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.

Examples

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

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 $2n$ -sided zonogon can be tiled by ${\tbinom {n}{2}}$ parallelograms.^[4] (For equilateral zonogons, a $2n$ -sided one can be tiled by ${\tbinom {n}{2}}$ 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.^[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

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.^[9]

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.^[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

^ a b Boltyanski, Vladimir; Martini, Horst; Soltan, P. S. (2012), Excursions into Combinatorial Geometry, Springer, p. 319, ISBN 9783642592379
^ 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
^ Alexandrov, A. D. (2005), Convex Polyhedra, Springer, p. 351, ISBN 9783540231585
^ Beck, József (2014), Probabilistic Diophantine Approximation: Randomness in Lattice Point Counting, Springer, p. 28, ISBN 9783319107417
^ Andreescu, Titu; Feng, Zuming (2000), Mathematical Olympiads 1998-1999: Problems and Solutions from Around the World, Cambridge University Press, p. 125, ISBN 9780883858035
^ 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
^ Monsky, Paul (1990), "A conjecture of Stein on plane dissections", Mathematische Zeitschrift, 205 (4): 583–592, doi:10.1007/BF02571264, MR 1082876, S2CID 122009844
^ 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
^ 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

[bms-1] Boltyanski, Vladimir; Martini, Horst; Soltan, P. S. (2012), Excursions into Combinatorial Geometry, Springer, p. 319, ISBN 9783642592379

[ys-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

[aa-3] Alexandrov, A. D. (2005), Convex Polyhedra, Springer, p. 351, ISBN 9783540231585

[beck-4] Beck, József (2014), Probabilistic Diophantine Approximation: Randomness in Lattice Point Counting, Springer, p. 28, ISBN 9783319107417

[mo-5] Andreescu, Titu; Feng, Zuming (2000), Mathematical Olympiads 1998-1999: Problems and Solutions from Around the World, Cambridge University Press, p. 125, ISBN 9780883858035

[gnf-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

[monsky-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

[ss-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

[af-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