24:
146:
116:
In three dimensions, it is not possible for a geometrically chiral polytope to have finitely many finite faces. For instance, the snub cube is vertex-transitive, but its flags have more than two orbits, and it is neither edge-transitive nor face-transitive, so it is not symmetric enough to meet the
129:, provide another interesting type of near-miss: they have two orbits of flags, but are mirror-symmetric, and not every adjacent pair of flags belongs to different orbits. However, despite the nonexistence of finite chiral three-dimensional polyhedra, there exist infinite three-dimensional chiral
103:
Geometrically chiral polytopes are relatively exotic compared to the more ordinary regular polytopes. It is not possible for a geometrically chiral polytope to be convex, and many geometrically chiral polytopes of note are
86:
For the purposes of this definition, the symmetry group of a polytope may be defined in either of two different ways: it can refer to the symmetries of a polytope as a geometric object (in which case the polytope is called
83:, as each vertex, edge, or face must be represented by flags in both orbits; however, it cannot be mirror-symmetric, as every mirror symmetry of the polytope would exchange some pair of adjacent flags.
454:
212:, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 4, Providence, RI: American Mathematical Society, pp. 493–516,
95:). Chirality is meaningful for either type of symmetry but the two definitions classify different polytopes as being chiral or nonchiral.
239:
414:
365:
497:
286:
44:
130:
16:
This article is about polytopes with full rotational symmetry. For polytopes lacking mirror symmetry see, see
118:
91:) or it can refer to the symmetries of the polytope as a combinatorial structure (the automorphisms of an
43:
is a polytope that is as symmetric as possible without being mirror-symmetric, formalized in terms of the
176:
In four dimensions, there are a geometrically chiral finite polytopes. One example is Roli's cube, a
126:
542:
537:
479:
400:
374:
336:
266:
449:
360:
92:
72:
36:
506:
463:
423:
384:
295:
248:
23:
520:
475:
437:
396:
262:
217:
516:
471:
433:
392:
258:
213:
205:
80:
76:
64:
52:
313:
Bracho, Javier; Hubard, Isabel; Pellicer, Daniel (2014), "A Finite Chiral 4-polytope in
63:
The more technical definition of a chiral polytope is a polytope that has two orbits of
445:
68:
48:
145:
531:
177:
122:
105:
483:
404:
270:
492:
234:
201:
363:; Schulte, Egon; Weiss, Asia Ivić (2007), "Semisymmetric graphs from polytopes",
28:
428:
388:
299:
284:
Pellicer, Daniel (2012). "Developments and open problems on chiral polytopes".
31:
under its automorphism group form two orbits, colored here in black and yellow.
511:
253:
412:
Hubard, Isabel; Weiss, Asia Ivić (2005), "Self-duality of chiral polytopes",
17:
467:
71:, with adjacent flags in different orbits. This implies that it must be
210:
Applied
Geometry and Discrete Mathematics (The Victor Klee Festschrift)
379:
181:
341:
22:
204:; Weiss, Asia Ivić (1991), "Chiral polytopes", in Gritzmann, P.;
140:
157:
237:(2004), "Chiral polyhedra in ordinary space. I",
452:(2008), "Constructions for chiral polytopes",
8:
229:
227:
455:Journal of the London Mathematical Society
510:
493:"Cayley graphs and symmetric 4-polytopes"
427:
378:
340:
252:
491:Monson, Barry; Ivić Weiss, Asia (2008),
193:
319:Discrete & Computational Geometry
7:
117:formal definition of chirality. The
240:Discrete and Computational Geometry
133:of types {4,6}, {6,4}, and {6,6}.
14:
144:
415:Journal of Combinatorial Theory
366:Journal of Combinatorial Theory
99:Geometrically chiral polytopes
1:
498:Ars Mathematica Contemporanea
287:Ars Mathematica Contemporanea
121:and their duals, such as the
559:
429:10.1016/j.jcta.2004.11.012
389:10.1016/j.jcta.2006.06.007
300:10.26493/1855-3974.183.8a2
15:
512:10.26493/1855-3974.79.919
254:10.1007/s00454-004-0843-x
180:on the skeleton of the
51:of the polytope on its
331:Monson, Barry (2021),
119:quasiregular polyhedra
32:
26:
127:rhombic dodecahedron
89:geometrically chiral
468:10.1112/jlms/jdm093
112:In three dimensions
69:group of symmetries
448:; Hubard, Isabel;
156:. You can help by
137:In four dimensions
37:abstract polytopes
33:
458:, Second Series,
174:
173:
93:abstract polytope
73:vertex-transitive
550:
523:
514:
486:
440:
431:
407:
382:
346:
345:
344:
328:
322:
321:
316:
310:
304:
303:
281:
275:
273:
256:
231:
222:
220:
198:
169:
166:
148:
141:
35:In the study of
558:
557:
553:
552:
551:
549:
548:
547:
528:
527:
490:
450:Pisanski, TomaĹľ
446:Conder, Marston
444:
411:
361:Pisanski, TomaĹľ
359:Monson, Barry;
358:
355:
353:Further reading
350:
349:
330:
329:
325:
314:
312:
311:
307:
283:
282:
278:
233:
232:
225:
200:
199:
195:
190:
170:
164:
161:
154:needs expansion
139:
114:
101:
81:face-transitive
77:edge-transitive
61:
41:chiral polytope
21:
12:
11:
5:
556:
554:
546:
545:
540:
530:
529:
526:
525:
505:(2): 185–205,
488:
462:(1): 115–129,
442:
422:(1): 128–136,
409:
373:(3): 421–435,
354:
351:
348:
347:
333:On Roli's Cube
323:
305:
294:(2): 333–354.
276:
223:
192:
191:
189:
186:
172:
171:
151:
149:
138:
135:
131:skew polyhedra
113:
110:
100:
97:
60:
57:
49:symmetry group
13:
10:
9:
6:
4:
3:
2:
555:
544:
541:
539:
536:
535:
533:
522:
518:
513:
508:
504:
500:
499:
494:
489:
485:
481:
477:
473:
469:
465:
461:
457:
456:
451:
447:
443:
439:
435:
430:
425:
421:
417:
416:
410:
406:
402:
398:
394:
390:
386:
381:
376:
372:
368:
367:
362:
357:
356:
352:
343:
338:
334:
327:
324:
320:
309:
306:
301:
297:
293:
289:
288:
280:
277:
272:
268:
264:
260:
255:
250:
246:
242:
241:
236:
235:Schulte, Egon
230:
228:
224:
219:
215:
211:
207:
206:Sturmfels, B.
203:
202:Schulte, Egon
197:
194:
187:
185:
183:
179:
178:skew polytope
168:
159:
155:
152:This section
150:
147:
143:
142:
136:
134:
132:
128:
124:
123:cuboctahedron
120:
111:
109:
107:
98:
96:
94:
90:
84:
82:
78:
74:
70:
66:
58:
56:
54:
50:
46:
42:
38:
30:
27:The flags of
25:
19:
502:
496:
459:
453:
419:
418:, Series A,
413:
380:math/0606469
370:
369:, Series A,
364:
332:
326:
318:
308:
291:
285:
279:
247:(1): 55–99,
244:
238:
209:
196:
175:
165:January 2024
162:
158:adding to it
153:
115:
102:
88:
85:
62:
40:
34:
29:Heawood map
532:Categories
342:2102.08796
188:References
67:under its
59:Definition
543:Polytopes
538:Chirality
18:Chirality
484:14658184
405:10203794
271:13098983
208:(eds.),
125:and the
521:2466196
476:2389920
438:2144859
397:2310743
263:2060817
218:1116373
47:of the
519:
482:
474:
436:
403:
395:
269:
261:
216:
182:4-cube
79:, and
45:action
480:S2CID
401:S2CID
375:arXiv
337:arXiv
267:S2CID
65:flags
53:flags
184:.
106:skew
39:, a
507:doi
464:doi
424:doi
420:111
385:doi
371:114
317:",
296:doi
249:doi
160:.
534::
517:MR
515:,
501:,
495:,
478:,
472:MR
470:,
460:77
434:MR
432:,
399:,
393:MR
391:,
383:,
335:,
290:.
265:,
259:MR
257:,
245:32
243:,
226:^
214:MR
108:.
75:,
55:.
524:.
509::
503:1
487:.
466::
441:.
426::
408:.
387::
377::
339::
315:â„ť
302:.
298::
292:5
274:.
251::
221:.
167:)
163:(
20:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
