# The Problem with the Shape

More than anything else, the phenotype of an organism determines how it interacts with the environment. It’s subject to natural selection, and may help to unravel the underlying evolutionary processes. So shape traits are key elements in many ecological and biological studies.

Commonly, basic parameters like distances, areas, angles, or derived ratios are used to describe and compare the shapes of organisms. These parameters usually work well in organisms with a regular body plan. The shape of irregular organisms – such as many plants, fungi, sponges or corals – is mainly determined by environmental factors and often lacks the distinct landmarks needed for traditional morphometric methods. The application of these methods is problematic and shapes are more often categorised than actually measured.

As scientists though, we favour independent statistical analyses, and there’s an urgent need for reliable shape characterisation based on numerical approaches. So, scientists often determine complexity parameters such as surface/volume ratios, rugosity, or the level of branching. However, these parameters all share the same drawback: they are delineated to a univariate number, taking information from one or few spatial scales and because of this essential information is lost.

This problem is founded in the analytical tools we tend to use – the classical Euclidean geometry. Classical Euclidean geometry, as we know it from school, is based on a set of ideal shapes (e.g. rectangles, triangles, or circles), of which we can calculate distances, angles, or areas, based on simple mathematical formulas. Nature provides at best imperfect representations of these though, limiting the application of classical geometric approaches to natural structures and living organisms.

# A Potential Solution

One attempt to summarize the highly complex patterns found in nature comes from fractal geometry. If you take a closer look at a fern leaf, you’ll see that the little leaves have the same shape as the whole fern leaf. The same principle applies to fractals – mathematical sets that exhibit a repetitive pattern across all scales. The self-similar growth form of a fern resembles a fractal. ©Jessica Reichert

To understand the novelty of fractal geometry approaches we’ll go back to the fern leaf. Before we can start to measure the outline of that leaf, we have to define the scale we’re working at. If we’re interested in the shape of the entire leaf – for example, to address competition for space – we would probably use a cm scale and measure the outline of the big leaf. If we’re more interested in the smaller attributes – studying, for instance, habitat availability for small insects – we’d want to address the smaller leaves, and measure the outline on a mm scale.

These measurements are examples for classical Euclidian approaches. Fractal geometry, however, combines the information across spatial scales. The highly complex spatial patterns are summarized to a univariate or multivariate dataset. In our fern leaf example, this dataset now combines the information from all spatial scales that are within the preset frame of the analysis. This is a big advantage as we can then either analyse the whole fern at once or select the respective scales of interest in a second step.

The mathematical background behind this approach is derived from the hypothetical, self-similar shapes – the fractals. To determine their complexity, mathematicians have developed various methods to calculate the fractal dimension D. Mandelbrot’s famous synthesis of fractals (1983) introduced the measuring principle to almost all parts of science and has proven successful not only for true fractals but also for studying real-world phenomena. In biology for example, fractal dimension analyses are used to address ecological and evolutionary questions (e.g., as an objective, scale-independent descriptor of shape and measure of complexity).

# The Maths Behind the Measuring Principle of Fractal Dimensions Measuring principle of the Minkowski-Bouligand method. The placement of the spheres is defined by the vertices of the 3D mesh.

Different methods have been developed to determine the fractal dimension D (e.g., Box-counting, Minkowski-Bouligand dimension, or Hausdorff dimension; Schroeder 1991). Applying, for example, the Minkowski-Bouligand method at the 3D level, the object is dilated using spheres of every possible radius r. The influence volume V(r) of the spheres depends on the level of their interaction and produces a characteristic pattern for the object. These values are then either combined to a single value of fractal dimension: D; or given as a log-log curve of the influence volume V as a function of r: V(r). This means that, D includes information from various spatial scales (r) and synthesizes them to an absolute measure, with increasing D indicating a higher complexity.

# Applying 3D Fractal Dimension Analyses to Study the shape of Irregular Organisms

In ‘The power of 3D fractal dimensions for comparative shape and structural complexity analyses of irregularly shaped organisms’, we tested the potential of 3D fractal dimension analyses as a shape descriptor for irregular organisms. We characterised the shapes of a set of stony corals applying the Minkowski-Bouligand method. The models were generated using the handheld 3D scanner Artec Spider.

As model taxa we used six widespread species with varying complexity, belonging to the genera Acropora, Pocillopora, and Porites. For each species, we studied six to eight individuals, from three origin coral colonies, accounting for variation within species. To assess the performance of this novel approach, we addressed interspecific differences as well as changes in shape over a time of eleven weeks. We compared the performance of fractal dimensions to traditional measures such as surface-volume ratios and rugosity.

We were able to show that 3D fractal dimension analyses can be used to quantify both interspecific variations and changes over time in irregularly shaped organisms like stony corals. Compared to traditional methods, fractal dimensions performed at least as well at the interspecific level and considerably better at the intraspecific level over time.

# The Power of Infinity and its Application for Future Studies

In contrast to many other complexity and shape measures, fractal dimension analyses are easily computed and aren’t dependent on the orientation of the study object. This makes them less error-prone and more user-friendly. Based on our results, we can conclude that 3D fractal dimension analyses are an efficient and easily applicable method for shape quantification and complexity characterization of stony corals. It’s highly likely that this is will be the case for other irregularly shaped organisms, too. We see their strength in the application to detect small variations in shape, for example due to environmental change or competitive pressure. At the species level, fractal dimensions may provide new opportunities for semi-automatic or automatic species determination based on 3D morphological images. Selected applications of fractal dimension analyses addressing different levels. Animated Pythagoras tree modified after Marek Fišer (2017).

The applications of 3D fractal dimension analyses are almost as infinite as a fractal itself. They can be used to study other taxa, including organisms with a more regular body plan such as insects or mollusks. Besides their application at the level of individual organisms, fractal analyses can be applied to the ecosystem scale for studying productivity, biodiversity, or resilience (Bradbury & Reichelt 1983; Martin-Garin et al. 2007; Thistle et al. 2010; Kamal et al. 2014).

We’d encourage everybody to explore the potential of fractal dimension analyses for answering ecological, taxonomic, and evolutionary questions in a wide range of organisms. Support for anxious anti-mathematicians could come from our fractal dimension toolbox. Ultimately, we see fractal dimension analyses as an essential compound to provide a more integrative understanding of morphological variation within ecological and evolutionary contexts.

To find out more about 3D fractal dimensions read our Methods in Ecology and Evolution article ‘The power of 3D fractal dimensions for comparative shape and structural complexity analyses of irregularly shaped organisms