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

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## Abstract

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## 1. Introduction

#### A Historical Anecdote

“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.”

## 2. Geometric Properties of Seed Silhouettes

## 3. Quantification of Seed Shape by Comparison with Models

#### 3.1. Quantification by Comparison with Known Geometric Figures as Models

#### 3.2. Closed Curves from Fourier Equations as Models

#### 3.3. Curvature Analysis

## 4. Seed Shape Diversity Linked to Embryogeny and Fruit Development

#### 4.1. Embryogeny

#### 4.2. Variations in Seed Shape Related to Fruit Development

## 5. Variations in Seed Shape Related to Environmental Adaptations

## 6. Silene as a Model System for Seed Geometry

#### 6.1. Fourier Analysis of Seeds: Calculation of Equations Corresponding to New Models

#### 6.2. A Classification According to the Geometric Properties of the Seeds

## 7. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**Left**) The image of a seed of A. natalia (Balslev & A.J. Hend.) Barfod (Arecaceae). (

**Right**) the corresponding convex Hull.

**Figure 2.**To obtain a Fourier equation corresponding to the silhouette of a seed of Aphandra natalia reference points are taken along the seed profile, and the information related to the coordinates corresponding to the points is transformed into Fourier coefficients. The entire process is done in Mathematica 12.3

^{®}. From left to right: The seed with points marked (bar equals 0.5 cm) and the curves corresponding to Fourier equations with 2, 6, 10, and 20 harmonics, respectively.

**Figure 3.**Superposition of the convex Hull with the silhouette image of A. natalia (in red,

**left**) and with the silhouette obtained with 20 harmonics (

**right**). Partial concavities correspond with small white areas between the silhouette and the convex Hull.

**Figure 4.**(

**Left**) The silhouette of the seed image of A. natalia given in Figure 1 with points marked on it. The series of points start in the red point marked and go counter-clockwise. (

**Right**) curvature values corresponding to the points along the silhouette.

**Figure 5.**Seed images and their corresponding silhouettes from species with diverse ovule types. Above: Orthotropous: Piper nudilimbum C.DC. (Piperaceae); Chloranthus erectus (Buch.-Ham.) Verdc. (Chloranthaceae). Anatropous: Cabomba aquatica Aubl. (Cabombaceae); Canna indica L. (Cannaceae). Below: Amphitropous: Stellaria media (L.) Vill. (Caryophyllaceae). Campylotropous: Hibiscus sabdariffa L. (Malvaceae); Silene noctiflora L. (Caryophyllaceae); Gypsophila elegans M. Bieb. (Caryophyllaceae). The images were obtained and modified from [57], except S. media (https://plants.usda.gov/home/plantProfile?symbol=STME2; accessed on 12 April 2021).

**Figure 6.**Seed silhouettes with lune-like and polygonal shapes of Peganum harmala L. (Nitrariaceae).

**Figure 7.**Dorsal and lateral average silhouettes for the images of 20 seeds of S. apetala AJ283, S. conica AJ300, and S. dioica Pol.

**Figure 8.**Dorsal and lateral average silhouettes for the images of 20 seeds of Silene apetala AJ283, S. conica AJ300, and S. dioica Pol. with the corresponding models superimposed in red. Models LM5, LM1, and LM4 resemble, respectively, the lateral views of S. apetala AJ283, S. conica AJ300, and S. dioica Pol. DM5 and DM3 resemble the dorsal views of S. conica AJ283 and S. dioica Pol. The dorsal view of S. apetala AJ283 does not adjust to any model described.

**Figure 9.**Silhouettes of seeds of S. apetala AJ283. L: Lateral view; D: Dorsal view; SI: Seed image; AS: Average silhouette of 20 seeds; M: Model calculated, LM5; H: Curves obtained by Fourier equations with 4, 12, and 20 harmonics.

**Figure 10.**Silhouettes of seeds of S. conica AJ300. L: Lateral view; D: Dorsal view; SI: Seed image; AS: Average silhouette of 20 seeds; M: Models calculated LM1 and DM5; H: Curves obtained by Fourier equations with 4, 8, and 12 harmonics.

**Figure 11.**Silhouettes of seeds of S. dioica Pol. L: Lateral view; D: Dorsal view; SI: Seed image; AS: Average silhouette of 20 seeds; M: Models calculated LM2 and DM3; H: Curves obtained by Fourier equations with 4, 8, and 12 harmonics.

**Figure 12.**Average silhouettes for the dorsal views of 20 seeds of S. apetala AJ283, S. conica AJ300, and S. dioica Pol. with the corresponding convex hull superimposed.

**Figure 13.**Average silhouettes for the lateral views of 20 seeds of S. apetala AJ283 (smooth seed), S. conica AJ300 (rugose seed), and S. dioica Pol. (echinate seed), with the corresponding convex hull.

**Table 1.**Values of solidity and perimeter ratio (perimeter of the convex hull/ perimeter of the image) in three seeds for the dorsal 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). N indicates the number of seeds analyzed.

N | Perimeter | Convex Hull Perimeter | Perimeter Ratio | Solidity | |
---|---|---|---|---|---|

S. apetala AJ283 | 3 | 1.01 ^{b} (6.02) | 0.86 ^{a} (6.02) | 0.852 ^{a} (0.76) | 0.834 ^{a} (1.78) |

S. conica AJ300 | 3 | 0.88 ^{a} (2.78) | 0.85 ^{a} (2.96) | 0.967 ^{c} (0.31) | 0.972 ^{b} (0.06) |

S. dioica Pol. | 3 | 1.13 ^{c} (1.90) | 1.03 ^{b} (1.10) | 0.914 ^{b} (1.54) | 0.968 ^{b} (0.32) |

**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). N indicates the number of seeds analyzed.

N | Perimeter | Convex Hull Perimeter | Perimeter Ratio | Solidity | |
---|---|---|---|---|---|

S. apetala AJ283 | 3 | 0.53 ^{a} (1.31) | 0.49 ^{a} (1.55) | 0.924 ^{b} (0.33) | 0.949 ^{a} (0.69) |

S. conica AJ300 | 3 | 0.90 ^{b} (0.61) | 0.88 ^{b} (1.00) | 0.975 ^{c} (1.01) | 0.970 ^{c} (0.16) |

S. dioica Pol. | 3 | 1.25 ^{c} (3.51) | 1.12 ^{c} (4.02) | 0.897 ^{a} (1.17) | 0.961 ^{b} (0.24) |

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## Share and Cite

**MDPI and ACS Style**

Cervantes, E.; Rodríguez-Lorenzo, J.L.; Gutiérrez del Pozo, D.; Martín-Gómez, J.J.; Janousek, B.; Tocino, Á.; Juan, A.
Seed Silhouettes as Geometric Objects: New Applications of Elliptic Fourier Transform to Seed Morphology. *Horticulturae* **2022**, *8*, 974.
https://doi.org/10.3390/horticulturae8100974

**AMA Style**

Cervantes E, Rodríguez-Lorenzo JL, Gutiérrez del Pozo D, Martín-Gómez JJ, Janousek B, Tocino Á, Juan A.
Seed Silhouettes as Geometric Objects: New Applications of Elliptic Fourier Transform to Seed Morphology. *Horticulturae*. 2022; 8(10):974.
https://doi.org/10.3390/horticulturae8100974

**Chicago/Turabian Style**

Cervantes, Emilio, José Luis Rodríguez-Lorenzo, Diego Gutiérrez del Pozo, José Javier Martín-Gómez, Bohuslav Janousek, Ángel Tocino, and Ana Juan.
2022. "Seed Silhouettes as Geometric Objects: New Applications of Elliptic Fourier Transform to Seed Morphology" *Horticulturae* 8, no. 10: 974.
https://doi.org/10.3390/horticulturae8100974