Parallelogon

Not to be confused with zonogon, a polygon with parallel opposite sides, that does not necessarily tile a plane but must be convex.

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.

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). ^{[1] [2]}

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]

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			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

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.
2. Grünbaum, Branko (2010-12-01). "The Bilinski Dodecahedron and Assorted Parallelohedra, Zonohedra, Monohedra, Isozonohedra, and Otherhedra" . The Mathematical Intelligencer. 32 (4): 5–15. doi:10.1007/s00283-010-9138-7. hdl: 1773/15593 . ISSN   1866-7414. S2CID   120403108. PDF

• 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

External links

• Fedorov's Five Parallelohedra