# Online Girih Tiling Editor

Girih tiles are a set of five tiles that were historically used in the creation of ornaments for decoration of buildings in Islamic architecture. This browser app allows you to:

• Lay Girih patterns on an infinite canvas.
• Color and style your design.
• Export as vector images.

# PLATONIC SOLIDS

 Tetrahedron Cube Octahedron Dodecahedron Icosahedron Four faces Six faces Eight faces Twelve faces Twenty faces (Animation) (Animation) (Animation) (Animation) (Animation)
Polyhedron Vertices Edges Faces Schläfli symbol Vertex configuration
tetrahedron 4 6 4 {3, 3} 3.3.3
cube 8 12 6 {4, 3} 4.4.4
octahedron 6 12 8 {3, 4} 3.3.3.3
dodecahedron 20 30 12 {5, 3} 5.5.5
icosahedron 12 30 20 {3, 5} 3.3.3.3.3

## Cartesian coordinates

For Platonic solids centered at the origin, simple Cartesian coordinates are given below. The Greek letter φ is used to represent the golden ratio 1 + √5/2.

Cartesian coordinates
Figure Tetrahedron Octahedron Cube Icosahedron Dodecahedron
Faces 4 8 6 20 12
Vertices 4 6 (2 × 3) 8 12 (4 × 3) 20 (8 + 4 × 3)
Orientation
set
1 2 1 2 1 2
Coordinates (1, 1, 1)
(1, −1, −1)
(−1, 1, −1)
(−1, −1, 1)
(−1, −1, −1)
(−1, 1, 1)
(1, −1, 1)
(1, 1, −1)
(±1, 0, 0)
(0, ±1, 0)
(0, 0, ±1)
(±1, ±1, ±1) (0, ±1, ±φ)
(±1, ±φ, 0)
φ, 0, ±1)
(0, ±φ, ±1)
φ, ±1, 0)
(±1, 0, ±φ)
(±1, ±1, ±1)
(0, ±1/φ, ±φ)
1/φ, ±φ, 0)
φ, 0, ±1/φ)
(±1, ±1, ±1)
(0, ±φ, ±1/φ)
φ, ±1/φ, 0)
1/φ, 0, ±φ)
Image

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# REGULAR TESSELLATIONS

The three regular tessellations of the plane are closely related to the Platonic solids. Indeed, one can view the Platonic solids as regular tessellations of the sphere. This is done by projecting each solid onto a concentric sphere. The faces project onto regular spherical polygons which exactly cover the sphere. There are three possibilities:

{4, 4} {3, 6} {6, 3) 44 36 63 (t=1, e=1) (t=1, e=1) (t=1, e=1) *442 (3)p6m *632p4m

 {3,3} Defect 180° {3,4} Defect 120° {3,5} Defect 60° {3,6} Defect 0° {4,3} Defect 90° {4,4} Defect 0° {5,3} Defect 36° {6,3} Defect 0° A vertex needs at least 3 faces, and an angle defect. A 0° angle defect will fill the Euclidean plane with a regular tiling. By Descartes’ theorem, the number of vertices is 720°/defect.

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