Parallelogon

A parallelogon is a polygon such that images of the polygon will tile the plane when fitted together along entire sides, without rotation.[1]

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 must have an even number of sides and opposite sides must be equal in length and parallel (hence the name). A less obvious corollary is that all parallelogons have either four or six sides;[1] a four-sided parallelogon is called a parallelogram. In general a parallelogon has 180-degree rotational symmetry around its center.

The faces of a parallelohedron are parallelogons.

Two polygonal types

Quadrilateral and hexagonal parallelogons each have varied geometric symmetric forms. In general they all have central inversion symmetry, order 2. Every convex parallelogon is a zonogon, but hexagonal parallelogons enable the possibility of nonconvex polygons.

Sides	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.

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)

References

Aleksandr Danilovich Alexandrov (2005) [1950]. Convex Polyhedra. Translated by N.S. Dairbekov; S.S. Kutateladze; A.B. Sosinsky. Springer. p. 351. ISBN 3-540-23158-7. ISSN 1439-7382.

The facts on file: Geometry handbook, Catherine A. Gorini, 2003, ISBN 0-8160-4875-4, p.117
Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. list of 107 isohedral tilings, p.473-481
Fedorov's Five Parallelohedra

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[aleksandrov-1] Aleksandr Danilovich Alexandrov (2005) [1950]. Convex Polyhedra. Translated by N.S. Dairbekov; S.S. Kutateladze; A.B. Sosinsky. Springer. p. 351. ISBN 3-540-23158-7. ISSN 1439-7382.