Fig. 1: Type-I and type-II hyperbolic lattices.

a One brunch of a two-sheet hyperboloid (\({t}^{2}-{x}^{2}-{y}^{2}=1\)) in (2 + 1)-dimensional (x,y,t) Minkowski spacetime is mapped onto a Poincaré disk (z = 0) by stereographic projection through the point (0, 0, −1) (green dot). b Poincaré disk with unit radius. c A type-I hyperbolic lattice with {8, 3}-tessellation constructed by discretizing the Poincaré disk with octagons. d One-sheet hyperboloid (\({u}^{2}+{v}^{2}-{z}^{2}=1\)) in (1 + 2)-dimensional (u,v,z) Minkowski spacetime is mapped onto a set of overlapped line segments at z axis by stereographic projection through the point (0,1,0) (green dot). Inset displays a Poincaré band configuration formed by extending the line segments along angular position φ. e Poincaré ring with unit outer radius, inner radius \({r}_{{in}}\), and characteristic radius \({r}_{h}\) generated by wrapping and gluing the Poincaré band in d. f A type-II hyperbolic lattice with {\({r}_{h}\), 8, 3}-tessellation constructed by discretizing the Poincaré ring with octagons.