Seed Silhouettes as Geometric Objects: New Applications of Elliptic Fourier Transform to Seed Morphology

: Historically, little attention has been paid to the resemblance between seed silhouettes to geometric ﬁgures. Cardioid and derivatives, ellipses, heart curves, lemniscates, lenses, lunes, ovals, superellipses, waterdrops, and other ﬁgures can be used to describe seed shape, as well as models for quantiﬁcation. Algebraic expressions representing the average silhouettes for a group of seeds are available, and their shape can be described and quantiﬁed by comparison with geometric models. Bidimensional closed-plane ﬁgures resulting from the representation of Fourier equations can be used as models for shape analysis. Elliptic Fourier Transform equations reproduce the seed silhouettes for any closed-plane curve corresponding to the contour of the image of a seed. We review the geometric properties of the silhouettes from seed images and discuss them in the context of seed development, plant taxonomy, and environmental adaptation. Silene is proposed as a model for the study of seed morphology. Three groups have been recently deﬁned among Silene species based on the structure of their seed silhouettes, and their geometric properties are discussed. Using models based on Fourier Transform equations is useful in Silene species where the seeds are homogenous in shape but don’t adjust to described ﬁgures.


Introduction
Seed shape description is usually mixed with size observations (see, for example, [1]). Measurements derived from size often distract attention from the absence of quantification data from the seed shape itself. Size measurements have been a constant in seed science, finding published databases of both seed size and weight [2,3]. On the other hand, the descriptions of seed shape contain terms without correspondence to precisely defined geometric objects (calycine, cuneiform, globoid, globose, globular, obovate, piriform, renifo, etc.), lacking in shape measurements. Seed shape quantification is infrequent in the botanical literature, and only a preliminary seed shape dataset has been recently published [4]. Nevertheless, the seeds of many plant species resemble geometric objects, and their images can be described by their similarity to geometric figures. This property can also be used in the process of seed shape quantification that may be useful for phenotype characterization.
The seeds of the Silene species resemble in their lateral views the cardioid and related figures, which can be used as models for the quantification of seed shape. The models

A Historical Anecdote
The absence of data based on seed shape quantification linked to geometric models may be due to the influence of animal biologists and paleontologists on plant morphologists. For example, in an article published in the journal Human Evolution, Professor Dwight Read stated [22]: "I first consider-and then discard as inadequate-two commonly used representations of form. The first one makes the strong theoretical assumption that the empirical form can be idealized and replaced by a geometric figure." Nevertheless, it is not incorrect to compare seed images with geometric figures. On the contrary, looking at seed images, or at their silhouettes, as geometric objects is a direct way to achieve mathematical accuracy in seed shape description. This is required to combine morphology with genetics, ecology, or taxonomy.

Geometric Properties of Seed Silhouettes
Solidity is an important property of closed-plane curves. It is related to convexity and expresses the ratio of two areas: the area of an object to the area of its convex Hull [23] ( Figure 1). The convex Hull is the smallest convex set that contains a plane figure.
A figure is convex when the segment that joins any pair of its points is contained. Then solidity of any convex figure equals one. The tangent at any point of a convex curve leaves the complete figure on one side of it. In addition, a closed plane curve is convex if, and only if, its signed curvature does not change sign [23].
Solidity is linked to the ratio of perimeter/area in plane figures and represents an estimation of the surface/volume ratio of the corresponding tridimensional objects. In seeds, surface/volume ratio has a role in regulating metabolic aspects from the gas exchange and oxygen availability to nutrient uptake or water imbibition and evaporation [24]. Given the importance of this ratio, we may consider solidity as an important property of seeds and their relative value as a candidate for taxonomic plant studies.
A measurement related to solidity is the ratio between the perimeter of a plane curve and the perimeter of the corresponding convex hull. Comparing this measurement with the solidity value gives information about the irregularities on a surface. High values of solidity may be combined with low perimeter ratios in curves with many small surface irregularities.

Quantification of Seed Shape by Comparison with Models
Seed shape quantification can be done by estimating the percentage surface shared between the seed image and a given model [13,25]. The resulting measurement has been termed the J index. The models used can be either a canonical geometric figure or any other mathematically defined closed curve, such as those from Fourier equations. In both cases, the process requires a manual seed image-model adjustment, which does not impede achieving objective and reproducible values. Seed shape quantification by comparing seed images with canonical models has been applied to various plant species [5][6][7][8][9][11][12][13][14][15][16][17][25][26][27][28][29][30][31][32][33][34].
Multivariate analysis based on Fourier coefficients has been used for shape analysis in Zoology and Palaeontology [35,36], as well as for leaf shape description on trees [37] and seed shape comparison in diverse species [38][39][40][41][42][43]. Nevertheless, the continuous, closed curves represented by Fourier equations can also be used as models for seed shape quantification.

Quantification by Comparison with Known Geometric Figures as Models
The cardioid is a good model for the description of seed shape in the model legume Lotus japonicus (Regel) K. Larsen [26], species of Capparis Tourn. ex L. (Capparaceae) [27], Searsia tripartita (Ucria) Moffett (Anacardiaceae) (as Rhus tripartita (Ucria) Grande, [28]), and, also in many species of Silene L. (Caryophyllaceae) [5][6][7][8][9]. Slight modifications in the cardioid resulted in curves similar to the seed silhouettes of Medicago truncatula Gaertn. (Fabaceae) and Arabidopsis thaliana (L.) Heynh. (Brassicaceae) [26,29,30]. Mean values of percent similarity (J index) superior to 90 for samples containing at least 20 well-oriented seeds have been reported for many species of Silene choosing the cardioid as a model [5][6][7][8][9]. Seed shape has been described for several species in the Arecaceae. Seed images in this family resemble a compendium of geometric figures serving as models for shape quantification [11]. General equations with variable coefficients corresponding to ellipses, ovals, lemniscates, superellipses, and others allowed us to compare figures with different proportions adjusted to the seed shape of different species [11]. Seeds resembling some of the models described for Arecaceae, such as lenses and superellipses, were also found in Vitaceae [12]. The introduction of further variations in the equation of the ellipse resulted

Quantification of Seed Shape by Comparison with Models
Seed shape quantification can be done by estimating the percentage surface shared between the seed image and a given model [13,25]. The resulting measurement has been termed the J index. The models used can be either a canonical geometric figure or any other mathematically defined closed curve, such as those from Fourier equations. In both cases, the process requires a manual seed image-model adjustment, which does not impede achieving objective and reproducible values. Seed shape quantification by comparing seed images with canonical models has been applied to various plant species [5][6][7][8][9][11][12][13][14][15][16][17][25][26][27][28][29][30][31][32][33][34].
Multivariate analysis based on Fourier coefficients has been used for shape analysis in Zoology and Palaeontology [35,36], as well as for leaf shape description on trees [37] and seed shape comparison in diverse species [38][39][40][41][42][43]. Nevertheless, the continuous, closed curves represented by Fourier equations can also be used as models for seed shape quantification.

Quantification by Comparison with Known Geometric Figures as Models
The cardioid is a good model for the description of seed shape in the model legume Lotus japonicus (Regel) K. Larsen [26], species of Capparis Tourn. ex L. (Capparaceae) [27], Searsia tripartita (Ucria) Moffett (Anacardiaceae) (as Rhus tripartita (Ucria) Grande, [28]), and, also in many species of Silene L. (Caryophyllaceae) [5][6][7][8][9]. Slight modifications in the cardioid resulted in curves similar to the seed silhouettes of Medicago truncatula Gaertn. (Fabaceae) and Arabidopsis thaliana (L.) Heynh. (Brassicaceae) [26,29,30]. Mean values of percent similarity (J index) superior to 90 for samples containing at least 20 well-oriented seeds have been reported for many species of Silene choosing the cardioid as a model [5][6][7][8][9]. Seed shape has been described for several species in the Arecaceae. Seed images in this family resemble a compendium of geometric figures serving as models for shape quantification [11]. General equations with variable coefficients corresponding to ellipses, ovals, lemniscates, superellipses, and others allowed us to compare figures with different proportions adjusted to the seed shape of different species [11]. Seeds resembling some of the models described for Arecaceae, such as lenses and superellipses, were also found in Vitaceae [12]. The introduction of further variations in the equation of the ellipse resulted in curves well adapted to seed shape in members of this latter family, including the agricultural grape varieties, and seeds in other species having more complex shapes [12,31,32].

Closed Curves from Fourier Equations as Models
For seed shape quantification, Fourier curves reproducing the shape of seed images represent an alternative to canonical geometric figures. Fourier analysis is a mathematical method for reducing complex curves into their component spatial frequencies. The parametric functions representing a closed plane curve may be approximated by sums of trigonometric functions and their truncated Fourier expansions [37,44]. In particular, when the parametric components are piecewise linear functions where A 0 and C 0 define the mean size of the contour, and the coefficients a n , b n , c n , d n are calculated as The position of any point on the outline is approximated by the displacement of a point traveling around a series of superimposed and successively smaller ellipses, corresponding to successive harmonics. For each harmonic, two Fourier coefficients are computed for both the x-and y-projections resulting in a total number of coefficients of 4N, where N is the number of harmonics used to fit the outline. Departing with basic ellipses obtained with four coefficients, the complexity of the figure increases. The coefficients of the lower order correspond to the overall shape, and the higher order harmonics correspond to the smaller details of the outline [37,44].
Based on the elliptic Fourier analysis described above, it is possible to obtain a figure that reproduces the silhouette of a given seed or the average silhouette for a population of seeds. For example, Figure 2 presents the images corresponding to Fourier equations for a seed silhouette of Aphandra natalia (Balslev & A.J. Hend.) Barfod (Arecaceae) with 2, 6, 10, and 20 harmonics (a Mathematica code for obtaining the Fourier expansion and the corresponding figure from any set of points approximating a closed-plane curve is available at Zenodo; see Supplementary Materials). In the example of the seed of A. natalia shown in Figure 1, the images corresponding to the equations calculated with different harmonics are represented. For convex figures, a low number of harmonics may give an accurate shape representation. The representation of partial concavities requires a larger number of harmonics. Figure 3 shows the superposition of the convex Hull (see later) with the silhouette obtained with 20 harmonics.

Curvature Analysis
The trigonometrical polynomials obtained as approximations to the parametric components allow for calculating the curvature values along the curve ( [45][46][47][48]; Figure 4). The maximum curvature value at the apex is observed. In the example of the seed of A. natalia shown in Figure 1, the images corresponding to the equations calculated with different harmonics are represented. For convex figures, a low number of harmonics may give an accurate shape representation. The representation of partial concavities requires a larger number of harmonics. Figure 3 shows the superposition of the convex Hull (see later) with the silhouette obtained with 20 harmonics. In the example of the seed of A. natalia shown in Figure 1, the images corresponding to the equations calculated with different harmonics are represented. For convex figures, a low number of harmonics may give an accurate shape representation. The representation of partial concavities requires a larger number of harmonics. Figure 3 shows the superposition of the convex Hull (see later) with the silhouette obtained with 20 harmonics.

Curvature Analysis
The trigonometrical polynomials obtained as approximations to the parametric components allow for calculating the curvature values along the curve ( [45][46][47][48]; Figure 4). The maximum curvature value at the apex is observed.

Curvature Analysis
The trigonometrical polynomials obtained as approximations to the parametric components allow for calculating the curvature values along the curve ( [45][46][47][48]; Figure 4). The maximum curvature value at the apex is observed.

Embryogeny
Davis [19] and Johri and collaborators [20] reviewed how the categories of orthotropous, anatropous, hemianatropous, amphitropous, and campylotropous ovules were distributed among 315 families of angiosperms. Of all of them, 204 had anatropous, 20 orthotropous, 13 hemianatropous, and 11 campylotropous or amphitropous ovules [19,20]. Both the orthotropous and the anatropous types are found in the basal angiosperms and were proposed to have independent origins [49]. In addition, multiple ovule types with differences among genera, or even species, were found in 67 families [19,20]. This number could probably be underestimated, considering that many families were incompletely studied, and their taxonomic value throughout families supports their study [50]. The four main different types of ovules can be classified into two different groups (anatropousorthotropous vs. campylotropous-amphitropous), depending on the aspect of the nucellus. Anatropous and orthotropous are both characterized by a straight nucellus, which is bent or curved for the campylotropous and amphitropous ovules [51]. These differences might influence seed development, specifically on certain features of the seed morphology.
However, two-three types of ovules often occur within some families, such as Fabaceae and Caryophyllaceae, which mostly includes campylotropous or anatropous ovules [51], though rare amphitropous ones are also present [54]. The family Arecaceae is a notable example of a high intra-familiar diversity of ovule type. The different Arecaceae subfamilies can show anatropous, orthotropous, hemianatropous, and campylotropous

Embryogeny
Davis [19] and Johri and collaborators [20] reviewed how the categories of orthotropous, anatropous, hemianatropous, amphitropous, and campylotropous ovules were distributed among 315 families of angiosperms. Of all of them, 204 had anatropous, 20 orthotropous, 13 hemianatropous, and 11 campylotropous or amphitropous ovules [19,20]. Both the orthotropous and the anatropous types are found in the basal angiosperms and were proposed to have independent origins [49]. In addition, multiple ovule types with differences among genera, or even species, were found in 67 families [19,20]. This number could probably be underestimated, considering that many families were incompletely studied, and their taxonomic value throughout families supports their study [50]. The four main different types of ovules can be classified into two different groups (anatropous-orthotropous vs. campylotropous-amphitropous), depending on the aspect of the nucellus. Anatropous and orthotropous are both characterized by a straight nucellus, which is bent or curved for the campylotropous and amphitropous ovules [51]. These differences might influence seed development, specifically on certain features of the seed morphology.
However, two-three types of ovules often occur within some families, such as Fabaceae and Caryophyllaceae, which mostly includes campylotropous or anatropous ovules [51], though rare amphitropous ones are also present [54]. The family Arecaceae is a notable example of a high intra-familiar diversity of ovule type. The different Arecaceae subfamilies can show anatropous, orthotropous, hemianatropous, and campylotropous ovules [55], which would be associated with a high diversity of seed shapes [11]. In this respect, the preliminary observations between ovule and seed shape suggest that the anatropous and or-thotropous ovules develop oval and/or ellipse-shaped seeds. In contrast, campylotropous and amphitropous ovules seem more similar to a C-shaped morphology related to the cardioid seed shape ( Figure 5). Nevertheless, variations in seed shape are also related to changes in developmental processes during seed maturation, with the presence of extended hilum [20,56]. Modifications based on the funiculus in species of Opuntia Mill. (Cactaceae) led Archibald to give the denomination of circinotropous specifically to this genus [56]. ovules [55], which would be associated with a high diversity of seed shapes [11]. In this respect, the preliminary observations between ovule and seed shape suggest that the anatropous and orthotropous ovules develop oval and/or ellipse-shaped seeds. In contrast, campylotropous and amphitropous ovules seem more similar to a C-shaped morphology related to the cardioid seed shape ( Figure 5). Nevertheless, variations in seed shape are also related to changes in developmental processes during seed maturation, with the presence of extended hilum [20,56]. Modifications based on the funiculus in species of Opuntia Mill. (Cactaceae) led Archibald to give the denomination of circinotropous specifically to this genus [56].

Variations in Seed Shape Related to Fruit Development
The relationships between seed shape and possible ecological, functional, and evolutionary correlation have been less studied, mainly due to the difficulty of accurately defining and quantifying seed shape [58]. Seed shape represents a remarkable feature since it maximizes the efficiency of packing, dispersal, landing, and seedlings establishment [1,10]. The number of seeds per fruit was positively correlated to the number of ovules for many Fabaceae species [59]. Recently, the number of seeds is also strongly correlated even with the flower size [60]. Therefore, the final seed shape depends on the number of ovules and their distribution within the ovary, hence, the proximity among ovules during their development. Seed shape is also related to the structure of mature fruits that may, themselves, be similar to geometric figures. However, this aspect has received more attention from an informational point of view than strictly academic in the literature [61]. The seeds are closely bound within the fruits, and fruit shape might determine the final seed shape for many genera.
For some plants, seeds have different shapes depending on their position within the mature fruit. A notable example corresponds to the maize (Zea mays L., Poaceae), in which the seeds from the most distal row have a more spherical shape than those that remain immersed in the cob. In other cases, seeds also adopt the shape of different types of polyhedrons, whose silhouettes in bi-dimensional images are polygons, or their shape resembles what is known as "lune" (moon). In geometry, lunes are plane figures bounded by two circular arcs of unequal radium [62]. The seeds of Peganum harmala L. (Nitrariaceae)

Variations in Seed Shape Related to Fruit Development
The relationships between seed shape and possible ecological, functional, and evolutionary correlation have been less studied, mainly due to the difficulty of accurately defining and quantifying seed shape [58]. Seed shape represents a remarkable feature since it maximizes the efficiency of packing, dispersal, landing, and seedlings establishment [1,10]. The number of seeds per fruit was positively correlated to the number of ovules for many Fabaceae species [59]. Recently, the number of seeds is also strongly correlated even with the flower size [60]. Therefore, the final seed shape depends on the number of ovules and their distribution within the ovary, hence, the proximity among ovules during their development. Seed shape is also related to the structure of mature fruits that may, themselves, be similar to geometric figures. However, this aspect has received more attention from an informational point of view than strictly academic in the literature [61]. The seeds are closely bound within the fruits, and fruit shape might determine the final seed shape for many genera.
For some plants, seeds have different shapes depending on their position within the mature fruit. A notable example corresponds to the maize (Zea mays L., Poaceae), in which the seeds from the most distal row have a more spherical shape than those that remain immersed in the cob. In other cases, seeds also adopt the shape of different types of polyhedrons, whose silhouettes in bi-dimensional images are polygons, or their shape resembles what is known as "lune" (moon). In geometry, lunes are plane figures bounded by two circular arcs of unequal radium [62]. The seeds of Peganum harmala L. (Nitrariaceae) represent a remarkable example of these lune-like and polygonal shapes of the seeds within the same fruit (see Figure 6). represent a remarkable example of these lune-like and polygonal shapes of the seeds within the same fruit (see Figure 6).

Variations in Seed Shape Related to Environmental Adaptations
Variation in seed shape occurs along all taxonomic levels, and fruit and ovary development could be considered important factors on the origin of the shape variations in a given species, as we have reported in the previous section, but not the only ones. On the one hand, different seed morphotypes characterized by different aspect ratios (AR) corresponding to the ratio length/width have been observed in wild populations of Silene diclinis (Lag.) M. Laínz (Caryophyllaceae) [5], Echinocactus platyacanthus Link & Otto (Cactaceae) [33], and diverse species of Capparis Tourn. ex L. (Capparaceae) [27] (data not shown). On the other hand, the process of plant cultivation during hundreds of generations in wheat (Triticum L., Poaceae) has resulted in more rounded seed forms. The old cultivars were characterized by models based on lenses (AR = 3.2), whereas ellipse-based models (AR = 1.8) better define more recent cultivars [34]. Another relevant case corresponded to the remarkable intraspecific variation in seed shape among cultivars of Vitis vinifera L. [31,32].
Many aspects of seed shape can be explained by particular adaptations in some species of the family Hydrangeaceae (Hydrangea integrifolia Hayata, H. barbara (L.) Bernd Schulz) [63], the family Alzateaceae (Alzatea verticillata Ruiz & Pav.), the family Petrosaviaceae (Petrosavia sinii (K.Krause) Gagnep.), and the family Ericaceae (Orthilia secunda (L.) House), whose seeds are fusiform undulated, and resemble the seeds of the family Orchidaceae [64]. These seed forms could be the result of adaptations to seed dispersal. In addition, certain morphological structures of the seed have also been associated with hydrochory and zoochory, as has been reported for the Cactaceae [65].

Variations in Seed Shape Related to Environmental Adaptations
Variation in seed shape occurs along all taxonomic levels, and fruit and ovary development could be considered important factors on the origin of the shape variations in a given species, as we have reported in the previous section, but not the only ones. On the one hand, different seed morphotypes characterized by different aspect ratios (AR) corresponding to the ratio length/width have been observed in wild populations of Silene diclinis (Lag.) M. Laínz (Caryophyllaceae) [5], Echinocactus platyacanthus Link & Otto (Cactaceae) [33], and diverse species of Capparis Tourn. ex L. (Capparaceae) [27] (data not shown). On the other hand, the process of plant cultivation during hundreds of generations in wheat (Triticum L., Poaceae) has resulted in more rounded seed forms. The old cultivars were characterized by models based on lenses (AR = 3.2), whereas ellipse-based models (AR = 1.8) better define more recent cultivars [34]. Another relevant case corresponded to the remarkable intraspecific variation in seed shape among cultivars of Vitis vinifera L. [31,32].
Many aspects of seed shape can be explained by particular adaptations in some species of the family Hydrangeaceae (Hydrangea integrifolia Hayata, H. barbara (L.) Bernd Schulz) [63], the family Alzateaceae (Alzatea verticillata Ruiz & Pav.), the family Petrosaviaceae (Petrosavia sinii (K.Krause) Gagnep.), and the family Ericaceae (Orthilia secunda (L.) House), whose seeds are fusiform undulated, and resemble the seeds of the family Orchidaceae [64]. These seed forms could be the result of adaptations to seed dispersal. In addition, certain morphological structures of the seed have also been associated with hydrochory and zoochory, as has been reported for the Cactaceae [65].

Silene as a Model System for Seed Geometry
Seed images from 95 populations belonging to 52 species of Caryophyllaceae (49 species of Silene and three related species belonging to the genera Atocion Adans. and Viscaria Bernh.) were classified according to the layout of their silhouettes in three groups: smooth, rugose, and echinate [5]. Hereafter, we expose the application of the Fourier analysis to the average silhouettes of representative seeds for each of these three groups and a description  Figure 7 shows the images used in the analysis, which correspond to the average silhouettes of the species S. apetala, S. conica, and S. dioica (named hereafter S. apetala AJ283, S. conica AJ300 and S. dioica Pol., respectively). Each one was calculated from 20 seed images of a specific population of the corresponding species in both dorsal and lateral views.

Silene as a Model System for Seed Geometry
Seed images from 95 populations belonging to 52 species of Caryophyllaceae (49 species of Silene and three related species belonging to the genera Atocion Adans. and Viscaria Bernh.) were classified according to the layout of their silhouettes in three groups: smooth, rugose, and echinate [5]. Hereafter, we expose the application of the Fourier analysis to the average silhouettes of representative seeds for each of these three groups and a description of their general morphological properties. Figure 7 shows the images used in the analysis, which correspond to the average silhouettes of the species S. apetala, S. conica, and S. dioica (named hereafter S. apetala AJ283, S. conica AJ300 and S. dioica Pol., respectively). Each one was calculated from 20 seed images of a specific population of the corresponding species in both dorsal and lateral views. All the silhouettes resemble some geometric models previously described for this genus [5][6][7][8][9], except for the dorsal silhouette of S. apetala AJ283 (Figure 8). Models LM5, LM1, and LM2 resemble, respectively, the lateral views of S. apetala AJ283, S. conica AJ300, and S. dioica Pol.; while models DM5 and DM3 resemble the dorsal views of S. conica AJ300 and S. dioica Pol., respectively. Fourier analysis provided new models for the quantitative description of seed imaging. The models can be estimated both for seed images that do not resemble known geometric figures, such as the average seed silhouette of S. apetala AJ283 in their dorsal views (Figures 7 and 8), and also for seed images that do not resemble accurately enough the available models. An equation fitting convex images, like the average dorsal silhouette of S. dioica Pol. (Figure 8), can be obtained with a reduced number of harmonics. On the other hand, equations fitting more complex curves, like the average dorsal silhouettes of the seeds of S. conica AJ300 and S. apetala AJ283, require more harmonics. All the silhouettes resemble some geometric models previously described for this genus [5][6][7][8][9], except for the dorsal silhouette of S. apetala AJ283 (Figure 8). Models LM5, LM1, and LM2 resemble, respectively, the lateral views of S. apetala AJ283, S. conica AJ300, and S. dioica Pol.; while models DM5 and DM3 resemble the dorsal views of S. conica AJ300 and S. dioica Pol., respectively.

Silene as a Model System for Seed Geometry
Seed images from 95 populations belonging to 52 species of Caryophyllaceae (49 species of Silene and three related species belonging to the genera Atocion Adans. and Viscaria Bernh.) were classified according to the layout of their silhouettes in three groups: smooth, rugose, and echinate [5]. Hereafter, we expose the application of the Fourier analysis to the average silhouettes of representative seeds for each of these three groups and a description of their general morphological properties. Figure 7 shows the images used in the analysis, which correspond to the average silhouettes of the species S. apetala, S. conica, and S. dioica (named hereafter S. apetala AJ283, S. conica AJ300 and S. dioica Pol., respectively). Each one was calculated from 20 seed images of a specific population of the corresponding species in both dorsal and lateral views. All the silhouettes resemble some geometric models previously described for this genus [5][6][7][8][9], except for the dorsal silhouette of S. apetala AJ283 (Figure 8). Models LM5, LM1, and LM2 resemble, respectively, the lateral views of S. apetala AJ283, S. conica AJ300, and S. dioica Pol.; while models DM5 and DM3 resemble the dorsal views of S. conica AJ300 and S. dioica Pol., respectively. Fourier analysis provided new models for the quantitative description of seed imaging. The models can be estimated both for seed images that do not resemble known geometric figures, such as the average seed silhouette of S. apetala AJ283 in their dorsal views (Figures 7 and 8), and also for seed images that do not resemble accurately enough the available models. An equation fitting convex images, like the average dorsal silhouette of S. dioica Pol. (Figure 8), can be obtained with a reduced number of harmonics. On the other hand, equations fitting more complex curves, like the average dorsal silhouettes of the seeds of S. conica AJ300 and S. apetala AJ283, require more harmonics. Fourier analysis provided new models for the quantitative description of seed imaging. The models can be estimated both for seed images that do not resemble known geometric figures, such as the average seed silhouette of S. apetala AJ283 in their dorsal views (Figures 7 and 8), and also for seed images that do not resemble accurately enough the available models. An equation fitting convex images, like the average dorsal silhouette of S. dioica Pol. (Figure 8), can be obtained with a reduced number of harmonics. On the other hand, equations fitting more complex curves, like the average dorsal silhouettes of the seeds of S. conica AJ300 and S. apetala AJ283, require more harmonics.

Fourier Analysis of Seeds: Calculation of Equations Corresponding to New Models
After applying Fourier analysis, we got the equations adjusting to the average silhouettes of the dorsal and lateral views of S. apetala AJ283, S. conica AJ300, and S. dioica Pol. Several points between 100 and 200 were taken along the silhouettes, and the corresponding Fourier equations were obtained. Figure 9 shows the results of the Fourier analysis for the lateral and dorsal seed views of the sample S. apetala AJ283, using 4, 12, and 20 harmonics. With 12 harmonics, the resulting figures reproduced well the lateral views of the seeds, while in the dorsal views, higher similarity with the average silhouette was obtained with 20 harmonics (Figure 9).

Fourier Analysis of Seeds: Calculation of Equations Corresponding to New Models
After applying Fourier analysis, we got the equations adjusting to the average silhouettes of the dorsal and lateral views of S. apetala AJ283, S. conica AJ300, and S. dioica Pol. Several points between 100 and 200 were taken along the silhouettes, and the corresponding Fourier equations were obtained. Figure 9 shows the results of the Fourier analysis for the lateral and dorsal seed views of the sample S. apetala AJ283, using 4, 12, and 20 harmonics. With 12 harmonics, the resulting figures reproduced well the lateral views of the seeds, while in the dorsal views, higher similarity with the average silhouette was obtained with 20 harmonics (Figure 9). In the case of the sample S. conica AJ300, the results of the Fourier analysis yielded reasonably good models with 12 harmonics for both the lateral and the dorsal views (Figure 10). In S. dioica Pol., the Fourier analysis achieved remarkably good models with only 4 harmonics for the dorsal and lateral views ( Figure 11); however, for the lateral view, the adjustment notably improved up to 12 harmonics. In the case of the sample S. conica AJ300, the results of the Fourier analysis yielded reasonably good models with 12 harmonics for both the lateral and the dorsal views ( Figure 10).

Fourier Analysis of Seeds: Calculation of Equations Corresponding to New Models
After applying Fourier analysis, we got the equations adjusting to the average silhouettes of the dorsal and lateral views of S. apetala AJ283, S. conica AJ300, and S. dioica Pol. Several points between 100 and 200 were taken along the silhouettes, and the corresponding Fourier equations were obtained. Figure 9 shows the results of the Fourier analysis for the lateral and dorsal seed views of the sample S. apetala AJ283, using 4, 12, and 20 harmonics. With 12 harmonics, the resulting figures reproduced well the lateral views of the seeds, while in the dorsal views, higher similarity with the average silhouette was obtained with 20 harmonics (Figure 9). In the case of the sample S. conica AJ300, the results of the Fourier analysis yielded reasonably good models with 12 harmonics for both the lateral and the dorsal views (Figure 10). In S. dioica Pol., the Fourier analysis achieved remarkably good models with only 4 harmonics for the dorsal and lateral views ( Figure 11); however, for the lateral view, the adjustment notably improved up to 12 harmonics. In S. dioica Pol., the Fourier analysis achieved remarkably good models with only 4 harmonics for the dorsal and lateral views ( Figure 11); however, for the lateral view, the adjustment notably improved up to 12 harmonics. The seeds of S. dioica Pol. could be represented by dorsal and lateral models (named DM3 and LM2, respectively, Figure 11), and similarly, those of S. conica AJ300 by the models DM5 and LM1 ( Figure 10). However, for S. apetala AJ283, good modeling resulted from the combination of the lateral model LM5 and a new dorsal model (DM) (Figure 9). In the first case, seeds of S. dioica Pol. are convex in both views, whereas the seeds of S. conica AJ300 present concavities in both the lateral and dorsal views. Finally, the sample of S. apetala AJ283 is characterized by marked concavities, especially on the dorsal view, and a good model could be obtained by Fourier transform analysis with 20 harmonics.

A Classification According to the Geometric Properties of the Seeds
Concerning the classification of Silene seeds in three groups (smooth, rugose, and echinate [5]), differences between these groups were found in all the measurements (area, perimeter, length, width, aspect ratio, circularity, roundness, and solidity). Solidity was the most conserved index among them, being unique with no differences for the lateral views in these three groups of seeds. Regarding the dorsal view, the highest values of solidity corresponded to the species with echinate seeds (most species of this group belong to S. subg. Behenantha) and the lowest solidity to the smooth seeds (most of the species of this group are included in S. subg. Silene). These higher solidity values were associated with the rounded shape of seeds and the lack of a dorsal channel that was notably marked in the smooths seeds [5,7]. Figure 12 represents the silhouettes of the dorsal view for the smooth, rugose, and echinate seeds. The images show the average silhouettes of 20 seeds from one species population with their corresponding convex hull superimposed.
While there is a remarkable difference between the average silhouette and the convex hull in S. apetala, there is little difference between S. conica and S. dioica. The values of solidity corresponding to the three sample seeds are 0.834 (S. apetala), 0.972 (S. conica), and 0.968 (S. dioica). In the same images, the values of the ratio perimeter of the convex hull/ perimeter are 0.852 (S. apetala), 0.967 (S. conica), and 0.914 (S. dioica). The values of solidity obtained here agree with the results obtained for the populations of different species [5]. At the same time, the calculations of the perimeter ratio are shown here for the first time. Table 1 contains the mean value for solidity and perimeter ratio in the dorsal views of three individual seeds of each species. Statistics were done on IBM SPSS Statistics v28 (SPSS 2021) and R software v. 4.1.2 [66]. Non-parametric Kruskal-Wallis tests were applied to compare populations, followed by stepwise step-down comparisons by the ad hoc procedure developed by Campbell and Skillings [67]; p values inferior to 0.05 were considered significant. The coefficient of variation was calculated as CVtrait = standard deviation trait/mean trait × 100 [68]. The seeds of S. dioica Pol. could be represented by dorsal and lateral models (named DM3 and LM2, respectively, Figure 11), and similarly, those of S. conica AJ300 by the models DM5 and LM1 ( Figure 10). However, for S. apetala AJ283, good modeling resulted from the combination of the lateral model LM5 and a new dorsal model (DM) (Figure 9). In the first case, seeds of S. dioica Pol. are convex in both views, whereas the seeds of S. conica AJ300 present concavities in both the lateral and dorsal views. Finally, the sample of S. apetala AJ283 is characterized by marked concavities, especially on the dorsal view, and a good model could be obtained by Fourier transform analysis with 20 harmonics.

A Classification According to the Geometric Properties of the Seeds
Concerning the classification of Silene seeds in three groups (smooth, rugose, and echinate [5]), differences between these groups were found in all the measurements (area, perimeter, length, width, aspect ratio, circularity, roundness, and solidity). Solidity was the most conserved index among them, being unique with no differences for the lateral views in these three groups of seeds. Regarding the dorsal view, the highest values of solidity corresponded to the species with echinate seeds (most species of this group belong to S. subg. Behenantha) and the lowest solidity to the smooth seeds (most of the species of this group are included in S. subg. Silene). These higher solidity values were associated with the rounded shape of seeds and the lack of a dorsal channel that was notably marked in the smooths seeds [5,7]. Figure 12 represents the silhouettes of the dorsal view for the smooth, rugose, and echinate seeds. The images show the average silhouettes of 20 seeds from one species population with their corresponding convex hull superimposed.   While there is a remarkable difference between the average silhouette and the convex hull in S. apetala, there is little difference between S. conica and S. dioica. The values of solidity corresponding to the three sample seeds are 0.834 (S. apetala), 0.972 (S. conica), and 0.968 (S. dioica). In the same images, the values of the ratio perimeter of the convex hull/ perimeter are 0.852 (S. apetala), 0.967 (S. conica), and 0.914 (S. dioica). The values of solidity obtained here agree with the results obtained for the populations of different species [5]. At the same time, the calculations of the perimeter ratio are shown here for the first time. Table 1 contains the mean value for solidity and perimeter ratio in the dorsal views of three individual seeds of each species. Statistics were done on IBM SPSS Statistics v28 (SPSS 2021) and R software v. 4.1.2 [66]. Non-parametric Kruskal-Wallis tests were applied to compare populations, followed by stepwise step-down comparisons by the ad hoc procedure developed by Campbell and Skillings [67]; p values inferior to 0.05 were considered significant. The coefficient of variation was calculated as CVtrait = standard deviation trait/mean trait × 100 [68]. In the lateral views ( Figure 13), values of solidity for three sample seeds are 0.949 for S. apetala AJ283, 0.970 for S. conica AJ300, and 0.961 for S. dioica Pol. (Table 2). Meanwhile, the values of the ratio perimeter of the convex hull/ perimeter are 0.907 (S. apetala AJ283), 0.864 (S. conica AJ300), and 0.803 (S. dioica Pol.), and there are differences both in solidity as well as in the perimeter ratio (Table 2). Here, the protrusions are also reduced by the obtention of the average silhouette, and the effect is more notable in the seeds of S. dioica Pol. than in S. apetala AJ283 or S. conica AJ300.  In the lateral views ( Figure 13), values of solidity for three sample seeds are 0.949 for S. apetala AJ283, 0.970 for S. conica AJ300, and 0.961 for S. dioica Pol. (Table 2). Meanwhile, the values of the ratio perimeter of the convex hull/ perimeter are 0.907 (S. apetala AJ283), 0.864 (S. conica AJ300), and 0.803 (S. dioica Pol.), and there are differences both in solidity as well as in the perimeter ratio (Table 2). Here, the protrusions are also reduced by the obtention of the average silhouette, and the effect is more notable in the seeds of S. dioica Pol. than in S. apetala AJ283 or S. conica AJ300.  Table 2. Values of solidity and perimeter ratio (perimeter of the convex hull/ perimeter of the image) in the average silhouettes and as mean of three seeds for the lateral views of the seed images of S. apetala AJ283, S. conica AJ300 and S. dioica Pol. Values marked with the same superscript letter in each column correspond to populations that do not differ significantly at p < 0.05 (Campbell and Skillings's test   In summary and regarding dorsal views, a notable concavity was observed in the basal side of the silhouettes of S. apetala AJ283, corresponding with the dorsal fold expressed by the term "dorso canaliculata", being this characteristic predominant in the group of smooth seeds, except for S. littorea [5]. Thus, a concave region corresponding to the dorsal side of the seed is more pronounced in S. apetala AJ283, intermediate in S. conica AJ300, and does not exist in S. dioica Pol. and the other members of S. sect. Melandrium, whose dorsal views are convex [5].

Conclusions
The shape of seeds is related to ovule types, fruit shape, and developmental conditions. Quantification of seed shape is required for the phenotypical characterization in genetics as well as for the application of seed morphology in taxonomy. An accurate and quantitative form description can be achieved by comparison with geometric figures. Silene seeds show inter-specific variation and constitute a good model for studying morphological variation. Models used for Silene include geometric curves derived from the cardioid, but in some cases (dorsal views), the seeds have a conserved shape that does not fit these models. Images obtained from Fourier Transform analysis can be used as models for seeds that don't adjust well to canonical geometric figures. Morphological aspects of their silhouettes can be of interest in taxonomy. Nevertheless, seeds are tri-dimensional figures, and the Fourier transform is a step to obtain an accurate representation of seeds but not the final step in the process. Funding: Project "CLU-2019-05-IRNASA/CSIC Unit of Excellence", funded by the Junta de Castilla y León and co-financed by the European Union (ERDF "Europe drives our growth").