Plant Modeling

Section Plant Genetics | Radboud University Nijmegen | The Netherlands

A Botanical Perspective on Modeling Plants and Plant Shapes in Computer Graphics

Johan Gielis & Tom Gerats

A range of techniques and software packages is available for modeling different aspects of plant architecture. However, the structure of these "virtual plants" is only a gross abstraction of the enormous biological variation we encounter in real plants, both within and between individuals. Growth and development of plants can not just be represented using L-systems, but should include the complex interplay between environmental, genetic, physiological, biophysical and biomathematical factors. To properly model plant growth and development, such that it represents a plant’s variability in morphology, it is necessary to include all these factors into the modeling. Such a dynamic modeling will complement the successes of molecular biology and contribute to the "in silico" study of plants.

One of the reasons that computer-based modeling so far is hardly more than a gross abstraction of shape, is that most computer graphics representations of shape are simply meaningless for the study of real plants, for which a botanical knowledge-based approach is needed. For knowledge-based modeling, the particular choice of a method is crucial, and different models, which take into account biological information are needed.

In this paper we discuss progress on the use of the Superformula (Gielis J., 2003, American Journal of Botany, 90(3) 333-338), and its potential for knowledge-based modeling in botany. Our biomathematical approach allows us to develop new, testable hypotheses in plants. Fusion of plant organs is taken as a starting point to study evolutionay patterns in plants from mathematical properties only. As an example we focus on flowers and floral organs. We demonstrate how the Superformula allows an improved understanding of botanical shapes in relation to optimization and evolutionary trends in plants, and discuss consequences for plant modeling.

Keywords: plant modeling, leaves, flowers, superformula

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Introduction

Computer graphics is a powerful technology using a range of tools, more or less determined by their computational efficiency and demands of hardware and software. This range of tools is thus also available in computer graphics to model plants and plant parts (for a comprehensive review see [1] and references therein). Such models are called virtual plants and can mimic real plants to a high degree of accuracy. The focus of computer graphics has been largely on the result, the image on the screen. Plants are modeled based on formal grammars (L-systems), but other approaches such as CSG are also used [1].

Virtual plants definitely can be useful as a supportive model and many have been used successfully for morphometric studies in which shapes have to be compared, e.g. for taxonomic studies using leaf shapes [2],  to assess the influence of CO2 on plant growth [3], or to track growth and development of flowers. However, the notion of abstract plants is much more useful than ‘virtual plants’, since it abstracts the plant as a living and growing organism, and takes in account molecular, biophysical, physiological and mathematical aspects.

Abstract plants will reflect a much deeper knowledge about plant growth and morphology, and this requires radically new models for plant growth and development, different from virtual plants. For knowledge-based approaches it is crucial to select and use models which provide useful information for the botanist, at every level, from the description of cells and organs, to the actual construction of abstract plants. This is similar to knowledge-based image analysis, in which the most crucial step is image synthesis, which is based on specialized knowledge, not computer efficiency [5]. The models used must be botanical in their own right, and they should be able to link the mathematical model itself to plant growth, development and evolution. Current models used in morphometrics cannot be used for this. In the grand project of building the “in silico plant”, the aim of current systems biology approaches [4], virtual plants yet form only a small part.

For modeling virtual plants, basically any method can be used. Most of the models, including commercially available plant modelling software, is based on software related motivation, and lack any botanical motivation [1]. Botanical motivation is seen mainly in the modelling of branching and growth habits following the classic work of Aristid Lindenmayer on L-systems, but none of the existing techniques show any botanical motivation to model individual organs and parts. In geometric morphometrics outline models are approximated with radial basis functions (elliptic Fourier descriptors) and in other cases ad hoc procedures (programming to mimic real leaves) are used. These models provide no insight whatsoever in growth and development.

In abstract plants, the ideal and perfect model, should be able to capture generality and individuality, and combine symmetry with asymmetry, a qualitative with a precise description. Besides, the model should also give insight in growth and development, or issues on optimisation and developments. In addition, the model should also be able to answer questions from Plant Morphology, on how certain homologous shapes can transform into other organs. Such a mathematical model for plants should be a model from which already many conjectures (for example about the invariants, efficiency, or differential equations and boundary conditions) could be derived which can be proven mathematically, and which are confirmed by empirical evidence, in the same way physics works. Models like this are much less known in botany. In the development of biological models, we still need to go a long way.

We have proposed a new approach to model plants and other natural and man-made shapes [6]-[8]. The Superformula, as an extension of superellipses, takes into account important biological characteristics.  Many natural and abstract shapes can be described by this single formula. More specifically, the unit circle could be modeled into superpolygons with rational or irrational symmetries, also in 3-D [9]. This has not only allowed us to model plant shapes, but also to develop new frameworks for interpreting shapes (instead of approximating a shape using a Fourier series [2], or a cactus cross section as a collection of triangles on a circle [10]).

One of its main advantages is that the formula provides a radial and an angular function in one. It includes valuable biological information regarding symmetry, shape and size, in one single equation and is more general than many existing methods. For example, Fourier descriptors of same shapes always require a specific number of terms, since base functions for Fourier series are circle, sine and cosine. If the base functions would be extended with Superformular metrics – trigonometric functions can be defined on any supershape - the number of terms would decrease, in many cases to a single term only [9]. Since the formula is an implicit function, any point inside, on or outside the boundary is described as well. This allows to describe leaves and plant organs in great detail, with detailed height maps for venation and microvenation, so that not only boundary (or single parameter boundary descriptors) but also landmarks and pseudolandmarks can be used, making it an integrative model for morphometrics in botany, also for highly variable leaves.

In this paper we address the use of the Superformula to describe flower shapes as modified leaf structures, and create a link with evolutionary strategies and developmental biophysical aspects. Indeed, by modifying the metrics of for example trigonometric functions [7], flowers and leaves can be modelled (see Materials and Methods section and Fig. 1). The relation between shapes of flowers and leaves on the one hand, and trigonometric functions like sine and cosine on the other, was first postulated by Grandus in the 17th century [11]. The generalized trigonometric functions, based on the Superformula, allow to model a much wider range of  flower shapes [7], [8]. In botany, leaves are regarded as structures which can take many different shapes, as diverse as vegetative leaves, floral parts, spines or tendrils. In addition, leaf structures are compound structures, parts of which are involved in special plant forms, such as succulent stems [12].

As natural organs, flowers are highly specialised structures which occur in angiophytes, a group of seed plants (spermatophytes), and they differentiate flowering plants from all other plants. Flowers are most advanced in the angiosperms. Flowers are considered a key innovation in evolution. Key innovations are new characters which have acquired an indispensable biological role, either developmental or ecological, and which are conserved evolutionary, while changing under adaptive pressures [13]. They are uniaxial (non-branching) structures, which have become stabilized, determined structures. Most flowers are either polysymmetric (radial symmetry) or zygomorphic, although more complicated symmetries are possible [13]. Evolution has resulted in a wide range of flower shapes, colours and patterns.  In many cases co-evolution with pollinators has resulted in the development of complex variations on the basic design, like in complicated olfactory patterns in orchids, or in the reduction of the perianth in wind pollinated flowers.

The sporogenic floral structures which occur only in angiosperms, and which were a key innovation in the evolution of angiosperms, are the carpels (exhibiting postgenital fusion of epidermal cells) and the anthers with four sporangia (organized in theca). Other key innovations in the evolution of angiosperms include the advent of petals and sepals, syncarpy (the congenital fusion of carpels), sympetaly and floral tubes. Such key innovations provided an enormous potential for further diversification of flowers, since this implied among other things that flowers could develop more stable structures. At another level, flowers play and experiment with number and arrangement of flower organs. The more such structures are co-organized, the more flexibility in shapes is possible.

The molecular background to flower initiation and flower formation is unfolding from an initially rather simple model, the basics of which remain largely unchallenged. The original ABC model for floral organ initiation and development [14], [15] has received considerable attention, and many refinements and additions have been reported, detailing  the ABC model to a high degree [16]. From one viewpoint, a molecular system consisting of interacting sets of transcription factors acting on a range of downstream target genes, provides many opportunities to generate genetic diversity which ultimately leads to the development of stable structures.  From a different point of view however, it can indeed be argued that a molecular toolbox just provides the tools, and what is really being stabilised are biophysical constraints; these may play a far more crucial role in ultimately defining shape and structure [17].

It is no understatement to say that our current understanding of flowers and flower formation contains major gaps. Overwhelming empirical evidence may point out a decisive role for a range of genes, but in most cases such approaches are basically describing parts of the blue print. The system “flower and flower formation” is a very complex one with a very intricate interplay between physical, biophysical, chemical, biochemical and biological factors inside and outside the plant. About the biophysical aspects of flower formation very little is yet known.

In this paper we will further explore the theoretical background of the Superformula and its potential to model flower shapes. In particular, it will be shown how a simple model of addition and multiplication is sufficient to describe a wide variety of flowers, and how these findings fit into models of evolution and development of flowers. In addition, while the focus is on flowers and fusion as key innovations in evolution, also the adaptive significance of fusion of Superformular shapes in succulents and cacti will be discussed.

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Materials & Methods

Supertrigonometric functions and Superformular flowers

A Superformula-distance SF(Φ) or r(Φ) in polar coordinates is given in [7]. In this Superformula there are three distinct groups of parameters. The first one, m, defines the symmetry. The second group, the exponents n1,2,3 define the basic shape and determine whether the shape is inscribed in or circumscribes the unit circle, while the third group, a and b, defines its size. Each shape can be represented in a six-dimensional space (R6) called SF-space, with its six parameters. The classical Euclidean space with Euclidean metric can be considered as a subset of this SF-space, corresponding to the subset of zerogons or circles. The Superformula can not only be used to deform the unit circle, but can modify the graphs and distances of functions as well (r = SF * f(Φ)). The functions can be any function, such as a constant function or a logarithmic function, or a trigonometric function as shown in Equation 1.

Equation 1
Equation 1

Alternatively, this allows to define Supertrigonometric functions, as projection functions onto the horizontal and vertical axes (super- and subcosines, and super-and subsines). For each value of the Superformula, such generalized trigonometric functions can be defined. They are in fact a different graphical expression of Equation 1.

Supershapes, defined by the Superformula or Equation 1 can also be combined in various ways, under specific rules of addition and multiplication. Other examples of combinations are blending functions, or recursive functions. A well-known example is the classical cardioid (r =1+cos f) function, as well as lemniscates and limacon.

Figure 1
Figure 1: Different graphical expressions for Equation 2 as a function of the angle Φ. Upper right: polar view. In the lower graph the upper curve is the distance from origin to boundary, and the lower curve is a generalized trigonometric function as a projection function.

Calculating area and moment of inertia

Since the Superformula is a closed form expression in polar coordinates and distances, area and moment of inertia can be calculated by evaluating the integral of the square and the fourth power of the SF respectively. For given values of n this formula always generates the same area. When n = 2, we can calculate the area of a zerogon or circle, which will simply be πR2. In fact, the formula for area is for a superpolygon what π is for a circle. It can be shown (and calculated) that for given values of n, the value of the integral is independent from the value of m.

The integral to calculate Ip (polar moment of inertia) uses the fourth power of the Superformula. Also here for given values of n, the values of Ip are constant, irrespective of the value of m. In both cases m can be non-integer, and then the total area calculated for this non-integer polygons has to be divided by the number of rotations. It can be proven that both area and Ip are constant irrespective of m (for m > 0), for given a, b and exponents.

Equation 2
Equation 2
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Results

Superfolding in flower buds and developing flowers

In figure 2 distinct supershapes are shown, which can be understood as transformed or deformed supercircles. For m = 4 and all exponents n equal this corresponds to Minkowski-metrics. The best known example is the Manhattan distance (for p=1) defined on the inscribed square (Fig. 2a), the Euclidean distance (for p = 2) defined on the circle (Fig. 2b), and the max-metric, defined on the circumscribed square (for n to infinity). Such distances and metrics can be generalized for all supershapes, for any positive real m. They can be determined from Equation 1. In these supershapes the function which is moderated is the unit circle, a constant function f(φ) = 1. Equation 1 sets geometric constraints and acts as a bounding box for the unit circle. Only for m = 0 or n2 = n3 = 2 the result is the classic Euclidean circle.

Figure 2
Figure 2: Superellipses, Superpentagons and non-Euclidean distances.
Figure 3
Figure 3: Flower buds at different developmental stages. 3a. Capparis yco, and 3b. Deherania smaragdina with tetramerous and pentamerous symmetry respectively, already in the earliest stages of bud development. 3c. Orbea variegata, and 3d. Platycodon grandiflorus flowers prior to flower opening.
Figure 4
Figure 4: Flowers with reflexed petals (rhododendron, rose, Tradescantia ohiensis and Codonopsis species) forming superpolygons.

Superpolygonal shapes occur in a wide range of plant forms and organs such as stems [7], [8]. Such shapes are observed in many flower buds and flowers (Figure 3 and 4). The basic shapes of flowers are determined when the flower buds (flower primordia) are formed. These flower primordia are packed into a superpolygon, e.g. in a square or a pentagon, optimizing the use of the sparse space. During development flowers prove to be true masters in the art of unfolding. The neatly packed flowers often unfold in a very spectacular way.

It is precisely the 2-D shape which determines the way the buds are closed. 3-D shapes based on the Superformula to model 3-D flowers can be created in a number of ways, using implicit or parametric functions. This has also been done to model 3-D starfish models but can also be based on differential equations.

Supertrigonometric Functions: the multiplication case

Perianth, corolla and calyx

During the transition from the vegetative to the flowering state, inflorescences, flowers and flower parts develop along the flowering axis. These flowers are either borne solitary at the end of the flowering stalk as in tulips or cut roses, or individual flowers are combined in inflorescences, as in hyacinth or the spikelets of grasses. The various structures within flowers, including the male and female flowers parts, are modified leaves. In a perfect flower these parts are arranged at the distal end of the flowering axis in three whorls around the carpels. Already early in development this can be observed microscopically. The first, outermost whorl, the calyx, is formed by the sepals, while the second whorl – the corolla – is formed by the petals. Calyx and corolla both have a protective function but the corolla mainly functions as the plant’s billboard.  Sepals are generally small and inconspicuous while petals are generally larger and coloured.

The calyx is not always smaller than the corolla. If sepals and petals are almost the same size, the flowers are said to have a perigon. Sepals and petals are then given a common name, tepals.  In several bulb species, the sepals and petals are almost equally large and coloured, as in tulips and daylilies, with two trimerous whorls. Such colourful tepals are called petaloid. Calyx and corolla can also be equally small, inconspicuous and without colour. Such flowers are seen in sedges and rushes.  These groups of plants are wind pollinated and have no need for colourful flowers to attract pollinators. Such tepals are called sepaloid. The two innermost whorls consist of male and female floral parts. The third whorl – the androecium – consists of the male flowering parts, the anthers or stamens. The innermost whorl is the gynoecium, consisting of (a) pistil(s) with carpel, style and stigma.

Flowers in general are either radially symmetric or monosymmetric. In radially symmetric or polysymmetric flowers, also called actinomorphic flowers, two “laws” can be observed [18]. The first is the law of equidistance, which states that all leaves within one whorl are equally spaced, including stamens and carpels. The second law states that the whorls are rotated respective to the subsequent whorls so that the leaves of one whorl fills the gaps created by the leaves, or leaf structures, of the previous whorl. Such actinomorphic flowers exist in all kinds of forms and sizes. Also the number of symmetries can differ between flowers. Trimerous symmetry is observed in various bulb plants. Tetramerous or fourfold symmetry is typical of Cruciferae, such as cabbages, mustards and Arabidopsis thaliana.  Pentamerous flowers, with fivefold symmetry, are very common in several plant families such as Rosaceae – the family of roses, apples and berries – and Solanaceae, with species as tomato,  petunia and potato.

In its original formulation, the Superformula is a deformation of a function, using the multiplication operator [7]. This function can be a constant function but also a trigonometric function, and can thus be used to describe flowers as well. In Fig. 5 several shapes are shown of flowers inscribed in a superpolygon. The superpolygons describe the internal distances as well as the space constraints which lead to the given shape during development. Many of the flowers with individual petals (choripetalous flowers) develop in one plane, as individual leaflets of a rose curve (Fig. 5a, 5c, 5d). They have exposed stamens and pistils.

In other cases petals are fused to some extent (Fig. 5b, Fig. 6; Fig. 7d). Fusion of petals generally precedes the development of a three-dimensional central tube or keel, in which the male and female organs are buried (Fig. 5b, Fig. 6). The various symmetries in flowers are genetically determined (see Fig. 6) [19], but also biophysical constraints are important [17]. In tobacco, the effect of a transgene may introduce new symmetries, but the bounding boxes remain superpolygonal (Fig. 6), combining both gene actions and biophysics.

Figure 5
Figure 5: Various naturally occurring flowers, their polar graphs (upper row) and the constraining superpolygons (middle row). From left to right: Tradescantia, Nicotiana sp., strawberry, Geranium.
Figure 6
Figure 6: Different symmetries of tobacco flowers as the pleiotropic effect of gene action [19].
Figure 7
Figure 7: Superpolygonal instances of leaves. 7a  Leaflets of Marsilea inscribed in supercircle; 7b Sepals of Hydrangea with attractive function forming a supercircle; 7c Sepals of rose with protective function; 7d  Corona of Frerea indica.

Not only petals are arranged in a superpolygonal arrangement; there are many  examples of other leaf structures as well. Leaflets terminating a petiole, arranged in supercircular arrangement, are characteristic for water ferns Marsilea quadrifolia (Fig. 7a). In the transition from vegetative growth to flowers, the first whorl - the calyx - consists of sepals. They are generally small, compared to the petals, but can also be showy, as in Hydrangea’s where the flowers at the periphery of the inflorescences are large sepals, inscribed in a supercircle (Fig. 7b). In sepals the original phyllotaxy of the vegetative leaves is often still visible. In rose, the spiral 2/5 phyllotaxy is observed in non-integer superpolygonal symmetry with m = 5/2 in the sepals (Fig. 7c). In the highly specialised flowers of Stapeliads – small succulents – the petals are fused to form a corona, again in a superpentagon (Fig. 7d).

 
Zygomorphic or monosymmetric flowers

A radially symmetric (or polysymmetric) flower as described above, is inscribed in a superpolygon closing in one rotation. This superpolygon is symmetric with respect to rotation and so is the resulting flower.  The flowers of many plants are not radially symmetric. Only one axis of symmetry slices from top to bottom through the middle of the flower. Such zygomorphic flowers are typical for orchids, snapdragons, pansies and various labiate flowers, for example in sage, rosemary and thyme. Moreover, a large number of flowers seems to be radially symmetric but are in fact zygomorphic. The zygomorphy is caused by a distinct upper and lower part of the flower as in Crinum x powellii, tobacco, Petunia and Tibouchina. Very often this zygomorphy is caused by gravitation. Often such zygomorphic flowers can be reversed to actinomorphy in circumstances where net gravitation is zero, as can be simulated in slowly rotating clinostats.

The simplest shape that closes in one rotation and introduces both symmetry and asymmetry is a one-angle or monogon (Eq. 1 with m = 1). This shape introduces both symmetry (mirror symmetry) and asymmetry (up and down) as in bird eggs, or tears. In this sense the bounding superpolygon is a monogon, causing zygomorphy in the flower (Fig. 8).

Figure 8
Figure 8. Zygomorphic flowers of Viola tricolor.
Vegetative leaves

Petals and sepals are leaf-derived structures which occur in flowers. In many flowers and developmental stages of these superpolygons, described by the Superformula, provide a bounding box. Such bounding boxes or constraining functions are also observed in leaf blades of vegetative leaves.  As an example cardioid and reniform leaves are shown in Fig. 9, in which the original cardioid (Fig. 9 left) is inscribed in superpolygons (Eq.1). A trigonometric function is not only a projection function, but it is also a circle with the center on the shape itself [11]. In flat organs like petals or vegetative leaves, such trigonometric functions grow with one point attached. For superformular trigonometric equations this is similar.

A cardioid r = 1+ cos Φ is in fact an example of combining shapes, since it is the sum of a unit circle with a second unit circle with its center on the vertex. In all these plants the petiole is attached to the leaf blade in the origin of the cardioid.  There are also plants where the petiole is attached in the middle of a circle, e.g. in floating leaves of Victoria amazonica, or plants where the petiole is slightly out of centre as in Tropaeolum species.

This allows for modelling development of leaves as a concentric development. Already Galilei showed that the cosine functions described the velocity of a marble rolling down a specified slope. In the same way the concentric development of leaves shown in Figure 9, is simply described by a one parameter change, namely the radius (amplitude) of the cosine function.

Figure 9
Figure 9: Leaf blades of plants restricted by Superformular distances. From left to right: Hydrochoris morsus-ranae, Fagopyrum tataricum, Polygonum convolvulus.

Supertrigonometric functions: the addition case

The Superformula flowers above are based on multiplication. The Superformula can be used not only for multiplication as with the original Superformula (Eq 1), but also under rules of addition with or without blending. Under multiplication, the original function is bounded by bounding functions. Here we limit ourselves to showing how fusion of petals can occur. The enclosing pentagon/circle creates a center which is caused by the fusion of the petals. Actually, in real plants, such fusion is a prerequisite for the formation of 3-dimensional folding of flowers, as observed in bell shaped flowers. Either one or both of the functions can be moderated by the Superformula.

Such three dimensional folding is observed in many flowers, such as tobacco (figure 6), Campanulaceae (fig 4d), and stapeliads (Fig. 7d). The folding can be very gradual as in stapeliads or more abrupt, as in Thunbergia.

Figure 10
Figure 10: Black-eyed Susan (Thunbergia alata); Fusion of the petals creates a central (black) keel which is a 3-dimensional fold concealing the stamens and pistil. The keel itself can be a few centimetres deep.

Superpolygonal Optimization

Flowers are highly specialised structures with many variations on a common theme. The models described above, are not only useful for descriptive purposes and to understand flower shapes from an evolutionary point of view, but they also allow us to think about optimising certain functions. In this case we focus mainly on area use efficiency. It can be done for perimeter or moment of inertia or other aspects of optimization.

The constraining of space by the Superformula, compared to the Euclidean distance, has many positive effects, for example the equidistant positioning and spacing of flower organs such as petals and stamens, as well as developing gynoecia in the “corners” of the superpolygons (Fig. 3a and 3b). Such constraining functions also can provide directed and concerted development of the flowers. Inscribing flowers in superpolygons shows how petals are efficiently packed in a limited area. This can be done by defining the area efficiency ratio (AER) as the ratio of the area of the flower to the area of the bounding superpolygon.

Figure 11
Figure 11. Area Efficiency Ratio (AER) of flowers compared to bounding functions.  In the blue curve the bounding function is a circle and in the green curve the bounding function is a superpentagon with m= 5 and n = 1. Right side of curve is trigonometric function, inscribed in circle (lower line) or superpentagon (upper curve). At the extreme left are flowers with completely fused petals approximating the bounding function. Examples are circular flowers of Petunia (lower curve) and Frerea (upper curve).

In the case of the classic rose curve inscribed in a Euclidean circle (a zerogon), the AER ratio is low. Indeed, when we consider the rose curve as inscribed in a circle, the AER is only 50% (or 50% of lost space).  In this particular ground state, the Euclidean point of view, there are no constraints acting on the developable function, and all directions are the same. In contrast, when the area of the petals of flowers in Fig. 5 is compared to the area of the underlying superpolygons, the area efficiency ratio is over 90%. A similarly very efficient area use is observed in the square arrangements of leaves in sepals of various Hydrangea’s and in leaflets of the Water Fern Marsilea quadrifolia (Marsilaceae), which are inscribed in a supercircle. The highest efficiency is seen in Asclepiad flowers like Frerea (Fig. 7d), in which the space use is close to 100% since it completely approximates a superpolygon.

Possible direct consequences of the improved area use through bounding or constraining functions are not only the improved area use itself; other advantages are increased second moments of inertia, and less free perimeter of petals. The effect of blending as observed in flowers of Asclepiads (Huernia, Stapelia, Frerea) has an important influence on all these properties. As shown in the graph below, blending not only increases area and area use efficiency, but also shortens perimeter. In such fusions, either the trigonometric function, or the superformular bounding function gets more weight, depending on a fusion or blending parameter alpha. The effect is seen in Figure 11 where the zero state corresponds to the bounding function (e.g. Frerea indica).

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Discussion

Knowledge-based digital plants and plant shapes

Based on genomics and proteomics, plant scientists are currently building ‘in silico plants’ [4]. This project relies almost exclusively on data obtained from molecular and physiological studies. Computer graphics at present only play a role in the morphometric sense, as supporting data. In these in silico plants neither biomaths (except for bio-informatics), nor biophysics are included. This is the same lacuna as mentioned in the introduction, namely, that only one side of the transduction chain is presently covered by virtual models of plants. Although attempts are made to connect plant shape and shape development with gene action using classical computer graphics approaches, there is a great need for better models of plant shape, which have a clear botanical background and which will allow us to go beyond the descriptive, morphometric approach.

In this paper we have shown that the Superformula is not only an excellent model for describing shapes, but it also allows us to understand shapes in an evolutionary and developmental sense. In particular, the equations proper can be used to study plant form and optimisation. We have shown, for example, that these shapes generate new insights into optimisation. Moreover, fusion infers new biophysical properties.

The decoupling of the function and its distance functions defined by the Superformula, creates a duality which is reminiscent of dualities in physics such as action-reaction. The intricate interplay between these functions provide for innovative approaches to study biophysics in flower development, since the distance functions provide exact boundary conditions for differential equations. Fusion in leaves for example, gives new exciting biophysical properties (results not shown). The various pictures prove how well Superformula flowers correspond to real life flowers. The Superformula provides a logical and powerful framework to describe and understand flower shapes, and allow building computer graphical models for flowers and leaves based on botanical knowledge.

Most flower shapes can thus be cast into simple algebraic addition and multiplication rules. At different developmental levels superpolygons play an important  role as bounding functions. In addition, the functions, for example fusion, lead to distinct three dimensional development of the petals to form a tube or keel. Examples of this can be seen in Thunbergia alata, in which the dark centre in fact is a tube containing stamens and pistils, and in members of the Solanaceae (petunia, potato, tobacco, see also Fig. 5b). In contrast, in the choripetalous state of separate petals, flowers are generally more flat and stamens and pistils exposed as for example in Geranium, strawberry and Tradescantia (Fig 7).  We have also shown how shapes can be understood in the framework of optimisation. Improved area use efficiency, increased strength through a higher moment of inertia, and shorter perimeter due to fusion, are different aspects of shape optimisation observed in succulent plants.

Aspects of biophysics and optimisation

Using the Superformula framework as a botanically-driven knowledge model, also requires rethinking existing concepts, especially regarding symmetry. In the Superformular framework every shape has its own particular symmetry, and within its own right these symmetries are the O(2) symmetry of the circle, i.e. infinite symmetry. In current research concepts, symmetry breaking is widely used in which a potentially infinite symmetry O(2), breaks into much lower symmetries. This is the case not only in physics, but also in biology.  In a physical sense symmetry breaking is associated with irreversibility.

But in a Superformular sense, all symmetries being equivalent, they simply represent different points in SF-space. In flowers a journey through superformular space is possible in various directions, both in an evolutionary and a developmental sense. For example, zygomorphy, which occurs when the circumscribing superpolygon is a monogon (see Viola Fig.7b) has occurred in plant evolution independently in many different plant families. In Asteraceae both  evolution from and reversal to zygomorphy are possible. In Linaria vulgaris a switch from zygomorphy to actinomorphy depends only on the methylation status of a single gene [20].

For all shapes, simple optimisation aspects can be considered. Evolutionary patterns such as fusion, result in, among other aspects, improved area use, shorter petal perimeters, increased second moments of area. In addition, as the Superformula allows to define all shapes in their internal metrics, the shapes are all circles, and provide minimal curves (or minimal surfaces in three dimensions). This leads to new insights in why leaf organs occur so often as flat organs. In addition, it should be remarked that the Superformula includes a range of invariants [7]. A change from an inscribed square to a pentagon with slightly concave sides such as Fig. 5 middle row left, both with exponents all equal to one, keeps area and moment of inertia constant.

A very interesting biophysical aspect is that fusion in petals leads to new, emerging biophysical properties. Fusion of petals, for example, creates a stable centre. Displacement modes under perturbation of such fused shapes resemble closely 3-D shapes of flowers. The connection between 2-D shapes and displacement modes may possibly be linked to co-evolutionary patterns in flowers. Connecting displacement modes with distance maps in 2-D shapes of flowers provides an easy way to design fully three dimensional flowers, based on actual biophysical constraints and forces.

The ideas about optimisation and results regarding stability of shapes of flowers, is also observed in physical systems such as snowflakes. Here we have a fixed molecular symmetry, but the growth and actual shape of the snow crystal is dependent on a range of parameters and influences. Despite the large variation, many snow crystals can also be described using fusion as described for flowers (Fig. 12).

Figure 12
Figure 12: Snow crystals, with hexagonal symmetry.

Conclusion

In this paper we have demonstrated the power of the Superformula approach to model shapes in plants, concentrating on flowers and leaves. While the foucs was on 2-D descriptors, it is obvious that 3-D models can be obtained in a variety of ways. In addition, the Superformula defines not only the shape boundary, but any point inside this boundary as well. The Superformula was already used to develop such models in starfish [21]. This allows the description of leaves and flowers in great details, both outlines and inside of the leaf defining venation patterns. Also for geometric morphometrics, the Superformula will be very valuable because of its potential for providing a very precise description of leaves. In addition, it can account for highly variable leaves, and single parameter descriptors such as ratio of perimeter to surface area can easily be deduced (see figure 11). Most interestingly, it can account for fusion of leaves and leaflets and our models thus provides an integrated solution for geometric morphometrics with the potential to solve the current major problems of modelling [22].

Furthermore we have shown that the Superformula allows for models in which a link is made between descriptive and developmental-evolutionary factors. The biophysical aspects allow to develop simple 3-D description of flowers based on 2-D shapes of flowers. The molding of the functions by constraining functions create the stage for a detailed study of the biophysics of flower and leaf development, both in asymmetrical and symmetrical flowers and organs.

In the future we will further analyse how simple rules of addition and multiplication may lead to strategies in floral evolution. More specifically, evolution of the flowers of Stapeliads, are linked with fusion of Superformular bounding or constraining functions with trigonometric functions. The convergence between a descriptive model and biophysical processes in flowers is based on emergent biophysical properties of two dimensional shapes of flowers.  This approach goes beyond the current state of the art of plant modelling in computer graphics, and allows to introduce and develop botanical knowledge in computer graphics beyond the merely morphometrical aspects. Our models are botanically based models which will be much more useful for building in silico or abstract plants, than the variety of software motivated computer graphics models for plants and plant parts that have been used so far.

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