A geometric realization of an abstract polyhedron {\cal P} is a mapping that sends an i-face to an open set of dimension i. This work adapts a method based on Wythoff construction to generate a full rank realization of an abstract regular polyhedron from its automorphism group Γ. The method entails finding a real orthogonal representation of Γ of degree 3 and applying its image to suitably chosen (not necessarily connected) open sets in space. To demonstrate the use of the method, it is applied to the abstract polyhedra whose automorphism groups are isomorphic to the non-crystallographic Coxeter group H3.

中文翻译：

---

具有自同构群H3的抽象规则多面体的几何实现。

抽象多面体{\ cal P}的几何实现是一种将i面发送到尺寸为i的开放集的映射。这项工作采用了一种基于Wythoff构造的方法，以从其自同构群Γ生成抽象规则多面体的完整秩实现。该方法需要找到度3的Γ的实正交表示和应用其的图像到适当选择的在空间（不一定是连接的）打开套。为了证明该方法的使用，将其应用于抽象多面体，该抽象多面体的自同构基团与非晶体Coxeter基团H3同构。