Evidence That Supertriangles Exist in Nature from the Vertical Projections of <i>Koelreuteria paniculata</i> Fruit
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...
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MDPI AG
2021-12-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/1/23 |
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| author | Yuping Li Brady K. Quinn Johan Gielis Yirong Li Peijian Shi |
| author_facet | Yuping Li Brady K. Quinn Johan Gielis Yirong Li Peijian Shi |
| author_sort | Yuping Li |
| collection | DOAJ |
| description | 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 <i>Koelreuteria paniculata</i> 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<sub>adj</sub>) 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<sub>adj</sub> 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. |
| first_indexed | 2024-03-10T00:26:54Z |
| format | Article |
| id | doaj.art-a981d23cfdf947e49a02cc7308aa207a |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T00:26:54Z |
| publishDate | 2021-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-a981d23cfdf947e49a02cc7308aa207a2023-11-23T15:32:11ZengMDPI AGSymmetry2073-89942021-12-011412310.3390/sym14010023Evidence That Supertriangles Exist in Nature from the Vertical Projections of <i>Koelreuteria paniculata</i> FruitYuping Li0Brady K. Quinn1Johan Gielis2Yirong Li3Peijian Shi4College of Horticulture, Jinling Institute of Technology, Nanjing 210038, ChinaFisheries and Oceans Canada, St. Andrews, NB E5B 0E4, CanadaDepartment of Biosciences Engineering, University of Antwerp, B-2020 Antwerp, BelgiumBamboo Research Institute, Nanjing Forestry University, Nanjing 210037, ChinaBamboo Research Institute, Nanjing Forestry University, Nanjing 210037, ChinaMany 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 <i>Koelreuteria paniculata</i> 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<sub>adj</sub>) 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<sub>adj</sub> 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.https://www.mdpi.com/2073-8994/14/1/23Gielis equationgoodness of fitnatural geometrypolar coordinateradial symmetry |
| spellingShingle | Yuping Li Brady K. Quinn Johan Gielis Yirong Li Peijian Shi Evidence That Supertriangles Exist in Nature from the Vertical Projections of <i>Koelreuteria paniculata</i> Fruit Symmetry Gielis equation goodness of fit natural geometry polar coordinate radial symmetry |
| title | Evidence That Supertriangles Exist in Nature from the Vertical Projections of <i>Koelreuteria paniculata</i> Fruit |
| title_full | Evidence That Supertriangles Exist in Nature from the Vertical Projections of <i>Koelreuteria paniculata</i> Fruit |
| title_fullStr | Evidence That Supertriangles Exist in Nature from the Vertical Projections of <i>Koelreuteria paniculata</i> Fruit |
| title_full_unstemmed | Evidence That Supertriangles Exist in Nature from the Vertical Projections of <i>Koelreuteria paniculata</i> Fruit |
| title_short | Evidence That Supertriangles Exist in Nature from the Vertical Projections of <i>Koelreuteria paniculata</i> Fruit |
| title_sort | evidence that supertriangles exist in nature from the vertical projections of i koelreuteria paniculata i fruit |
| topic | Gielis equation goodness of fit natural geometry polar coordinate radial symmetry |
| url | https://www.mdpi.com/2073-8994/14/1/23 |
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