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.
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 | |
  |
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.
| 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 | ||
|---|---|---|---|---|
|
|
|
|
|
| Regular hexagon p6m (*632) |
Elongated rhombus cmm (2*22) |
Elongated parallelogram p2 (2222) | ||
See also
- Parallelohedron - Dimensional extension of parallelogons into 3D
References
- ^ 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.
- 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
- Fedorov's Five Parallelohedra