Hyperbolic small dodecahedral honeycomb in the context of Hyperbolic metric

In mathematics, a finite subdivision rule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. Subdivision rules in a sense are generalizations of regular geometric fractals. Instead of repeating exactly the same design over and over, they have slight variations in each stage, allowing a richer structure while maintaining the elegant style of fractals. Subdivision rules have been used in architecture, biology, and computer science, as well as in the study of hyperbolic manifolds. Substitution tilings are a well-studied type of subdivision rule.

In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant negative sectional curvature, often taken to be  −1 for simplicity. It is homogeneous, and satisfies the stronger property of being a symmetric space. There are many ways to construct it as an open subset of ${\displaystyle \mathbb {R} ^{n}}$ with an explicitly written Riemannian metric; such constructions are referred to as models. Hyperbolic 2-space, H, which was the first instance studied, is also called the hyperbolic plane.

It is also sometimes referred to as Lobachevsky space or Bolyai–Lobachevsky space after the names of the author who first published on the topic of hyperbolic geometry. Sometimes the qualificative "real" is added to distinguish it from complex hyperbolic spaces.