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 2 n {\displaystyle 2n} -sided zonogon can be tiled by ( n 2 ) {\displaystyle {\tbinom {n}{2}}} parallelograms.[4] (For equilateral zonogons, a 2 n {\displaystyle 2n} -sided one can be tiled by ( n 2 ) {\displaystyle {\tbinom {n}{2}}} rhombi.) In this tiling, there is a parallelogram for each pair of slopes of sides in the 2 n {\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 n {\displaystyle n} -sided zonogon, at most 2 n − − --> 3 {\displaystyle 2n-3} pairs of vertices can be at unit distance from each other. There exist n {\displaystyle n} -sided zonogons with 2 n − − --> O ( n ) {\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.
If a regular polygon has an even number of sides, its center is a center of symmetry of the polygon
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