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 zonogon is a centrally-symmetric,

convex polygon In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is ...

. 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 In geometry, a zonohedron is a convex polyhedron that is point symmetry, centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski addition, Minkows ...

Examples

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...

is a zonogon if and only if it has an even number of sides. Thus, the square, regular hexagon, and regular octagon are all zonogons. The four-sided zonogons are the square, the

rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...

s, the

rhombi In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhom ...

, and the

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 ...

Tiling and equidissection

The four-sided and six-sided zonogons are

parallelogon In geometry, a parallelogon is a polygon with Parallel (geometry), parallel opposite sides (hence the name) that can Tessellation, tile a Plane (geometry), plane by Translation (geometry), translation (Rotation (mathematics), rotation is not per ...

s, able to tile the plane by translated copies of themselves, and all convex parallelogons have this form. Every

2n

-sided zonogon can be tiled by

\tbinom

s. (For equilateral zonogons, a

2n

-sided one can be tiled by

\tbinom

.) In this tiling, there is a parallelogram for each pair of slopes of sides in the

2n

-sided zonogon. At least three of the zonogon's vertices must be vertices of only one of the parallelograms in any such tiling. For instance, the regular octagon can be tiled by two squares and four 45° rhombi. In a generalization of Monsky's theorem, proved that no zonogon has an equidissection into an odd number of equal-area triangles.

Other properties

In an

n

-sided zonogon, at most

2n-3

pairs of vertices can be at unit distance from each other. There exist

n

-sided zonogons with

2n-O(\sqrt)

unit-distance pairs.

Related shapes

Zonogons are the two-dimensional analogues of three-dimensional

zonohedra In geometry, a zonohedron is a convex polyhedron that is point symmetry, centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski addition, Minkows ...

and higher-dimensional zonotopes. As such, each zonogon can be generated as the

Minkowski sum In geometry, the Minkowski sum of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'': A + B = \ The Minkowski difference (also ''Minkowski subtraction'', ''Minkowsk ...

of a collection of line segments in the plane. If no two of the generating line segments are parallel, there will be one pair of parallel edges for each line segment. Every face of a zonohedron is a zonogon, and every zonogon is the face of at least one zonohedron, the prism over that zonogon. Additionally, every planar cross-section through the center of a centrally-symmetric polyhedron (such as a zonohedron) is a zonogon.

References

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