In geometry, a parallelogon is a polygon with parallel opposite sides (hence the name) that can tile a plane by translation (rotation is not permitted).[1]
Parallelogons have an even number of sides and opposite sides that are equal in length. A less obvious corollary is that parallelogons can only have either four or six sides; Parallelogons have 180-degree rotational symmetry around the center.
A four-sided parallelogon is called a parallelogram.
The faces of a parallelohedron (the three dimensional analogue) are called parallelogons.[1]
Quadrilateral and hexagonal parallelogons each have varied geometric symmetric forms. They all have central inversion symmetry, order 2. Every convex parallelogon is a zonogon, but hexagonal parallelogons enable the possibility of nonconvex polygons.
| Sides | Examples | Name | Symmetry | |
|---|---|---|---|---|
| 4 | Parallelogram | Z2, order 2 | ||
| Rectangle & rhombus | Dih2, order 4 | |||
| Square | Dih4, order 8 | |||
| 6 | Elongated parallelogram | Z2, order 2 | ||
| Elongated rhombus | Dih2, order 4 | |||
| Dih6, order 12 | ||||
A parallelogram can tile the plane as a distorted square tiling while a hexagonal parallelogon can tile the plane as a distorted regular hexagonal tiling.