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).[1][2]

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

Parallelogons have an even number of sides and opposite sides that are equal in length. A less obvious corollary is that parallelogons can only have either four or six sides;[1] Parallelogons have 180-degree rotational symmetry around the center.

A four-sided parallelogon is called a parallelogram.

The faces of a parallelohedron (the three dimensional analogue) are called 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.

SidesExamplesNameSymmetry
4 ParallelogramZ2, order 2
Rectangle & rhombusDih2, order 4
SquareDih4, 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
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