Parallelogon

File:Parallelogons as 2 or 3 vectors.png

A parallelogon is constructed by two or three pairs of parallel line segments. The vertices and edges on the interior of the hexagon are suppressed.

There are five Bravais lattices in two dimensions, related to the parallelogon tessellations by their five symmetry variations.

A parallelogon is a polygon such that images of the polygon under translations only tile the plane when fitted together along entire sides.

A parallelogon must have an even number of sides and opposite sides must be equal in length and parallel (hence the name). A less obvious restriction is that a parallelogon can only have four or six sides; a four-sided parallelogon is a parallelogram. In general a parallelogon has 180-degree rotational symmetry around its center.

Two polygonal types

Quadrilateral and hexagonal parallelogons each have varied geometric symmetric forms. In general they all have central inversion symmetry, order 2. Hexagonal parallelogons enable the possibility of nonconvex polygons.

Sides	Examples		Name	Symmetry
4	60px		Parallelogram	Z2, order 2
	60px 60px		Rectangle & rhombus	Dih2, order 4
	40px		Square	Dih4, order 8
6		50px 60px 60px	Elongated parallelogram	Z2, order 2
	50px50px	50px50px	Elongated rhombus	Dih2, order 4
	50px		Regular hexagon	Dih6, order 12

Geometric variations

Parallelogram can tile the plane as a distorted square tiling while hexagonal parallelogon can tiling the plane as a distorted regular hexagonal tiling.

Parallelogram tilings

1 length		2 lengths
Right	Skew	Right	Skew
Square p4m (*442)	140px Rhombus cmm (2*22)	Rectangle pmm (*2222)	Parallelogram p2 (2222)

Hexagonal parallelogon tilings

1 length	2 lengths		3 lengths
		140px		140px
Regular hexagon p6m (*632)	Elongated rhombus cmm (2*22)		Elongated parallelogram p2 (2222)

See also

• Parallelohedron - Dimensional extension of parallelogons into 3D

References

• The facts on file: Geometry handbook, Catherine A. Gorini, 2003, ISBN 0-8160-4875-4, p.117
• list of 107 isohedral tilings, p.473-481
• Fedorov's Five Parallelohedra

• Aleksandr Danilovich Aleksandrov Convex Polyhedra p351