We study chiral polyhedra in 3-dimensional geometries (Euclidean, hyperbolic, and projective) in a unified manner. This extends to hyperbolic and projective spaces some structural results in the classification of chiral polyhedra in Euclidean 3-space given in 2005 by Schulte. Then, we describe a way to produce examples with helical faces based on a classic Petrie–Coxeter construction, yielding a new family in \(\mathbb {S}^3\) which is described exhaustively.

中文翻译：

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3 维几何中的手性多面体和来自 Petrie-Coxeter 构造

我们以统一的方式研究 3 维几何（欧几里得、双曲线和射影）中的手性多面体。这扩展到双曲线和射影空间，舒尔特在 2005 年给出的欧几里得 3 空间中手性多面体分类的一些结构结果。然后，我们描述了一种基于经典 Petrie-Coxeter 构造生成具有螺旋面的示例的方法，在\(\mathbb {S}^3\)中产生一个新的家族，该家族被详尽描述。