Parallelogon

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).¹²

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

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.

Parallelogram tilings

1 length		2 lengths
Right	Skew	Right	Skew
Square p4m (*442)	Rhombus cmm (2*22)	Rectangle pmm (*2222)	Parallelogram p2 (2222)

Hexagonal parallelogon tilings

1 length	2 lengths		3 lengths
Regular hexagon p6m (*632)	Elongated rhombus cmm (2*22)		Elongated parallelogram p2 (2222)