geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, 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 a ...

. Equivalently, it is a convex polygon whose sides can be grouped into

parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of ...

pairs with equal lengths and opposite orientations.

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 p ...

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 plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containi ...

s, the

rhombi In plane Euclidean geometry, a rhombus (plural 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 ...

, and the

parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...

Tiling and equidissection

The four-sided and six-sided zonogons are

parallelogon 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 an even number of sides and opposite sides that are equal in length. A ...

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 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 In geometry, Monsky's theorem states that it is not possible to dissect a square into an odd number of triangles of equal area. In other words, a square does not have an odd equidissection. The problem was posed by Fred Richman in the '' American ...

, proved that no zonogon has an

equidissection In geometry, an equidissection is a partition of a polygon into triangles of equal area. The study of equidissections began in the late 1960s with Monsky's theorem, which states that a square cannot be equidissected into an odd number of triang ...

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 centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in ...

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

Minkowski sum In geometry, the Minkowski sum (also known as dilation) 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'', i.e., the set : A + B = \. Analogously, the Minkowski ...

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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