The Poincaré disk model: the entire hyperbolic plane fits inside the unit disk, with the boundary at infinity. Straight lines are arcs of circles meeting the boundary at right angles. Regular tilings {p, q} — p-sided polygons, q meeting at each vertex — exist if 1/p + 1/q < 1/2. Euclid only allows {3,6}, {4,4}, {6,3}. Hyperbolic geometry gives you all the others.
p (sides) 5
q (meeting) 4
depth 5
rotation 0°
why each tile is the same size
In the hyperbolic metric, every tile in the picture has identical area. The optical shrinking toward the boundary is the Poincaré model lying about distance — the boundary of the disk is infinitely far away in the hyperbolic metric. The geometry M.C. Escher famously borrowed for Circle Limit III and IV: an infinite tessellation drawn in a finite space, with tiles shrinking only because of the projection.
Why this matters beyond aesthetics: hyperbolic geometry is the third classical geometry, alongside Euclidean and spherical. It's the geometry of negatively-curved surfaces. Modern theoretical physics uses it in: holography (AdS/CFT — bulk space is hyperbolic), error-correcting codes (hyperbolic surface codes are the best known fault-tolerant quantum memories), and the geometry of phylogenetic tree spaces. The {4, 5} tiling is the one Escher drew.