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Parallelogon

Polygon able to tessellate edge-to-edge, without rotation

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.

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		colspan=2	Examples		Name
[[File:Parallelogon parallelogram.png	60px]]	Parallelogram	Z2, order 2
[[File:Parallelogon rectangle.png	60px]] [[File:Parallelogon rhombus.png	60px]]	Rectangle & rhombus	Dih2, order 4
[[File:Parallelogon square.png	40px]]	Square	Dih4, order 8
[[File:Hexagonal parallelogon.png	60px]]	[[File:Parallelogon general hexagon.png	50px]] [[File:Concave hexagonal parallelogon.png	60px]] [[File:Concave hexagonal parallelogon2.png	60px]]	Elongatedparallelogram
[[File:Elongated hexagonal parallelogon.png	50px]][[File:Vertex elongated hexagonal parallelogon.png	50px]]	[[File:Bow-tie hexagon.png	50px]][[File:Bow-tie hexagon2.png	50px]]	Elongated rhombus
[[File:Regular hexagonal parallelogon.png	50px]]	Regularhexagon	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.

1 length	2 lengths	Right	Skew	Right	Skew
[[File:Lattice of squares.svg	140px]]Squarep4m (*442)	[[File:Lattice of rhombuses.svg	140px]]Rhombuscmm (2*22)	[[File:Lattice of rectangles.svg	140px]]Rectanglepmm (*2222)	[[File:Lattice of rhomboids.svg	140px]]Parallelogramp2 (2222)

1 length	2 lengths	3 lengths
[[File:Isohedral tiling p6-13.svg	140px]]	[[File:Isohedral tiling p6-12.svg	140px]]	[[File:Isohedral tiling p4-22-concave.png	140px]]
Regular hexagonp6m (*632)	Elongated rhombuscmm (2*22)	Elongated parallelogramp2 (2222)

References

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

References

Grünbaum, Branko. (2010-12-01). "The Bilinski Dodecahedron and Assorted Parallelohedra, Zonohedra, Monohedra, Isozonohedra, and Otherhedra". The Mathematical Intelligencer.
Aleksandr Danilovich Alexandrov. (2005). "Convex Polyhedra". Springer.

Info: Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

types-of-polygons

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