42:
803:
789:
33:
869:
855:
836:
822:
782:
796:
465:
479:
446:
453:
417:
848:
815:
439:
862:
829:
583:
472:
526:
486:
432:
567:
533:
505:
493:
519:
512:
547:
161:, repeating the Petrie dual operation twice returns to the original surface embedding. Unlike the usual dual graph (which is an embedding of a generally different graph in the same surface) the Petrie dual is an embedding of the same graph in a generally different surface.
219:
with eight vertices and twelve edges, embedded onto a sphere with six square faces) has four hexagonal faces, the equators of the cube. Topologically, it forms an embedding of the same graph onto a torus.
250:, {4,3}, has 8 vertices, 12 edges, and 4 skew hexagons, colored red, green, blue and orange here. With an Euler characteristic of 0, it can also be seen in the four hexagonal faces of the
139:
112:
61:. Seen from the solid's 5-fold symmetry axis it looks like a regular decagon. Every pair of consecutive sides belongs to one pentagon (but no triple does).
294:, {3,5}, has 12 vertices, 30 edges, and 6 skew decagonal faces, and Euler characteristic of −12, related to the hyperbolic tiling as type {10,5}
283:, {5,3}, has 20 vertices, 30 edges, and 6 skew decagonal faces, and Euler characteristic of −4, related to the hyperbolic tiling as type {10,3}
268:, {3,4}, has 6 vertices, 12 edges, and 4 skew hexagon faces. It has an Euler characteristic of −2, and has a mapping to the hyperbolic
1012:
944:
961:
613:
802:
788:
255:
181:
41:
269:
68:
939:, Encyclopedia of Mathematics and its Applications, vol. 92, Cambridge University Press, p. 192,
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32:
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51:
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117:
970:
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438:
251:
165:
984:
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861:
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908:
216:
142:
76:
900:
471:
209:
97:
84:
1006:
975:
959:
Jones, G. A.; Thornton, J. S. (1983), "Operations on maps, and outer automorphisms",
903:(2000), "Bridges between geometry and graph theory", in Gorini, Catherine A. (ed.),
525:
485:
431:
907:, MAA Notes, vol. 53, Washington, DC: Math. Assoc. America, pp. 174–194,
566:
532:
504:
492:
146:
55:
518:
201:
158:
511:
149:
representation of the embedding by twisting every edge of the embedding.
80:
624:, {5,5/2}, has 12 vertices, 30 edges, and 10 skew hexagon faces with an
58:
17:
665:, {5/2,3}, has 20 vertices, 30 edges, and 6 skew decagram faces with
639:, {5/2,5}, has 12 vertices, 30 edges, and 10 skew hexagon faces with
546:
231:, {3,3}, has 4 vertices, 6 edges, and 3 skew square faces. With an
996:
Octahedral symmetry is order 48, Coxeter number is 6, 48/(2×6)=4
83:
with all faces disks) is another embedded graph that has the
168:, and together generate the group of these operations.
223:
The regular maps obtained in this way are as follows.
164:
Surface duality and Petrie duality are two of the six
120:
100:
133:
106:
650:, {3,5/2}, has 12 vertices, 30 edges, and 6 skew
239:, of 1, it is topologically identical to the
8:
538:
94:, and the Petrie dual of an embedded graph
215:For example, the Petrie dual of a cube (a
974:
125:
119:
99:
672:
301:
933:McMullen, Peter; Schulte, Egon (2002),
883:
928:
926:
87:of the first embedding as its faces.
7:
891:
889:
887:
663:petrial great stellated dodecahedron
637:petrial small stellated dodecahedron
141:. It can be obtained from a signed
90:The Petrie dual is also called the
25:
612:There are also 4 petrials of the
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40:
31:
962:Journal of Combinatorial Theory
176:Applying the Petrie dual to a
1:
775:(one blue decagram outlined)
976:10.1016/0095-8956(83)90065-5
841:
808:
778:
763:
727:
710:
498:
458:
428:
408:
365:
336:
1029:
936:Abstract Regular Polytopes
622:petrial great dodecahedron
50:The Petrie polygon of the
770:
767:
764:
648:petrial great icosahedron
424:
421:
412:
409:
1013:Topological graph theory
614:Kepler–Poinsot polyhedra
270:order-4 hexagonal tiling
134:{\displaystyle G^{\pi }}
69:topological graph theory
721:{3,5/2} , {10/3,5/2}
674:Regular star petrials
135:
108:
184:. The number of skew
136:
109:
916:. See in particular
724:{5/2,3} , {10/3,3}
626:Euler characteristic
281:petrial dodecahedron
233:Euler characteristic
118:
98:
715:{5,5/2} , {6,5/2}
675:
304:
292:petrial icosahedron
229:petrial tetrahedron
673:
302:
266:petrial octahedron
178:regular polyhedron
131:
104:
875:
874:
718:{5/2,5} , {6,5}
610:
609:
303:Regular petrials
172:Regular polyhedra
166:Wilson operations
107:{\displaystyle G}
16:(Redirected from
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905:Geometry at work
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768:10 skew hexagons
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488:
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467:
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448:
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425:6 skew decagons
422:4 skew hexagons
419:
362:{3,5} , {10,5}
359:{5,3} , {10,3}
305:
252:hexagonal tiling
188:-gonal faces is
140:
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35:
21:
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897:Pisanski, Tomaž
895:
894:
885:
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706:
705:great stellated
704:
699:
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691:small stellated
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272:, as type {6,4}
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217:bipartite graph
174:
157:Like the usual
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143:rotation system
121:
116:
115:
114:may be denoted
96:
95:
85:Petrie polygons
65:
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47:
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413:3 skew squares
411:
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378:
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364:
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360:
357:
354:
353:{3,4} , {6,4}
351:
348:
347:{4,3} , {6,3}
345:
342:
341:{3,3} , {4,3}
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309:
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212:of the group.
210:coxeter number
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77:embedded graph
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969:(2): 93–103,
968:
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955:
952:
948:
946:9780521814966
942:
938:
937:
929:
927:
923:
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901:Randić, Milan
898:
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752:
748:
745:
741:
738:
734:
732:
728:
723:
720:
717:
714:
711:
707:dodecahedron
702:
695:
693:dodecahedron
688:
686:dodecahedron
681:
678:
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664:
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631:
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386:
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328:dodecahedron
325:
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60:
57:
53:
43:
34:
19:
992:
966:
965:, Series B,
960:
954:
935:
904:
757:
750:
743:
742:(12,30,10),
736:
735:(12,30,10),
700:icosahedron
666:
662:
655:
647:
640:
636:
629:
621:
611:
402:
395:
388:
381:
374:
333:icosahedron
313:tetrahedron
291:
280:
265:
248:petrial cube
247:
236:
228:
222:
214:
205:
197:
193:
189:
185:
175:
163:
156:
147:ribbon graph
91:
89:
72:
66:
52:dodecahedron
918:p. 181
756:(20,30,6),
749:(12,30,6),
654:faces with
401:(12,30,6),
394:(20,30,6),
323:octahedron
202:group order
182:regular map
180:produces a
73:Petrie dual
878:References
843:Animation
500:Animation
387:(6,12,4),
380:(8,12,4),
243:, {4,3}/2.
159:dual graph
153:Properties
773:decagrams
729:(v,e,f),
373:(4,6,3),
367:(v,e,f),
241:hemi-cube
127:π
1007:Category
652:decagram
632:, of -8.
542:figures
254:as type
196:, where
81:manifold
79:(on a 2-
985:0733017
913:1782654
771:6 skew
712:Symbol
703:Petrial
696:Petrial
689:Petrial
682:Petrial
658:of -12.
591:= {6,4}
575:= {6,3}
559:= {4,3}
557:{4,3}/2
540:Related
338:Symbol
331:Petrial
326:Petrial
321:Petrial
316:Petrial
311:Petrial
208:is the
200:is the
92:Petrial
59:decagon
18:Petrial
983:
943:
911:
810:Image
765:Faces
758:χ
751:χ
744:χ
737:χ
731:χ
669:of -4.
643:of -8.
603:{10,5}
597:{10,3}
460:Image
410:Faces
405:= −12
403:χ
396:χ
389:χ
382:χ
375:χ
369:χ
204:, and
75:of an
71:, the
760:= -4
753:= -12
698:great
684:great
679:Name
593:(4,0)
587:{6,4}
577:(2,0)
571:{6,3}
561:(2,0)
551:{4,3}
318:cube
308:Name
258:(2,0)
256:{6,3}
54:is a
941:ISBN
746:= -8
739:= -8
661:The
646:The
635:The
620:The
398:= −4
391:= −2
290:The
279:The
264:The
246:The
227:The
56:skew
971:doi
384:= 0
377:= 1
145:or
67:In
1009::
981:MR
979:,
967:35
925:^
909:MR
899:;
886:^
628:,
616::
555:=
235:,
192:/2
973::
920:.
667:χ
656:χ
641:χ
630:χ
605:3
599:5
589:3
573:3
553:3
355:3
349:4
343:3
298:.
296:3
287:.
285:5
276:.
274:3
261:.
237:χ
206:h
198:g
194:h
190:g
186:h
123:G
102:G
20:)
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