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). PDF
Parallelogons have four or six sides, opposite sides that are equal in length, and 180-degree rotational symmetry around the center. A four-sided parallelogon is a parallelogram.
The three-dimensional analogue of a parallelogon is a parallelohedron. All faces of a parallelohedron are parallelogons.
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. SidesExamplesNameSymmetry4ParallelogramZ2, order 2 Rectangle & rhombusDih2, order 4SquareDih4, order 86 Elongatedparallelogram Z2, order 2 Elongated rhombusDih2, order 4 Regularhexagon 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.
+ Parallelogram tilings1 length2 lengthsRightSkewRightSkewSquarep4m (*442)Rhombuscmm (2*22)Rectanglepmm (*2222)Parallelogramp2 (2222)
+ Hexagonal parallelogon tilings1 length2 lengths3 lengthsRegular hexagonp6m (*632)Elongated rhombuscmm (2*22)Elongated parallelogramp2 (2222)
References
The facts on file: Geometry handbook, Catherine A. Gorini, 2003, , p. 117 list of 107 isohedral tilings, p. 473-481
External links Fedorov's Five Parallelohedra
The surprising 3D parallelo-hexagon at Dynamic Geometry Sketches