A generic geometric transformation that unifies a wide range of natural and abstract shapes
A generic geometric transformation that unifies a wide range of natural and abstract shapes
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DOI：
10.3732/ajb.90.3.333
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发表时间：
2003-03-01
影响因子：
3
通讯作者：
Gielis, J
中科院分区：
文献类型：
--
作者：
Gielis, J
为了研究植物和其他生物的形态，有几种可用的数学工具，其中大多数是通用工具，没有考虑到有价值的生物信息。在这篇报告中，我提出了一种新的几何方法来建模和理解各种抽象的、自然的和人造的形状。从圆的概念出发，我展示了各种各样的形状都可以用一个简单的几何方程——超级公式来描述。修改参数允许生成各种自然多边形。例如，将方程应用于对数或三角函数会修改这些函数和所有相关图的度量。作为一个统一的框架，所有这些形状在其内部度量上都被证明是圆，并且超级公式提供了形状的欧几里得测量与内部非欧几里得度量之间的精确数学关系。超越欧几里得的圆圈和毕达哥拉斯的尺度，揭示了一种研究自然形式和现象的新颖而有力的方法。
To study forms in plants and other living organisms, several mathematical tools are available, most of which are general tools that do not take into account valuable biological information. In this report I present a new geometrical approach for modeling and understanding various abstract, natural, and man-made shapes. Starting from the concept of the circle, I show that a large variety of shapes can be described by a single and simple geometrical equation, the Superformula. Modification of the parameters permits the generation of various natural polygons. For example, applying the equation to logarithmic or trigonometric functions modifies the metrics of these functions and all associated graphs. As a unifying framework, all these shapes are proven to be circles in their internal metrics, and the Superformula provides the precise mathematical relation between Euclidean measurements and the internal non-Euclidean metrics of shapes. Looking beyond Euclidean circles and Pythagorean measures reveals a novel and powerful way to study natural forms and phenomena.