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][2]
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;[1] 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.[2]
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 | |||
Regular hexagon |
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.
1 length | 2 lengths | ||
---|---|---|---|
Right | Skew | Right | Skew |
Square p4m (*442) |
Rhombus cmm (2*22) |
Rectangle pmm (*2222) |
Parallelogram p2 (2222) |
1 length | 2 lengths | 3 lengths | ||
---|---|---|---|---|
Regular hexagon p6m (*632) |
Elongated rhombus cmm (2*22) |
Elongated parallelogram p2 (2222) |