Faculty of Life Sciences, University of Manchester, Michael Smith Building, Oxford Road, Manchester M13 9PT, UK

Abstract

Background

Studies of symmetric structures have made important contributions to evolutionary biology, for example, by using fluctuating asymmetry as a measure of developmental instability or for investigating the mechanisms of morphological integration. Most analyses of symmetry and asymmetry have focused on organisms or parts with bilateral symmetry. This is not the only type of symmetry in biological shapes, however, because a multitude of other types of symmetry exists in plants and animals. For instance, some organisms have two axes of reflection symmetry (biradial symmetry; e.g. many algae, corals and flowers) or rotational symmetry (e.g. sea urchins and many flowers). So far, there is no general method for the shape analysis of these types of symmetry.

Results

We generalize the morphometric methods currently used for the shape analysis of bilaterally symmetric objects so that they can be used for analyzing any type of symmetry. Our framework uses a mathematical definition of symmetry based on the theory of symmetry groups. This approach can be used to divide shape variation into a component of symmetric variation among individuals and one or more components of asymmetry. We illustrate this approach with data from a colonial coral that has ambiguous symmetry and thus can be analyzed in multiple ways. Our results demonstrate that asymmetric variation predominates in this dataset and that its amount depends on the type of symmetry considered in the analysis.

Conclusions

The framework for analyzing symmetry and asymmetry is suitable for studying structures with any type of symmetry in two or three dimensions. Studies of complex symmetries are promising for many contexts in evolutionary biology, such as fluctuating asymmetry, because these structures can potentially provide more information than structures with bilateral symmetry.

Background

Morphological symmetry results from the repetition of parts in different orientations or positions and is widespread in the body plans of most organisms. For example, the human body is bilaterally symmetric in external appearance because the same anatomical parts are repeated on the left and right sides. Likewise, many flowers are radially symmetric because sets of petals and other organs are repeated in circular patterns. The evolution of morphological symmetry is of interest in its own right

Because bilateral symmetry is the most widespread and simplest type of symmetry, it has been the most studied in various contexts

Whereas bilateral symmetry has been the focus of most studies, it is not the only kind of symmetry in living organisms (Figure

Some types of symmetry found in the structure of living organisms

**Some types of symmetry found in the structure of living organisms**. A. An alga (

In this paper, we generalize the approach of Mardia et al.

This paper first reviews the mathematical definition of symmetry and the theory of symmetry groups

Results

Mathematical definition of symmetry

Many types of symmetry exist in nature besides the familiar bilateral symmetry. Some organisms, such as the green alga

Instead, we use a definition of symmetry that was developed in a mathematical context

In geometry, symmetry is defined as invariance of an object to a particular transformation that can be applied to it, such that the object is the same before and after transformation

The set of symmetry transformations that define the symmetry group of the equilateral triangle

**The set of symmetry transformations that define the symmetry group of the equilateral triangle**. This symmetry group includes six symmetry transformations: the identity, rotations of order 3, and combinations of reflection with rotations.

The symmetry transformations of an object jointly characterize its symmetry. For instance, bilateral symmetry is defined by two symmetry transformations: reflection and the identity. The set of symmetry transformations of an object forms a special kind of set called a symmetry group [e.g.

The order of a group is the number of distinct elements in the group. Some symmetry groups are of finite order, because the repeated application of the same transformation eventually maps the object onto itself and thus produces only a finite number of different transformations (e.g. a rotation by 120° can only be applied three times to yield the rotations by 120°, 240° and 360° or 0°). These are called finite symmetry groups. All finite symmetry groups in two and three dimensions consist of rotations and reflections (and their combinations; e.g. Figure

Matching symmetry and object symmetry

In their paper on shape analysis of bilateral symmetry, Mardia et al.

Object symmetry concerns structures that are symmetric as a whole

Because object symmetry requires the entire object to be invariant under the transformations in the symmetry groups, some limitations are imposed on the types of symmetry for which object symmetry is possible. For instance, an object cannot be invariant to transformations involving a change of scale. Likewise, a finite object cannot be invariant to transformations involving translations because at least the ends of the object would not be invariant. Moreover, for objects with distinct landmarks (i.e. excluding circles and spheres without any identifiable landmarks), rotations are limited to those by an angle of 360°/

Shape analysis of matching symmetry

Bilateral symmetry

For studies of structures with matching symmetry, the analysis starts with a pair of separate landmark configurations from the left and right sides of each individual in the sample. First, the landmark configurations from one side are reflected. Then, a generalized Procrustes fit superimposes all configurations and produces an overall mean shape [e.g.

The variation around the consensus is decomposed into a component of variation among individuals and a component of left-right asymmetry. The variation among individuals is computed from the average shapes of the left and right sides for each individual in the sample. Asymmetry is quantified from the shape differences between the left and right sides for each individual. The average asymmetry is interpreted as directional asymmetry and the individual variation in asymmetry as fluctuating asymmetry [for details see

To quantify the different effects, we can use the decomposition of the sum of squared Procrustes distances of the configurations from the overall mean shape

In this equation, the expression ^{2}(**a**, **b**) denotes the squared Procrustes (tangent) distance between two landmark configurations **a **and **b **(landmark configurations are written as vectors of coordinates after Procrustes superimposition and projection to tangent space). The landmark configuration **x**
_{is }

Note that the decomposition of the sum of squared Procrustes distances and Procrustes ANOVA usually are only preliminary steps to quantify the relative amounts of variation at the different levels. In particular, reducing shape variation to a scalar distance measure ignores the patterning of variation in the multidimensional space. To extract information from those patterns, the preliminary assessments based on Procrustes distances are usually followed by multivariate analyses of individual variation and asymmetry that can address a wide range of biological questions

Other types of symmetry

Matching symmetry can be generalized directly from bilateral symmetry to any type of symmetry. The main difference is the number and arrangement of the parts that are considered for each individual in the sample, which also results in different components of asymmetry. Instead of a pair of parts on the left and right sides, there may be more than two parts, which can be arranged in many different ways. For example, the flower of Figure

The partitioning of the components of variation extends the type of analysis used for bilateral symmetry. A component of variation among individuals is computed from the averages of the landmark configurations of parts for each individual. Directional asymmetry can be visualised by comparing the mean shape for each of the repeated parts to the grand mean across all parts. Fluctuating asymmetry is the individual variation of the deviations of each repeated part from the individual average of all parts.

The decomposition of the sum of squared Procrustes distances of the shapes of the landmark configurations from the overall mean shape is also a direct extension of the formula for bilateral symmetry (equation 1):

In this equation, the summation over the parts for each individual is not limited to the left and right sides, but can accommodate any number of repeated parts,

The decomposition in equation (2) is a "minimal" version in which the asymmetry, the variation among parts of each individual, is included as a single component (contributing to both fluctuating and directional asymmetry). For example, an analysis of variation in a sample of radially symmetric flowers can use the variation among petal shapes to extract a component of directional asymmetry (deviation of the mean shape of petals at each position from the overall mean of all petals) and a component of fluctuating asymmetry (individual deviations from the mean shape of the respective petal).

Depending on the number and arrangement of parts, however, it may be possible to decompose asymmetry further into multiple components. Such a more complex decomposition is possible for the example of the alga in Figure

As in the case of bilateral symmetry, a variety of multivariate analyses can be used to investigate the patterns of variation in the differences or averages among repeated parts. This is a potentially rich, but so far unexplored area for morphometric studies.

Parts with object symmetry

For many structures that have matching symmetry overall, the repeated parts themselves show bilateral object symmetry. This applies to the vertebrae in a vertebral column (Figure

Shape analysis of object symmetry

Bilateral symmetry

Because the shape analysis for object symmetry is somewhat more complex than for matching symmetry, we review the established method for analyzing bilateral object symmetry

For each specimen, a reflected copy of the original configuration of landmarks is produced (e.g. by changing the signs for one of the coordinates of all landmarks). Because the reflection brings the landmarks from the right side onto the left side and vice versa, the paired landmarks are relabelled by exchanging the labels between left and right paired landmarks of the reflected copy to make them compatible with the paired landmarks of the original configuration. Finally, a Procrustes fit superimposes both the original configurations of landmarks and their reflected and relabelled copies

The resulting consensus configuration is symmetric under reflection

The variation around the consensus is decomposed into a component of symmetric variation and a component of asymmetry

Mardia et al. [

Note that this expression is identical to the one for bilateral matching symmetry (equation 1) except for the subscript

The patterns of covariation in the symmetric and symmetric components of variation can be further explored in multivariate analyses to answer a wide range of questions. Such analyses can use principal component analysis

It is useful at this point to reconsider this established method from the new perspective of symmetry groups. For bilateral symmetry, the symmetry group consists of just two symmetry transformations: the identity and a reflection. The unchanged and reflected copies of each configuration that are included in the Procrustes fit can therefore be interpreted as a set of copies of each original configuration to which the whole set of symmetry transformations has been applied. The relabelling serves to map each landmark to the corresponding one after the transformation: for the reflection, each paired landmark is mapped to its equivalent on the opposite side and each median landmark is mapped to itself (for the identity, every landmark is mapped to itself).

This reasoning can also be used to understand why the Procrustes average of the unchanged copy and the copy that has been reflected and relabelled must be symmetric. The pair of unchanged and reflected copies of each configuration is invariant under reflection because the two transformations that were applied to the original configuration (identity and reflection) constitute the entire symmetry group. If we combine any given symmetry transformation with each of the transformations in the symmetry group, the resulting transformations must also be elements of the symmetry group. Therefore, the set of all the resulting transformations is the same as the original symmetry group. As a consequence, the Procrustes average of the complete set of copies must be the same before and after the transformation, and therefore is symmetric.

These explanations from the perspective of symmetry groups may seem an overly complicated manner of describing the procedure and of justifying why the consensus is symmetric, but this new reasoning has the advantage of offering a direct way to generalize the analysis to more complex types of symmetry.

Generalization to other types of symmetry

Using the reasoning based on symmetry groups, it is possible to extend the method for analyzing object symmetry to any type of symmetry that is associated with a finite symmetry group. All finite symmetry groups can be generated by reflection, rotation or a combination of both. For configurations in two dimensions, a symmetry group can contain only a single rotation about a central point. This rotation can have different orders, the number of repeated steps it takes to cover a full circle (i.e. for a rotation of order

Enumeration of all finite symmetry groups in 3D space, with the Schoenflies and orbifold notations and the order of each group

**Schoenflies**

**Orbifold**

**Order**

**Comments**

C_{n}

Rotational symmetry of order

C_{nv}

*

2

Rotational symmetry of order

C_{nh}

2

Rotational symmetry of order

S_{2n}

2

Rotational symmetry of order 2

D_{n}

22

2

Dihedral symmetry: rotational symmetry of order

D_{nd}

2*

2

Antiprismatic symmetry: Rotation symmetry of order

D_{nh}

*22

4

Prismatic symmetry: rotational symmetry of order

T

332

12

Tetrahedral symmetry, rotations only

T_{d}

*332

24

Complete tetrahedral symmetry, including reflection

T_{h}

3*2

24

Pyritohedral symmetry

O

432

24

Octahedral symmetry, rotations only (also applies to cube)

O_{h}

*432

48

Complete octahedral symmetry, icluding reflection (also applies to cube)

I

532

60

Icosahedral symmetry, rotations only

I_{h}

*532

120

Complete icosahedral symmetry, including reflection

Bilateral symmetry can be viewed as a special case of C_{nv }or C_{nh }with a rotation of order 1.

The idea for shape analysis with object symmetry of any type is to assemble a data set containing copies of each original landmark configuration to which all the transformations in the symmetry group have been applied with the appropriate relabelling, and then to perform a generalized Procrustes fit on this data set. As for bilateral symmetry, the resulting consensus shape is symmetric and it is possible to extract components of symmetric and asymmetric variation by computing the appropriate averages or differences between landmark configurations after the Procrustes fit.

Because all finite symmetry groups consist of rotations and reflections, it is sufficient to consider the steps to produce the transformed and relabelled copies for reflection and rotation. For reflection, there are paired and unpaired landmarks. The reflection itself can be carried out by changing the sign of one coordinate for all landmarks (e.g. all

The generalized Procrustes fit of the combined data set produces a consensus shape that is invariant under all the transformations in the symmetry group, that is, a completely symmetric shape. Because the set of transformations used to produce the combined dataset is the complete symmetry group, applying any of the symmetry transformations leaves the set unchanged as a whole, and therefore does not alter the average shape resulting from the Procrustes fit. Therefore, the consensus shape is invariant under all the symmetry transformations, and is thus perfectly symmetric. This is true both for the consensus shapes for individual specimens (the Procrustes consensus for the set of the original and transformed copies of just one specimen at a time) and for the combined Procrustes fit of all specimens jointly (all copies for all individuals).

The final alignment of landmark configurations in the generalized Procrustes fit is obtained by an ordinary Procrustes fit of each configuration to the consensus shape

The Procrustes fit of the transformed and relabelled copies of a single triangle to the symmetric consensus

**The Procrustes fit of the transformed and relabelled copies of a single triangle to the symmetric consensus**. The diagram shows the symmetric mean shape (bold solid triangle) and six copies of the triangle that have been transformed and relabelled using six symmetry transformations: the identity, rotations of order 3, and combinations of reflection with rotations (i.e. this is the same symmetry group as in Figure 2). Copies of the triangle for which the transformation does or does not include reflection about the vertical axis are distinguished by dashed and dotted lines.

Because of the symmetry of the consensus configuration, the Procrustes distance between it and every transformed and relabelled copy must be the same for each specimen included in the analysis. This applies both for the consensus configuration for each particular specimen and for the grand mean across all copies of all specimens included in an analysis (and the distance to the particular specimen's consensus shape is less than or equal to the distance to the overall consensus shape). These regularities have direct consequences for the structure of variation in the shape tangent space. Recall that usually the mean shape is chosen as the tangent point

It is possible to extract and quantify components of symmetric and asymmetric variation by averaging the transformed and relabelled copies or by computing differences between them. The decomposition of the total sum of squared Procrustes distances in a sample into these components of symmetric and asymmetric variation is a direct extension of the decomposition for bilateral symmetry (equation 3):

The difference to equation (3) for bilateral symmetry is that any symmetry group can be accommodated. Accordingly, the subscript **x**
_{is }

A component of symmetric variation among individuals can be computed from the variation among the average shapes for the complete set of copies for each specimen. Directional asymmetry can be obtained as the difference of the average shape of all original configurations from the overall Procrustes consensus of the original and transformed copies. Fluctuating asymmetry can be computed from the variation of the individual asymmetries. These computations are direct extensions of those for bilateral object symmetry

The symmetric and asymmetric components of variation occupy mutually orthogonal subspaces of the shape tangent space, again extending the situation found for bilateral symmetry

Here, we only work out the dimensionalities of these subspaces for symmetry groups containing a single rotation with or without reflection. Together with bilateral symmetry, these are all the possible types of complex object symmetry for two-dimensional data (i.e. all finite symmetry groups in 2D). For three-dimensional data, however, these two types do not cover all possibilities for complex object symmetry (cf. Table _{n }
_{
nv
}), but they are by far the most widespread types of complex symmetry in the shapes of organisms and their parts (radial and disymmetric floral symmetries, the body plans of cnidarians and echinoderms, diatom cells, etc.).

For the types of symmetry involving a single rotation, the configuration of landmarks can be divided into sectors, which are the repeated units (Figure

Sectors and types of landmarks for complex object symmetry with a rotation and reflection

**Sectors and types of landmarks for complex object symmetry with a rotation and reflection**. To compute the dimensionalities of the different components of shape space, it is helpful to subdivide the configuration of landmarks into sectors and to distinguish different types of landmarks. The diagram shows an example of symmetry under rotation of order 4 and reflection. Therefore, the configuration can be divided into four sectors: the regions that correspond to each other when the rotation is applied (sector boundaries are indicated by solid black lines). If the symmetry also includes reflection, as in this example, the arrangement of landmark in each sector is also bilaterally symmetric about the midline or mid-plane of each sector (dashed lines). Several types of landmarks can be distinguished. There may be a landmark in the centre of rotation or, for 3D data, there may be multiple landmarks of the axis of rotation (

For symmetry involving rotation only, shape variation has a symmetric and an asymmetric component (Table

Number of dimensions in the different components of shape space under object symmetry with rotation only or with rotation and reflection, for landmark data in two and three dimensions

**2D**

**3D**

Symmetry under rotation only:

Symmetric

2

3

Asymmetric

2

3

Symmetry under rotation and reflection:

Completely symmetric

2

=

3

Reflection symmetry only

2

=

If

If

Rotational symmetry only

2

=

3

Completely asymmetric

2

=

If

If

Notation: For rotational symmetry of order

If the symmetry group includes both rotation and reflection (with the reflection plane containing the rotation axis), there is a totally symmetric component of variation and the asymmetric component can be subdivided into a component symmetric under reflection only, a component symmetric under rotation only, and a component that is symmetric under neither rotation nor reflection (Table

For types of symmetry groups where the asymmetric component of shape variation can be subdivided into multiple complementary subspaces (e.g. the group of rotation and reflection, lower part of Table

Because the symmetric and asymmetric components of shape variation are orthogonal subspaces of shape tangent space, they can also be identified and characterized through a principal component analysis (PCA) of the Procrustes-superimposed data in the combined sample of the original and transformed copies of landmark configurations. This approach can be used for any type of symmetry. It is a direct extension of the method for bilateral symmetry, where it can be shown that the symmetric and asymmetric components of variation in the combined sample do not covary

Procrustes ANOVA for complex symmetry

To quantify the different components of variation, we offer an extension to the Procrustes ANOVA for bilateral symmetry

Procrustes ANOVA for complex matching symmetry

For Procrustes ANOVAs of complex symmetries in the framework of matching symmetry, the units of analysis are the repeated parts (e.g. petals of flowers) on which landmarks are recorded. Landmark configurations for all parts of all individuals are included in the analysis. The simplest structure for the Procrustes ANOVA is then to include individuals and repeated parts as factors, and the main difference to the corresponding analysis for bilateral symmetry is that the factor of repeated parts may have more than two levels (e.g. five petals instead of two sides).

The interpretation of effects in the Procrustes ANOVA, in this case, closely follows that for bilateral symmetry. The main effect of individuals represents individual variation; because it is computed from averaging of all repeated parts in each individual, this component of variation is "symmetrized". The main effect of repeated parts represents directional asymmetry, the average asymmetry in the sample. Because of the complex symmetry, directional asymmetry is not just the difference between left and right averages, as for bilateral symmetry, but it represents the variation among the means of the different parts, averaged over all individuals. For the example of a flower with five petals, this means that directional asymmetry is represented by the shape differences of the five mean shapes of the petals from the overall mean shape. Finally, the individual × repeated-part effect represents fluctuating asymmetry, the individual variation in the differences among the repeated parts (just as fluctuating asymmetry, in the two-factor design for bilateral symmetry, is the individual variation in left-right differences

If there is a more complex structure in the arrangement of repeated parts, however, this can be accommodated by dividing the factor for repeated parts into multiple factors that capture aspects of the differences between repeated parts. We call these collectively "asymmetry factors". For instance, if four parts are arranged as quadrants (Figure

The interpretation of the effects in the Procrustes ANOVA is somewhat more intricate in this case because there are additional effects to be considered. As before, the main effect for individuals represents the variation among individuals and is symmetrized completely. The main effects of the asymmetry factors (e.g. dorsal-ventral and left-right) represent directional asymmetry in each particular aspect, symmetrized with respect to the other aspects (e.g. directional dorsal-ventral asymmetry, symmetrized for left-right differences, or directional left-right asymmetry, symmetrized for dorsal-ventral differences). This means that each of the aspects of symmetry is associated with a component of directional asymmetry that isolates this particular aspect and "averages out" other asymmetries. An additional component of directional asymmetry comes from the interaction between asymmetry factors, such as the interaction between the dorsal-ventral and the left-right effect. This component of directional asymmetry represents deviations of each part from the asymmetry expected from adding together the separate components of asymmetry (e.g. how the average shape of each quadrant differs from the asymmetry expected by adding the effects of the dorsal-ventral and the left-right asymmetries). Overall, directional asymmetry is divided into three components (or more, if there are more than two asymmetry factors). The interaction effects between the factor for individuals and the asymmetry factors represent components of fluctuating asymmetry. Again, there are several of these components: the two-way interactions between individuals and each of the asymmetry factors represent fluctuating asymmetry in just that aspect of asymmetry (e.g. in the dorsal-ventral or left-right components separately, symmetrized for the other aspects of asymmetry) and the three-way interaction between individuals and both asymmetry factors represents fluctuating asymmetry of each individual part as deviations from the added effects of the two asymmetry factors (deviations of each part from the asymmetry expected for that quadrant by adding the dorsal-ventral and left-right asymmetries). In summary, directional and fluctuating asymmetry are divided into components that represent particular aspects of asymmetry and can be used to address specific biological questions about the organisms under study.

Statistical testing in the context of Procrustes ANOVA for complex matching symmetries is similar to Procrustes ANOVA for bilateral matching symmetry

Procrustes ANOVA for complex object symmetry

As for matching symmetry, the Procrustes ANOVA for object symmetry with complex types of symmetry provides a similar decomposition of the variation into symmetric variation and one or more components of asymmetry. For any type of symmetry, it is possible to conduct Procrustes ANOVA with a single asymmetry factor that includes all transformations in the symmetry group as its levels. This follows the decomposition of the sum of squared Procrustes distances (equation 4).

Depending on the type of symmetry of the objects under study, it may be possible to use two or more transformations in the symmetry group as separate asymmetry factors in the ANOVA model. Specifically, the transformations used as asymmetry factors in the Procrustes ANOVA must be a set of generators of the symmetry group (a set of transformations that, under repeated application and in combination with each other, produce all the transformations in the symmetry group). Because some symmetry groups have multiple sets of generators, there may be an element of choice for the investigator. For instance, for the algal cell in Figure

As in the Procrustes ANOVA for matching symmetry, the main effect of individuals represents among-individual variation; shape changes associated with this effect are completely symmetric because it is averaged over all transformed and relabelled copies of each individual configuration. The main effects of the asymmetry factors represent the respective components of directional asymmetry (e.g. average asymmetries with respect to rotations or reflections), each symmetrized with regard to the effects of all other asymmetry factors. The interaction effects between asymmetry factors characterize the components of directional asymmetry (those features of directional asymmetry that result from combinations of the different asymmetry factors). The interaction effects between the factor for individuals and the asymmetry factors represent the different components of fluctuating asymmetry (if more than one asymmetry factor is included, there are multiple components of fluctuating asymmetry).

Whether multiple asymmetry factors can be used in the Procrustes ANOVA depends on the type of symmetry. We only illustrate this in more detail for symmetries consisting of a rotation with or without reflection, which cover the vast majority of biological structures with complex symmetry. The components of variation of the effects of individuals, the one or more asymmetry factors and their interaction effects occupy different subspaces of the shape tangent space. The dimensionality of those subspaces (Table

If the symmetry group includes only a rotation around a single point or axis, the Procrustes ANOVA contains this rotation as the only asymmetry factor (Table

Degrees of freedom in the Procrustes ANOVA for object symmetry with rotation only in two and three dimensions

**Effect**

**2D**

**3D**

Individual

(

(

Rotation

2

3

Rotation × individual

(

(

Measurement error

(

(

Notation: For rotational symmetry of order

If the symmetry group includes rotation and reflection, the Procrustes ANOVA includes both of them as asymmetry factors, which leads to a more complex decomposition of the total variation (Table

Degrees of freedom in the Procrustes ANOVA for object symmetry with rotation and reflection in two and three dimensions

**Effect**

**2D**

**3D**

Individual

(

= (

(

Rotation

2

=

If

If

Reflection

2

3

Rotation × reflection

2

=

If

If

Rotation × individual

(

= (

If

If

Reflection × individual

(

= (

(

Rotation × reflection × individual

(

= (

If

If

Measurement error

(

= (

(

Notation: For rotational symmetry of order

Testing in the framework of Procrustes ANOVA for complex symmetries is similar to Procrustes ANOVA for bilateral symmetry

Finally, a note of caution about terminology may be useful. To count as symmetric under rotation for the Procrustes ANOVA, a shape change must be symmetric under rotation of the full order. But these may not be the only components of variation that are rotationally symmetric if the order of rotation is not a prime number. For instance, analyses involving rotation of order 6 may include shape features that are symmetric under rotations of order 2 or 3; these will be counted as shape changes that are asymmetric under rotation, because they are not symmetric under rotation of order 6. Likewise, reflections are defined in relation to a particular axis or plane of reflection; shape changes that are symmetric under reflection about a different axis or plane will therefore not be considered as symmetric under reflection for the purposes of the Procrustes ANOVA. Because these partial symmetries concern one or more of the components of fluctuating asymmetry, a PCA of the covariance matrices for the respective effects can reveal this additional structure in the data. The order of rotation and axis or plane of reflection should be chosen to relate to the anatomical features and the biological question under investigation. All these considerations are best illustrated by an example.

Case study: Shape analysis of symmetry and asymmetry in a colonial coral

Our case study concerns the symmetry and asymmetry of corallites in a colonial coral (

Schematic representation of a corallite with the landmarks used in this study

**Schematic representation of a corallite with the landmarks used in this study**. The septa colored in dark grey belong to the first cycle, the ones in light grey to the second cycle, and those in black to the third cycle. The ring in white represents the mural structure of the corallite.

We characterize the shape of 50 corallites from a single colony by a configuration of 48 landmarks (Figure

In all analyses, we use a full Procrustes fit of all transformed and relabelled copies of the landmark configurations and the data are projected into the tangent space by orthogonal projection

In addition, principal component analysis (PCA) is also used to separate different components of symmetric and asymmetric shape variation and to display the corresponding shape changes

Analysis 1: Reflection and rotation of order 2

The first analysis uses the type of symmetry with two perpendicular axes of reflection symmetry, also known as biradial symmetry or disymmetry (e.g. Figure

The shape tangent space consists of four subspaces: a component that is symmetric under both rotation and reflection (about both horizontal and vertical axes), a component symmetric under rotation by 180° (but not under any reflection), a component symmetric under reflection about the vertical axis and, finally, a component that is not symmetric under either rotation or reflection about the vertical axis (but due to the constraints imposed by the Procrustes fit, it is symmetric under reflection about the horizontal axis). Each of these subspaces has 23 dimensions (Table

The Procrustes ANOVA includes individuals, rotation by 180° and reflection about the vertical axis as the factors (Table

Procrustes ANOVA for the coral example, with a symmetry group consisting of reflection and rotation of order 2

**Source**

**Degrees of freedom**

**Sums of squares**

**Mean squares**

**
F
**

**
P
**

Individual

1127

4.9380

0.0043815

1.82

< 0.000001

Rotation

23

0.096360

0.0041896

4.20

< 0.000001

Reflection

23

0.16058

0.0069818

1.34

0.13

Rotation × reflection

23

0.072451

0.0031501

3.17

< 0.000001

Rotation × individual

1127

1.1251

0.0009983

4.95

< 0.000001

Reflection × individual

1127

5.8839

0.0052209

25.87

< 0.000001

Rotation × reflection × individual

1127

1.1184

0.0009924

4.92

< 0.000001

[Total FA]

3381

8.1274

0.0024039

Imaging error

4600

0.92842

0.0002018

1.26

< 0.000001

Digitizing error

9200

1.4789

0.0001607

The main effect of individuals is tested against the mean square for the total fluctuating asymmetry ("Total FA": pooling sums of squares and degrees of freedom across all three subspaces with asymmetric variation: rotation × individual, reflection × individual and rotation × reflection × individual). This total asymmetry is not used otherwise in the analysis.

In accordance with the structure of the shape space, the PCA produces PCs associated with four types of shape changes that differ in their symmetries. The PC1 is symmetric under rotations of order 2, but not under reflection (Figure

Analysis 1: Decomposition of shape variation for symmetry with respect to reflection and rotation of order 2

**Analysis 1: Decomposition of shape variation for symmetry with respect to reflection and rotation of order 2**. This figure shows examples of PCs that account for the maximum of variance for each category of shape variation. Each diagram shows the symmetric consensus (open circles and dotted lines) and the differences between the consensus and the other configuration (solid circles and solid lines) represent the shape change associated with the respective PC by an arbitrary amount of + 0.1 units of Procrustes distance. The percentages represent the part of the total shape variation for which each PC accounts. A. Asymmetric component, symmetric under rotation of order 2. B. Symmetric component. C. Asymmetric component, symmetric relative to reflection about the horizontal axis. D. Asymmetric component, symmetric under reflection about the vertical axis.

Because each PC can be unambiguously allocated to one of the components of shape space, it is possible to count the dimensions in the respective subspaces and to add up the corresponding eigenvalues to quantify the amount of total variance for each type of symmetry (note that this also includes the imaging and digitizing errors, which are not separated according to components by the Procrustes ANOVA). The 23 PCs that are symmetric under rotation by 180° account for nearly half the total shape variation (45.48% of the total variance). The 23 completely symmetric PCs account for 35.38% of the total variance, 9.09% of the total variance are apportioned to the 23 PCs that are symmetric under reflection about the horizontal axis (asymmetric under both rotation and reflection about the vertical axis), and 9.05% of the total variance are allocated to the 23 PCs that are symmetric with respect to reflection about the vertical axis. These numbers of PCs correspond to the numbers of dimensions in the respective subspaces.

Analysis 2: Rotation of order 6

For the analysis of symmetry under rotation of order 6, the symmetry group contains six symmetry transformations: rotations by 60°, 120°, 180°, 240°, 300°, and 360° (the latter is the same as the identity).

The analysis partitions the shape tangent space into a 14-dimensional symmetric component and a 78-dimensional component that is not symmetric under rotations of order 6. In the Procrustes ANOVA, the main effect of rotation and the rotation × individual interaction dominate both the sums of squares and mean squares (Table

Procrustes ANOVA for the coral example, with a symmetry group consisting of rotation of order 6

**Source**

**Degrees of freedom**

**Sums of squares**

**Mean squares**

**
F
**

**
P
**

Individual

686

0.79714

0.001162

0.24

> 0.999999

Rotation

78

7.6145

0.097622

20.04

< 0.000001

Rotation × individual

3822

18.616

0.0048708

16.08

< 0.000001

Imaging error

4600

1.3934

0.0003029

1.25

< 0.000001

Digitizing error

9200

2.2213

0.0002414

The 92 PCs are divided into four types of shape changes. The first category includes the PC1, which is symmetric under rotations of order 2 (Figure

Analysis 2: Decomposition of shape variation for symmetry with respect to rotation of order 6

**Analysis 2: Decomposition of shape variation for symmetry with respect to rotation of order 6**. This figure shows examples of PCs that account for the maximum of variance for each category of shape variation. Each diagram shows the symmetric consensus (open circles and dotted lines) and the differences between the consensus and the other configuration (solid circles and solid lines) represent the shape change associated with the respective PC by an arbitrary amount of + 0.1 units of Procrustes distance. The percentages represent the part of the total shape variation for which each PC accounts. A. Asymmetric component, symmetric under rotation of order 2. B. Asymmetric component, symmetric under rotation of order 3. C. Symmetric component. D. Totally asymmetric component.

In total, 32 PCs are symmetric under rotation of order 2 and account for most of the variance (82.63% of the total variance), 16 are symmetric under rotation of order 3 (7.15% of the total variance), 14 are symmetric under rotation of order 6 (the symmetric component of the Procrustes ANOVA) and represent 3.43% of the total variance, and 30 are completely asymmetric (6.79% of the total variance). Above all, it is notable that the PCA further subdivides the 78 dimensions of the asymmetric component of variation of the Procrustes ANOVA.

Analysis 3: Reflection and rotation of order 6

For the analysis of symmetry under reflection and rotation of order 6, the symmetry group includes twelve symmetry transformations: each of the six rotations of the preceding analysis is now included with or without reflection.

There are four subspaces: a 7-dimensional subspace of completely symmetric shape changes, a 7-dimensional subspace of shape changes that are symmetric under rotations of order 6 but not under reflection, a 39-dimensional subspace of shape changes that are symmetric under reflection about the vertical axis but not under rotation of order 6, and a 39-dimensional shape space of completely asymmetric shape changes. The Procrustes ANOVA shows that, for both the sums of squares and mean squares (Table

Procrustes ANOVA for the coral example, with a symmetry group consisting of reflection and rotation of order 6

**Source**

**Degrees of freedom**

**Sums of squares**

**Mean squares**

**
F
**

**
P
**

Individual

343

1.510

0.0044038

0.49

> 0.999999

Rotation

39

7.6968

0.19735

20.31

< 0.000001

Reflection

7

0.008013

0.0011447

4.69

0.000049

Rotation × reflection

39

7.5323

0.19314

19.78

< 0.000001

Rotation × individual

1911

18.569

0.0097171

16.04

< 0.000001

Reflection × individual

343

0.083800

0.0002443

0.40

> 0.999999

Rotation × reflection × individual

1911

18.663

0.009766

16.11

< 0.000001

[Total FA]

4165

37.316

0.0089594

Imaging error

4600

2.7869

0.0006058

1.25

< 0.000001

Digitizing error

9200

4.4426

0.0004829

The main effect of individuals is tested against the mean square for the total fluctuating asymmetry ("Total FA": pooling sums of squares and degrees of freedom across all three subspaces with asymmetric variation: rotation × individual, reflection × individual and rotation × reflection × individual). This total asymmetry is not used otherwise in the analysis.

The PCs fall into eight distinct categories of shape changes (Figure

Analysis 3: Decomposition of shape variation for symmetry under reflection and rotation of order 6

**Analysis 3: Decomposition of shape variation for symmetry under reflection and rotation of order 6**. This figure shows examples of PCs that account for the maximum of variance for each category of shape variation. Each diagram shows the symmetric consensus (open circles and dotted lines) and the differences between the consensus and the other configuration (solid circles and solid lines) represent the shape change associated with the respective PC by an arbitrary amount of + 0.1 units of Procrustes distance. The percentages represent the part of the total shape variation for which each PC accounts. A. Asymmetric component, symmetric under reflection and rotation of order 2. B. Asymmetric component, symmetric under rotation of order 2 but not reflection. C. Asymmetric component, symmetric under reflection about the vertical axis and rotation of order 3. D. Asymmetric component, symmetric under reflection about the horizontal axis and rotation of order 3. E. Completely symmetric component. F. Asymmetric component, symmetric under reflection about the vertical axis. G. Asymmetric component, symmetric under reflection about the horizontal axis. H. Asymmetric component, symmetric under rotation of order 6.

Overall, patterns of shape variation symmetric under rotations of order 2 dominate the variation: 16 PCs are symmetric under rotation of order 2 and account for 41.3% of the total variance and 16 PCs are symmetric under both reflection and rotations of order 2 and account for a further 41.3% of the total variance. In addition, there are 15 PCs that are symmetric under reflection about the vertical axis and account for 3.4% of the total variance, and another 15 PCs are symmetric under reflection about the horizontal axis and also account for 3.4% of the total variance. There are 8 PCs that are symmetric under rotation of order 3 and reflection about the vertical axis and take up 3.5% of the total variance, whereas 8 PCs are symmetric under rotation of order 3 and reflection about the horizontal axis and account for 3.6% of the total variance. Finally, 7 PCs are completely symmetric and comprise 3.3% of the total variance, whereas 7 PCs are symmetric under rotation of order 6 and make up the remaining 0.2% of the total variance.

Discussion

In this paper, we have introduced a new and general framework for shape analysis of symmetric and asymmetric variation in a configuration of landmarks with any type of symmetry. We have presented methods that implement this framework and extend the methods widely used for analyzing shapes with bilateral symmetry

In our case study, the Procrustes ANOVAs and PCAs for the three analyses produce different decompositions of the shape variation into components of symmetric and asymmetric shape variation. A common feature, however, is that the majority of the shape variation is contained in the asymmetric component. In the first analysis, with symmetry under reflection and rotation of order 2, the PCA indicates that asymmetric variation accounts for 64.62% of the total variance. In the two other analyses, which include rotation of order 6, the components of asymmetric variation take up more than 96% of the total variance in the PCA. The large proportion of shape variation that is taken up by the asymmetric component of variation is in striking contrast to most studies of shape variation in structures with bilateral symmetry [e.g.

The PCAs reveal much more shape variation that is structured according to biradial symmetry (rotation of order 2 and reflection) than there is variation with hexagonal symmetry (rotation of order 6 and reflection). The components that are symmetric under rotation by 180° are consistently among the dominant PCs and account for about 40% of the total variance of shape (Figure

For our example analyses, only a fairly small proportion of shape variation is symmetric. Some patterns of symmetric variation among corallites might be due to differences in septal growth. The septa develop from the mural structure of the corallite towards the centre and also towards the outside of the corallite

Although the Procrustes ANOVAs and the PCAs both produce broadly similar results, these analyses differ substantially in how they partition the total variation into components. The Procrustes ANOVA allocates shape variation to various components that reflect the study design and data collection, such as directional and fluctuating asymmetry or imaging and digitizing error, even if several of these components are located in the same subspace. The components extracted by the Procrustes ANOVA, however, may be somewhat heterogeneous in their symmetries (e.g. the component of variation considered asymmetric under rotation of order 6 may contain shape changes that are symmetric under rotations of orders 2 or 3 or under reflections). In contrast, the PCA separates components according to the structure of the shape tangent space, and can therefore identify more classes of shape changes according to their symmetries. The PCA, however, does not consider other aspects of the data structure--importantly, it provides only a total characterization of asymmetry that does not distinguish between directional and fluctuating asymmetry. The PCA considers asymmetry as the total of the deviations from complete symmetry. PCA does not distinguish whether variation stems from consistent differences between repeated parts that are shared among individuals and are therefore directional asymmetry, or whether variation reflects individual differences in the deviations from symmetry and is therefore fluctuating asymmetry. Overall, therefore, Procrustes ANOVA and PCA give somewhat different perspectives on the variation in the data, which is most important for the interpretation of asymmetry.

The three analyses, based on different symmetry groups, produce different estimates of the consensus shape and different partitions of the observed shape variation into components with distinct types of symmetry. The number of these components is influenced primarily by the number of symmetry transformations for the type of symmetry considered and, accordingly, by the number of transformed copies of each landmark configuration that are included in the dataset. In the first analysis, only four transformed copies are included in the dataset, whereas six transformed copies are considered in the second analysis, and twelve transformed copies are included in the third analysis. More complex types of symmetry are associated with symmetry groups that consist of greater numbers of transformations, and thus produce more types of PCs.

These differences between analyses of the same data raise the question how the most appropriate type of symmetry should be chosen. The coral example was included specifically because of its ambiguous symmetry, so that several types of symmetry can be demonstrated with the same data, and this analysis should therefore not be viewed as a model for studies of complex symmetries in general [for such an example, see

Alternative approaches

A few different morphometric methods for the analysis of complex symmetries have been suggested in recent years

Potapova and Hamilton

Finally, the approach of Zabrodsky et al.

Conclusion

The approach we have introduced in this paper provides powerful morphometric tools for biologists to analyze symmetry and asymmetry in landmark data. The method generalizes the approach previously used for bilateral symmetry for studies of landmark configurations with any possible type of symmetry and provides a unified perspective on biological symmetry. Previous insights about the structure of the shape tangent space for bilateral symmetry

The use of complex asymmetries, where more than two repeated parts can be compared, avoids some of the problems of fluctuating asymmetry in organisms with bilateral symmetry

Methods

Our case study concerns the symmetry and asymmetry of corallites in a specimen of colonial coral (

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

YS and CPK jointly devised the methodology. YS collected the data and conducted the analyses for the case study. YS and CPK wrote the paper. Both authors read and approved the final manuscript.

Acknowledgements

We are grateful to past and present members of the Klingenberg lab, particularly Nicolas Navarro, as well as to Richard Abel and Sylvain Gerber, for valuable discussions and suggestions. We also thank John Kent, Fred Bookstein, Kanti Mardia and Ian Dryden for stimulating discussions about the mathematical details of the method, Henry McGhie (Manchester Museum) for access to the specimen and Bernard Lathuilière for providing reprints of papers and helpful discussion. Dean Adams and the anonymous reviewers provided constructive comments on earlier versions of the manuscript. This work was partly funded by a studentship for YS from the Biotechnology and Biological Sciences Research Council (UK).