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with a single mathematical model. Although it is an attempt to model our universe it takes place in a space with more dimensions than the four of spacetime that we are familiar with. In particular a number of string theories take place in a ten-dimensional space, adding an extra six dimensions. These
29:
is any space that has six dimensions, six degrees of freedom, and that needs six pieces of data, or coordinates, to specify a location in this space. There are an infinite number of these, but those of most interest are simpler ones that model some aspect of the environment. Of particular interest is
1309:
Although the space we live in is considered three-dimensional, there are practical applications for four-dimensional space. Quaternions, one of the ways to describe rotations in three dimensions, consist of a four-dimensional space. Rotations between quaternions, for interpolation, for example, take
2595:. It consisted of two three-dimensional vectors in a single object, which he used to describe ellipses in three dimensions. It has fallen out of use as other techniques have been developed, and the name bivector is now more closely associated with geometric algebra.
2288:
1838:
2303:
2038:, can be added subtracted and scaled like in other dimensions. Rather than use letters of the alphabet, higher dimensions usually use suffixes to designate dimensions, so a general six-dimensional vector can be written
1082:
908:
1230:
in six dimensions. These can be treated as six-dimensional vectors that transform linearly when changing frame of reference. Translations and rotations cannot be done this way, but are related to a twist by
1286:
the matrix is determined, up to a change in sign, by e.g. the six elements above the main diagonal. But this group is not linear, and it has a more complex structure than other applications seen so far.
2574:
1268:
to highlight the relationship between the quantities. A general particle moving in three dimensions has a phase space with six dimensions, too many to plot but they can be analysed mathematically.
1192:. Often these transformations are handled separately as they have very different geometrical structures, but there are ways of dealing with them that treat them as a single six-dimensional object.
1397:
1150:
973:
1576:
1298:, one before and one after the vector. These quaternion are unique, up to a change in sign for both of them, and generate all rotations when used this way, so the product of their groups,
1747:
1531:
2122:
1704:
1640:
746:
1805:
1776:
1669:
1605:
502:
150:
that are a fixed distance from the origin. This constraint reduces the number of coordinates needed to describe a point on the 6-sphere by one, so it has six dimensions. Such
148:
72:
702:
2000:{\displaystyle \mathbf {B} =B_{12}\mathbf {e} _{12}+B_{13}\mathbf {e} _{13}+B_{14}\mathbf {e} _{14}+B_{23}\mathbf {e} _{23}+B_{24}\mathbf {e} _{24}+B_{34}\mathbf {e} _{34}}
624:
789:
774:
687:
644:
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507:
431:
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289:
2482:{\displaystyle \left|\mathbf {a} \right\vert ={\sqrt {\mathbf {a} \cdot \mathbf {a} }}={\sqrt {{a_{1}}^{2}+{a_{2}}^{2}+{a_{3}}^{2}+{a_{4}}^{2}+{a_{5}}^{2}+{a_{6}}^{2}}}.}
804:
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1362:-valued representation of the electromagnetic field. Using this Maxwell's equations can be condensed from four equations into a particularly compact single equation:
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The rotation group in four dimensions, SO(4), has six degrees of freedom. This can be seen by considering the 4 Ă 4 matrix that represents a rotation: as it is an
1778:). They can also be related to general transformations in three dimensions through homogeneous coordinates, which can be thought of as modified rotations in
1008:
834:
2758:
1277:
1477:
are non-gravitational string theories in five and six dimensions that arise when considering limits of ten-dimensional string theory.
2506:
2686:
1707:
20:
1368:
1093:
1314:, which has three space dimensions and one time dimension is also four-dimensional, though with a different structure to
994:
The 6-sphere, or hypersphere in seven dimensions, is the six-dimensional surface equidistant from a point. It has symbol
820:
The 5-sphere, or hypersphere in six dimensions, is the five-dimensional surface equidistant from a point. It has symbol
1490:
A number of the above applications can be related to each other algebraically by considering the real, six-dimensional
1462:
extra dimensions are required by the theory, but as they cannot be observed are thought to be quite different, perhaps
919:
2861:
2811:
1536:
1713:
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2283:{\displaystyle \mathbf {a} \cdot \mathbf {b} =a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}+a_{4}b_{4}+a_{5}b_{5}+a_{6}b_{6}.}
2751:
2010:
They are the first bivectors that cannot all be generated by products of pairs of vectors. Those that can are
182:
2496:; with one corner at the origin, edges aligned to the axes and side length 1 the opposite corner could be at
2846:
154:
spaces are far more common than
Euclidean spaces, and in six dimensions they have far more applications.
1674:
1610:
2678:
1463:
206:
2034:
6-vectors are simply the vectors of six-dimensional
Euclidean space. Like other such vectors they are
1781:
1752:
1645:
1581:
124:
48:
2744:
2781:
2026:
rotations and correspond to non-simple bivectors that cannot be generated by single wedge product.
113:, not necessarily Euclidean ones, is six-dimensional. One example is the surface of the 6-sphere,
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in this space. As such it has the properties of all
Euclidean spaces, so it is linear, has a
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1474:
1189:
1177:
95:
87:
8:
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201:, constructed from fundamental symmetry domains of reflection, each domain defined by a
94:
between two 6-vectors is readily defined and can be used to calculate the metric. 6 Ă 6
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2707:
2294:
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198:
1294:
multiplication. Every rotation in four dimensions can be achieved by multiplying by a
277:
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2729:
2682:
1458:
1295:
1283:
419:
2011:
1607:
while the electromagnetic tensor discussed in the previous section is a bivector in
2796:
2725:
1327:
1207:
178:
83:
39:
35:
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published a work on vectors that included a six-dimensional quantity he called a
2023:
1473:
Since 1997 another string theory has come to light that works in six dimensions.
1315:
31:
1578:
for the set of bivectors in spacetime. The PlĂźcker coordinates are bivectors in
348:
2923:
2908:
2656:
Vector analysis: a text-book for the use of students of mathematics and physics
1350:. A second approach is to combine them in a single object, the six-dimensional
1339:
1335:
1232:
2019:
2015:
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1815:
1244:
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194:
110:
91:
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34:, in which 6-polytopes and the 5-sphere are constructed. Six-dimensional
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which is the length of the vector and so of the diagonal of the 6-cube.
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Aharony, Ofer (2000). "A brief review of "little string theories"".
2918:
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2109:
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1359:
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1211:
1077:{\displaystyle S^{6}=\left\{x\in \mathbb {R} ^{7}:\|x\|=r\right\}.}
903:{\displaystyle S^{5}=\left\{x\in \mathbb {R} ^{6}:\|x\|=r\right\}.}
490:
177:
in six dimensions is called a 6-polytope. The most studied are the
174:
106:
99:
2928:
913:
The volume of six-dimensional space bounded by this 5-sphere is
42:
are also studied, with constant positive and negative curvature.
1214:
velocity are combined into one six-dimensional object, called a
711:
609:
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2035:
1355:
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It can be used to find the angle between two vectors and the
1434:
1249:
1223:
79:
2736:
2492:
This can be used for example to calculate the diagonal of a
2569:{\displaystyle {\sqrt {1+1+1+1+1+1}}={\sqrt {6}}=2.4495,}
1334:
is generally thought of as being made of two things, the
1642:. Bivectors can be used to generate rotations in either
1710:(e.g. applying the exponential map of all bivectors in
90:
and a full set of vector operations. In particular the
1171:
19:"Sixth dimension" redirects here. For other uses, see
2509:
2306:
2125:
1841:
1832:= 6 components, and can be written most generally as
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is the bivector form of the electromagnetic tensor,
1392:{\displaystyle \partial \mathbf {F} =\mathbf {J} \,}
1145:{\displaystyle V_{7}={\frac {16\pi ^{3}r^{7}}{105}}}
1087:
The volume of the space bounded by this 6-sphere is
2654:Josiah Willard Gibbs, Edwin Bidwell Wilson (1901).
1260:Phase space is a space made up of the position and
1188:along the three coordinate axes and three from the
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1999:
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1634:
1599:
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1391:
1264:of a particle, which can be plotted together in a
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142:
66:
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1814:between pairs of 4-vectors. They therefore have
968:{\displaystyle V_{6}={\frac {\pi ^{3}r^{6}}{6}}}
98:can be used to describe transformations such as
2104:cannot be used in six dimensions; instead, the
1533:for the set of bivectors in Euclidean space or
1485:
1271:
205:. Each uniform polytope is defined by a ringed
1810:The bivectors arise from sums of all possible
1571:{\displaystyle \Lambda ^{2}\mathbb {R} ^{3,1}}
2752:
2611:
1290:Another way of looking at this group is with
241:(Displayed as orthogonal projections in each
1742:{\displaystyle \Lambda ^{2}\mathbb {R} ^{4}}
1526:{\displaystyle \Lambda ^{2}\mathbb {R} ^{4}}
1057:
1051:
998:, and the equation for the 6-sphere, radius
883:
877:
824:, and the equation for the 5-sphere, radius
45:Formally, six-dimensional Euclidean space,
2759:
2745:
1306:of SO(4), which must have six dimensions.
1278:Rotations in 4-dimensional Euclidean space
2711:
2018:. Other rotations in four dimensions are
1787:
1758:
1729:
1680:
1651:
1616:
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1552:
1513:
1494:in four dimensions. These can be written
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1388:
1038:
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130:
54:
2672:
1433:
1248:
2697:
2658:. Yale University Press. p. 481ff.
1466:to form a six-dimensional space with a
3043:
105:More generally, any space that can be
2740:
2014:and the rotations they generate are
1321:
1172:Transformations in three dimensions
239:Uniform polytopes in six dimensions
117:. This is the set of all points in
16:Geometric space with six dimensions
13:
2582:
1718:
1699:{\displaystyle \mathbb {R} ^{3,1}}
1635:{\displaystyle \mathbb {R} ^{3,1}}
1541:
1502:
1372:
1342:. They are both three-dimensional
74:, is generated by considering all
14:
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1962:
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1800:{\displaystyle \mathbb {R} ^{4}}
1771:{\displaystyle \mathbb {R} ^{4}}
1664:{\displaystyle \mathbb {R} ^{4}}
1600:{\displaystyle \mathbb {R} ^{4}}
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143:{\displaystyle \mathbb {R} ^{7}}
67:{\displaystyle \mathbb {R} ^{6}}
1195:
1166:
213:is a unique polytope from the D
2647:
2638:
2629:
2620:
2614:Perspectives of Modern Physics
2605:
2108:of two 6-vectors results in a
1238:
1:
2766:
2700:Classical and Quantum Gravity
2675:Clifford algebras and spinors
2666:
1176:In three dimensional space a
162:
2626:Lounesto (2001), pp. 109â110
2598:
2100:Of the vector operators the
2069:. Written like this the six
2029:
1486:Bivectors in four dimensions
1470:too small to be observable.
1272:Rotations in four dimensions
1218:. A similar object called a
1163:that contains the 6-sphere.
1159:, or 0.0369 of the smallest
986:that contains the 5-sphere.
982:, or 0.0807 of the smallest
274:
102:that keep the origin fixed.
7:
1749:generates all rotations in
1346:, related to each other by
989:
815:
157:
10:
3072:
2730:10.1088/0264-9381/17/5/302
2679:Cambridge University Press
2644:Lounesto (2001), pp. 86â89
1453:is an attempt to describe
1310:place in four dimensions.
1275:
1242:
1199:
181:, of which there are only
166:
18:
3027:
3006:
2942:
2880:
2834:
2823:
2774:
2673:Lounesto, Pertti (2001).
267:
255:
197:. A wider family are the
2112:with 15 dimensions. The
1296:pair of unit quaternions
2500:, the norm of which is
1002:, centre the origin is
828:, centre the origin is
183:three in six dimensions
119:seven-dimensional space
2612:Arthur Besier (1969).
2570:
2483:
2284:
2001:
1801:
1772:
1743:
1700:
1665:
1636:
1601:
1572:
1527:
1481:Theoretical background
1475:Little string theories
1446:
1393:
1352:electromagnetic tensor
1257:
1255:Van der Pol oscillator
1253:Phase portrait of the
1182:six degrees of freedom
1146:
1078:
969:
904:
207:CoxeterâDynkin diagram
144:
68:
2571:
2484:
2285:
2002:
1802:
1773:
1744:
1701:
1666:
1637:
1602:
1573:
1528:
1437:
1424:differential operator
1394:
1332:electromagnetic field
1252:
1147:
1079:
970:
905:
605:{3,3} = h{4,3,3,3,3}
145:
69:
27:Six-dimensional space
2943:Dimensions by number
2507:
2304:
2123:
1839:
1782:
1753:
1714:
1675:
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1611:
1582:
1537:
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1190:rotation group SO(3)
1178:rigid transformation
1094:
1009:
920:
835:
231:polytopes from the E
125:
49:
2722:2000CQGra..17..929A
1468:particular geometry
1439:CalabiâYau manifold
1348:Maxwell's equations
1155:which is 4.72477 Ă
978:which is 5.16771 Ă
246:
199:uniform 6-polytopes
21:The Sixth Dimension
2872:Degrees of freedom
2775:Dimensional spaces
2566:
2498:(1, 1, 1, 1, 1, 1)
2479:
2280:
2116:of two vectors is
2095:(0, 0, 0, 0, 0, 1)
2091:(0, 0, 0, 0, 1, 0)
2087:(0, 0, 0, 1, 0, 0)
2083:(0, 0, 1, 0, 0, 0)
2079:(0, 1, 0, 0, 0, 0)
2075:(1, 0, 0, 0, 0, 0)
1997:
1797:
1768:
1739:
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1523:
1455:general relativity
1447:
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1258:
1142:
1074:
965:
900:
238:
140:
64:
3038:
3037:
2847:Lebesgue covering
2812:Algebraic variety
2688:978-0-521-00551-7
2555:
2545:
2474:
2338:
1459:quantum mechanics
1284:orthogonal matrix
1140:
963:
813:
812:
179:regular polytopes
107:described locally
40:hyperbolic spaces
3063:
2835:Other dimensions
2829:
2797:Projective space
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2016:simple rotations
2012:simple bivectors
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1328:electromagnetism
1322:Electromagnetism
1206:In screw theory
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36:elliptical space
30:six-dimensional
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2787:Euclidean space
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1708:exponential map
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1316:Euclidean space
1280:
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1169:
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1126:
1120:
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32:Euclidean space
24:
17:
12:
11:
5:
3069:
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2924:Cross-polytope
2921:
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2909:Hyperrectangle
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2713:hep-th/9911147
2694:
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2665:
2662:
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2646:
2637:
2635:Aharony (2000)
2628:
2619:
2616:. McGraw-Hill.
2603:
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2008:
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1812:wedge products
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1422:is a suitable
1400:
1399:
1386:
1382:
1378:
1374:
1340:magnetic field
1336:electric field
1323:
1320:
1276:Main article:
1273:
1270:
1243:Main article:
1240:
1237:
1233:exponentiation
1200:Main article:
1197:
1194:
1173:
1170:
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1165:
1153:
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1123:
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487:
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345:
273:
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167:Main article:
164:
161:
159:
156:
137:
132:
61:
56:
15:
9:
6:
4:
3:
2:
3068:
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2927:
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2922:
2920:
2917:
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2914:Demihypercube
2912:
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2907:
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2902:
2900:
2897:
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2727:
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2696:
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2690:
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2680:
2677:. Cambridge:
2676:
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2119:
2118:
2117:
2115:
2111:
2107:
2106:wedge product
2103:
2102:cross product
2098:
2072:
2071:basis vectors
2042:
2037:
2027:
2025:
2021:
2017:
2013:
1992:
1980:
1976:
1972:
1967:
1955:
1951:
1947:
1942:
1930:
1926:
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1563:
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1557:
1545:
1518:
1506:
1493:
1478:
1476:
1471:
1469:
1465:
1460:
1456:
1452:
1451:string theory
1444:
1443:3D projection
1440:
1436:
1430:String theory
1427:
1425:
1417:
1412:
1406:
1380:
1365:
1364:
1363:
1361:
1357:
1353:
1349:
1345:
1344:vector fields
1341:
1337:
1333:
1329:
1319:
1317:
1313:
1307:
1305:
1301:
1297:
1293:
1288:
1285:
1279:
1269:
1267:
1266:phase diagram
1263:
1256:
1251:
1246:
1236:
1234:
1229:
1225:
1221:
1217:
1213:
1209:
1203:
1193:
1191:
1187:
1183:
1179:
1164:
1162:
1158:
1137:
1131:
1127:
1121:
1117:
1113:
1107:
1102:
1098:
1090:
1089:
1088:
1071:
1067:
1063:
1060:
1054:
1048:
1043:
1033:
1030:
1026:
1022:
1017:
1013:
1005:
1004:
1003:
1001:
997:
987:
985:
981:
960:
954:
950:
944:
940:
933:
928:
924:
916:
915:
914:
897:
893:
889:
886:
880:
874:
869:
859:
856:
852:
848:
843:
839:
831:
830:
829:
827:
823:
721:
713:
709:
619:
611:
607:
497:
492:
488:
426:
421:
417:
355:
350:
346:
284:
279:
275:
261:
249:
248:
245:of symmetry)
244:
243:Coxeter plane
236:
230:
223:
212:
208:
204:
203:Coxeter group
200:
196:
192:
188:
184:
180:
176:
170:
155:
153:
152:non-Euclidean
135:
120:
116:
112:
108:
103:
101:
97:
93:
89:
85:
81:
77:
59:
43:
41:
37:
33:
28:
22:
3029:
2995:
2979:
2934:Hyperpyramid
2899:Hypersurface
2792:Affine space
2782:Vector space
2703:
2699:
2674:
2655:
2649:
2640:
2631:
2622:
2613:
2607:
2592:
2586:
2578:
2491:
2292:
2099:
2040:
2033:
2009:
1816:
1809:
1706:through the
1489:
1472:
1464:compactified
1448:
1416:four-current
1410:
1404:
1401:
1325:
1308:
1304:double cover
1289:
1281:
1259:
1219:
1215:
1205:
1202:Screw theory
1196:Screw theory
1186:translations
1175:
1167:Applications
1156:
1154:
1086:
999:
995:
993:
979:
977:
912:
825:
821:
819:
486:{3,3,3,3,4}
415:{4,3,3,3,3}
344:{3,3,3,3,3}
217:family, and
172:
121:(Euclidean)
114:
104:
44:
26:
25:
3019:Codimension
2998:-dimensions
2919:Hypersphere
2802:Free module
2114:dot product
1449:In physics
1245:Phase space
1239:Phase space
425:6-orthoplex
195:6-orthoplex
111:coordinates
92:dot product
3056:6 (number)
3045:Categories
3014:Hyperspace
2894:Hyperplane
2667:References
2589:J.W. Gibbs
1302:Ă S, is a
1292:quaternion
496:6-demicube
211:6-demicube
169:6-polytope
163:6-polytope
3051:Dimension
2904:Hypercube
2882:Polytopes
2862:Minkowski
2857:Hausdorff
2852:Inductive
2817:Spacetime
2768:Dimension
2599:Footnotes
2331:⋅
2132:⋅
2030:6-vectors
2024:isoclinic
1719:Λ
1542:Λ
1503:Λ
1492:bivectors
1373:∂
1312:Spacetime
1222:combines
1118:π
1058:‖
1052:‖
1034:∈
941:π
884:‖
878:‖
860:∈
283:6-simplex
187:6-simplex
109:with six
100:rotations
3031:Category
3007:See also
2807:Manifold
2593:bivector
2587:In 1901
2110:bivector
1360:bivector
1262:momentum
1184:, three
990:6-sphere
816:5-sphere
707:{3,3,3}
235:family.
175:polytope
158:Geometry
96:matrices
2929:Simplex
2867:Fractal
2718:Bibcode
1414:is the
1228:torques
1208:angular
84:vectors
2886:shapes
2685:
2561:2.4495
2494:6-cube
2036:linear
2020:double
1827:
1402:where
1356:tensor
1330:, the
1224:forces
1220:wrench
1212:linear
1161:7-cube
984:6-cube
809:{3,3}
354:6-cube
209:. The
193:, and
191:6-cube
185:: the
88:metric
80:tuples
2990:Eight
2985:Seven
2965:Three
2842:Krull
2708:arXiv
2706:(5).
1358:- or
1216:twist
82:as 6-
2975:Five
2970:Four
2950:Zero
2884:and
2683:ISBN
2295:norm
2093:and
2073:are
2043:= (a
2022:and
1457:and
1418:and
1354:, a
1338:and
1226:and
1210:and
1180:has
224:and
76:real
38:and
2980:Six
2960:Two
2955:One
2726:doi
2063:, a
2059:, a
2055:, a
2051:, a
2047:, a
1807:.
1671:or
1326:In
1138:105
3047::
2724:.
2716:.
2704:17
2702:.
2681:.
2297:,
2097:.
2089:,
2085:,
2081:,
2077:,
1993:34
1981:34
1968:24
1956:24
1943:23
1931:23
1918:14
1906:14
1893:13
1881:13
1868:12
1856:12
1426:.
1318:.
1235:.
1114:16
770:=
719:22
668:=
617:21
546:=
228:22
221:21
189:,
173:A
78:6-
2996:n
2760:e
2753:t
2746:v
2732:.
2728::
2720::
2710::
2691:.
2564:,
2558:=
2553:6
2548:=
2543:1
2540:+
2537:1
2534:+
2531:1
2528:+
2525:1
2522:+
2519:1
2516:+
2513:1
2477:.
2470:2
2463:6
2459:a
2453:+
2448:2
2441:5
2437:a
2431:+
2426:2
2419:4
2415:a
2409:+
2404:2
2397:3
2393:a
2387:+
2382:2
2375:2
2371:a
2365:+
2360:2
2353:1
2349:a
2341:=
2335:a
2327:a
2321:=
2317:|
2313:a
2309:|
2278:.
2273:6
2269:b
2263:6
2259:a
2255:+
2250:5
2246:b
2240:5
2236:a
2232:+
2227:4
2223:b
2217:4
2213:a
2209:+
2204:3
2200:b
2194:3
2190:a
2186:+
2181:2
2177:b
2171:2
2167:a
2163:+
2158:1
2154:b
2148:1
2144:a
2140:=
2136:b
2128:a
2067:)
2065:6
2061:5
2057:4
2053:3
2049:2
2045:1
2041:a
1988:e
1977:B
1973:+
1963:e
1952:B
1948:+
1938:e
1927:B
1923:+
1913:e
1902:B
1898:+
1888:e
1877:B
1873:+
1863:e
1852:B
1848:=
1844:B
1824:2
1822:4
1817:C
1793:4
1788:R
1764:4
1759:R
1735:4
1730:R
1723:2
1692:1
1689:,
1686:3
1681:R
1657:4
1652:R
1628:1
1625:,
1622:3
1617:R
1593:4
1588:R
1564:1
1561:,
1558:3
1553:R
1546:2
1519:4
1514:R
1507:2
1445:)
1441:(
1420:â
1411:J
1405:F
1385:J
1381:=
1377:F
1300:S
1157:r
1132:7
1128:r
1122:3
1108:=
1103:7
1099:V
1072:.
1068:}
1064:r
1061:=
1055:x
1049::
1044:7
1039:R
1031:x
1027:{
1023:=
1018:6
1014:S
1000:r
996:S
980:r
961:6
955:6
951:r
945:3
934:=
929:6
925:V
898:.
894:}
890:r
887:=
881:x
875::
870:6
865:R
857:x
853:{
849:=
844:5
840:S
826:r
822:S
717:1
615:2
270:6
268:E
264:6
262:D
258:6
256:B
252:6
250:A
233:6
226:1
219:2
215:6
136:7
131:R
115:S
60:6
55:R
23:.
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