Evidence That Supertriangles Exist in Nature from the Vertical Projections of Koelreuteria paniculata Fruit

Symmetry

Peijian Shi¹  Yirong Li¹  Yuping Li²  Johan Gielis³  Brady K. Quinn⁴ 

[1] Bamboo Research Institute, Nanjing Forestry University, Nanjing 210037, China;College of Horticulture, Jinling Institute of Technology, Nanjing 210038, China;Department of Biosciences Engineering, University of Antwerp, B-2020 Antwerp, Belgium;Fisheries and Oceans Canada, St. Andrews, NB E5B 0E4, Canada;

关键词: Gielis equation;    goodness of fit;    natural geometry;    polar coordinate;    radial symmetry;   

DOI  :  10.3390/sym14010023

来源: DOAJ

【 摘 要 】

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 (RMSE_adj) of the two equations were always less than 0.08, and >70% were less than 0.05. For 57/100 fruit projections, the GE had a lower RMSE_adj 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.

【 授权许可】

Unknown