In geometry, the order-4 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of . It can also be called a pentapentagonal tiling in a bicolored quasiregular form.
This tiling represents a hyperbolic kaleidoscope of 5 mirrors meeting as edges of a regular pentagon. This symmetry by orbifold notation is called *22222 with 5 order-2 mirror intersections. In Coxeter notation can be represented as [5<sup>*</sup>,4], removing two of three mirrors (passing through the pentagon center) in the [5,4] symmetry.
The kaleidoscopic domains can be seen as bicolored pentagons, representing mirror images of the fundamental domain. This coloring represents the uniform tiling t1 and as a quasiregular tiling is called a pentapentagonal tiling.
This tiling is topologically related as a part of sequence of regular polyhedra and tilings with pentagonal faces, starting with the dodecahedron, with Schläfli symbol, and Coxeter diagram, progressing to infinity.
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol, and Coxeter diagram, with n progressing to infinity.
This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n).