740:
479:
206:
388:
297:
1449:
656:
572:
1043:
over the real numbers, the largest such algebra. Mathematically they can be specified by 8-tuplets of real numbers, so form an 8-dimensional vector space over the reals, with addition of vectors being the addition in the algebra. A normed algebra is one with a product that satisfies
1329:
921:
982:
1155:
1211:
1095:
794:
490:
730:
586:
710:
626:
495:
469:
308:
217:
814:
804:
784:
774:
764:
754:
720:
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670:
646:
636:
616:
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550:
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409:
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378:
368:
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348:
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287:
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809:
799:
789:
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769:
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715:
705:
695:
685:
675:
641:
631:
621:
611:
601:
591:
545:
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505:
464:
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232:
222:
847:
928:
1113:
1380:
1112:
additionally must be finite-dimensional, and have the property that every non-zero vector has a unique multiplicative inverse.
1349:
1261:
834:
or hypersphere in eight dimensions is the seven-dimensional surface equidistant from a point, e.g. the origin. It has symbol
1677:
1256:, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
1130:
1483:
1433:
1315:
1341:
1373:
1238:
1175:
109:
1050:
1468:
1322:
1366:
1264:
1222:
1403:
1292:
39:
1606:
1586:
1040:
1596:
1591:
1571:
1162:
1007:
1001:
133:
1338:
Concise
Encyclopedia of Supersymmetry And Noncommutative Structures in Mathematics and Physics
1601:
1581:
1576:
8:
1682:
1478:
1473:
1217:. It has also been proposed as a practical or pedagogical tool for doing calculations in
66:
128:, constructed from fundamental symmetry domains of reflection, each domain defined by a
1672:
1652:
1493:
1448:
1303:
1218:
125:
205:
73:
vector space, which has 16 real dimensions. It may also refer to an eight-dimensional
1488:
1345:
1311:
1257:
1226:
387:
1418:
1170:
1109:
560:
105:
59:
31:
1463:
1408:
1116:
prohibits such a structure from existing in dimensions other than 1, 2, 4, or 8.
1018:
55:
296:
65:
More generally the term may refer to an eight-dimensional vector space over any
1545:
1530:
70:
739:
1666:
1535:
1333:
1214:
1011:
744:
660:
576:
176:
159:
152:
145:
129:
1555:
1520:
1413:
1158:
1022:
51:
1640:
1423:
1296:
392:
121:
27:
20:
1635:
1515:
1166:
1125:
483:
137:
95:
1308:
On
Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry
104:
in eight dimensions is called an 8-polytope. The most studied are the
1616:
1525:
1438:
1389:
210:
113:
1010:
has been solved in eight dimensions, thanks to the existence of the
1540:
1503:
1428:
1034:
916:{\displaystyle S^{7}=\left\{x\in \mathbb {R} ^{8}:\|x\|=r\right\}.}
831:
478:
101:
78:
74:
1550:
989:
301:
117:
1507:
655:
571:
1358:
1165:'s work in the 1850s. This algebra is equivalent (that is,
1265:
Wiley::Kaleidoscopes: Selected
Writings of H.S.M. Coxeter
840:, with formal definition for the 7-sphere with radius
1178:
1133:
1053:
977:{\displaystyle V_{8}\,={\frac {\pi ^{4}}{24}}\,R^{8}}
931:
850:
1293:
Table of the
Highest Kissing Numbers Presently Known
925:
The volume of the space bounded by this 7-sphere is
54:, without any notion of distance. Eight-dimensional
1328:
1254:Kaleidoscopes: Selected Writings of H.S.M. Coxeter
1205:
1149:
1089:
976:
915:
173:Regular and uniform polytopes in eight dimensions
81:, or a variety of other geometric constructions.
1664:
1150:{\displaystyle \mathbb {C} \otimes \mathbb {H} }
132:. Each uniform polytope is defined by a ringed
1161:," are an eight-dimensional algebra dating to
1374:
175:(Displayed as orthogonal projections in each
58:is eight-dimensional space equipped with the
46:= 8, the set of all such locations is called
1084:
1078:
1075:
1069:
1063:
1054:
1021:. The kissing number in eight dimensions is
896:
890:
1302:
1381:
1367:
1336:, Warren; Bagger, Jonathan, eds. (2005),
1196:
1143:
1135:
995:
963:
942:
877:
1206:{\displaystyle C\ell _{2}(\mathbb {C} )}
1221:, and in that context goes by the name
1665:
1285:Regular and Semi-Regular Polytopes III
1362:
1278:Regular and Semi-Regular Polytopes II
1090:{\displaystyle \|xy\|\leq \|x\|\|y\|}
16:Geometric space with eight dimensions
1271:Regular and Semi Regular Polytopes I
1248:, 3rd Edition, Dover New York, 1973
50:. Often such spaces are studied as
13:
14:
1694:
1447:
1295:maintained by Gabriele Nebe and
1119:
812:
807:
802:
797:
792:
787:
782:
777:
772:
767:
762:
757:
752:
738:
728:
723:
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713:
708:
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673:
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654:
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619:
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604:
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584:
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498:
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275:
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265:
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255:
250:
245:
240:
235:
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225:
220:
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140:is a unique polytope from the D
69:, such as an eight-dimensional
1200:
1192:
1:
1388:
1232:
1225:(not to be confused with the
89:
1229:, which is 16-dimensional.)
1028:
1017:polytope and its associated
992:that contains the 7-sphere.
568:
202:
7:
1283:(Paper 24) H.S.M. Coxeter,
1276:(Paper 23) H.S.M. Coxeter,
1269:(Paper 22) H.S.M. Coxeter,
825:
124:. A broader family are the
84:
10:
1699:
1678:Multi-dimensional geometry
1306:; Smith, Derek A. (2003),
1032:
999:
559:
108:, of which there are only
93:
1649:
1628:
1564:
1502:
1456:
1445:
1396:
1223:Algebra of physical space
1108:in the algebra. A normed
737:
653:
569:
476:
385:
294:
203:
195:
189:
183:
110:three in eight dimensions
1041:normed division algebra
30:can be understood as a
1310:, A. K. Peters, Ltd.,
1207:
1163:William Rowan Hamilton
1151:
1091:
1008:kissing number problem
1002:Kissing number problem
996:Kissing number problem
978:
917:
134:Coxeter-Dynkin diagram
1208:
1152:
1092:
979:
918:
1565:Dimensions by number
1340:, Berlin, New York:
1176:
1131:
1051:
1039:The octonions are a
988:, or 0.01585 of the
929:
848:
165:polytopes from the E
1304:Conway, John Horton
984:which is 4.05871 ×
180:
126:uniform 8-polytopes
48:8-dimensional space
1494:Degrees of freedom
1397:Dimensional spaces
1219:special relativity
1203:
1147:
1087:
974:
913:
172:
1660:
1659:
1469:Lebesgue covering
1434:Algebraic variety
1355:(Second printing)
1351:978-1-4020-1338-6
1262:978-0-471-01003-6
1246:Regular Polytopes
1227:Spacetime algebra
1124:The complexified
1114:Hurwitz's theorem
961:
823:
822:
555:h{4,3,3,3,3,3,3}
106:regular polytopes
1690:
1457:Other dimensions
1451:
1419:Projective space
1383:
1376:
1369:
1360:
1359:
1354:
1320:
1244:H.S.M. Coxeter,
1212:
1210:
1209:
1204:
1199:
1191:
1190:
1171:Clifford algebra
1156:
1154:
1153:
1148:
1146:
1138:
1110:division algebra
1096:
1094:
1093:
1088:
983:
981:
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975:
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839:
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521:
517:
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512:
511:
507:
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502:
501:
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496:
492:
491:
481:
474:{3,3,3,3,3,3,4}
472:
471:
470:
466:
465:
461:
460:
456:
455:
451:
450:
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445:
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410:
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401:
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390:
383:{4,3,3,3,3,3,3}
381:
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292:{3,3,3,3,3,3,3}
290:
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288:
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181:
171:
60:Euclidean metric
23:, a sequence of
1698:
1697:
1693:
1692:
1691:
1689:
1688:
1687:
1663:
1662:
1661:
1656:
1645:
1624:
1560:
1498:
1452:
1443:
1409:Euclidean space
1392:
1387:
1352:
1318:
1235:
1195:
1186:
1182:
1177:
1174:
1173:
1142:
1134:
1132:
1129:
1128:
1122:
1052:
1049:
1048:
1037:
1031:
1015:
1004:
998:
968:
964:
952:
948:
946:
936:
932:
930:
927:
926:
881:
876:
875:
868:
864:
855:
851:
849:
846:
845:
835:
828:
818:
813:
808:
803:
798:
793:
788:
783:
778:
773:
768:
763:
758:
753:
751:
750:
748:
743:
734:
729:
724:
719:
714:
709:
704:
699:
694:
689:
684:
679:
674:
669:
667:
666:
664:
659:
650:
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615:
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605:
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583:
582:
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564:
554:
549:
544:
539:
534:
529:
524:
519:
514:
509:
504:
499:
494:
489:
487:
486:
482:
473:
468:
463:
458:
453:
448:
443:
438:
433:
428:
423:
418:
413:
408:
403:
398:
396:
395:
391:
382:
377:
372:
367:
362:
357:
352:
347:
342:
337:
332:
327:
322:
317:
312:
307:
305:
304:
300:
291:
286:
281:
276:
271:
266:
261:
256:
251:
246:
241:
236:
231:
226:
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216:
214:
213:
209:
199:
193:
187:
174:
168:
163:
156:
149:
143:
98:
92:
87:
56:Euclidean space
17:
12:
11:
5:
1696:
1686:
1685:
1680:
1675:
1658:
1657:
1650:
1647:
1646:
1644:
1643:
1638:
1632:
1630:
1626:
1625:
1623:
1622:
1614:
1609:
1604:
1599:
1594:
1589:
1584:
1579:
1574:
1568:
1566:
1562:
1561:
1559:
1558:
1553:
1548:
1546:Cross-polytope
1543:
1538:
1533:
1531:Hyperrectangle
1528:
1523:
1518:
1512:
1510:
1500:
1499:
1497:
1496:
1491:
1486:
1481:
1476:
1471:
1466:
1460:
1458:
1454:
1453:
1446:
1444:
1442:
1441:
1436:
1431:
1426:
1421:
1416:
1411:
1406:
1400:
1398:
1394:
1393:
1386:
1385:
1378:
1371:
1363:
1357:
1356:
1350:
1326:
1316:
1300:
1299:(lower bounds)
1290:
1289:
1288:
1281:
1274:
1251:
1250:
1249:
1239:H.S.M. Coxeter
1234:
1231:
1202:
1198:
1194:
1189:
1185:
1181:
1145:
1141:
1137:
1121:
1118:
1098:
1097:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1033:Main article:
1030:
1027:
1013:
1000:Main article:
997:
994:
971:
967:
960:
955:
951:
945:
939:
935:
912:
908:
904:
901:
898:
895:
892:
889:
884:
879:
874:
871:
867:
863:
858:
854:
827:
824:
821:
820:
746:
736:
662:
652:
578:
567:
566:
562:
557:
556:
475:
384:
293:
201:
200:
197:
194:
191:
188:
185:
166:
161:
154:
147:
141:
94:Main article:
91:
88:
86:
83:
15:
9:
6:
4:
3:
2:
1695:
1684:
1681:
1679:
1676:
1674:
1671:
1670:
1668:
1655:
1654:
1648:
1642:
1639:
1637:
1634:
1633:
1631:
1627:
1621:
1619:
1615:
1613:
1610:
1608:
1605:
1603:
1600:
1598:
1595:
1593:
1590:
1588:
1585:
1583:
1580:
1578:
1575:
1573:
1570:
1569:
1567:
1563:
1557:
1554:
1552:
1549:
1547:
1544:
1542:
1539:
1537:
1536:Demihypercube
1534:
1532:
1529:
1527:
1524:
1522:
1519:
1517:
1514:
1513:
1511:
1509:
1505:
1501:
1495:
1492:
1490:
1487:
1485:
1482:
1480:
1477:
1475:
1472:
1470:
1467:
1465:
1462:
1461:
1459:
1455:
1450:
1440:
1437:
1435:
1432:
1430:
1427:
1425:
1422:
1420:
1417:
1415:
1412:
1410:
1407:
1405:
1402:
1401:
1399:
1395:
1391:
1384:
1379:
1377:
1372:
1370:
1365:
1364:
1361:
1353:
1347:
1343:
1339:
1335:
1331:
1327:
1324:
1319:
1317:1-56881-134-9
1313:
1309:
1305:
1301:
1298:
1294:
1291:
1286:
1282:
1279:
1275:
1272:
1268:
1267:
1266:
1263:
1259:
1255:
1252:
1247:
1243:
1242:
1240:
1237:
1236:
1230:
1228:
1224:
1220:
1216:
1215:Pauli algebra
1187:
1183:
1179:
1172:
1168:
1164:
1160:
1159:biquaternions
1139:
1127:
1120:Biquaternions
1117:
1115:
1111:
1107:
1103:
1081:
1072:
1066:
1060:
1057:
1047:
1046:
1045:
1042:
1036:
1026:
1024:
1020:
1016:
1009:
1003:
993:
991:
987:
969:
965:
958:
953:
949:
943:
937:
933:
923:
910:
906:
902:
899:
893:
887:
882:
872:
869:
865:
861:
856:
852:
843:
838:
833:
749:
741:
665:
657:
581:
573:
565:
558:
485:
480:
394:
389:
303:
298:
212:
207:
182:
179:of symmetry)
178:
177:Coxeter plane
170:
164:
157:
150:
139:
135:
131:
130:Coxeter group
127:
123:
119:
115:
111:
107:
103:
97:
82:
80:
76:
72:
68:
63:
61:
57:
53:
52:vector spaces
49:
45:
41:
37:
33:
29:
26:
22:
1651:
1617:
1611:
1556:Hyperpyramid
1521:Hypersurface
1414:Affine space
1404:Vector space
1337:
1307:
1284:
1277:
1270:
1253:
1245:
1123:
1105:
1101:
1099:
1038:
1005:
985:
924:
841:
836:
829:
651:{3,3,3,3,3}
144:family, and
99:
64:
47:
43:
42:space. When
35:
28:real numbers
24:
18:
1641:Codimension
1620:-dimensions
1541:Hypersphere
1424:Free module
1332:, Steven ;
1297:Neil Sloane
1126:quaternions
393:8-orthoplex
122:8-orthoplex
77:such as an
40:dimensional
21:mathematics
1683:8 (number)
1667:Categories
1636:Hyperspace
1516:Hyperplane
1233:References
1167:isomorphic
484:8-demicube
138:8-demicube
96:8-polytope
90:8-polytope
1673:Dimension
1526:Hypercube
1504:Polytopes
1484:Minkowski
1479:Hausdorff
1474:Inductive
1439:Spacetime
1390:Dimension
1184:ℓ
1169:) to the
1140:⊗
1085:‖
1079:‖
1076:‖
1070:‖
1067:≤
1064:‖
1055:‖
1029:Octonions
950:π
897:‖
891:‖
873:∈
211:8-simplex
114:8-simplex
1653:Category
1629:See also
1429:Manifold
1342:Springer
1213:and the
1100:for all
1035:Octonion
832:7-sphere
826:7-sphere
735:{3,3,3}
169:family.
102:polytope
85:Geometry
79:8-sphere
75:manifold
32:location
1551:Simplex
1489:Fractal
1019:lattice
71:complex
1508:shapes
1348:
1334:Siegel
1330:Duplij
1323:Review
1314:
1260:
1157:, or "
990:8-cube
819:{3,3}
302:8-cube
158:, and
136:. The
120:, and
118:8-cube
112:: the
1612:Eight
1607:Seven
1587:Three
1464:Krull
67:field
1597:Five
1592:Four
1572:Zero
1506:and
1346:ISBN
1312:ISBN
1258:ISBN
1104:and
1006:The
830:The
1602:Six
1582:Two
1577:One
1321:. (
1023:240
844:of
62:.
34:in
19:In
1669::
1344:,
1325:).
1287:,
1280:,
1273:,
1241::
1025:.
1014:21
959:24
747:42
663:41
579:21
162:42
155:41
151:,
148:21
116:,
100:A
1618:n
1382:e
1375:t
1368:v
1201:)
1197:C
1193:(
1188:2
1180:C
1144:H
1136:C
1106:y
1102:x
1082:y
1073:x
1061:y
1058:x
1012:4
986:r
970:8
966:R
954:4
944:=
938:8
934:V
911:.
907:}
903:r
900:=
894:x
888::
883:8
878:R
870:x
866:{
862:=
857:7
853:S
842:r
837:S
745:1
661:2
577:4
563:8
561:E
198:8
196:D
192:8
190:B
186:8
184:A
167:8
160:1
153:2
146:4
142:8
44:n
38:-
36:n
25:n
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