{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,16]],"date-time":"2026-02-16T10:15:17Z","timestamp":1771236917254,"version":"3.50.1"},"reference-count":56,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,9,29]],"date-time":"2022-09-29T00:00:00Z","timestamp":1664409600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Considering both biological and non-biological polygonal shape organizations, in this paper we introduce a quantitative method which is able to determine informational entropy as spatial differences between heterogeneity of internal areas from simulation and experimental samples. According to these data (i.e., heterogeneity), we are able to establish levels of informational entropy using statistical insights of spatial orders using discrete and continuous values. Given a particular state of entropy, we establish levels of information as a novel approach which can unveil general principles of biological organization. Thirty-five geometric aggregates are tested (biological, non-biological, and polygonal simulations) in order to obtain the theoretical and experimental results of their spatial heterogeneity. Geometrical aggregates (meshes) include a spectrum of organizations ranging from cell meshes to ecological patterns. Experimental results for discrete entropy using a bin width of 0.5 show that a particular range of informational entropy (0.08 to 0.27 bits) is intrinsically associated with low rates of heterogeneity, which indicates a high degree of uncertainty in finding non-homogeneous configurations. In contrast, differential entropy (continuous) results reflect negative entropy within a particular range (\u22120.4 to \u22120.9) for all bin widths. We conclude that the differential entropy of geometrical organizations is an important source of neglected information in biological systems.<\/jats:p>","DOI":"10.3390\/e24101390","type":"journal-article","created":{"date-parts":[[2022,9,29]],"date-time":"2022-09-29T21:03:10Z","timestamp":1664485390000},"page":"1390","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Entropy as a Geometrical Source of Information in Biological Organizations"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5174-6046","authenticated-orcid":false,"given":"Juan","family":"Lopez-Sauceda","sequence":"first","affiliation":[{"name":"Consejo Nacional de Ciencia y Tecnolog\u00eda (CONACYT), Avenida Insurgentes Sur 1582, Colonia Cr\u00e9dito Constructor, Alcald\u00eda Benito Ju\u00e1rez, Mexico City 03940, Mexico"},{"name":"Departamento de Procesos Productivos, Universidad Aut\u00f3noma Metropolitana, Avenida de las Garzas No. 10, Colonia El Pante\u00f3n, Lerma de Villada 52005, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Philipp","family":"von B\u00fclow","sequence":"additional","affiliation":[{"name":"Departamento de Procesos Productivos, Universidad Aut\u00f3noma Metropolitana, Avenida de las Garzas No. 10, Colonia El Pante\u00f3n, Lerma de Villada 52005, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6072-8480","authenticated-orcid":false,"given":"Carlos","family":"Ortega-Laurel","sequence":"additional","affiliation":[{"name":"Departamento de Sistemas de Informaci\u00f3n y Comunicaciones, Universidad Aut\u00f3noma Metropolitana, Avenida de las Garzas No. 10, Colonia El Pante\u00f3n, Lerma de Villada 52005, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5770-7849","authenticated-orcid":false,"given":"Francisco","family":"Perez-Martinez","sequence":"additional","affiliation":[{"name":"Departamento de Sistemas de Informaci\u00f3n y Comunicaciones, Universidad Aut\u00f3noma Metropolitana, Avenida de las Garzas No. 10, Colonia El Pante\u00f3n, Lerma de Villada 52005, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kalina","family":"Miranda-Perkins","sequence":"additional","affiliation":[{"name":"Secretar\u00eda de Medio Ambiente y Recursos Naturales (SEMARNAT), Avenida Ej\u00e9rcito Nacional 223, Colonia An\u00e1huac, Alcald\u00eda Miguel Hidalgo, Mexico City 11320, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jos\u00e9 Gerardo","family":"Carrillo-Gonz\u00e1lez","sequence":"additional","affiliation":[{"name":"Consejo Nacional de Ciencia y Tecnolog\u00eda (CONACYT), Avenida Insurgentes Sur 1582, Colonia Cr\u00e9dito Constructor, Alcald\u00eda Benito Ju\u00e1rez, Mexico City 03940, Mexico"},{"name":"Departamento de Sistemas de Informaci\u00f3n y Comunicaciones, Universidad Aut\u00f3noma Metropolitana, Avenida de las Garzas No. 10, Colonia El Pante\u00f3n, Lerma de Villada 52005, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"12323","DOI":"10.1038\/s41598-017-12521-1","article-title":"Explorability and the Origin of Network Sparsity in Living Systems","volume":"7","author":"Busiello","year":"2017","journal-title":"Sci. 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