Zonogon

In geometry, a zonogon is a centrally-symmetric, convex polygon.^[1] Equivalently, it is a convex polygon whose sides can be grouped into parallel pairs with equal lengths and opposite orientations.

A regular polygon is a zonogon if and only if it has an even number of sides.^[2] 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.

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.^[3]

Every ${\displaystyle 2n}$-sided zonogon can be tiled by ${\displaystyle {\tbinom {n}{2}}}$ parallelograms.^[4] (For equilateral zonogons, a ${\displaystyle 2n}$-sided one can be tiled by ${\displaystyle {\tbinom {n}{2}}}$ rhombi.) In this tiling, there is a parallelogram for each pair of slopes of sides in the ${\displaystyle 2n}$-sided zonogon. At least three of the zonogon's vertices must be vertices of only one of the parallelograms in any such tiling.^[5] For instance, the regular octagon can be tiled by two squares and four 45° rhombi.^[6]

In a generalization of Monsky's theorem, Paul Monsky (1990) proved that no zonogon has an equidissection into an odd number of equal-area triangles.^[7][8]

In an ${\displaystyle n}$-sided zonogon, at most ${\displaystyle 2n-3}$ pairs of vertices can be at unit distance from each other. There exist ${\displaystyle n}$-sided zonogons with ${\displaystyle 2n-O({\sqrt {n}})}$ unit-distance pairs.^[9]

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.^[1] 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.