved into a cylinder, for example, the layer of the blood vessel, the prism fails because the outer area is larger than the inner area (Figure 15).

**Fig. 15.** A visualization of a layer curved into a cylinder. Note that for the segment marked in green, the outer layer and the inner layer have the same surface area, but for the segment marked in red, the outer layer has a larger area than the inner layer (What the Hell Is a Scutoid??—a REAL Explanation, n.d.).

Since the outer layer has a larger area than the inner layer, an arrangement of four cells in the inner layer will change their center locations by shifting outwards (Figure 16 (a)). The Voronoi pattern generated for the outer layer and the inner layer will then be different (Figure 16 (b)). Particularly, all the hexagons on the inner surface become pentagons on the outer surface and vice-versa. Accordingly, every cell is bounded by one hexagon on one side and by a pentagon on the other side, which is the shape of a scutoid. Scutoid shape takes into consideration the Voronoi pattern of the epithelial cells and the layer curvature at the same time, which explains the interest of scientists for the shape.

**Fig. 16**. (a) The arrangement of four cells’ generating sites. Two sites will shift outwards when they are on the outer layer. (b) The Voronoi pattern generated from the sites on the inner layer and the sites on the outer layer (What the Hell Is a Scutoid??—a REAL Explanation, n.d.).

**Voronoi pattern in two dimensions**

By definition of the Voronoi pattern, the distance between any point of a cell and its site must be closest compared to the other sites. This distance can be measured using Euclidean distance or Manhattan distance.

Euclidean distance is the length of a straight segment that connects two points, in other words, the smallest distance between two points. In Cartesian coordinates, the distance is defined as:

\begin{equation} l_{1,2}=\sqrt{\left(x_1-x_2\right)^2+\left(y_1-y_2\right)^2} \end{equation}

where (*x _{1}* ,

*y*) and (

_{1}*x*,

_{2}*y*) are the xy coordinates of point 1 and point 2 in the Cartesian system.

_{2}On the other hand, Manhattan distance (or rectilinear distance) is the sum of the absolute differences of their Cartesian coordinates. It is defined as:

\begin{equation} l_{1,2}=\lvert x_1-x_2\rvert+\lvert y_1-y_2\rvert \end{equation}

where (*x _{1}* ,

*y*) and (

_{1}*x*,

_{2}*y*) are defined as the xy coordinates of point 1 and point 2 in the Cartesian system just like above.

_{2}With the Euclidean distance, the Voronoi diagram can be applied in situations such as the airport location analysis and capital location analysis since the distances studied are linear. With the Manhattan distance, the Voronoi diagram can be applied in urban planning since most of the streets are orthogonal or parallel to each other, and the distance people can travel should be Manhattan distance (Figure 17).

**Fig. 17**. The same twenty randomly distributed points generate the Voronoi pattern with Euclidean distance (on the left), and the Manhattan distance (on the right). Image generated using Frederik Brasz’s Voronoi pattern generator (Voronoi Diagram Generator, n.d.).

**Application — Epidemic and Voronoi pattern**

An important application of the Voronoi pattern is in epidemiology. During the 1854 cholera epidemic in London, the physician John Snow used the Voronoi diagram to determine the infected water pump on Broad Street and control the epidemic.

Cholera usually infects humans through the water. People infected with cholera often have diarrhea. The bacterium can be transmitted through their highly liquid stool, commonly referred to as “rice-water”, contaminating water used by other people. If the contaminated water source is allowed to get into waterways, groundwater, or drinking water supplies, people living in an area could be infected. Cholera is rarely spread directly from person to person.

Snow divided the map of London using the Voronoi diagram with Manhattan distance. He marked every water pump as the “site” of the Voronoi diagram and generated “cells” that represent all the habitats on his map which are closest to each pump in Manhattan distance. Snow’s map shows that the area that contains the most of infected families is the area around the Broad Street pump and concludes that the Broad Street pump is the source of infection (Figure 18). The London government sealed the Broad Street pump, and the epidemic was quickly ended.

**Fig. 18**. The Broad Street section marked on John Snow’s map. Each dark bar represents a death at an address. The border marks points at equal distances from the Broad Street pump and another pump (Uncovering the Cause of Cholera, n.d.).

**Other Voronoi Applications**

Due to the Voronoi pattern’s feature that every point is closest to its own generating site, the Voronoi pattern can be applied in cartography. A Voronoi pattern on the map of a country or the world, taking the capitals as generating sites can give information on the distance between the capital and its territory. In most cases, the Voronoi boundaries coincide with the boundaries in the real world. The Voronoi pattern can also be applied to airport locations. A plane in an airport’s Voronoi cell is closest to that airport than the others (Figure 19).

**Fig. 19**. (a) A Voronoi pattern of the United States map where the sites are the state capitals. (b) A Voronoi pattern of the terrestrial globe where the sites are the country capitals (c) A Voronoi pattern of the airports in the world where the sites are their locations (Spherical Voronoi Diagram, n.d.).

**Geometric Analysis in the Study of Crab Morphology**

Throughout history, animals have been observed to respond to ecological changes (variation in vegetation and other physical factors such as climate and topography) by genetic differentiation or phenotypic plasticity (Freudiger et al., 2021). As these responses can include a variety of behavioral, physiological and structural adaptations, the morphology of these animals can provide information about their environment and their effect on population sizes.

To study the relationship between an aquatic organism’s growth and maturity relative to these environmental changes, researchers conduct age and growth assessments from annual growth bands in calcified structures. As crustaceans undergo molting by shedding their exoskeleton, the absence of permanent growth structures has limited the study of their mortality, growth and ecological niche. Similarly, as endangered or rare species cannot be dissected, any studies carried out with reference to reproductive organs or other small/ concealed structures have often resulted in the misclassification of these species.

To address these limitations, geometric morphometrics is used to assess variations in the size and shape of the carapace as they serve as indicators for taxonomic identifications, maturity instars and sexual dimorphism (Responte et al., 2015). Geometric morphometrics is an analytical tool that provides a quantitative description of variation in organisms. Such techniques involving the application of statistics, geometry and imaging/modeling have proved effective in studying the morphology and anatomy of crustaceans. As geometric morphometric data is dependent on the use of 2D Cartesian landmark coordinates, the carapace is typically modeled using 22 of these landmarks (Figure 20). Inter-landmark distances are calculated to draw comparisons between the structures of the male and female carapace (Figure 21).

**Fig. 20**. Depiction of the location of landmark points on the dorsal carapace surface (Adapted from Responte et al., 2015).

**Fig. 21. **Inter-landmark distances between landmark coordinates of (a) male and (b) female carapace (Adapted from Responte et al., 2015).

While numerous studies have been conducted to assess the physical and mechanical properties of the crab carapace, there is limited research about the protective role of carapace geometry during predatory attack. As a result, quantitative analyses are used to determine the effect of crab morphologies on impact resistance and plasticity. These characteristics describe the carapace’s ability to withstand large forces and its function as a biological armor. Recent studies have shown that carapaces with smaller arc lengths and shallow, evenly distributed grooves, effectively dissipate mechanical stress caused by impact loading, which in turn provides fracture resistance (Sayekti et al., 2020). In this study, five genera of crab species were collected from Sombu, Indonesia. The thickness, arc length, and topographical depth of their carapace, along with several other parameters, were measured. Arc length was determined by intersecting the closest fitting curve with the ventral face of the carapace and was used to calculate the carapace angle (Figure 22) The maximum distance between intersecting arcs was used to determine topographical depth, whereas the ratio of actual length of carapace surface and calculated arc length were used to denote 2D Wenzel roughness, which is a measure of surface roughness used to characterize hydrophobicity.

**Fig. 22**. Intersecting arcs and lines used to determine arc length, carapace angle and the topographical depth (maximum distance between dorsal and ventral arcs) (Adapted from Sayekti et al., 2020).

The euthanized crabs were prepared for impact testing by removing internal organs after dissection and supporting the underside of the carapace with silicone gel (Sayekti et al., 2020). The apex, which is the main target point for predatory attack, was marked and drop impact testing was carried out. Representative crab shells from each species were then scanned to conduct finite element (FE) simulations and design orphan meshes, ensuring that element properties matched the experimental morphometric data (Figure 23). A freefall impact simulation was then run and a steel ball, of mass 1.043g, accelerating from a height of 151cm with a projectile velocity of 5.44ms^{-1}, was dropped onto the carapace apex (Sayekti et al., 2020). To reduce other unknown variables, the steel ball was kinematically limited to a single degree of freedom. Furthermore, to accurately represent experimental testing conditions, the ventral edges of the carapace were limited to zero degree for both rotational and translational freedom. As the carapace exhibits bilateral symmetry, a symmetry-bound condition was applied (Figure 24).

**Fig. 23**. Steps involved to convert 3D scan to FE (finite element) model. (a) point cloud data is processed; (b) surface mesh is edited; (c) orphan mesh is generated; (d) FE model is defined (Based on Sayekti et al., 2020).

**Fig. 24. **Freefall impact simulation with established symmetry-bound conditions along plane BC (Adapted from Sayekti et al., 2020).

The morphological parameters were plotted as median values with error bars, based on either global parameters or surface parameters (Fig. 25). From the five species, *Thalamita* sp. and *C.* *hepatica* had similar high carapace arc lengths, however the latter showed a greater degree of thickness. Furthermore, while four of out five species had similar carapace angles, in the range of 20-25°, *C. hepatica* had the largest angle, nearly 40°. Likewise, this species also depicted the greatest topographical depth and Wenzel roughness, hence the carapace was considered to be less flat in comparison to other species (Sayekti et al., 2020).

**Fig. 25**. Morphological comparison between five crab species: (a) carapace arc length and thickness; (b) carapace arc length and angle; (c) topographical depth and 2D Wenzel roughness (Adapted on Sayekti et al., 2020).

The shear stress term, which represents a force acting coplanar to the cross-section of a material such that it results in deformation by slippage along the plane, was calculated using the individual shear stress components (*T _{1} *and

*T*), as shown in Equation 1 (Sayekti et al., 2020). Similarly, the critical stress, which refers to the product of pressure (

_{2}*p*) and friction coefficient (

_{c}*μ*), assumed to be 0.5, was determined at the interface, as shown in Equation 2.

\begin{equation} T_{eq}=\sqrt{T_1^2+T_2^2} \end{equation}

\begin{equation} T_{crit}=\mu p_c \end{equation}

Graphs of residual stiffness against the number of impact cycles for each species was plotted to provide a measure of dorsal face failure profiles (Figure 26). Although both *Thalamita* sp. and *C.* *hepatica* exhibit similar brittle fractures, only the carapace from the former species was completely penetrated from impact loading (Sayekti et al., 2020). Alternatively, *U. tetragonon*, *P. vespertilio *and *L. sanguineus *displayed the lowest levels of ductile denting (Sayekti et al., 2020). The data shows an overall negative correlation with regard to brittle-to-ductile failure characteristics and the residual stiffness of the different crab carapaces. The study concluded that the crab species that exhibited brittle failure, have carapaces with larger arc lengths and deeper grooves. Although these deep grooves allow for extended deformation, they also decrease stress dissipation through the carapace. On the contrary, carapaces with relatively low topographical depths and shallow grooves not only minimized the severity of localized stress concentrations, but also enabled stress dissipation and prevented premature brittle failure under impact loading. The geometrical analysis of impact resistance in crab carapaces plays a key role in understanding biomimetic structures such as aerospace skins and military armor.

**Fig. 26**. Graph of carapace residual stiffness plotted against the number of impact cycles for five different crab species (Adapted from Sayekti et al., 2020).

**Conclusion**

The carapace is an important biological structure whose shape allows it to perform a variety of functions. The most obvious function being physical protection. The turtle and its carapace are inseparable and prove to be an excellent choice of animal to analyze when examining the carapace. The design of the turtle’s carapace comes with challenges. How to design a protective self-righting armor both tough and ductile? The biological environment of the turtle provides a frame for this problem as the turtle’s carapace is Nature’s answer. The Gomboc shape is able to self-right while the arrangement of the scutes and their patterns are a result of the biological framework of the question, maintaining equilibrium between physical forces between cells and the chemical reactions. This requirement for equilibrium results in the Voronoi and Turing patterns. These mathematical ideas found in the shape and structure of a carapace prove to be the answer to a variety of problems. For instance, the Gomboc shape addresses the challenge of a self-righting body. Euclidean Voronoi patterns also answered the challenge of finding the infected water source during the 1854 cholera epidemic in London. These geometry ideas, found in the carapace, will only continue to appear both in nature and design.

**References **

All About Sea Turtles – Scientific Classification | SeaWorld Parks & Entertainment. (n.d.). https://seaworld.org/animals/all-about/sea-turtles/classification/

Ascarrunz, E., & Sánchez-Villagra, M. R. (2022). The macroevolutionary and developmental evolution of the turtle carapacial scutes. *Vertebrate Zoology,* 72, 29–46. https://doi.org/10.3897/vz.72.e76256

Cherepanov, G. O. (2013). Patterns of scute development in turtle shell: Symmetry and asymmetry. *Paleontological Journal*, 48(12), 1275–1283. https://doi.org/10.1134/s0031030114120028

Domokos, G. (2019). The gömböc pill. The Mathematical Intelligencer, 41(2), 9–11. https://doi.org/10.1007/s00283-019-09891-x

Domokos, G., & Várkonyi, P. L. (2007). Geometry and self-righting of turtles. *Proceedings of the Royal Society B: Biological Sciences, *275(1630), 11–17. https://doi.org/10.1098/rspb.2007.1188

Elsevier. (2001). Encyclopedia of materials: Science and technology.

Freudiger, A., Josi, D., Thünken, T., Herder, F., Flury, J. M., Marques, D. A., Taborsky, M., & Frommen, J. G. (2021). Ecological variation drives morphological differentiation in a highly social vertebrate. *Functional Ecology*, 35(10), 2266.

File:Bee on his alvear.jpg – Wikimedia Commons. (2012, October 7). https://commons.wikimedia.org/wiki/File:Bee_on_his_alvear.jpg

Girardeau, L. (n.d.). Close-up of giraffe print, Florida, USA. Getty Images. https://www.gettyimages.ca/detail/photo/close-up-of-giraffe-print-florida-usa-royalty-free-image/973946200

Gómez-Gálvez, P., Vicente-Munuera, P., Tagua, A., Forja, C., Castro, A. M., Letrán, M., Valencia-Expósito, A., Grima, C., Bermúdez-Gallardo, M., Serrano-Pérez-Higueras, S., Cavodeassi, F., Sotillos, S., Martín-Bermudo, M. D., Márquez, A., Buceta, J., & Escudero, L. M. (2018). Scutoids are a geometrical solution to three-dimensional packing of epithelia. *Nature Communications,* 9(1). https://doi.org/10.1038/s41467-018-05376-1

Javascript implementation of Steven J. Fortune’s algorithm to compute Voronoi diagrams. (n.d.). http://www.raymondhill.net/voronoi/rhill-voronoi.html

Jayasankar, A. K., Seidel, R., Naumann, J., Guiducci, L., Hosny, A., Fratzl, P., Weaver, J. C., Dunlop, J. W. C., & Dean, M. N. (2017). Mechanical behavior of idealized, Stingray-skeleton-inspired tiled composites as a function of geometry and material properties. *Journal of the Mechanical Behavior of Biomedical Materials*, 73, 86–101. https://doi.org/10.1016/j.jmbbm.2017.02.028

Maini, P. K., Woolley, T. E., Baker, R. E., Gaffney, E. A., & Lee, S. S. (2012). Turing’s model for biological pattern formation and the robustness problem. *Interface Focus*, 2(4), 487–496. https://doi.org/10.1098/rsfs.2011.0113

McMillan, D., & Harris, R. J. (2018). An Atlas of Comparative Vertebrate Histology. Elsevier Gezondheidszorg.

Merriam-Webster. (2022). Tessellation definition & meaning. Merriam-Webster. Retrieved November 30, 2022, from https://www.merriam-webster.com/dictionary/tessellation

Moustakas-Verho, J. E., & Cherepanov, G. O. (2015). The integumental appendages of the Turtle Shell: An evo-devo perspective. *Journal of Experimental Zoology Part B: Molecular and Developmental Evolution,* 324(3), 221–229. https://doi.org/10.1002/jez.b.22619

Moustakas-Verho, J. E., Zimm, R., Cebra-Thomas, J., Lempiäinen, N. K., Kallonen, A., Mitchell, K. L., Hämäläinen, K., Salazar-Ciudad, I., Jernvall, J., & Gilbert, S. F. (2014). The origin and loss of periodic patterning in the Turtle Shell. *Development,* 141(15), 3033–3039. https://doi.org/10.1242/dev.109041

Munafo, R. (2015, November 7). Gray-scott model at F=0.1100, K=0.0550. Gray-Scott Model at F 0.1100, k 0.0550 at MROB. Retrieved November 30, 2022, from http://mrob.com/pub/comp/xmorphia/F1100/F1100-k550.html

Pngkey. (n.d.). Download juvenile flatback sea turtle PNG image with no backgroud. pngkey.com. Retrieved November 30, 2022, from https://www.pngkey.com/maxpic/u2t4y3i1w7w7e6y3/

Responte, A. A., Torres, M. A. J., Tabugo, S. R. E., Manting, M. M. E., Demayo, C. G., & Gorospe, J. (2015). Describing variations in the carapace shape of the red-clawed crab perisesarma bidens. *Advances in Environmental Biology,* 9(19), 137–145.

Sayekti, P. R., Fahrunnida, Cerniauskas, G., Robert, C., Retnoaji, B., & Alam, P. (2020). The impact behaviour of crab carapaces in relation to morphology. *Materials* (1996-1944), 13(18).

Spherical Voronoi Diagram. (n.d.). https://www.jasondavies.com/maps/voronoi/

Summers, A. (2009). The living gömböc. The Living Gömböc | *Natural History Magazine.* Retrieved November 29, 2022, from https://www.naturalhistorymag.com/biomechanics/10309/the-living-gomboc

Turing, A. M. (1952). The chemical basis of morphogenesis. *Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences*, *237*(641), 37–72. https://doi.org/10.1098/rstb.1952.0012

Uncovering the cause of cholera. (n.d.). Plus Maths. https://plus.maths.org/content/uncovering-cause-cholera

Varkonyi, P. L., & Domokos, G. (2006). Static equilibria of rigid bodies: Dice, Pebbles, and the poincare-hopf theorem. *Journal of Nonlinear Science, *16(3), 255–281. https://doi.org/10.1007/s00332-005-0691-8

Vittadello, S. T., Leyshon, T., Schnoerr, D., & Stumpf, M. P. (2021). Turing pattern design principles and their robustness. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 379(2213). https://doi.org/10.1098/rsta.2020.0272

Voronoi Diagram Generator. (n.d.). Frederik Brasz. https://cfbrasz.github.io/Voronoi.html

What the Hell is a Scutoid??—a REAL Explanation. (n.d.). http://pwuth.blogspot.com/2018/08/wtf-is-a-scutoid.html