{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T12:42:56Z","timestamp":1776429776745,"version":"3.51.2"},"reference-count":30,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,12,24]],"date-time":"2021-12-24T00:00:00Z","timestamp":1640304000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Many natural radial symmetrical shapes (e.g., sea stars) follow the Gielis equation (GE) or its twin equation (TGE). A supertriangle (three triangles arranged around a central polygon) represents such a shape, but no study has tested whether natural shapes can be represented as\/are supertriangles or whether the GE or TGE can describe their shape. We collected 100 pieces of Koelreuteria paniculata fruit, which have a supertriangular shape, extracted the boundary coordinates for their vertical projections, and then fitted them with the GE and TGE. The adjusted root mean square errors (RMSEadj) of the two equations were always less than 0.08, and &gt;70% were less than 0.05. For 57\/100 fruit projections, the GE had a lower RMSEadj than the TGE, although overall differences in the goodness of fit were non-significant. However, the TGE produces more symmetrical shapes than the GE as the two parameters controlling the extent of symmetry in it are approximately equal. This work demonstrates that natural supertriangles exist, validates the use of the GE and TGE to model their shapes, and suggests that different complex radially symmetrical shapes can be generated by the same equation, implying that different types of biological symmetry may result from the same biophysical mechanisms.<\/jats:p>","DOI":"10.3390\/sym14010023","type":"journal-article","created":{"date-parts":[[2021,12,27]],"date-time":"2021-12-27T01:03:37Z","timestamp":1640567017000},"page":"23","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Evidence That Supertriangles Exist in Nature from the Vertical Projections of Koelreuteria paniculata Fruit"],"prefix":"10.3390","volume":"14","author":[{"given":"Yuping","family":"Li","sequence":"first","affiliation":[{"name":"College of Horticulture, Jinling Institute of Technology, Nanjing 210038, China"},{"name":"Bamboo Research Institute, Nanjing Forestry University, Nanjing 210037, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Brady K.","family":"Quinn","sequence":"additional","affiliation":[{"name":"Fisheries and Oceans Canada, St. Andrews, NB E5B 0E4, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4536-3839","authenticated-orcid":false,"given":"Johan","family":"Gielis","sequence":"additional","affiliation":[{"name":"Department of Biosciences Engineering, University of Antwerp, B-2020 Antwerp, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yirong","family":"Li","sequence":"additional","affiliation":[{"name":"Bamboo Research Institute, Nanjing Forestry University, Nanjing 210037, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4696-0130","authenticated-orcid":false,"given":"Peijian","family":"Shi","sequence":"additional","affiliation":[{"name":"Bamboo Research Institute, Nanjing Forestry University, Nanjing 210037, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"333","DOI":"10.3732\/ajb.90.3.333","article-title":"A general geometric transformation that unifies a wide range of natural and abstract shapes","volume":"90","author":"Gielis","year":"2003","journal-title":"Am. J. Bot."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Shi, P., Ratkowsky, D.A., and Gielis, J. (2020). The generalized Gielis geometric equation and its application. Symmetry, 12.","DOI":"10.3390\/sym12040645"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"751","DOI":"10.1007\/s11676-018-0770-y","article-title":"Comparison of seed morphology of two ginkgo cultivars","volume":"31","author":"Tian","year":"2020","journal-title":"J. Forest Res."},{"key":"ref_4","unstructured":"Lam\u00e9, G. (1818). Examen des Diff\u00e9rentes M\u00e9thodes Employ\u00e9es Pour R\u00e9soudre les Probl\u00e8mes de G\u00e9om\u00e9trie, V. Courcier."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Huang, W., Li, Y., Niklas, K.J., Gielis, J., Ding, Y., Cao, L., and Shi, P. (2020). A superellipse with deformation and its application in describing the cross-sectional shapes of a square bamboo. Symmetry, 12.","DOI":"10.3390\/sym12122073"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"4578","DOI":"10.1002\/ece3.1728","article-title":"Comparison of dwarf bamboos (Indocalamus sp.) leaf parameters to determine relationship between spatial density of plants and total leaf area per plant","volume":"5","author":"Shi","year":"2015","journal-title":"Ecol. Evol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"6798","DOI":"10.1002\/ece3.2407","article-title":"A geometrical model for testing bilateral symmetry of bamboo leaf with a simplified Gielis equation","volume":"6","author":"Lin","year":"2016","journal-title":"Ecol. Evol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"856","DOI":"10.3389\/fpls.2015.00856","article-title":"Capturing spiral radial growth of conifers using the superellipse to model tree-ring geometric shape","volume":"6","author":"Shi","year":"2015","journal-title":"Front. Plant Sci."},{"key":"ref_9","unstructured":"Gielis, J., and Shi, P. (Ann. N. Y. Acad. Sci., 2021). Universal equations\u2014A fresh perspective, Ann. N. Y. Acad. Sci., under\u00a0review."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1007\/s00022-015-0269-z","article-title":"Gielis\u2019 superformula and regular polygons","volume":"106","author":"Matsuura","year":"2015","journal-title":"J. Geom."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1016\/j.scienta.2018.06.034","article-title":"Leaf area estimation by considering leaf dimensions in olive tree","volume":"240","author":"Koubouris","year":"2018","journal-title":"Sci. Hortic."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Shi, P., Ratkowsky, D.A., Li, Y., Zhang, L., Lin, S., and Gielis, J. (2018). General leaf-area geometric formula exists for plants\u2014Evidence from the simplified Gielis equation. Forests, 9.","DOI":"10.3390\/f9110714"},{"key":"ref_13","first-page":"e00666","article-title":"Lamina shape does not correlate with lamina surface area: An analysis based on the simplified Gielis equation","volume":"19","author":"Su","year":"2019","journal-title":"Glob. Ecol. Conserv."},{"key":"ref_14","unstructured":"R Core Team (2020). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing. Available online: https:\/\/www.R-project.org\/."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"308","DOI":"10.1093\/comjnl\/7.4.308","article-title":"A simplex algorithm for function minimization","volume":"7","author":"Nelder","year":"1965","journal-title":"Comput. J."},{"key":"ref_16","first-page":"135","article-title":"Leaf shape simulation of castor bean and its application in nondestructive leaf area estimation","volume":"12","author":"Wei","year":"2019","journal-title":"Int. J. Agric. Biol. Eng."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/j.jtherbio.2010.05.005","article-title":"A comparison of different thermal performance functions describing temperature-dependent development rates","volume":"35","author":"Shi","year":"2010","journal-title":"J. Therm. Biol."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Spiess, A.-N., and Neumeyer, N. (2010). An evaluation of R squared as an inadequate measure for nonlinear models in pharmacological and biochemical research: A Monte Carlo approach. BMC Pharmacol., 10.","DOI":"10.1186\/1471-2210-10-6"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Chen, X., Jiang, K., Zhu, Y., Wang, X., and Yun, T. (2021). Individual tree crown segmentation directly from UAV-Borne LiDAR data using the PointNet of deep learning. Forests, 12.","DOI":"10.3390\/f12020131"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"382","DOI":"10.1086\/297175","article-title":"Inter- and intraspecific variation in fruit traits in co-occurring vertebrate-dispersed plants","volume":"155","author":"Obeso","year":"1994","journal-title":"Int. J. Plant Sci."},{"key":"ref_21","first-page":"103","article-title":"Exploration on modeling and application of 3D digital model of landscape plants","volume":"26","author":"Wang","year":"2019","journal-title":"Landsc. Archit."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1046\/j.1420-9101.1999.00040.x","article-title":"Asymmetry of wild mustard, Sinapis arvensis (Brassicaceae), in response to severe physiological stresses","volume":"12","author":"Roy","year":"1999","journal-title":"J. Evol. Biol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/j.compag.2005.09.004","article-title":"Plant species identification using Elliptic Fourier leaf shape analysis","volume":"50","author":"Meyer","year":"2006","journal-title":"Comput. Electron. Agric."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1415","DOI":"10.1109\/TPAMI.2010.184","article-title":"Shape analysis of elastic curves in Euclidean spaces","volume":"33","author":"Srivastava","year":"2011","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1016\/j.jtbi.2014.07.036","article-title":"Landmark-free statistical analysis of the shape of plant leaves","volume":"363","author":"Laga","year":"2014","journal-title":"J. Theor. Biol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"10421","DOI":"10.1002\/ece3.7845","article-title":"Using comprehensive machine-learning models to classify complex morphological characters","volume":"11","author":"Teng","year":"2021","journal-title":"Ecol. Evol."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Thompson, D.W. (1917). On Growth and Form, Cambridge University Press.","DOI":"10.5962\/bhl.title.11332"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Harte, J. (2011). Maximum Entropy and Ecology: A Theory of Abundance, Distribution, and Energetics, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780199593415.001.0001"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2991\/gaf.k.200124.001","article-title":"A note on spirals and curvature","volume":"1","author":"Gielis","year":"2020","journal-title":"Growth Form"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1007\/BF00985329","article-title":"Floral ontogeny and anatomy in Koelreuteria with special emphasis on monosymmetry and septal cavities","volume":"223","author":"Decraene","year":"2000","journal-title":"Plant Syst. Evol."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/1\/23\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:52:32Z","timestamp":1760169152000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/1\/23"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,24]]},"references-count":30,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,1]]}},"alternative-id":["sym14010023"],"URL":"https:\/\/doi.org\/10.3390\/sym14010023","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,12,24]]}}}