Contreras-Figueroa Gabriela, Hernández-Sandoval Luis, Aragón José L
Facultad de Ciencias Naturales, Universidad Autónoma de Querétaro, Av. de las Ciencias s/n, Juriquilla, Querétaro, 76230, Mexico.
Centro de Física Aplicada y Tecnología Avanzada, Universidad Nacional Autónoma de Mexico, Boulevard Juriquilla 3001, Juriquilla, Querétaro, 76230, Mexico.
Theor Biol Med Model. 2015 Nov 16;12:27. doi: 10.1186/s12976-015-0022-1.
The quantification of the spatial order of biological patterns or mosaics provides useful information as many properties are determined by the spatial distribution of their constituent elements. These are usually characterised by methods based on nearest neighbours distances, by the number of sides of cells, or by angles defined by the adjacent cells.
A measure of regularity in polygonal mosaics of different kinds in biological systems is proposed. It is based on the condition of eutacticity, expressed in terms of eutactic stars, which is closely related to regularity of polytopes. Thus it constitutes a natural measure of regularity. The proposed measure is tested with numerical and real data. Numerically is tested with a hexagonal lattice that is distorted progressively and with a non-periodic regular tiling. With real data, the distribution of oak trees in forests from three locations in the State of Querétaro, Mexico, and the spiral pattern of florets in a flowering plant are characterised.
The proposed measure performs well and as expected while tested with a numerical experiment, as well as when applied to a known non-periodic tiling of the plane. Concerning real data, the measure is sensitive to the degree of perturbation observed in the distribution of oak trees and detects high regularity in a phyllotactic pattern studied.
The measure here proposed has a clear geometrical meaning, establishing what regularity means, and constitute an advantageous general purposes alternative to analyse spatial distributions, capable to indicate the degree of regularity of a mosaic or an array of points.
生物模式或镶嵌图的空间秩序量化提供了有用信息，因为许多属性由其组成元素的空间分布决定。这些通常通过基于最近邻距离的方法、细胞边数或相邻细胞定义的角度来表征。
提出了一种衡量生物系统中不同类型多边形镶嵌图规则性的方法。它基于优形性条件，用优形星来表示，这与多面体的规则性密切相关。因此，它构成了一种自然的规则性度量。用数值和实际数据对所提出的度量进行了测试。在数值上，用逐渐变形的六边形晶格和非周期性规则平铺进行测试。对于实际数据，对墨西哥克雷塔罗州三个地点森林中橡树的分布以及一种开花植物小花的螺旋模式进行了表征。
所提出的度量在数值实验测试时表现良好且符合预期，在应用于已知的平面非周期性平铺时也是如此。关于实际数据，该度量对橡树分布中观察到的扰动程度敏感，并在所研究的叶序模式中检测到高度规则性。
这里提出的度量具有明确的几何意义，确定了规则性的含义，并构成了一种分析空间分布的有利通用替代方法，能够指示镶嵌图或点阵列的规则程度。
