Parallelogon
Polygon able to tessellate edge-to-edge, without rotation From Wikipedia, the free encyclopedia
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 four or six sides, opposite sides that are equal in length, and 180-degree rotational symmetry around the center.[1] A four-sided parallelogon is a parallelogram.
The three-dimensional analogue of a parallelogon is a parallelohedron. All faces of a parallelohedron are parallelogons.[2]
Two polygonal types
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 | ||
Geometric variations
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) | ||
References
External links
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