AI 中文总结
本文通过计算机辅助证明，枚举了有限的高贵多面体，得出除Stephhanoids和双四面体的无限集合外，恰好有146种高贵多面体的结论。
AI 中文摘要
我们通过计算机辅助证明，完整枚举了有限的高贵多面体，即同时具有顶点可递性和面可递性的多面体，该证明将高贵多面体的可能存在性与某些一元或二元三次多项式的根关联起来。这一过程通过对三维欧几里得空间中点群的轨道进行参数化实现，该参数化自然引出了给定点群某些轨道的临界性概念，以及轨道集合上的相关等价关系≡_c。我们证明，在该关系下给定的轨道等价类中，高贵面是等价的，且这类等价类的数量有限。这提供了有限数量的测试案例，可通过计算机搜索进行验证。总体而言，除了先前已知的Stephhanoids（冠状多面体）和双四面体的无限集合外，我们还发现恰好有146种高贵多面体。
英文摘要
We provide a complete enumeration of the finite polyhedra that are noble, that is, polyhedra that are vertex transitive and facet transitive, through a computer-assisted proof which relates the possible existence of noble polyhedra to the roots of certain univariate or bivariate cubic polynomials. This is done through a parametrization of the orbits of point groups in $\mathbb{R}^3$, which naturally leads to a notion of criticality for certain orbits of a given point group, and a related equivalence relation $\equiv_c$ on the set of orbits. We show that within a given equivalence class of orbits under this relation, noble facetings are equivalent, and furthermore that there are a finite number of such equivalence classes. This gives a finite number of test cases, which may be tested through computer search. In total, we find that there are exactly 146 noble polyhedra in addition to the previously known infinite sets of stephanoids (crown polyhedra) and disphenoids.
Comments34 pages, 6 figures. For associated GitHub repository, see https://github.com/Plasmath/noble-tools-revised
