geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a parallelogon is a

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

with

parallel Parallel may refer to: Mathematics * Parallel (geometry), two lines in the Euclidean plane which never intersect * Parallel (operator), mathematical operation named after the composition of electrical resistance in parallel circuits Science a ...

opposite sides (hence the name) that can

tile Tiles are usually thin, square or rectangular coverings manufactured from hard-wearing material such as ceramic, Rock (geology), stone, metal, baked clay, or even glass. They are generally fixed in place in an array to cover roofs, floors, wal ...

a plane by

translation Translation is the communication of the semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...

(

rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...

is not permitted). PDF
/ref> Parallelogons have four or six sides, opposite sides that are equal in length, and 180-degree

rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape (geometry), shape has when it looks the same after some rotation (mathematics), rotation by a partial turn (angle), turn. An object's degree of rotational s ...

around the center. A four-sided parallelogon is a

parallelogram In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram a ...

. The three-dimensional analogue of a parallelogon is a

parallelohedron In geometry, a parallelohedron or Fedorov polyhedron is a convex polyhedron that can be Translation (geometry), translated without rotations to fill Euclidean space, producing a Honeycomb (geometry), honeycomb in which all copies of the polyhed ...

. 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 In geometry, a point reflection (also called a point inversion or central inversion) is a geometric transformation of affine space in which every point (geometry), point is reflected across a designated inversion center, which remains Fixed p ...

symmetry, order 2. Every convex parallelogon is a

zonogon In geometry, a zonogon is a centrally-symmetric, convex polygon. Equivalently, it is a convex polygon whose sides can be grouped into parallel pairs with equal lengths and opposite orientations, the two-dimensional analog of a zonohedron. Ex ...

, but hexagonal parallelogons enable the possibility of nonconvex polygons.

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.

References

* ''The facts on file: Geometry handbook'', Catherine A. Gorini, 2003, , p. 117 * {{cite book, last1=Grünbaum, first1=Branko, author1-link=Branko Grünbaum, last2=Shephard, first2=G. C., title=Tilings and Patterns, location=New York, publisher=W. H. Freeman, year=1987, isbn=0-7167-1193-1, url-access=registration, url=https://archive.org/details/isbn_0716711931 list of 107 isohedral tilings, p. 473-481

External links

Fedorov's Five Parallelohedra
Types of polygons