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 pairs with equal lengths and opposite orientations.

Examples

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

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 rectangles, the rhombi, 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 eq ...

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 Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, sourc ...

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

rhombi.) 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 ''Ameri ...

, 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 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 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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