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.

Octagonal zonogon
Tessellation by irregular hexagonal zonogons
Regular octagon tiled by squares and rhombi

Examples

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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.

Tiling and equidissection

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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  -sided zonogon can be tiled by   parallelograms.[4] (For equilateral zonogons, a  -sided one can be tiled by   rhombi.) In this tiling, there is a parallelogram for each pair of slopes of sides in the  -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]

Other properties

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In an  -sided zonogon, at most   pairs of vertices can be at unit distance from each other. There exist  -sided zonogons with   unit-distance pairs.[9]

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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.

References

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  1. ^ a b Boltyanski, Vladimir; Martini, Horst; Soltan, P. S. (2012), Excursions into Combinatorial Geometry, Springer, p. 319, ISBN 9783642592379
  2. ^ Young, John Wesley; Schwartz, Albert John (1915), Plane Geometry, H. Holt, p. 121, If a regular polygon has an even number of sides, its center is a center of symmetry of the polygon
  3. ^ Alexandrov, A. D. (2005), Convex Polyhedra, Springer, p. 351, ISBN 9783540231585
  4. ^ Beck, József (2014), Probabilistic Diophantine Approximation: Randomness in Lattice Point Counting, Springer, p. 28, ISBN 9783319107417
  5. ^ Andreescu, Titu; Feng, Zuming (2000), Mathematical Olympiads 1998-1999: Problems and Solutions from Around the World, Cambridge University Press, p. 125, ISBN 9780883858035
  6. ^ Frederickson, Greg N. (1997), Dissections: Plane and Fancy, Cambridge University Press, Cambridge, p. 10, doi:10.1017/CBO9780511574917, ISBN 978-0-521-57197-5, MR 1735254
  7. ^ Monsky, Paul (1990), "A conjecture of Stein on plane dissections", Mathematische Zeitschrift, 205 (4): 583–592, doi:10.1007/BF02571264, MR 1082876, S2CID 122009844
  8. ^ Stein, Sherman; Szabó, Sandor (1994), Algebra and Tiling: Homomorphisms in the Service of Geometry, Carus Mathematical Monographs, vol. 25, Cambridge University Press, p. 130, ISBN 9780883850282
  9. ^ Ábrego, Bernardo M.; Fernández-Merchant, Silvia (2002), "The unit distance problem for centrally symmetric convex polygons", Discrete & Computational Geometry, 28 (4): 467–473, doi:10.1007/s00454-002-2882-5, MR 1949894