877:), 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.
210:
129:
386:
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
374:
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
239:
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
387:
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
720:+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
346:
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).
701:, 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).
362:
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,
259:, 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
534:
382:
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
688:
375:
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.
958:
317:
781:
1141:
1098:
1011:
985:
358:) of a projective polyhedron and covering spherical polyhedron are related: the symmetries of the projective polyhedron are naturally identified with the
697:
symmetry group of the covering spherical polyhedron; the full symmetry group of the spherical polyhedron is then just the direct product with
1479:
1457:
1435:
1410:
1256:
1222:
450:, 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
472:
1214:
434:
of points, which is not a projective concept, and is infrequently addressed in the literature, but has been defined, such as in (
430:-dimensional projective space is somewhat trickier, because the usual definition of polytopes in Euclidean space requires taking
1299:
721:
422:
Projective polytopes can be defined in higher dimension as tessellations of projective space in one less dimension. Defining
904:) is a 2-to-1-cover, and hence has an analogous Galois connection between subgroups. However, while discrete subgroups of O(
1499:
1143:
is an isomorphism but the two groups are subsets of different spaces, hence the isomorphism rather than an equality. See (
698:
912:) correspond to symmetry groups of spherical and projective polytopes, corresponding geometrically to the covering map
391:– equivalently, the compound of two tetrahedra – which has 8 vertices, 12 edges, and 8 faces, and vertex figure 3.3.3.
319:
of the sphere to the projective plane, and under this map, projective polyhedra correspond to spherical polyhedra with
896:) is a 2-to-1 cover, and hence there is a Galois connection between binary polyhedral groups and polyhedral groups, O(
595:
447:
1466:
Vives, Gilberto
Calvillo; Mayo, Guillermo Lopez (1991). Susana Gómez; Jean Pierre Hennart; Richard A. Tapia (eds.).
915:
378:
The correspondence between projective polyhedra and centrally symmetric spherical polyhedra can be extended to a
783:
and is instead a proper (index 2) subgroup, so there is a distinct notion of orientation-preserving isometries.
323:– the 2-fold cover of a projective polyhedron is a centrally symmetric spherical polyhedron. Further, because a
277:
1336:
881:
260:
218:
140:
The best-known examples of projective polyhedra are the regular projective polyhedra, the quotients of the
1345:
336:
197:
On the other hand, the tetrahedron does not have central symmetry, so there is no "hemi-tetrahedron". See
1108:+1) are equal subsets of the target (namely, the whole space), hence the equality, while the induced map
827:-gon (in the projective line), and accordingly the quotient groups, subgroups of PO(2) and PSO(2) are Dih
727:
415:
is a "locally projective polytope", but is not a globally projective polyhedron, nor indeed tessellates
32:
1017:, and thus there is no corresponding "binary polytope" for which subgroups of Pin are symmetry groups.
81:
1, while spherical polyhedra have Euler characteristic 2. The qualifier "globally" is to contrast with
70:
applies to both spherical and projective geometries, so the term carries some ambiguity for polyhedra.
1192:
1026:
388:
248:
78:
1350:
1031:
383:
343:
328:
244:
214:
159:
141:
133:
101:
36:
1394:
1182:
1055:
843:
690:
so the group of projective isometries can be identified with the group of rotational isometries.
451:
431:
97:
90:
44:
419:
manifold, as it is not locally
Euclidean, but rather locally projective, as the name indicates.
1111:
1475:
1471:
1453:
1449:
1431:
1427:
1406:
1252:
1218:
866:
408:
400:
379:
236:
171:
67:
59:
1402:
1242:
1148:
1074:
990:
1382:
1355:
1308:
1014:
858:(2), SO(2), O(2) – here going up to a 2-fold cover, rather than down to a 2-fold quotient.
320:
177:
109:
105:
63:
963:
190:
These can be obtained by taking the quotient of the associated spherical polyhedron by the
862:
466:
165:
1071:/equality distinction in this equation is because the context is the 2-to-1 quotient map
446:
The symmetry group of a projective polytope is a finite (hence discrete) subgroup of the
1196:
1248:
1238:
795:
724:
of the projective group). Further, in odd projective dimension (even vector dimension)
332:
226:
144:
1493:
1419:
1328:
1051:
256:
252:
232:
191:
1444:
McMullen, Peter; Schulte, Egon (December 2002), "6C. Projective
Regular Polytopes",
819:, these being subgroups of O(2) and SO(2), respectively. The projectivization of a 2
324:
272:
209:
28:
1373:
Bracho, Javier (2000-02-01). "Regular projective polyhedra with planar faces II".
1068:
136:
is a regular projective polyhedron with 3 square faces, 6 edges, and 4 vertices.
1294:
335:), both the spherical and the corresponding projective polyhedra have the same
847:
355:
74:
1213:. CBMS regional conference series in mathematics (4). AMS Bookstore. p.
851:
217:
is a projective polyhedron, and the only uniform projective polyhedron that
152:
1397:; Smith, Derek Alan (2003-02-07), "3.7 The Projective or Elliptic Groups",
1313:
20:
1177:
Schulte, Egon; Weiss, Asia Ivic (2006), "5 Topological classification",
693:
Thus in particular the symmetry group of a projective polyhedron is the
1386:
1359:
1187:
412:
148:
1468:
Advances in numerical partial differential equations and optimization
40:
247:
is topologically a projective polyhedron, as can be verified by its
128:
529:{\displaystyle \mathbf {RP} ^{n}=\mathbf {P} (\mathbf {R} ^{n+1}),}
208:
127:
1054:, finite and discrete sets are identical – infinite sets have an
1327:
Arocha, Jorge L.; Bracho, Javier; Montejano, Luis (2000-02-01).
461:-dimensional real projective space is the projectivization of (
263:
in
Euclidean three-space as a uniform traditional polyhedron.
880:
These symmetry groups should be compared and contrasted with
66:, a synonym for "spherical polyhedron". However, the term
117:
1470:. Fifth United States-Mexico Workshop. SIAM. pp.
231:
Note that the prefix "hemi-" is also used to refer to
198:
113:
1448:(1st ed.), Cambridge University Press, pp.
1179:
Problems on
Polytopes, Their Groups, and Realizations
1114:
1077:
993:
966:
918:
730:
598:
475:
280:
1295:"The construction of self-dual projective polyhedra"
58:, referring to the projective plane as (projective)
1329:"Regular projective polyhedra with planar faces I"
1135:
1092:
1005:
979:
952:
775:
682:
528:
342:For example, the 2-fold cover of the (projective)
311:
1151:) for an example of this distinction being made.
790: = 1 (polygons), the symmetries of a 2
683:{\displaystyle PO(2k+1)=PSO(2k+1)\cong SO(2k+1)}
194:(identifying opposite points on the sphere).
50:Projective polyhedra are also referred to as
8:
371:below for elaboration and other dimensions.
1144:
953:{\displaystyle S^{n}\to \mathbf {RP} ^{n},}
112:. This is elaborated and extended below in
1244:Noneuclidean tesselations and their groups
809:), with rotational group the cyclic group
312:{\displaystyle S^{2}\to \mathbf {RP} ^{2}}
147:, as well as two infinite classes of even
16:Plane tiling corresponding to a polyhedron
1349:
1312:
1186:
1113:
1076:
992:
971:
965:
941:
933:
923:
917:
729:
597:
540:-dimensional projective space is denoted
536:so the projective orthogonal group of an
508:
503:
494:
485:
477:
474:
435:
303:
295:
285:
279:
243:Of these uniform hemipolyhedra, only the
201:below on how the tetrahedron is treated.
1209:Coxeter, Harold Scott Macdonald (1970).
457:The relevant dimensions are as follows:
1293:Archdeacon, Dan; Negami, Seiya (1993),
1169:
1043:
251:and visually obvious connection to the
846:of subgroups occurs for the square of
96:Non-overlapping projective polyhedra (
426:-dimensional projective polytopes in
7:
592:} decomposes as a product, and thus
86:
368:
351:
118:relation with traditional polyhedra
77:of the projective plane, they have
776:{\displaystyle PSO(2k)\neq PO(2k)}
186:Hemi-hosohedron, {2,2p}/2, p>=1
35:. These are projective analogs of
14:
1274:, 1969, Second edition, sec 21.3
409:Abstract polytope: Local topology
267:Relation with spherical polyhedra
199:relation with spherical polyhedra
114:relation with spherical polyhedra
937:
934:
504:
495:
481:
478:
299:
296:
183:Hemi-dihedron, {2p,2}/2, p>=1
85:projective polyhedra, which are
47:– tessellations of the toroids.
1300:Journal of Combinatorial Theory
1121:
1081:
960:there is no covering space of
929:
770:
761:
749:
740:
712: + 1 is odd, then O(
677:
662:
650:
635:
620:
605:
520:
499:
291:
1:
1401:, A K Peters, Ltd., pp.
699:reflection through the origin
1424:Geometry and the imagination
1399:On quaternions and octonions
407:projective polytopes" – see
823:-gon (in the circle) is an
448:projective orthogonal group
1516:
1446:Abstract Regular Polytopes
1422:; Cohn-Vossen, S. (1999),
224:
1426:, AMS Bookstore, p.
1136:{\displaystyle SO\to PSO}
403:, one instead refers to "
255:. It is 2-covered by the
1375:Aequationes Mathematicae
1337:Aequationes Mathematicae
1272:Introduction to geometry
882:binary polyhedral groups
1145:Conway & Smith 2003
1093:{\displaystyle O\to PO}
1006:{\displaystyle n\geq 2}
869:between subgroups of O(
75:cellular decompositions
39:– tessellations of the
1314:10.1006/jctb.1993.1059
1137:
1094:
1007:
981:
954:
873:) and subgroups of PO(
777:
684:
530:
337:abstract vertex figure
313:
222:
137:
52:elliptic tessellations
1138:
1095:
1008:
982:
980:{\displaystyle S^{n}}
955:
842:. Note that the same
778:
685:
531:
436:Vives & Mayo 1991
314:
225:Further information:
221:in Euclidean 3-space.
212:
131:
33:real projective plane
25:projective polyhedron
1500:Projective polyhedra
1112:
1075:
1027:Spherical polyhedron
991:
964:
916:
728:
596:
584:+1) = SO(2
580:+1 is odd), then O(2
548:+1) = P(O(
473:
389:stellated octahedron
278:
249:Euler characteristic
79:Euler characteristic
1395:Conway, John Horton
1197:2006math......8397S
1032:Toroidal polyhedron
1013:) as the sphere is
716:+1) = O(2
552:+1)) = O(
432:convex combinations
411:. For example, the
384:polyhedral compound
329:local homeomorphism
245:tetrahemihexahedron
215:tetrahemihexahedron
142:centrally symmetric
102:spherical polyhedra
37:spherical polyhedra
1387:10.1007/PL00000122
1360:10.1007/PL00000128
1286:General references
1211:Twisted honeycombs
1133:
1090:
1056:accumulation point
1003:
977:
950:
844:commutative square
773:
680:
526:
452:fundamental domain
401:abstract polytopes
399:In the context of
309:
271:There is a 2-to-1
240:projective plane.
223:
138:
91:abstract polyhedra
62:, by analogy with
45:toroidal polyhedra
1481:978-0-89871-269-8
1459:978-0-521-81496-6
1437:978-0-8218-1998-2
1412:978-1-56881-134-5
1258:978-0-12-465450-1
1224:978-0-8218-1653-0
1181:, pp. 9–13,
900:) → PO(
867:Galois connection
722:central extension
380:Galois connection
237:uniform polyhedra
172:Hemi-dodecahedron
100:1) correspond to
89:in the theory of
68:elliptic geometry
60:elliptic geometry
1507:
1485:
1462:
1440:
1415:
1390:
1369:
1367:
1366:
1353:
1333:
1323:
1322:
1321:
1316:
1279:
1268:
1262:
1261:
1235:
1229:
1228:
1206:
1200:
1199:
1190:
1174:
1152:
1142:
1140:
1139:
1134:
1099:
1097:
1096:
1091:
1065:
1059:
1048:
1015:simply connected
1012:
1010:
1009:
1004:
986:
984:
983:
978:
976:
975:
959:
957:
956:
951:
946:
945:
940:
928:
927:
892:) → O(
786:For example, in
782:
780:
779:
774:
689:
687:
686:
681:
576:+1 = 2
535:
533:
532:
527:
519:
518:
507:
498:
490:
489:
484:
465:+1)-dimensional
331:(in this case a
321:central symmetry
318:
316:
315:
310:
308:
307:
302:
290:
289:
178:Hemi-icosahedron
110:central symmetry
106:convex polyhedra
64:spherical tiling
56:elliptic tilings
1515:
1514:
1510:
1509:
1508:
1506:
1505:
1504:
1490:
1489:
1488:
1482:
1465:
1460:
1443:
1438:
1418:
1413:
1393:
1372:
1364:
1362:
1351:10.1.1.498.9945
1331:
1326:
1319:
1317:
1292:
1288:
1283:
1282:
1269:
1265:
1259:
1239:Magnus, Wilhelm
1237:
1236:
1232:
1225:
1208:
1207:
1203:
1176:
1175:
1171:
1166:
1161:
1156:
1155:
1110:
1109:
1073:
1072:
1066:
1062:
1049:
1045:
1040:
1023:
989:
988:
967:
962:
961:
932:
919:
914:
913:
887:
863:lattice theorem
861:Lastly, by the
857:
841:
832:
818:
804:
726:
725:
594:
593:
502:
476:
471:
470:
467:Euclidean space
454:for the group.
444:
397:
395:Generalizations
294:
281:
276:
275:
269:
229:
207:
166:Hemi-octahedron
145:Platonic solids
126:
104:(equivalently,
23:, a (globally)
17:
12:
11:
5:
1513:
1511:
1503:
1502:
1492:
1491:
1487:
1486:
1480:
1463:
1458:
1441:
1436:
1420:Hilbert, David
1416:
1411:
1391:
1381:(1): 160–176.
1370:
1324:
1307:(1): 122–131,
1289:
1287:
1284:
1281:
1280:
1263:
1257:
1251:, p. 65,
1249:Academic Press
1230:
1223:
1201:
1188:math/0608397v1
1168:
1167:
1165:
1162:
1160:
1157:
1154:
1153:
1132:
1129:
1126:
1123:
1120:
1117:
1089:
1086:
1083:
1080:
1060:
1042:
1041:
1039:
1036:
1035:
1034:
1029:
1022:
1019:
1002:
999:
996:
974:
970:
949:
944:
939:
936:
931:
926:
922:
885:
855:
854:– Spin(2), Pin
837:
828:
813:
799:
796:dihedral group
772:
769:
766:
763:
760:
757:
754:
751:
748:
745:
742:
739:
736:
733:
679:
676:
673:
670:
667:
664:
661:
658:
655:
652:
649:
646:
643:
640:
637:
634:
631:
628:
625:
622:
619:
616:
613:
610:
607:
604:
601:
562:
561:
525:
522:
517:
514:
511:
506:
501:
497:
493:
488:
483:
480:
443:
442:Symmetry group
440:
396:
393:
369:symmetry group
352:symmetry group
333:local isometry
306:
301:
298:
293:
288:
284:
268:
265:
227:Hemipolyhedron
206:
203:
188:
187:
184:
181:
175:
169:
163:
125:
122:
15:
13:
10:
9:
6:
4:
3:
2:
1512:
1501:
1498:
1497:
1495:
1483:
1477:
1473:
1469:
1464:
1461:
1455:
1451:
1447:
1442:
1439:
1433:
1429:
1425:
1421:
1417:
1414:
1408:
1404:
1400:
1396:
1392:
1388:
1384:
1380:
1376:
1371:
1361:
1357:
1352:
1347:
1343:
1339:
1338:
1330:
1325:
1315:
1310:
1306:
1302:
1301:
1296:
1291:
1290:
1285:
1277:
1273:
1267:
1264:
1260:
1254:
1250:
1246:
1245:
1240:
1234:
1231:
1226:
1220:
1216:
1212:
1205:
1202:
1198:
1194:
1189:
1184:
1180:
1173:
1170:
1163:
1158:
1150:
1146:
1130:
1127:
1124:
1118:
1115:
1107:
1103:
1087:
1084:
1078:
1070:
1064:
1061:
1057:
1053:
1047:
1044:
1037:
1033:
1030:
1028:
1025:
1024:
1020:
1018:
1016:
1000:
997:
994:
972:
968:
947:
942:
924:
920:
911:
907:
903:
899:
895:
891:
884:– just as Pin
883:
878:
876:
872:
868:
864:
859:
853:
849:
845:
840:
836:
831:
826:
822:
817:
812:
808:
803:
797:
793:
789:
784:
767:
764:
758:
755:
752:
746:
743:
737:
734:
731:
723:
719:
715:
711:
707:
702:
700:
696:
691:
674:
671:
668:
665:
659:
656:
653:
647:
644:
641:
638:
632:
629:
626:
623:
617:
614:
611:
608:
602:
599:
591:
587:
583:
579:
575:
571:
567:
559:
555:
551:
547:
543:
542:
541:
539:
523:
515:
512:
509:
491:
486:
468:
464:
460:
455:
453:
449:
441:
439:
437:
433:
429:
425:
420:
418:
414:
410:
406:
402:
394:
392:
390:
385:
381:
376:
372:
370:
366:
361:
357:
353:
350:Further, the
348:
345:
340:
338:
334:
330:
326:
322:
304:
286:
282:
274:
266:
264:
262:
258:
257:cuboctahedron
254:
253:Roman surface
250:
246:
241:
238:
234:
233:hemipolyhedra
228:
220:
216:
211:
205:Hemipolyhedra
204:
202:
200:
195:
193:
192:antipodal map
185:
182:
179:
176:
173:
170:
167:
164:
161:
158:
157:
156:
154:
150:
146:
143:
135:
130:
123:
121:
119:
115:
111:
107:
103:
99:
94:
92:
88:
84:
80:
76:
71:
69:
65:
61:
57:
53:
48:
46:
42:
38:
34:
30:
26:
22:
1467:
1445:
1423:
1398:
1378:
1374:
1363:. Retrieved
1344:(1): 55–73.
1341:
1335:
1318:, retrieved
1304:
1303:, Series B,
1298:
1278:, p. 386-388
1276:Regular maps
1275:
1271:
1266:
1243:
1233:
1210:
1204:
1178:
1172:
1105:
1104:+1) and PO(2
1101:
1063:
1050:Since PO is
1046:
909:
905:
901:
897:
893:
889:
879:
874:
870:
860:
838:
834:
829:
824:
820:
815:
810:
806:
801:
794:-gon is the
791:
787:
785:
717:
713:
709:
705:
703:
694:
692:
589:
585:
581:
577:
573:
572:is even (so
569:
565:
563:
557:
553:
549:
545:
537:
462:
458:
456:
445:
427:
423:
421:
416:
404:
398:
377:
373:
364:
359:
349:
341:
325:covering map
273:covering map
270:
242:
235:, which are
230:
196:
189:
139:
95:
82:
72:
55:
51:
49:
29:tessellation
24:
18:
1069:isomorphism
865:there is a
805:(of order 4
588:+1)×{±
1365:2010-04-15
1320:2010-04-15
1159:References
848:Spin group
695:rotational
356:isometries
1346:CiteSeerX
1270:Coxeter,
1164:Footnotes
1122:→
1082:→
998:≥
930:→
908:) and PO(
852:Pin group
753:≠
654:≅
344:hemi-cube
292:→
180:, {3,5}/2
174:, {5,3}/2
168:, {3,4}/2
162:, {4,3}/2
160:Hemi-cube
153:hosohedra
134:hemi-cube
1494:Category
1241:(1974),
1021:See also
360:rotation
261:immerses
219:immerses
124:Examples
21:geometry
1450:162–165
1193:Bibcode
1100:– PSO(2
1052:compact
413:11-cell
405:locally
367:}. See
149:dihedra
108:) with
98:density
87:defined
83:locally
31:of the
1478:
1456:
1434:
1409:
1348:
1255:
1221:
556:+1)/{±
43:– and
41:sphere
1472:43–49
1332:(PDF)
1183:arXiv
1149:p. 34
1038:Notes
987:(for
327:is a
27:is a
1476:ISBN
1454:ISBN
1432:ISBN
1407:ISBN
1253:ISBN
1219:ISBN
1067:The
850:and
833:and
354:(of
213:The
151:and
132:The
116:and
1428:147
1383:doi
1356:doi
1309:doi
798:Dih
704:If
564:If
544:PO(
438:).
417:any
73:As
54:or
19:In
1496::
1474:.
1452:,
1430:,
1405:,
1403:34
1379:59
1377:.
1354:.
1342:59
1340:.
1334:.
1305:59
1297:,
1247:,
1217:.
1215:11
1191:,
1147:,
708:=2
568:=2
560:}.
469:,
363:{±
339:.
155::
120:.
93:.
1484:.
1389:.
1385::
1368:.
1358::
1311::
1227:.
1195::
1185::
1131:O
1128:S
1125:P
1119:O
1116:S
1106:k
1102:k
1088:O
1085:P
1079:O
1058:.
1001:2
995:n
973:n
969:S
948:,
943:n
938:P
935:R
925:n
921:S
910:n
906:n
902:n
898:n
894:n
890:n
888:(
886:±
875:n
871:n
856:+
839:r
835:C
830:r
825:r
821:r
816:r
814:2
811:C
807:r
802:r
800:2
792:r
788:n
771:)
768:k
765:2
762:(
759:O
756:P
750:)
747:k
744:2
741:(
738:O
735:S
732:P
718:k
714:n
710:k
706:n
678:)
675:1
672:+
669:k
666:2
663:(
660:O
657:S
651:)
648:1
645:+
642:k
639:2
636:(
633:O
630:S
627:P
624:=
621:)
618:1
615:+
612:k
609:2
606:(
603:O
600:P
590:I
586:k
582:k
578:k
574:n
570:k
566:n
558:I
554:n
550:n
546:n
538:n
524:,
521:)
516:1
513:+
510:n
505:R
500:(
496:P
492:=
487:n
482:P
479:R
463:n
459:n
428:n
424:k
365:I
305:2
300:P
297:R
287:2
283:S
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
