Parallelohedron
Polyhedra that fill space (without holes) using only translation

Written by Paul Bourke
September 2002

In the past (before x-tay diffraction) determining polyhedra that could tile space with only translations was of great interest to crystallographers. Convex polyhedra with this property were called parallelohedra, an understanding and categorising of crystal structures involved finding the parallelohedron of the structure.

2D

In two dimensions there are only 2 parallelohedra which tile the plane, the quadrilateral (square as a special case) and hexagon.

3D

In 3 dimensions there are exactly 5 parallelohedra, they are the cube, hexagonal prism, truncated octahedron, rhombic dodecahedron, and the elongated rhombic dodecahedron. Thee were first isolated by Fedorov around 1920.

Each polyhedra is illustrated in the table below along with the coordinates of each polygonally bound planar surface. The first number in the face list is the number of vertices in that face, this is followed by that many (x,y,z) vertices. Constants (a and b) are used to simplify the vertex components.

Cube

 
4 -1 -1 -1  1 -1 -1  1 -1  1 -1 -1  1 
4 -1 -1 -1 -1 -1  1 -1  1  1 -1  1 -1 
4 -1 -1  1  1 -1  1  1  1  1 -1  1  1 
4 -1  1 -1 -1  1  1  1  1  1  1  1 -1 
4  1 -1 -1  1  1 -1  1  1  1  1 -1  1 
4 -1 -1 -1 -1  1 -1  1  1 -1  1 -1 -1
Hexagonal Prism
 a = sqrt(3)/2
b = 1/2

4  a  b -b  a  b  b  a -b  b  a -b -b
4  0  1  b  0  1 -b  a  b -b  a  b  b
4 -a  b -b -a  b  b  0  1  b  0  1 -b
4 -a -b  b -a -b -b -a  b -b -a  b  b
4  0 -1  b  0 -1 -b -a -b -b -a -b  b
4  0 -1 -b  a -b -b  a -b  b  0 -1  b
6 -a  b  b  0  1  b  a  b  b  a -b  b  0 -1  b  -a -b  b
6 -a  b -b  0  1 -b  a  b -b  a -b -b  0 -1 -b  -a -b -b
Truncated Octahedron
 a = 0.5

4  1  0 -a  1 -a  0  1  0  a  1  a  0 
4 -1 -a  0 -1  0  a -1  a  0 -1  0 -a 
4  0  a -1 -a  0 -1  0 -a -1  a  0 -1 
4 -a  0  1  0 -a  1  a  0  1  0  a  1 
4  a  1  0  0  1  a -a  1  0  0  1 -a 
4  0 -1  a -a -1  0  0 -1 -a  a -1  0 
6 -a  0  1  0 -a  1  0 -1  a -a -1  0 -1 -a  0 -1  0  a
6 -a  0  1 -1  0  a -1  a  0 -a  1  0  0  1  a  0  a  1 
6  0  a  1  0  1  a  a  1  0  1  a  0  1  0  a  a  0  1
6  a  0  1  1  0  a  1 -a  0  a -1  0  0 -1  a  0 -a  1
6 -a  0 -1  0 -a -1  0 -1 -a -a -1  0 -1 -a  0 -1  0 -a
6 -a  0 -1 -1  0 -a -1  a  0 -a  1  0  0  1 -a  0  a -1
6  0  a -1  0  1 -a  a  1  0  1  a  0  1  0 -a  a  0 -1
6  a  0 -1  1  0 -a  1 -a  0  a -1  0  0 -1 -a  0 -a -1
Rhombic Dodecahedron
 a = 1/2

4 1  0  0  a  a -a  0  0 -1  a -a -a
4 1  0  0  a -a  a  0  0  1  a  a  a
4 0  0  1 -a -a  a -1  0  0 -a  a  a
4 0  0 -1 -a  a -a -1  0  0 -a -a -a
4 0  1  0  a  a -a  1  0  0  a  a  a
4 0  1  0  a  a  a  0  0  1 -a  a  a
4 0  1  0 -a  a  a -1  0  0 -a  a -a
4 0  1  0 -a  a -a  0  0 -1  a  a -a
4 0 -1  0 -a -a -a -1  0  0 -a -a  a
4 0 -1  0  a -a -a  0  0 -1 -a -a -a
4 0 -1  0  a -a  a  1  0  0  a -a -a
4 0 -1  0 -a -a  a  0  0  1  a -a  a
Elongated Rhombic Dodecahedron
 a = 1/2
b = 3/2

4  1  0 -a  a  a -1  0  0 -b  a -a -1 
4  1  0  a  a -a  1  0  0  b  a  a  1 
4  0  0  b -a -a  1 -1  0  a -a  a  1 
4  0  0 -b -a  a -1 -1  0 -a -a -a -1 
4  0 -1  a -a -a  1  0  0  b  a -a  1 
4  0  1  a  a  a  1  0  0  b -a  a  1 
4  0 -1 -a  a -a -1  0  0 -b -a -a -1 
4  0  1 -a -a  a -1  0  0 -b  a  a -1 
6 -a -a  1  0 -1  a  0 -1 -a -a -a -1 -1  0 -a -1  0 a
6 -a  a  1  0  1  a  0  1 -a -a  a -1 -1  0 -a -1  0 a
6  a  a  1  1  0  a  1  0 -a  a  a -1  0  1 -a  0  1 a
6  a -a  1  1  0  a  1  0 -a  a -a -1  0 -1 -a  0 -1 a