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 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 | Examples | Name | Symmetry | |
|---|---|---|---|---|
| 4 |  |
Parallelogram | Z2, order 2 | |
 60px |
Rectangle & rhombus | Dih2, order 4 | ||
 |
Square | Dih4, order 8 | ||
| 6 |  |
50px 60px  |
Elongated parallelogram |
Z2, order 2 |
 50px |
50px |
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.
| 1 length | 2 lengths | ||
|---|---|---|---|
| Right | Skew | Right | Skew |
 Square p4m (*442) |
 Rhombus cmm (2*22) |
 Rectangle pmm (*2222) |
 Parallelogram p2 (2222) |
| 1 length | 2 lengths | 3 lengths | ||
|---|---|---|---|---|
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|
|
|
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| Regular hexagon p6m (*632) |
Elongated rhombus cmm (2*22) |
Elongated parallelogram p2 (2222) | ||
References
- ↑ 1.0 1.1 Aleksandr Danilovich Alexandrov (2005) [1950]. Convex Polyhedra. Springer. p. 351. ISBN 3-540-23158-7. https://books.google.com/books?id=R9vPatr5aqYC&pg=PA351.
- 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. https://archive.org/details/isbn_0716711931. list of 107 isohedral tilings, p.473-481
- Fedorov's Five Parallelohedra
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