{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T17:54:08Z","timestamp":1778781248959,"version":"3.51.4"},"reference-count":30,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2020,12,14]],"date-time":"2020-12-14T00:00:00Z","timestamp":1607904000000},"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 cross-sectional shapes of plants have been found to approximate a superellipse rather than an ellipse. Square bamboos, belonging to the genus Chimonobambusa (Poaceae), are a group of plants with round-edged square-like culm cross sections. The initial application of superellipses to model these culm cross sections has focused on Chimonobambusa quadrangularis (Franceschi) Makino. However, there is a need for large scale empirical data to confirm this hypothesis. In this study, approximately 750 cross sections from 30 culms of C. utilis were scanned to obtain cross-sectional boundary coordinates. A superellipse exhibits a centrosymmetry, but in nature the cross sections of culms usually deviate from a standard circle, ellipse, or superellipse because of the influences of the environment and terrain, resulting in different bending and torsion forces during growth. Thus, more natural cross-sectional shapes appear to have the form of a deformed superellipse. The superellipse equation with a deformation parameter (SEDP) was used to fit boundary data. We find that the cross-sectional shapes (including outer and inner rings) of C. utilis can be well described by SEDP. The adjusted root-mean-square error of SEDP is smaller than that of the superellipse equation without a deformation parameter. A major finding is that the cross-sectional shapes can be divided into two types of superellipse curves: hyperellipses and hypoellipses, even for cross sections from the same culm. There are two proportional relationships between ring area and the product of ring length and width for both the outer and inner rings. The proportionality coefficients are significantly different, as a consequence of the two different superellipse types (i.e., hyperellipses and hypoellipses). The difference in the proportionality coefficients between hyperellipses and hypoellipses for outer rings is greater than that for inner rings. This work informs our understanding and quantifying of the longitudinal deformation of plant stems for future studies to assess the influences of the environment on stem development. This work is also informative for understanding the deviation of natural shapes from a strict rotational symmetry.<\/jats:p>","DOI":"10.3390\/sym12122073","type":"journal-article","created":{"date-parts":[[2020,12,14]],"date-time":"2020-12-14T22:30:52Z","timestamp":1607985052000},"page":"2073","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":37,"title":["A Superellipse with Deformation and Its Application in Describing the Cross-Sectional Shapes of a Square Bamboo"],"prefix":"10.3390","volume":"12","author":[{"given":"Weiwei","family":"Huang","sequence":"first","affiliation":[{"name":"Bamboo Research Institute, College of Biology and the Environment, Nanjing Forestry University, Nanjing 210037, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yueyi","family":"Li","sequence":"additional","affiliation":[{"name":"Bamboo Research Institute, College of Biology and the Environment, Nanjing Forestry University, Nanjing 210037, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Karl J.","family":"Niklas","sequence":"additional","affiliation":[{"name":"Plant Biology Section, School of Integrative Plant Science, Cornell University, Ithaca, NY 14853, USA"}],"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":"Yongyan","family":"Ding","sequence":"additional","affiliation":[{"name":"Bamboo Research Institute, College of Biology and the Environment, Nanjing Forestry University, Nanjing 210037, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Li","family":"Cao","sequence":"additional","affiliation":[{"name":"Bamboo Research Institute, College of Biology and the Environment, 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, College of Biology and the Environment, Nanjing Forestry University, Nanjing 210037, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,14]]},"reference":[{"key":"ref_1","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_2","unstructured":"Gielis, J. (2003). Inventing the Circle: The Geometry of Nature, Geniaal Press."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Gielis, J. (2017). The Geometrical Beauty of Plants, Atlantis Press.","DOI":"10.2991\/978-94-6239-151-2"},{"key":"ref_4","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_5","first-page":"20","article-title":"Supervormen in bamboo","volume":"14","author":"Gielis","year":"1997","journal-title":"Newsletter of the Belgian Bamboo Society"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Liese, W., and K\u00f6hl, M. (2015). Bamboo: The Plant and its Uses, Springer.","DOI":"10.1007\/978-3-319-14133-6"},{"key":"ref_7","first-page":"1","article-title":"The taxonomy and cultivation of Chimonobambusa Makino","volume":"11","author":"Wen","year":"1994","journal-title":"J. Amer. Bamboo Soc."},{"key":"ref_8","unstructured":"Chen, S., Li, D., Zhu, G., Wu, L., Lu, S., Liu, L., Wang, Z., Sun, B., Zhu, Z., and Xia, N. (2006). The Flora of China (Volume 22\u2014Poaceae), Missouri Botanical Garden Press."},{"key":"ref_9","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_10","doi-asserted-by":"crossref","first-page":"308","DOI":"10.1093\/comjnl\/7.4.308","article-title":"A simplex method for function minimization","volume":"7","author":"Nelder","year":"1965","journal-title":"Comput. J."},{"key":"ref_11","unstructured":"R Core Team (2019). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing. Available online: https:\/\/www.R-project.org\/."},{"key":"ref_12","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_13","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_14","unstructured":"Ratkowsky, D.A. (1990). Handbook of Nonlinear Regression Models, Marcel Dekker."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Hsu, J.C. (1996). Multiple Comparisons: Theory and Methods, Chapman and Hall\/CRC.","DOI":"10.1201\/b15074"},{"key":"ref_16","first-page":"115","article-title":"An extension of Shapiro and Wilk\u2019s W test for normality to large samples","volume":"31","author":"Royston","year":"1982","journal-title":"J. R. Stat. Soc. Ser. C Appl. Stat."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1073","DOI":"10.1007\/s00468-019-01843-4","article-title":"Leaf area-length allometry and its implications in leaf-shape evolution","volume":"33","author":"Shi","year":"2019","journal-title":"Trees Struct. Funct."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1481","DOI":"10.1002\/ajb2.1560","article-title":"Nondestructive estimation of leaf area for 15 species of vines with different leaf shapes","volume":"107","author":"Yu","year":"2020","journal-title":"Am. J. Bot."},{"key":"ref_19","unstructured":"Weisstein, E. (2020, November 01). Superellipse. MathWorld, Available online: https:\/\/mathworld.wolfram.com\/Superellipse.html."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1038\/032391d0","article-title":"The Square Bamboo","volume":"32","year":"1885","journal-title":"Nature"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"S11","DOI":"10.1016\/S1672-6529(09)60212-2","article-title":"The impact of tissue morphology, cross-section and turgor pressure on the mechanical properties of the leaf petiole in plants","volume":"7","author":"Faisal","year":"2010","journal-title":"J. Bionic Eng."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1080\/10255842.2016.1233328","article-title":"The twist-to-bend compliance of the Rheum rhabarbarum petiole: Integrated computations and experiments","volume":"20","author":"Faisal","year":"2017","journal-title":"Comput. Method. Biomec."},{"key":"ref_23","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_24","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_25","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_26","doi-asserted-by":"crossref","unstructured":"Shi, P., Liu, M., Yu, X., Gielis, J., and Ratkowsky, D.A. (2019). Proportional relationship between leaf area and the product of leaf length and width of four types of special leaf shapes. Forests, 10.","DOI":"10.3390\/f10020178"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Wang, P., Ratkowsky, D.A., Xiao, X., Yu, X., Su, J., Zhang, L., and Shi, P. (2018). Taylor\u2019s power law for leaf bilateral symmetry. Forests, 9.","DOI":"10.3390\/f9080500"},{"key":"ref_28","first-page":"e01152","article-title":"Mean-variance relationships of leaf bilateral asymmetry for 35 species of plants and their implications","volume":"23","author":"Guo","year":"2020","journal-title":"Glob. Ecol. Conser."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Shi, P., Niinemets, U., Hui, C., Niklas, K.J., Yu, X., and Holscher, D. (2020). Leaf bilateral symmetry and the scaling of the perimeter vs. the surface area in 15 vine species. Forests, 11.","DOI":"10.3390\/f11020246"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Thompson, D.W. (1917). On Growth and Form, Cambridge University Press.","DOI":"10.5962\/bhl.title.11332"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/12\/2073\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:44:40Z","timestamp":1760179480000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/12\/2073"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,14]]},"references-count":30,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2020,12]]}},"alternative-id":["sym12122073"],"URL":"https:\/\/doi.org\/10.3390\/sym12122073","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,12,14]]}}}