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.

File:2d-bravais.svg

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

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).

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.

Sides	Examples		Name	Symmetry
4	File:Parallelogon parallelogram.png		Parallelogram	Z2, order 2
	File:Parallelogon rectangle.png File:Parallelogon rhombus.png		Rectangle & rhombus	Dih2, order 4
	File:Parallelogon square.png		Square	Dih4, order 8
6	File:Hexagonal parallelogon.png	File:Parallelogon general hexagon.png File:Concave hexagonal parallelogon.png File:Concave hexagonal parallelogon2.png	Elongated parallelogram	Z2, order 2
	File:Elongated hexagonal parallelogon.pngFile:Vertex elongated hexagonal parallelogon.png	File:Bow-tie hexagon.pngFile:Bow-tie hexagon2.png	Elongated rhombus	Dih2, order 4
	File:Regular hexagonal parallelogon.png		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.

Parallelogram tilings

1 length		2 lengths
Right	Skew	Right	Skew
File:Lattice of squares.svg Square p4m (*442)	File:Lattice of rhombuses.svg Rhombus cmm (2*22)	File:Lattice of rectangles.svg Rectangle pmm (*2222)	File:Lattice of rhomboids.svg Parallelogram p2 (2222)

Hexagonal parallelogon tilings

1 length	2 lengths		3 lengths
File:Isohedral tiling p6-13.svg	File:Isohedral tiling p6-12.png	File:Isohedral tiling p4-22-concave.png	File:Isohedral tiling p6-7.svg	File:Isohedral tiling p4-22-concave2.svg
Regular hexagon p6m (*632)	Elongated rhombus cmm (2*22)		Elongated parallelogram p2 (2222)

References

• The facts on file: Geometry handbook, Catherine A. Gorini, 2003, Template:ISBN, p. 117
• list of 107 isohedral tilings, p. 473-481

External links

• Fedorov's Five Parallelohedra