There are few cases where numbers or types of possible phenotypes are known, although vast state spaces have been postulated. Rarely applied in this context, graph theory and topology enable enumeration of possible phenotypes and evolutionary transitions. Here, we generate polyhedral calyx graphs for the Late Cretaceous, stemless crinoids Marsupites testudinarius and Uintacrinus socialis (Uintacrinoidea Zittel) revealing structural similarities to carbon fullerene and fulleroid molecules (respectively). The U. socialis calyx incorporates numerous plates (e.g. graph vertices |V| ≥ 197), which are small, light, low‐density and have four to eight sides. Therefore, the corresponding number of possible plate arrangements (number of polyhedral graphs) is large (≫1 × 1014). Graph vertices representing plates with sides >6 introduce negative Gaussian curvature (surface saddle points) and topological instability, increasing buckling risk. However, observed numbers of vertices for Uintacrinus do not allow more stable pentaradial configurations. In contrast, the Marsupites calyx dual graph has 17 faces that are pentagonal or hexagonal. Therefore, it is structurally identical to a carbon fullerene, specifically C30‐D5h. Corresponding graph restrictions result in constraint to only three structural options (fullerene structures C30‐C2v 1, C30‐C2v 2 and C30‐D5h). Further restriction to pentaradial symmetry allows only one possibility: the Marsupites phenotype. This robust, stable topology is consistent with adaptation to predation pressures of the Mesozoic marine revolution. Consequently, the most plausible evolutionary pathway between unitacrinoid phenotypes was a mixed heterochronic trade‐off to fewer, larger calyx plates. Therefore, topological limitations radically constrained uintacrinoid skeletal possibilities but thereby aided evolution of a novel adaptive phenotype.

中文翻译：

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白垩纪海百合的富勒烯样结构揭示了拓扑有限的骨骼可能性

尽管已经假设了巨大的状态空间，但很少有可能表型的数量或类型是已知的。图论和拓扑很少应用于这种情况，能够枚举可能的表型和进化转变。在这里，我们为晚白垩世、无茎海棠 Marupites testudinarius 和 Uintacrinus socialis (Uintacrinoidea Zittel) 生成多面体花萼图，揭示了碳富勒烯和富勒烯分子（分别）的结构相似性。U. socialis 花萼包含许多板块（例如图形顶点|V| ≥ 197），它们小、轻、低密度并且有四到八个边。因此，对应的可能的板排列数量（多面体图的数量）很大（≫1 × 1014）。表示带边的板的图形顶点 > 6 引入负高斯曲率（表面鞍点）和拓扑不稳定性，增加屈曲风险。然而，观察到的 Uintacrinus 顶点数量不允许更稳定的五边形配置。相比之下，Marsupites 花萼对偶图有 17 个五边形或六边形面。因此，它在结构上与碳富勒烯相同，特别是 C30-D5h。相应的图限制导致仅约束三个结构选项（富勒烯结构 C30-C2v 1、C30-C2v 2 和 C30-D5h）。对五径向对称性的进一步限制只允许一种可能性：Marsupites 表型。这种稳健、稳定的拓扑结构与适应中生代海洋革命的捕食压力是一致的。最后，unitacrinoid 表型之间最合理的进化途径是对更少、更大的花萼板的混合异时权衡。因此，拓扑限制从根本上限制了 uintacrinoid 骨骼的可能性，但从而有助于新的适应性表型的进化。