The images of D’Arcy Wentworth Thompson’s book “On Growth and Form” got an iconic status and became influential for biometrics and other mathematical approaches to organismic form. In particular, this is true for those of the chapter on the theory of transformation, which even has an impact on art and humanities. Based on his approach, Thompson formulated far-reaching conclusions with a partly anti-Darwinian stance. Here, we use the example of Thompson’s transformation of crab carapaces to test to what degree the transformation of grids, landmarks, and shapes result in congruent images. For comparison, we applied the same series of tests to digitized carapaces of real crabs. Both approaches show similar results. Only the simple transformations show a reasonable form of congruence. In particular, the transformations to majoid spider crabs reveal a complicated transformation of grids with partly crossing lines. By contrast, the carapace of the lithodid species is relatively easily created despite the fact that it is no brachyuran, but evolved a spider crab-like shape convergently from a hermit crab ancestor.

中文翻译：

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D'Arcy W. Thompson 的笛卡尔变换：批判性评价

达西·温特沃斯·汤普森 (D'Arcy Wentworth Thompson) 的著作《论生长与形态》(On Growth and Form) 的图像获得了标志性的地位，并对生物识别学和其他有机体形态的数学方法产生了影响。尤其是转化理论一章的那些，甚至对艺术和人文学科都有影响。基于他的方法，汤普森以部分反达尔文主义的立场制定了影响深远的结论。在这里，我们使用 Thompson 的蟹壳变换示例来测试网格、地标和形状的变换在多大程度上会产生一致的图像。为了进行比较，我们对真实螃蟹的数字化甲壳进行了相同系列的测试。两种方法都显示出相似的结果。只有简单的变换才显示出合理的同余形式。特别是，对majoid蜘蛛蟹的转换揭示了网格的复杂转换，其中部分交叉线。相比之下，石斑鱼的甲壳相对容易形成，尽管它不是短尾龙，而是从寄居蟹祖先会聚进化出蜘蛛蟹状的形状。