Pentagonal Orthobirotunda Pattern

Girih patterns are decorative Islamic designs in which star shapes and polygons are connected with interlacing straight lines. Girih (Persian for “knot”) can be painstakingly created with a straightedge and compass. About nine hundred years ago Islamic architects developed an ingenious method for generating girih patterns simply with tiles. The girih tiles are five decorated polygons used to create girih patterns systematically without the need for complex drafting. We can extend the girih system into three dimensions by applying the concept to the faces of polyhedra.

Here are twelve pentagonal orthobirotundi, with girih patterns.

Girih Icosidodecahedron and Pentagonal Orthobirotunda

These two girih polyhedra are a 32-faced icosidodecahedron and the related pentagonal orthobirotunda. They each include 12 pentagon and 20 triangle faces. The pentagons are decorated with the girih tile pattern. A similar pattern continues across the equilateral triangles.

Icosidodecahedron Wireframe Girih Pattern

Icosidodecahedron with Girih Pattern

Icosidodecahedron

Pentagonal Orthobirotunda Wireframe Girih Pattern

Pentagonal Orthobirotunda with Girih Pattern

Pentagonal Orthobirotunda

Chalice: Concave Girih Polyhedra with Decagons and Pentagons

These girih polyhedra have 32 faces. They include 2 decagon and 10 pentagon faces, as well as triangles. The decagon and pentagon are decorated with the girih tile pattern. A similar pattern continues across the triangles. The triangles are equilateral.

These polyhedra are related to an icosidodecahedron and pentagonal orthobirotunda. You can split an icosidodecahedron in half (making two pentagonal rotunda) and reattached the halves at pentagons to create the chalice.

Chalice Wireframe Girih Pattern

Girih Polyhedra with Decagons and Elongated Hexagons

These girih polyhedra have 32 faces. They're related to the quasiregular polyhedron, the icosidodecahedron. They include 2 decagon and 10 elongated hexagon faces, as well as triangles. The decagon and elongated hexagon are decorated with the girih tile pattern. A similar pattern continues across the triangles.

 32 Faces with Girih Pattern

32 faces

 32 Faces with Wireframe Girih Pattern

Girih Polyhedra with Bow Tie Hexagon

These girih polyhedra are heptakaidecahedrons, but they include bow tie concave hexagons where my first version had elongated hexagons, and the second version had rhombi. The pentagon and elongated hexagon are decorated with the girih tile pattern. A similar pattern continues across the triangles.

 Heptakaidecahedron with Girih Pattern

 Heptakaidecahedron

 Heptakaidecahedron Wireframe Girih Pattern

Girih Polyhedra with Rhombus

These girih polyhedra are heptakaidecahedrons, but they include rhombi where my first version had elongated hexagons. The pentagon and rhombus are decorated with the girih tile pattern. A similar pattern continues across the triangles.

Heptakaidecahedron with Girih Pattern

Heptakaidecahedron

Heptakaidecahedron Wireframe Girih Pattern

Girih Polyhedra Pattern

This pattern is from twenty heptakaidecahedrons decorated with girih tile strapwork, and aligned side-by-side. The heptakaidecahedron is a seventeen sided, geometric solid with two girih tile pentagonal faces, five girih tile elongated hexagon faces, and ten isosceles triangles. In this pattern, the three dimensional solids are in five rows of four each heptakaidecahedrons with each alternating row rotated by 180 degrees, and viewed in perspective.

These are the same heptakaidecahedrons at a different scale, and rotating.