Zonogon

Octagonal zonogon

Tessellation by irregular hexagonal zonogons

Regular octagon tiled by squares and rhombi

In geometry, a zonogon is a centrally-symmetric, convex polygon.Template:R 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.

Examples

A regular polygon is a zonogon if and only if it has an even number of sides.Template:R Thus, the square, regular hexagon, and regular octagon are all zonogons. The four-sided zonogons are the square, the rectangles, the rhombi, and the parallelograms.

Tiling and equidissection

The four-sided and six-sided zonogons are parallelogons, able to tile the plane by translated copies of themselves, and all convex parallelogons have this form.Template:R

Every $2n$-sided zonogon can be tiled by $(\genfrac{}{}{0ex}{}{n}{2})$ parallelograms.Template:R (For equilateral zonogons, a $2n$-sided one can be tiled by $(\genfrac{}{}{0ex}{}{n}{2})$ rhombi.) 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.Template:R For instance, the regular octagon can be tiled by two squares and four 45° rhombi.Template:R

In a generalization of Monsky's theorem, Template:Harvs proved that no zonogon has an equidissection into an odd number of equal-area triangles.Template:R

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 $2n-O(\sqrt{n})$ unit-distance pairs.Template:R

Related shapes

Zonogons are the two-dimensional analogues of three-dimensional zonohedra and higher-dimensional zonotopes. As such, each zonogon can be generated as the Minkowski sum of a collection of line segments in the plane.Template:R 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.