888:), in particular of finite subgroups. Under this connection, symmetry groups of centrally symmetric polytopes correspond to symmetry groups of the corresponding projective polytope, while symmetry groups of spherical polytopes without central symmetry correspond to symmetry groups of degree 2 projective polytopes (tilings that cover projective space twice), whose cover (corresponding to the adjunction of the connection) is a compound of two polytopes – the original polytope and its central inverse.
221:
140:
397:
of a non-centrally symmetric polyhedron, together with its central inverse (a compound of 2 polyhedra). This geometrizes the Galois connection at the level of finite subgroups of O(3) and PO(3), under which the adjunction is "union with central inverse". For example, the tetrahedron is not centrally
385:
Spherical polyhedra without central symmetry do not define a projective polyhedron, as the images of vertices, edges, and faces will overlap. In the language of tilings, the image in the projective plane is a degree 2 tiling, meaning that it covers the projective plane twice – rather than 2 faces in
250:
having some faces that pass through the center of symmetry. As these do not define spherical polyhedra (because they pass through the center, which does not map to a defined point on the sphere), they do not define projective polyhedra by the quotient map from 3-space (minus the origin) to the
398:
symmetric, and has 4 vertices, 6 edges, and 4 faces, and vertex figure 3.3.3 (3 triangles meeting at each vertex). Its image in the projective plane has 4 vertices, 6 edges (which intersect), and 4 faces (which overlap), covering the projective plane twice. The cover of this is the
731:+2) does not decompose as a product, and thus the symmetry group of the projective polytope is not simply the rotational symmetries of the spherical polytope, but rather a 2-to-1 quotient of the full symmetry group of the corresponding spherical polytope (the spherical group is a
357:
is the (spherical) cube. The hemi-cube has 4 vertices, 3 faces, and 6 edges, each of which is covered by 2 copies in the sphere, and accordingly the cube has 8 vertices, 6 faces, and 12 edges, while both these polyhedra have a 4.4.4 vertex figure (3 squares meeting at a vertex).
712:, which is the kernel on passage to projective space. The projective plane is non-orientable, and thus there is no distinct notion of "orientation-preserving isometries of a projective polyhedron", which is reflected in the equality PSO(3) = PO(3).
373:
symmetries of the spherical polyhedron, while the full symmetry group of the spherical polyhedron is the product of its rotation group (the symmetry group of the projective polyhedron) and the cyclic group of order 2,
270:, and can be realized as the quotient of the spherical cuboctahedron by the antipodal map. It is the only uniform (traditional) polyhedron that is projective – that is, the only uniform projective polyhedron that
545:
393:
including all spherical polyhedra (not necessarily centrally symmetric) if the classes are extended to include degree 2 tilings of the projective plane, whose covers are not polyhedra but rather the
699:
386:
the sphere corresponding to 1 face in the projective plane, covering it twice, each face in the sphere corresponds to a single face in the projective plane, accordingly covering it twice.
969:
328:
792:
1152:
1109:
1022:
996:
369:) of a projective polyhedron and covering spherical polyhedron are related: the symmetries of the projective polyhedron are naturally identified with the
708:
symmetry group of the covering spherical polyhedron; the full symmetry group of the spherical polyhedron is then just the direct product with
1490:
1468:
1446:
1421:
1267:
1233:
461:, PO, and conversely every finite subgroup of PO is the symmetry group of a projective polytope by taking the polytope given by images of a
483:
1225:
445:
of points, which is not a projective concept, and is infrequently addressed in the literature, but has been defined, such as in (
441:-dimensional projective space is somewhat trickier, because the usual definition of polytopes in Euclidean space requires taking
1310:
732:
433:
Projective polytopes can be defined in higher dimension as tessellations of projective space in one less dimension. Defining
915:) is a 2-to-1-cover, and hence has an analogous Galois connection between subgroups. However, while discrete subgroups of O(
1510:
1154:
is an isomorphism but the two groups are subsets of different spaces, hence the isomorphism rather than an equality. See (
709:
923:) correspond to symmetry groups of spherical and projective polytopes, corresponding geometrically to the covering map
402:– equivalently, the compound of two tetrahedra – which has 8 vertices, 12 edges, and 8 faces, and vertex figure 3.3.3.
330:
of the sphere to the projective plane, and under this map, projective polyhedra correspond to spherical polyhedra with
907:) is a 2-to-1 cover, and hence there is a Galois connection between binary polyhedral groups and polyhedral groups, O(
606:
458:
1477:
Vives, Gilberto
Calvillo; Mayo, Guillermo Lopez (1991). Susana Gómez; Jean Pierre Hennart; Richard A. Tapia (eds.).
926:
389:
The correspondence between projective polyhedra and centrally symmetric spherical polyhedra can be extended to a
794:
and is instead a proper (index 2) subgroup, so there is a distinct notion of orientation-preserving isometries.
334:– the 2-fold cover of a projective polyhedron is a centrally symmetric spherical polyhedron. Further, because a
288:
1347:
892:
271:
229:
151:
The best-known examples of projective polyhedra are the regular projective polyhedra, the quotients of the
1356:
347:
208:
On the other hand, the tetrahedron does not have central symmetry, so there is no "hemi-tetrahedron". See
1119:+1) are equal subsets of the target (namely, the whole space), hence the equality, while the induced map
838:-gon (in the projective line), and accordingly the quotient groups, subgroups of PO(2) and PSO(2) are Dih
738:
426:
is a "locally projective polytope", but is not a globally projective polyhedron, nor indeed tessellates
43:
1028:, and thus there is no corresponding "binary polytope" for which subgroups of Pin are symmetry groups.
92:
1, while spherical polyhedra have Euler characteristic 2. The qualifier "globally" is to contrast with
81:
applies to both spherical and projective geometries, so the term carries some ambiguity for polyhedra.
1203:
1037:
399:
259:
89:
1361:
1042:
394:
354:
339:
255:
225:
170:
152:
144:
112:
47:
1405:
1193:
1066:
854:
701:
so the group of projective isometries can be identified with the group of rotational isometries.
462:
442:
108:
101:
55:
430:
manifold, as it is not locally
Euclidean, but rather locally projective, as the name indicates.
1122:
1486:
1482:
1464:
1460:
1442:
1438:
1417:
1263:
1229:
877:
419:
411:
390:
247:
182:
78:
70:
1413:
1253:
1159:
1085:
1001:
1393:
1366:
1319:
1025:
869:(2), SO(2), O(2) – here going up to a 2-fold cover, rather than down to a 2-fold quotient.
331:
188:
120:
116:
74:
974:
201:
These can be obtained by taking the quotient of the associated spherical polyhedron by the
873:
477:
176:
1082:/equality distinction in this equation is because the context is the 2-to-1 quotient map
457:
The symmetry group of a projective polytope is a finite (hence discrete) subgroup of the
1207:
1259:
1249:
806:
735:
of the projective group). Further, in odd projective dimension (even vector dimension)
343:
237:
155:
1504:
1430:
1339:
1062:
267:
263:
243:
202:
1455:
McMullen, Peter; Schulte, Egon (December 2002), "6C. Projective
Regular Polytopes",
830:, these being subgroups of O(2) and SO(2), respectively. The projectivization of a 2
17:
335:
283:
220:
39:
1384:
Bracho, Javier (2000-02-01). "Regular projective polyhedra with planar faces II".
1079:
147:
is a regular projective polyhedron with 3 square faces, 6 edges, and 4 vertices.
1305:
346:), both the spherical and the corresponding projective polyhedra have the same
858:
366:
85:
1224:. CBMS regional conference series in mathematics (4). AMS Bookstore. p.
862:
228:
is a projective polyhedron, and the only uniform projective polyhedron that
163:
1408:; Smith, Derek Alan (2003-02-07), "3.7 The Projective or Elliptic Groups",
1324:
31:
1188:
Schulte, Egon; Weiss, Asia Ivic (2006), "5 Topological classification",
704:
Thus in particular the symmetry group of a projective polyhedron is the
1397:
1370:
1198:
423:
159:
1479:
Advances in numerical partial differential equations and optimization
51:
258:
is topologically a projective polyhedron, as can be verified by its
139:
540:{\displaystyle \mathbf {RP} ^{n}=\mathbf {P} (\mathbf {R} ^{n+1}),}
219:
138:
1065:, finite and discrete sets are identical – infinite sets have an
1338:
Arocha, Jorge L.; Bracho, Javier; Montejano, Luis (2000-02-01).
472:-dimensional real projective space is the projectivization of (
274:
in
Euclidean three-space as a uniform traditional polyhedron.
891:
These symmetry groups should be compared and contrasted with
77:, a synonym for "spherical polyhedron". However, the term
128:
1481:. Fifth United States-Mexico Workshop. SIAM. pp.
242:
Note that the prefix "hemi-" is also used to refer to
209:
124:
1459:(1st ed.), Cambridge University Press, pp.
1190:
Problems on
Polytopes, Their Groups, and Realizations
1125:
1088:
1004:
977:
929:
741:
609:
486:
291:
1306:"The construction of self-dual projective polyhedra"
69:, referring to the projective plane as (projective)
1340:"Regular projective polyhedra with planar faces I"
1146:
1103:
1016:
990:
963:
786:
693:
539:
353:For example, the 2-fold cover of the (projective)
322:
1162:) for an example of this distinction being made.
801: = 1 (polygons), the symmetries of a 2
694:{\displaystyle PO(2k+1)=PSO(2k+1)\cong SO(2k+1)}
205:(identifying opposite points on the sphere).
61:Projective polyhedra are also referred to as
8:
382:below for elaboration and other dimensions.
1155:
964:{\displaystyle S^{n}\to \mathbf {RP} ^{n},}
123:. This is elaborated and extended below in
1255:Noneuclidean tesselations and their groups
820:), with rotational group the cyclic group
323:{\displaystyle S^{2}\to \mathbf {RP} ^{2}}
158:, as well as two infinite classes of even
27:Plane tiling corresponding to a polyhedron
1360:
1323:
1197:
1124:
1087:
1003:
982:
976:
952:
944:
934:
928:
740:
608:
551:-dimensional projective space is denoted
547:so the projective orthogonal group of an
519:
514:
505:
496:
488:
485:
446:
314:
306:
296:
290:
254:Of these uniform hemipolyhedra, only the
212:below on how the tetrahedron is treated.
1220:Coxeter, Harold Scott Macdonald (1970).
468:The relevant dimensions are as follows:
1304:Archdeacon, Dan; Negami, Seiya (1993),
1180:
1054:
262:and visually obvious connection to the
857:of subgroups occurs for the square of
107:Non-overlapping projective polyhedra (
437:-dimensional projective polytopes in
7:
603:} decomposes as a product, and thus
97:
379:
362:
129:relation with traditional polyhedra
88:of the projective plane, they have
787:{\displaystyle PSO(2k)\neq PO(2k)}
197:Hemi-hosohedron, {2,2p}/2, p>=1
46:. These are projective analogs of
25:
1285:, 1969, Second edition, sec 21.3
420:Abstract polytope: Local topology
278:Relation with spherical polyhedra
210:relation with spherical polyhedra
125:relation with spherical polyhedra
948:
945:
515:
506:
492:
489:
310:
307:
194:Hemi-dihedron, {2p,2}/2, p>=1
96:projective polyhedra, which are
58:– tessellations of the toroids.
1311:Journal of Combinatorial Theory
1132:
1092:
971:there is no covering space of
940:
781:
772:
760:
751:
723: + 1 is odd, then O(
688:
673:
661:
646:
631:
616:
531:
510:
302:
1:
1412:, A K Peters, Ltd., pp.
710:reflection through the origin
1435:Geometry and the imagination
1410:On quaternions and octonions
418:projective polytopes" – see
834:-gon (in the circle) is an
459:projective orthogonal group
1527:
1457:Abstract Regular Polytopes
1433:; Cohn-Vossen, S. (1999),
235:
1437:, AMS Bookstore, p.
1147:{\displaystyle SO\to PSO}
414:, one instead refers to "
266:. It is 2-covered by the
1386:Aequationes Mathematicae
1348:Aequationes Mathematicae
1283:Introduction to geometry
893:binary polyhedral groups
1156:Conway & Smith 2003
1104:{\displaystyle O\to PO}
1017:{\displaystyle n\geq 2}
880:between subgroups of O(
86:cellular decompositions
50:– tessellations of the
1325:10.1006/jctb.1993.1059
1148:
1105:
1018:
992:
965:
884:) and subgroups of PO(
788:
695:
541:
348:abstract vertex figure
324:
233:
148:
63:elliptic tessellations
1149:
1106:
1019:
993:
991:{\displaystyle S^{n}}
966:
853:. Note that the same
789:
696:
542:
447:Vives & Mayo 1991
325:
236:Further information:
232:in Euclidean 3-space.
223:
142:
44:real projective plane
36:projective polyhedron
1511:Projective polyhedra
1123:
1086:
1038:Spherical polyhedron
1002:
975:
927:
739:
607:
595:+1) = SO(2
591:+1 is odd), then O(2
559:+1) = P(O(
484:
400:stellated octahedron
289:
260:Euler characteristic
90:Euler characteristic
18:Projective polyhedra
1406:Conway, John Horton
1208:2006math......8397S
1043:Toroidal polyhedron
1024:) as the sphere is
727:+1) = O(2
563:+1)) = O(
443:convex combinations
422:. For example, the
395:polyhedral compound
340:local homeomorphism
256:tetrahemihexahedron
226:tetrahemihexahedron
153:centrally symmetric
113:spherical polyhedra
48:spherical polyhedra
1398:10.1007/PL00000122
1371:10.1007/PL00000128
1297:General references
1222:Twisted honeycombs
1144:
1101:
1067:accumulation point
1014:
988:
961:
855:commutative square
784:
691:
537:
463:fundamental domain
412:abstract polytopes
410:In the context of
320:
282:There is a 2-to-1
251:projective plane.
234:
149:
102:abstract polyhedra
73:, by analogy with
56:toroidal polyhedra
1492:978-0-89871-269-8
1470:978-0-521-81496-6
1448:978-0-8218-1998-2
1423:978-1-56881-134-5
1269:978-0-12-465450-1
1235:978-0-8218-1653-0
1192:, pp. 9–13,
911:) → PO(
878:Galois connection
733:central extension
391:Galois connection
248:uniform polyhedra
183:Hemi-dodecahedron
111:1) correspond to
100:in the theory of
79:elliptic geometry
71:elliptic geometry
16:(Redirected from
1518:
1496:
1473:
1451:
1426:
1401:
1380:
1378:
1377:
1364:
1344:
1334:
1333:
1332:
1327:
1290:
1279:
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1239:
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1211:
1210:
1201:
1185:
1163:
1153:
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1107:
1102:
1076:
1070:
1059:
1026:simply connected
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1021:
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997:
995:
994:
989:
987:
986:
970:
968:
967:
962:
957:
956:
951:
939:
938:
903:) → O(
797:For example, in
793:
791:
790:
785:
700:
698:
697:
692:
587:+1 = 2
546:
544:
543:
538:
530:
529:
518:
509:
501:
500:
495:
476:+1)-dimensional
342:(in this case a
332:central symmetry
329:
327:
326:
321:
319:
318:
313:
301:
300:
189:Hemi-icosahedron
121:central symmetry
117:convex polyhedra
75:spherical tiling
67:elliptic tilings
21:
1526:
1525:
1521:
1520:
1519:
1517:
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1501:
1500:
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1375:
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1362:10.1.1.498.9945
1342:
1337:
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1303:
1299:
1294:
1293:
1280:
1276:
1270:
1250:Magnus, Wilhelm
1248:
1247:
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1219:
1218:
1214:
1187:
1186:
1182:
1177:
1172:
1167:
1166:
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1120:
1084:
1083:
1077:
1073:
1060:
1056:
1051:
1034:
1000:
999:
978:
973:
972:
943:
930:
925:
924:
898:
874:lattice theorem
872:Lastly, by the
868:
852:
843:
829:
815:
737:
736:
605:
604:
513:
487:
482:
481:
478:Euclidean space
465:for the group.
455:
408:
406:Generalizations
305:
292:
287:
286:
280:
240:
218:
177:Hemi-octahedron
156:Platonic solids
137:
115:(equivalently,
34:, a (globally)
28:
23:
22:
15:
12:
11:
5:
1524:
1522:
1514:
1513:
1503:
1502:
1498:
1497:
1491:
1474:
1469:
1452:
1447:
1431:Hilbert, David
1427:
1422:
1402:
1392:(1): 160–176.
1381:
1335:
1318:(1): 122–131,
1300:
1298:
1295:
1292:
1291:
1274:
1268:
1262:, p. 65,
1260:Academic Press
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1234:
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866:
865:– Spin(2), Pin
848:
839:
824:
810:
807:dihedral group
783:
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777:
774:
771:
768:
765:
762:
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747:
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690:
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453:Symmetry group
451:
407:
404:
380:symmetry group
363:symmetry group
344:local isometry
317:
312:
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279:
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238:Hemipolyhedron
217:
214:
199:
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26:
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14:
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2:
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1027:
1011:
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1005:
983:
979:
958:
953:
935:
931:
922:
918:
914:
910:
906:
902:
895:– just as Pin
894:
889:
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851:
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842:
837:
833:
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401:
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361:Further, the
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341:
337:
333:
315:
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285:
277:
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273:
269:
268:cuboctahedron
265:
264:Roman surface
261:
257:
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249:
245:
244:hemipolyhedra
239:
231:
227:
222:
216:Hemipolyhedra
215:
213:
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206:
204:
203:antipodal map
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49:
45:
41:
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33:
19:
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1385:
1374:. Retrieved
1355:(1): 55–73.
1352:
1346:
1329:, retrieved
1315:
1314:, Series B,
1309:
1289:, p. 386-388
1287:Regular maps
1286:
1282:
1277:
1254:
1244:
1221:
1215:
1189:
1183:
1116:
1115:+1) and PO(2
1112:
1074:
1061:Since PO is
1057:
920:
916:
912:
908:
904:
900:
890:
885:
881:
871:
849:
845:
840:
835:
831:
826:
821:
817:
812:
805:-gon is the
802:
798:
796:
728:
724:
720:
716:
714:
705:
703:
600:
596:
592:
588:
584:
583:is even (so
580:
576:
574:
568:
564:
560:
556:
548:
473:
469:
467:
456:
438:
434:
432:
427:
415:
409:
388:
384:
375:
370:
360:
352:
336:covering map
284:covering map
281:
253:
246:, which are
241:
207:
200:
150:
106:
93:
83:
66:
62:
60:
40:tessellation
35:
29:
1080:isomorphism
876:there is a
816:(of order 4
599:+1)×{±
1376:2010-04-15
1331:2010-04-15
1170:References
859:Spin group
706:rotational
367:isometries
1357:CiteSeerX
1281:Coxeter,
1175:Footnotes
1133:→
1093:→
1009:≥
941:→
919:) and PO(
863:Pin group
764:≠
665:≅
355:hemi-cube
303:→
191:, {3,5}/2
185:, {5,3}/2
179:, {3,4}/2
173:, {4,3}/2
171:Hemi-cube
164:hosohedra
145:hemi-cube
1505:Category
1252:(1974),
1032:See also
371:rotation
272:immerses
230:immerses
135:Examples
32:geometry
1461:162–165
1204:Bibcode
1111:– PSO(2
1063:compact
424:11-cell
416:locally
378:}. See
160:dihedra
119:) with
109:density
98:defined
94:locally
42:of the
1489:
1467:
1445:
1420:
1359:
1266:
1232:
567:+1)/{±
54:– and
52:sphere
1483:43–49
1343:(PDF)
1194:arXiv
1160:p. 34
1049:Notes
998:(for
338:is a
38:is a
1487:ISBN
1465:ISBN
1443:ISBN
1418:ISBN
1264:ISBN
1230:ISBN
1078:The
861:and
844:and
365:(of
224:The
162:and
143:The
127:and
1439:147
1394:doi
1367:doi
1320:doi
809:Dih
715:If
575:If
555:PO(
449:).
428:any
84:As
65:or
30:In
1507::
1485:.
1463:,
1441:,
1416:,
1414:34
1390:59
1388:.
1365:.
1353:59
1351:.
1345:.
1316:59
1308:,
1258:,
1228:.
1226:11
1202:,
1158:,
719:=2
579:=2
571:}.
480:,
374:{±
350:.
166::
131:.
104:.
1495:.
1400:.
1396::
1379:.
1369::
1322::
1238:.
1206::
1196::
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1139:S
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1012:2
1006:n
984:n
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897:±
886:n
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867:+
850:r
846:C
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527:1
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511:(
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503:=
498:n
493:P
490:R
474:n
470:n
439:n
435:k
376:I
316:2
311:P
308:R
298:2
294:S
20:)
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