28:
39:
20:
419:
of a collection of line segments in the plane. 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.
265:-sided zonogon. At least three of the zonogon's vertices must be vertices of only one of the parallelograms in any such tiling. For instance, the regular octagon can be tiled by two squares and four 45° rhombi.
81:
is a zonogon if and only if it has an even number of sides. Thus, the square, regular hexagon, and regular octagon are all zonogons. The four-sided zonogons are the square, the
401:
237:
170:
668:
339:
263:
196:
129:
359:
310:
420:
Additionally, every planar cross-section through the center of a centrally-symmetric polyhedron (such as a zonohedron) is a zonogon.
697:Ábrego, Bernardo M.; Fernández-Merchant, Silvia (2002), "The unit distance problem for centrally symmetric convex polygons",
596:
699:
682:
558:
532:
503:
450:
485:
105:, able to tile the plane by translated copies of themselves, and all convex parallelogons have this form.
740:
364:
201:
134:
473:
If a regular polygon has an even number of sides, its center is a center of symmetry of the polygon
489:
580:
574:
674:
548:
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466:
440:
522:
722:
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8:
269:
66:
245:
178:
111:
647:
344:
295:
678:
651:
592:
554:
528:
499:
446:
242:.) In this tiling, there is a parallelogram for each pair of slopes of sides in the
708:
664:
631:
584:
58:
27:
718:
639:
602:
78:
42:
415:
and higher-dimensional zonotopes. As such, each zonogon can be generated as the
281:
62:
713:
550:
Mathematical
Olympiads 1998-1999: Problems and Solutions from Around the World
518:
734:
588:
524:
Probabilistic
Diophantine Approximation: Randomness in Lattice Point Counting
416:
172:
90:
673:, Carus Mathematical Monographs, vol. 25, Cambridge University Press,
102:
31:
619:
273:
635:
341:
pairs of vertices can be at unit distance from each other. There exist
65:. Equivalently, it is a convex polygon whose sides can be grouped into
19:
412:
82:
38:
50:
239:
86:
411:
Zonogons are the two-dimensional analogues of three-dimensional
670:
Algebra and Tiling: Homomorphisms in the
Service of Geometry
439:
Boltyanski, Vladimir; Martini, Horst; Soltan, P. S. (2012),
696:
438:
622:(1990), "A conjecture of Stein on plane dissections",
206:
139:
367:
347:
318:
298:
248:
204:
181:
137:
114:
69:
pairs with equal lengths and opposite orientations.
465:Young, John Wesley; Schwartz, Albert John (1915),
395:
353:
333:
304:
257:
231:
190:
164:
123:
16:Convex polygon with pairs of equal, parallel sides
579:, Cambridge University Press, Cambridge, p.
732:
546:
464:
222:
209:
155:
142:
572:
284:into an odd number of equal-area triangles.
96:
553:, Cambridge University Press, p. 125,
663:
484:
101:The four-sided and six-sided zonogons are
712:
37:
26:
18:
733:
618:
547:Andreescu, Titu; Feng, Zuming (2000),
442:Excursions into Combinatorial Geometry
277:
700:Discrete & Computational Geometry
612:
434:
432:
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511:
287:
13:
566:
429:
213:
146:
14:
752:
690:
657:
540:
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406:
396:{\displaystyle 2n-O({\sqrt {n}})}
280:) proved that no zonogon has an
232:{\displaystyle {\tbinom {n}{2}}}
165:{\displaystyle {\tbinom {n}{2}}}
175:. (For equilateral zonogons, a
131:-sided zonogon can be tiled by
34:by irregular hexagonal zonogons
573:Frederickson, Greg N. (1997),
390:
380:
1:
423:
576:Dissections: Plane and Fancy
7:
198:-sided one can be tiled by
72:
45:tiled by squares and rhombi
10:
757:
714:10.1007/s00454-002-2882-5
624:Mathematische Zeitschrift
445:, Springer, p. 319,
97:Tiling and equidissection
667:; Szabó, Sandor (1994),
589:10.1017/CBO9780511574917
527:, Springer, p. 28,
471:, H. Holt, p. 121,
312:-sided zonogon, at most
268:In a generalization of
397:
355:
335:
306:
259:
233:
192:
166:
125:
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35:
24:
403:unit-distance pairs.
398:
361:-sided zonogons with
356:
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260:
234:
193:
167:
126:
41:
30:
22:
494:, Springer, p.
365:
345:
334:{\displaystyle 2n-3}
316:
296:
246:
202:
179:
135:
112:
59:centrally-symmetric
636:10.1007/BF02571264
393:
351:
331:
302:
258:{\displaystyle 2n}
255:
229:
227:
191:{\displaystyle 2n}
188:
162:
160:
124:{\displaystyle 2n}
121:
47:
36:
25:
741:Types of polygons
598:978-0-521-57197-5
486:Alexandrov, A. D.
388:
354:{\displaystyle n}
305:{\displaystyle n}
220:
153:
23:Octagonal zonogon
748:
726:
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716:
694:
688:
687:
661:
655:
654:
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610:
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570:
564:
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509:
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491:Convex Polyhedra
482:
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360:
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288:Other properties
270:Monsky's theorem
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130:
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122:
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567:
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541:
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409:
383:
366:
363:
362:
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343:
342:
317:
314:
313:
297:
294:
293:
290:
274:Paul Monsky
247:
244:
243:
221:
208:
207:
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199:
180:
177:
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154:
141:
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132:
113:
110:
109:
99:
79:regular polygon
75:
43:Regular octagon
17:
12:
11:
5:
754:
744:
743:
728:
727:
707:(4): 467–473,
689:
683:
665:Stein, Sherman
656:
630:(4): 583–592,
611:
597:
565:
559:
539:
533:
510:
504:
477:
468:Plane Geometry
457:
451:
427:
425:
422:
408:
407:Related shapes
405:
392:
387:
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370:
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324:
321:
301:
289:
286:
282:equidissection
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224:
219:
216:
211:
187:
184:
173:parallelograms
157:
152:
149:
144:
120:
117:
98:
95:
91:parallelograms
74:
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63:convex polygon
15:
9:
6:
4:
3:
2:
753:
742:
739:
738:
736:
724:
720:
715:
710:
706:
702:
701:
693:
686:
684:9780883850282
680:
676:
672:
671:
666:
660:
653:
649:
645:
641:
637:
633:
629:
625:
621:
615:
608:
604:
600:
594:
590:
586:
582:
578:
577:
569:
562:
560:9780883858035
556:
552:
551:
543:
536:
534:9783319107417
530:
526:
525:
520:
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507:
505:9783540231585
501:
497:
493:
492:
487:
481:
474:
470:
469:
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454:
452:9783642592379
448:
444:
443:
435:
433:
428:
421:
418:
417:Minkowski sum
414:
404:
385:
377:
374:
371:
368:
348:
328:
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319:
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285:
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266:
252:
249:
241:
217:
214:
185:
182:
174:
150:
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118:
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106:
104:
103:parallelogons
94:
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88:
84:
80:
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68:
64:
60:
56:
52:
44:
40:
33:
29:
21:
704:
698:
692:
669:
659:
627:
623:
620:Monsky, Paul
614:
575:
568:
549:
542:
523:
519:Beck, József
513:
490:
480:
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467:
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441:
410:
291:
267:
107:
100:
76:
54:
48:
32:Tessellation
424:References
89:, and the
83:rectangles
652:122009844
413:zonohedra
375:−
326:−
735:Category
521:(2014),
488:(2005),
73:Examples
67:parallel
51:geometry
723:1949894
644:1082876
607:1735254
276: (
55:zonogon
721:
681:
675:p. 130
650:
642:
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449:
292:In an
240:rhombi
108:Every
87:rhombi
85:, the
648:S2CID
57:is a
679:ISBN
593:ISBN
555:ISBN
529:ISBN
500:ISBN
447:ISBN
278:1990
53:, a
709:doi
632:doi
628:205
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496:351
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