Ingeometry,azonogonis acentrally-symmetric,convex polygon.[1]Equivalently, it is a convex polygon whose sides can be grouped intoparallelpairs with equal lengths and opposite orientations.
Examples
editAregular polygonis 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, therectangles,therhombi,and theparallelograms.
Tiling and equidissection
editThe four-sided and six-sided zonogons areparallelogons,able to tile the plane by translated copies of themselves, and all convex parallelogons have this form.[3]
Every-sided zonogon can be tiled byparallelograms.[4](For equilateral zonogons, a-sided one can be tiled byrhombi.) 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 ofMonsky's theorem,Paul Monsky(1990) proved that no zonogon has anequidissectioninto an odd number of equal-area triangles.[7][8]
Other properties
editIn an-sided zonogon, at mostpairs of vertices can be at unit distance from each other. There exist-sided zonogons with unit-distance pairs.[9]
Related shapes
editZonogons are the two-dimensional analogues of three-dimensionalzonohedraand higher-dimensional zonotopes. As such, each zonogon can be generated as theMinkowski sumof 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
edit- ^abBoltyanski, Vladimir; Martini, Horst; Soltan, P. S. (2012),Excursions into Combinatorial Geometry,Springer, p. 319,ISBN9783642592379
- ^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,ISBN9783540231585
- ^Beck, József(2014),Probabilistic Diophantine Approximation: Randomness in Lattice Point Counting,Springer, p. 28,ISBN9783319107417
- ^Andreescu, Titu; Feng, Zuming (2000),Mathematical Olympiads 1998-1999: Problems and Solutions from Around the World,Cambridge University Press, p. 125,ISBN9780883858035
- ^Frederickson, Greg N. (1997),Dissections: Plane and Fancy,Cambridge University Press, Cambridge, p.10,doi:10.1017/CBO9780511574917,ISBN978-0-521-57197-5,MR1735254
- ^Monsky, Paul(1990), "A conjecture of Stein on plane dissections",Mathematische Zeitschrift,205(4): 583–592,doi:10.1007/BF02571264,MR1082876,S2CID122009844
- ^Stein, Sherman;Szabó, Sandor (1994),Algebra and Tiling: Homomorphisms in the Service of Geometry,Carus Mathematical Monographs, vol. 25, Cambridge University Press,p. 130,ISBN9780883850282
- ^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,MR1949894