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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

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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

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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

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  1. ^ a b 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. ^ a b 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
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📚 Artikel Terkait di Wikipedia

Hexagon

g2 hexagons, with opposite sides parallel are also called hexagonal parallelogons. Each subgroup symmetry allows one or more degrees of freedom for irregular

Conway criterion

rotations—about the midpoints of two adjacent edges in the case of a hexagonal parallelogon, and about the midpoint of an edge and one of its vertices in the case

Parallelogram

parallelepiped is a three-dimensional figure whose six faces are parallelograms. parallelogon, generalisation encompassing hexagons as well as quadrilaterals zonogon

Parallelotope

parallelepiped and parallelogram A generalization of a parallelohedron and parallelogon, this includes all parallelohedra in the first sense Zonotope This disambiguation

Point reflection

2D examples Hexagonal parallelogon Octagon

Hexagonal tiling

represent the lattice positions. Single-color (1-tile) lattices are parallelogon hexagons. Other isohedrally-tiled topological hexagonal tilings are seen

Triangular tiling

plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilateral triangle is 60 degrees

List of Greek and Latin roots in English/A–G

ἀλλήλων (allḗlōn) allele, allelomorph, allelotaxis, parallel, parallelism, parallelogon, parallelogram alph- A, a Greek Α, α, ἄλφα (álpha) alphabet, alphabetic