Abstract
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 >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.
Highlights
IntroductionR and φ are the polar radius and polar angle of a curve transcribing the outline of a shape, respectively; A, B, n1, n2, and n3 are parameters that need to be estimated; and m is a positive integer that determines the number of angles of the curve generated in [0, 2π)
Gielis [1] proposed a polar coordinate equation to describe natural shapes, especially symmetrical shapes, as follows: r(φ) = 1 cos m φ n2 + 1 sin mφ n3 − 1 n1 (1) A B4Here, r and φ are the polar radius and polar angle of a curve transcribing the outline of a shape, respectively; A, B, n1, n2, and n3 are parameters that need to be estimated; and m is a positive integer that determines the number of angles of the curve generated in [0, 2π)
The four-parameter Gielis equation with m = 3 (GE) and the five-parameter twin Gielis equation with m = 3 (TGE) can both describe the supertriangular shapes of the vertical projections of the studied golden rain tree fruit well
Summary
R and φ are the polar radius and polar angle of a curve transcribing the outline of a shape, respectively; A, B, n1, n2, and n3 are parameters that need to be estimated; and m is a positive integer that determines the number of angles of the curve generated in [0, 2π). This equation can be rewritten [2,3] as: r(φ) = γ cos m φ n2 + k sin m φ n3 −1 β (2).
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