plants Article Geometric Models for Seed Shape Description and Quantification in the Cactaceae José Javier Martín-Gómez 1 , Diego Gutiérrez del Pozo 2 , Ángel Tocino 3 and Emilio Cervantes 1, *   Citation: Martín-Gómez, J.J.; del Pozo, D.G.; Tocino, Á.; Cervantes, E. Geometric Models for Seed Shape Description and Quantification in the Cactaceae. Plants 2021, 10, 2546. https://doi.org/10.3390/ plants10112546 Academic Editor: Bertrand Dubreucq Received: 10 November 2021 Accepted: 22 November 2021 Published: 22 November 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Instituto de Recursos Naturales y Agrobiología del Consejo Superior de Investigaciones Científicas (IRNASA-CSIC), Cordel de Merinas, 40, E-37008 Salamanca, Spain; jjavier.martin@irnasa.csic.es 2 Herbario Amazónico del Ecuador ECUAMZ, Universidad Estatal Amazónica UEA, Carretera Tena a Puyo Km. 44, Puyo EC-150950, Ecuador; diego.gutierrez.pozo@gmail.com 3 Departamento de Matemáticas, Facultad de Ciencias, Universidad de Salamanca, Plaza de la Merced 1-4, E-37008 Salamanca, Spain; bacon@usal.es * Correspondence: emilio.cervantes@irnasa.csic.es Abstract: Seed shape in species of the Cactaceae is described by comparison with geometric models. Three new groups of models are presented, two for symmetric seeds, and a third group for asymmetric seeds. The first two groups correspond, respectively, to superellipses and the combined equations of two semi-ellipses. The third group contains models derived from the representation of polar equations of Archimedean spirals that define the shape of asymmetric seeds in genera of different subfamilies. Some of the new models are geometric curves, while others are composed with a part resulting from the average silhouettes of seeds. The application of models to seed shape quantification permits the analysis of variation in seed populations, as well as the comparison of shape between species. The embryos of the Cactaceae are of the peripheral type, strongly curved and in contact with the inner surface of the seed coat. A relationship is found between seed elongation and the models, in which the genera with elongated seeds are represented by models with longer trajectories of the spiral. The analysis of seed shape opens new opportunities for taxonomy and allows quantification of seed shape in species of the Cactaceae. Keywords: Archimedean spiral; cactaceae; geometry; models; morphology; seed shape; spiral 1. Introduction The Cactaceae Juss., with about 2000 species distributed in ca. 120 genera, are char- acterized by succulent photosynthetic stems. The family has been included in the order Caryophyllales Juss. ex Bercht. & J.Presl [1], in the suborder Portulacineae Thorne [2] which included Cactaceae, Portulacaceae, Didiereaceae, and Basellaceae, as well as other succulent plant families (Anacampserotaceae, Halophytaceae, Montiaceae, and Talinaceae) [3–5]. Cacti are common in the arid ecosystems of America, in the southern States of USA and Mexico, north-eastern Brazil, and along the slopes of the Andes from Ecuador to Argentina and Chile. Many species also live in humid forests as epiphytes. All known species are originally American with the only exception of Rhipsalis baccifera (J.S. Muell.) Stearn, found also in local populations in tropical Africa and Sri Lanka. The Cactaceae have been divided into four subfamilies: Cactoideae Eaton, Maihue- nioideae P. Fearn, Opuntioideae Burnett, and Pereskioideae Engelm. The Cactoideae and Opuntioideae contain, respectively, 100 and 9 genera [4,5]. Maihuenia (Phil.) Britton & Rose, with two cushion-forming, mucilaginous species, is the only genus in the Maihuenioideae. Pereskia Mill, together with Leuenbergeria Lodé in the Pereskioideae, are woody with large, glossy leaves and long thorns on the stem, and not all are succulents. The Cactoideae, with ten tribes, is the most diverse subfamily of globular and columnar cacti including all the iconic species in American dessert biomes [5]. The Opuntioideae correspond to the prickly pear cacti and relatives, with seeds enclosed by a bony aril that is considered a Plants 2021, 10, 2546. https://doi.org/10.3390/plants10112546 https://www.mdpi.com/journal/plants