2802:. For example, brane gas cosmology attempts to explain why there are three dimensions of space using topological and thermodynamic considerations. According to this idea it would be since three is the largest number of spatial dimensions in which strings can generically intersect. If initially there are many windings of strings around compact dimensions, space could only expand to macroscopic sizes once these windings are eliminated, which requires oppositely wound strings to find each other and annihilate. But strings can only find each other to annihilate at a meaningful rate in three dimensions, so it follows that only three dimensions of space are allowed to grow large given this kind of initial configuration.
2574:
121:
2563:
2462:
2514:
2525:
46:
70:
2551:
2473:
3999:
2734:
2499:
1807:
2893:
purposes of data efficiency, visual simplicity, or cognitive efficiency, and is acceptable if the distinction between the representation and the represented is understood but can cause confusion if information users assume that the digital shape is a perfect representation of reality (i.e., believing that roads really are lines).
2417:, the basic directions in which we can move are up/down, left/right, and forward/backward. Movement in any other direction can be expressed in terms of just these three. Moving down is the same as moving up a negative distance. Moving diagonally upward and forward is just as the name of the direction implies
2729:
also promotes 11D spacetime = 7D hyperspace + 4 common dimensions. To date, no direct experimental or observational evidence is available to support the existence of these extra dimensions. If hyperspace exists, it must be hidden from us by some physical mechanism. One well-studied possibility is
2892:
correlates with tendencies in spatial cognition. For example, asking the distance between two cities presumes a conceptual model of the cities as points, while giving directions involving travel "up," "down," or "along" a road imply a one-dimensional conceptual model. This is frequently done for
2888:, a representation of a real-world phenomenon may have a different (usually lower) dimension than the phenomenon being represented. For example, a city (a two-dimensional region) may be represented as a point, or a road (a three-dimensional volume of material) may be represented as a line. This
1897:
that are needed in order to have an intersection with the variety that is reduced to a finite number of points (dimension zero). This definition is based on the fact that the intersection of a variety with a hyperplane reduces the dimension by one unless if the hyperplane contains the variety.
2794:
by their endpoints, whereas the closed strings that mediate the gravitational interaction are free to propagate into the whole spacetime, or "the bulk". This could be related to why gravity is exponentially weaker than the other forces, as it effectively dilutes itself as it propagates into a
2607:" for this reason, but that is not to imply that it is a spatial dimension. A temporal dimension is one way to measure physical change. It is perceived differently from the three spatial dimensions in that there is only one of it, and that we cannot move freely in time but subjectively move
2122:
2870:(2-dimensional) usually represented as a line that closes at its endpoints, representing the boundary of a two-dimensional region. The software is expected to use this boundary to partition 2-dimensional space into an interior and exterior.
2757:
interactions, based on the realization that gravity propagating in small, compact extra dimensions is equivalent to gauge interactions at long distances. In particular when the geometry of the extra dimensions is trivial, it reproduces
2789:
are dynamical extended objects of various dimensionalities predicted by string theory that could play this role. They have the property that open string excitations, which are associated with gauge interactions, are confined to the
2124:
of prime ideals related by inclusion. It is strongly related to the dimension of an algebraic variety, because of the natural correspondence between sub-varieties and prime ideals of the ring of the polynomials on the variety.
2308:, permitting an inductive definition based on the dimension of the boundaries of open sets. Moreover, the boundary of a discrete set of points is the empty set, and therefore the empty set can be taken to have dimension -1.
1504:, is of dimension one, because the position of a point on a curve is determined by its signed distance along the curve to a fixed point on the curve. This is independent from the fact that a curve cannot be embedded in a
1962:
2258:
of points (such as a finite collection of points) to be 0-dimensional. By dragging a 0-dimensional object in some direction, one obtains a 1-dimensional object. By dragging a 1-dimensional object in a
1991:, and its dimension as variety agrees with its dimension as stack. There are however many stacks which do not correspond to varieties, and some of these have negative dimension. Specifically, if
2669:. Time is different from other spatial dimensions as time operates in all spatial dimensions. Time operates in the first, second and third as well as theoretical spatial dimensions such as a
2777:
In addition to small and curled up extra dimensions, there may be extra dimensions that instead are not apparent because the matter associated with our visible universe is localized on a
1859:
Conversely, in algebraically unconstrained contexts, a single complex coordinate system may be applied to an object having two real dimensions. For example, an ordinary two-dimensional
2056:
1810:
The complex plane can be mapped to the surface of a sphere, called the
Riemann sphere, with the complex number 0 mapped to one pole, the unit circle mapped to the equator, and a
1496:
The dimension is an intrinsic property of an object, in the sense that it is independent of the dimension of the space in which the object is or can be embedded. For example, a
1982:
3422:
2762:. However, at sufficiently high energies or short distances, this setup still suffers from the same pathologies that famously obstruct direct attempts to describe
1818:
The dimension of a manifold depends on the base field with respect to which
Euclidean space is defined. While analysis usually assumes a manifold to be over the
1485:
that are needed for defining the position of a point that is constrained to be on the object. For example, the dimension of a point is zero; the dimension of a
1905:
being a finite union of algebraic varieties, its dimension is the maximum of the dimensions of its components. It is equal to the maximal length of the chains
2174:
has an open refinement (a second open cover in which each element is a subset of an element in the first cover) such that no point is included in more than
65:; the cube is three-dimensional (3D) and bounded by two-dimensional squares; the tesseract is four-dimensional (4D) and bounded by three-dimensional cubes.
3930:
3170:
2913:
1249:
2425:
of up and forward. In its simplest form: a line describes one dimension, a plane describes two dimensions, and a cube describes three dimensions. (See
2770:, of the kind that string theory is intended to provide. In particular, superstring theory requires six compact dimensions (6D hyperspace) forming a
4591:
3628:
2880:
consisting of connected polygon faces. The software is expected to use this surface to partition 3-dimensional space into an interior and exterior.
3644:
2573:
3871:
3847:
3820:
3796:
3768:
3742:
3689:
3534:
3454:
3394:
2774:. Thus Kaluza-Klein theory may be considered either as an incomplete description on its own, or as a subset of string theory model building.
1908:
1890:
349:
3521:. Developments in Marketing Science: Proceedings of the Academy of Marketing Science. Springer International Publishing. pp. 38â46.
2835:. Different vector systems use a wide variety of data structures to represent shapes, but almost all are fundamentally based on a set of
2860:(1-dimensional) usually represented as an ordered list of points sampled from a continuous line, whereupon the software is expected to
2705:
In physics, three dimensions of space and one of time is the accepted norm. However, there are theories that attempt to unify the four
4775:
4454:
4033:
1298:(1D) because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A
4234:
3983:
3331:
2614:
The equations used in physics to model reality do not treat time in the same way that humans commonly perceive it. The equations of
1876:
1827:
1442:
315:
3492:
Rylov, Yuri A. (2007). "Non-Euclidean method of the generalized geometry construction and its application to space-time geometry".
4656:
2941:
2919:
1602:
is an example of a four-dimensional object. Whereas outside mathematics the use of the term "dimension" is as in: "A tesseract
1327:
4898:
4893:
3923:
3180:
2907:
2832:
1242:
1196:
802:
261:
3475:
4507:
4439:
2567:
1378:
3411:
3360:
4883:
4532:
3025:
2730:
that the extra dimensions may be "curled up" at such tiny scales as to be effectively invisible to current experiments.
2529:
2334:
joined at their faces with a complicated surface, then the dimension of the object is the dimension of those triangles.
1643:
4770:
4018:
3810:
3728:
3283:
3070:
2847:
2823:
Several types of digital systems are based on the storage, analysis, and visualization of geometric shapes, including
2578:
2503:
2430:
2154:
2392:. This cardinality is called the dimension of the Hilbert space. This dimension is finite if and only if the space's
4581:
4401:
3666:
3446:
3266:
2562:
2011:
1390:
1217:
827:
39:
2721:
requires 10 spacetime dimensions, and originates from a more fundamental 11-dimensional theory tentatively called
4888:
4253:
3916:
3293:
3150:
2806:
2674:
1235:
4755:
4837:
4709:
4416:
3953:
3901:
2145:
1671:
204:
3223:
2771:
2742:
2681:, as an infinitely small point can have no change and therefore no time. Just as when an object moves through
2117:{\displaystyle {\mathcal {P}}_{0}\subsetneq {\mathcal {P}}_{1}\subsetneq \cdots \subsetneq {\mathcal {P}}_{n}}
4807:
4494:
4411:
4381:
4161:
4156:
4136:
3788:
3213:
3175:
3075:
2936:
2540:
2518:
1750:
1377:
that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an
1358:
630:
310:
167:
1788:
are in some senses the most difficult. This state of affairs was highly marked in the various cases of the
1621:, substantial development of a higher-dimensional geometry only began in the 19th century, via the work of
4765:
4621:
4576:
4146:
4141:
4121:
3145:
3000:
2824:
2782:
2670:
2631:
2361:
2251:
1789:
1681:
1626:
1362:
1303:
706:
417:
295:
180:
3624:
4847:
4802:
4282:
4227:
4151:
4131:
4126:
2971:
2828:
2622:, and equations of quantum mechanics are typically symmetric if both time and other quantities (such as
2451:
2338:
1764:, the theory of manifolds is characterized by the way dimensions 1 and 2 are relatively elementary, the
1450:
1315:
1299:
1295:
478:
439:
398:
393:
246:
2630:) are reversed. In these models, the perception of time flowing in one direction is an artifact of the
2235:
is empty. This definition of covering dimension can be extended from the class of normal spaces to all
1318:(2D) because two coordinates are needed to specify a point on it – for example, both a
1095:
842:
4822:
4750:
4636:
4502:
4464:
4396:
3861:
3778:
3598:
3552:"The Space-Time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics"
3278:
2746:
2524:
2129:
1638:
1446:
1146:
1069:
917:
822:
344:
239:
153:
35:
2550:
1662:
The rest of this section examines some of the more important mathematical definitions of dimension.
4699:
4522:
4512:
4361:
4346:
4302:
4028:
4023:
3298:
3288:
2836:
2818:
2726:
2682:
2615:
2488:
2461:
2365:
2349:
2305:
2247:
1988:
1885:
may be defined in various equivalent ways. The most intuitive way is probably the dimension of the
1856:, in which x and y are both real numbers; hence, the complex dimension is half the real dimension.
1586:
1570:
1566:
1454:
1346:
1307:
1279:
1275:
1151:
1008:
862:
767:
657:
528:
518:
381:
256:
251:
234:
209:
197:
149:
144:
125:
1489:
is one, as a point can move on a line in only one direction (or its opposite); the dimension of a
120:
4878:
4832:
4689:
4542:
4356:
4292:
4202:
4043:
3998:
3857:
3571:
3493:
3115:
3050:
2951:
2926:
2718:
2706:
2654:
2650:
2513:
2422:
1967:
1761:
1708:
1544:
1532:. When trying to generalize to other types of spaces, one is faced with the question "what makes
1477:
of a point that moves on this object. In other words, the dimension is the number of independent
1474:
1398:
1394:
1374:
1110:
837:
677:
305:
229:
219:
190:
175:
3514:
1630:
4827:
4735:
4596:
4571:
4386:
4297:
4277:
4038:
3867:
3843:
3837:
3816:
3792:
3764:
3738:
3719:
3685:
3530:
3450:
3390:
3208:
2960:
2642:
2385:
2369:
1882:
1811:
1801:
1618:
1410:
1339:
1181:
1171:
1100:
969:
947:
897:
872:
807:
731:
386:
278:
224:
185:
50:
3461:
3382:
4842:
4740:
4517:
4484:
4469:
4351:
4220:
3968:
3677:
3606:
3563:
3522:
3387:
The Shape of Inner Space: String Theory and the
Geometry of the Universe's Hidden Dimensions
3110:
3060:
3045:
2995:
2759:
2710:
2627:
2042:
1823:
1634:
1573:, but there are also other answers to that question. For example, the boundary of a ball in
1490:
1366:
1287:
1161:
902:
612:
490:
425:
283:
268:
133:
2326:
is nontrivial. Intuitively, this can be described as follows: if the original space can be
4812:
4760:
4704:
4684:
4586:
4474:
4341:
4312:
4013:
3958:
3754:
3699:
3632:
3479:
3271:
3015:
2946:
2799:
2763:
2690:
2678:
2666:
2646:
2393:
2236:
2038:
2004:
1964:
of sub-varieties of the given algebraic set (the length of such a chain is the number of "
1831:
1512:
1505:
1486:
1438:
1434:
1382:
1331:
1291:
584:
457:
447:
290:
273:
214:
3335:
1688:
the number of coordinates necessary to specify any vector. This notion of dimension (the
1156:
1125:
1059:
1029:
907:
852:
847:
787:
3602:
2360:
and, unlike the dimensions considered above, can also have non-integer real values. The
4852:
4817:
4797:
4714:
4547:
4537:
4527:
4449:
4421:
4406:
4391:
4307:
4095:
4080:
3303:
3250:
3090:
3065:
2931:
2255:
2018:
1864:
1850:
1728:
1652:
1458:
1414:
1212:
1186:
1120:
1064:
937:
817:
797:
777:
682:
2472:
4872:
4789:
4694:
4606:
4479:
4085:
3610:
3245:
3228:
2861:
2767:
2608:
2381:
1902:
1886:
1754:
1720:
1622:
1191:
1176:
1105:
922:
882:
832:
607:
570:
537:
375:
371:
3889:
30:
This article is about the dimension of a space. For the dimension of an object, see
4857:
4661:
4646:
4611:
4459:
4444:
4105:
4070:
3963:
3203:
3125:
2754:
2357:
2312:
2133:
1819:
1731:
1677:
1402:
1130:
1079:
892:
747:
662:
452:
78:
62:
45:
2372:
that work for highly irregular sets and attain non-integer positive real values.
2197:
a manifold, this coincides with the dimension mentioned above. If no such integer
3827:
3782:
3758:
3732:
3472:
3440:
1342:(3D) because three coordinates are needed to locate a point within these spaces.
17:
4745:
4719:
4641:
4330:
4269:
4190:
3973:
3897:
3833:
3806:
3526:
3100:
3030:
2885:
2466:
2389:
2046:
1704:
1689:
1482:
1470:
1283:
1267:
1166:
1039:
857:
792:
720:
692:
667:
1397:
describe spacetime with matter and gravity. 10 dimensions are used to describe
69:
4626:
4185:
4065:
3784:
Surfing through
Hyperspace: Understanding Higher Universes in Six Easy Lessons
3760:
Beyond the Third
Dimension: Geometry, Computer Graphics, and Higher Dimensions
3515:"Definitions for The Fourth Dimension: A Proposed Time Classification System1"
3356:
3165:
3160:
3155:
3120:
3105:
3095:
2877:
2781:
subspace. Thus, the extra dimensions need not be small and compact but may be
2714:
2623:
2619:
2320:
2171:
1894:
1648:
1024:
1003:
993:
983:
942:
887:
782:
772:
672:
523:
73:
The first four spatial dimensions, represented in a two-dimensional picture.
2657:), which treats perceived space and time as components of a four-dimensional
4552:
4075:
3988:
3519:
Proceedings of the 1988 Academy of
Marketing Science (AMS) Annual Conference
3140:
2662:
2604:
1843:
1806:
1599:
1478:
1370:
1323:
1034:
752:
715:
579:
551:
99:
58:
2368:
is a variant of the same idea. In general, there exist more definitions of
2279:
direction. The inductive dimension of a topological space may refer to the
3589:
Brandenberger, R.; Vafa, C. (1989). "Superstrings in the early universe".
4631:
4616:
4090:
4053:
3978:
3703:
3309:
3233:
3198:
3193:
2722:
2658:
2327:
1724:
1656:
1406:
1319:
1115:
1074:
1044:
932:
927:
877:
602:
561:
509:
403:
366:
112:
3575:
4325:
4287:
4100:
3714:
3674:
Computational and
Algorithmic Linear Algebra and n-Dimensional Geometry
3551:
3055:
3005:
2864:
the intervening shape of the line as straight- or curved-line segments.
2786:
2750:
2733:
2635:
2353:
2331:
2164:
1957:{\displaystyle V_{0}\subsetneq V_{1}\subsetneq \cdots \subsetneq V_{d}}
1777:
are simplified by having extra space in which to "work"; and the cases
1595:, they turn out to be different when one looks at more general spaces.
1430:
1386:
1263:
1049:
762:
556:
500:
300:
4651:
4243:
3567:
3498:
3130:
3035:
2876:(3-dimensional) represented using a variety of strategies, such as a
2685:
in space, it also moves through positions in time. In this sense the
2673:. Time is not however present in a single point of absolute infinite
1860:
1685:
1501:
1311:
998:
988:
867:
812:
687:
650:
638:
593:
546:
464:
129:
85:
2745:
presented 5D including an extra dimension of space. At the level of
2498:
1893:. Another intuitive way is to define the dimension as the number of
61:. The square is two-dimensional (2D) and bounded by one-dimensional
2239:
merely by replacing the term "open" in the definition by the term "
4057:
3020:
2791:
2732:
2686:
2477:
2426:
2414:
2287:, and is based on the analogy that, in the case of metric spaces,
2262:, one obtains a 2-dimensional object. In general, one obtains an (
1805:
1565:
such small balls. This observation leads to the definition of the
1497:
1350:
1054:
978:
912:
757:
361:
356:
68:
44:
3681:
2352:
is useful for studying structurally complicated sets, especially
1473:, the dimension of an object is, roughly speaking, the number of
3080:
2410:
1420:
The concept of dimension is not restricted to physical objects.
1354:
1335:
645:
495:
92:
54:
31:
4216:
3912:
2725:
which subsumes five previously distinct superstring theories.
2388:, and any two such bases for a particular space have the same
3421:, Progr. Math., vol. 201, BirkhĂ€user, pp. 349â359,
1659:
in 1843 marked the beginning of higher-dimensional geometry.
1326:
are required to locate a point on the surface of a sphere. A
3908:
3482:, Boston University Department of Mathematics and Statistics
2634:(we perceive time as flowing in the direction of increasing
2603:, is a dimension of time. Time is often referred to as the "
2103:
2080:
2063:
3839:
Hyperspace, a
Scientific Odyssey Through the 10th Dimension
4212:
1606:", mathematicians usually express this as: "The tesseract
3812:
The Fourth
Dimension: Toward a Geometry of Higher Reality
2809:
if all fields are equally free to propagate within them.
2396:
is finite, and in this case the two dimensions coincide.
3556:
Proceedings of the
American Academy of Arts and Sciences
1727:
can be calculated. A connected topological manifold is
84:
Two parallel line segments can be connected to form a
1617:
Although the notion of higher dimensions goes back to
2059:
1970:
1911:
1792:, in which four different proof methods are applied.
1365:
but not the one that was found necessary to describe
3734:
The Annotated Flatland: A Romance of Many Dimensions
4788:
4728:
4677:
4670:
4562:
4493:
4430:
4374:
4321:
4268:
4261:
4178:
4114:
4052:
4006:
3946:
2798:Some aspects of brane physics have been applied to
2641:The best-known treatment of time as a dimension is
2136:is finite if and only if its Krull dimension is 0.
1700:to distinguish it from other notions of dimension.
1543:-dimensional?" One answer is that to cover a fixed
2116:
1976:
1956:
1508:of dimension lower than two, unless it is a line.
3513:Lane, Paul M.; Lindquist, Jay D. (May 22, 2015).
1401:(6D hyperspace + 4D), 11 dimensions can describe
1282:) is informally defined as the minimum number of
91:Two parallel squares can be connected to form a
2884:Frequently in these systems, especially GIS and
98:Two parallel cubes can be connected to form a
4228:
3924:
3667:"1. Systems of Simultaneous Linear Equations"
2356:. The Hausdorff dimension is defined for all
1753:, the dimension is also the dimension of the
1409:(7D hyperspace + 4D), and the state-space of
1243:
8:
3649:Essentials of Geographic Information Systems
3550:Wilson, Edwin B.; Lewis, Gilbert N. (1912).
2315:, the dimension of an object is the largest
1707:case, this generalizes to the notion of the
3334:. Curious.astro.cornell.edu. Archived from
2209:is said to be infinite, and one writes dim
4674:
4265:
4235:
4221:
4213:
3931:
3917:
3909:
3171:Rotations in 4-dimensional Euclidean space
2846:(0-dimensional), a single coordinate in a
2766:. Therefore, these models still require a
2409:Classical physics theories describe three
1250:
1236:
965:
484:
119:
108:
3497:
3439:Hurewicz, Witold; Wallman, Henry (2015).
3419:European Congress of Mathematics Volume I
2839:corresponding to the spatial dimensions:
2108:
2102:
2101:
2085:
2079:
2078:
2068:
2062:
2061:
2058:
1969:
1948:
1929:
1916:
1910:
1863:, when given a complex metric, becomes a
1822:, it is sometimes useful in the study of
77:Two points can be connected to create a
4592:Covariance and contravariance of vectors
2435:
1719:The uniquely defined dimension of every
1692:of a basis) is often referred to as the
1569:and its more sophisticated variant, the
1429:frequently occur in mathematics and the
1385:first approximates the universe without
3323:
1610:", or: "The dimension of the tesseract
1204:
1138:
1087:
1016:
968:
730:
592:
569:
536:
508:
111:
34:. For the dimension of a quantity, see
3705:Flatland: A Romance of Many Dimensions
3381:Yau, Shing-Tung; Nadis, Steve (2010).
1357:are different categories and refer to
350:Straightedge and compass constructions
3359:. Mathworld.wolfram.com. 2014-02-27.
2813:In computer graphics and spatial data
2737:Illustration of a CalabiâYau manifold
1987:Each variety can be considered as an
1891:Regular point of an algebraic variety
7:
2269:)-dimensional object by dragging an
1585:and this leads to the notion of the
1361:. That conception of the world is a
2665:, and in the special, flat case as
2170:for which the following holds: any
2045:is the maximal length of chains of
4455:Tensors in curvilinear coordinates
3442:Dimension Theory (PMS-4), Volume 4
1647:, and Hamilton's discovery of the
1644:Theorie der vielfachen KontinuitÀt
1330:is a two-dimensional space on the
25:
1877:Dimension of an algebraic variety
316:Noncommutative algebraic geometry
3997:
2805:Extra dimensions are said to be
2572:
2561:
2549:
2523:
2512:
2497:
2471:
2460:
1680:is the number of vectors in any
27:Property of a mathematical space
3763:. Scientific American Library.
3676:. World Scientific Publishing.
3428:from the original on 2006-01-17
3363:from the original on 2014-03-25
1589:. While these notions agree on
1328:two-dimensional Euclidean space
3181:Fourth dimension in literature
2833:Geographic information systems
2749:, KaluzaâKlein theory unifies
2620:symmetric with respect to time
2203:exists, then the dimension of
2163:is defined to be the smallest
1369:. The four dimensions (4D) of
709:- / other-dimensional
1:
4508:Exterior covariant derivative
4440:Tensor (intrinsic definition)
3517:. In Bahn, Kenneth D. (ed.).
3389:. Basic Books. pp. 60â.
2413:: from a particular point in
1746:is the manifold's dimension.
4533:Raising and lowering indices
3611:10.1016/0550-3213(89)90037-0
2446:Example co-ordinate systems
2332:higher-dimensional triangles
2311:Similarly, for the class of
1740:-space, in which the number
1559:, one needs on the order of
1338:, a cylinder or a sphere is
4771:Gluon field strength tensor
3842:. Oxford University Press.
3527:10.1007/978-3-319-17046-6_8
3284:Hyperspace (disambiguation)
3071:Cartesian coordinate system
2986:List of topics by dimension
2848:Cartesian coordinate system
2795:higher-dimensional volume.
2545:
2456:
2431:Cartesian coordinate system
2222:has dimension â1, i.e. dim
2181:elements. In this case dim
2155:Lebesgue covering dimension
1977:{\displaystyle \subsetneq }
1834:instead. A complex number (
1413:is an infinite-dimensional
1391:pseudo-Riemannian manifolds
4915:
4582:Cartan formalism (physics)
4402:Penrose graphical notation
3539:– via Springer Link.
3447:Princeton University Press
3410:Fantechi, Barbara (2001),
3267:Dimension (data warehouse)
2890:dimensional generalization
2816:
2336:
1995:is a variety of dimension
1874:
1867:of one complex dimension.
1799:
1669:
40:Dimension (disambiguation)
29:
4254:Glossary of tensor theory
4250:
4199:
3995:
3708:. London: Seely & Co.
3332:"Curious About Astronomy"
3294:Multidimensional analysis
3151:Convex regular 4-polytope
3101:Stereoscopy (3-D imaging)
2285:large inductive dimension
2281:small inductive dimension
2275:-dimensional object in a
2049:in it, a chain of length
1814:mapped to the other pole.
1553:by small balls of radius
4838:Gregorio Ricci-Curbastro
4710:Riemann curvature tensor
4417:Van der Waerden notation
3902:University of Nottingham
3665:Murty, Katta G. (2014).
3383:"4. Too Good to be True"
3146:Fourth spatial dimension
2671:fourth spatial dimension
2146:normal topological space
1751:differentiable manifolds
1672:Dimension (vector space)
205:Non-Archimedean geometry
4808:Elwin Bruno Christoffel
4741:Angular momentum tensor
4412:Tetrad (index notation)
4382:Abstract index notation
3815:. Courier Corporation.
3789:Oxford University Press
3214:Curse of dimensionality
3176:Fourth dimension in art
3076:List of uniform tilings
2937:Isoperimetric dimension
2254:as follows. Consider a
1359:absolute space and time
311:Noncommutative geometry
4622:Levi-Civita connection
3651:, Saylor Academy, 2012
3412:"Stacks for everybody"
3357:"MathWorld: Dimension"
3001:Zero-dimensional space
2783:large extra dimensions
2738:
2632:laws of thermodynamics
2530:Latitude and longitude
2118:
1978:
1958:
1815:
1627:William Rowan Hamilton
1424:High-dimensional space
1363:four-dimensional space
1286:needed to specify any
279:Discrete/Combinatorial
106:
66:
49:From left to right: a
38:. For other uses, see
4899:Mathematical concepts
4894:Geometric measurement
4848:Jan Arnoldus Schouten
4803:Augustin-Louis Cauchy
4283:Differential geometry
3888:Copeland, Ed (2009).
3779:Pickover, Clifford A.
3026:Graph (combinatorics)
2914:physics and chemistry
2829:Computer-aided design
2825:illustration software
2736:
2701:Additional dimensions
2339:dimension of a scheme
2330:into a collection of
2328:continuously deformed
2119:
1979:
1959:
1809:
1457:, independent of the
1451:Hamiltonian mechanics
262:Discrete differential
72:
48:
4823:Carl Friedrich Gauss
4756:stressâenergy tensor
4751:Cauchy stress tensor
4503:Covariant derivative
4465:Antisymmetric tensor
4397:Multi-index notation
4115:Dimensions by number
3863:Hiding in the Mirror
3279:Dimensional analysis
2837:geometric primitives
2747:quantum field theory
2130:algebra over a field
2057:
1968:
1909:
1881:The dimension of an
1755:tangent vector space
1639:Habilitationsschrift
1443:configuration spaces
36:Dimensional analysis
4884:Physical quantities
4700:Nonmetricity tensor
4555:(2nd-order tensors)
4523:Hodge star operator
4513:Exterior derivative
4362:Transport phenomena
4347:Continuum mechanics
4303:Multilinear algebra
3858:Krauss, Lawrence M.
3755:Banchoff, Thomas F.
3625:Brane Gas Cosmology
3603:1989NuPhB.316..391B
3299:Space-filling curve
3289:Intrinsic dimension
3224:KaluzaâKlein theory
2903:Degrees of freedom
2819:Geometric primitive
2779:(3 + 1)-dimensional
2772:CalabiâYau manifold
2743:KaluzaâKlein theory
2727:Supergravity theory
2616:classical mechanics
2557:(three-dimensional)
2411:physical dimensions
2366:Minkowski dimension
2350:Hausdorff dimension
2344:Hausdorff dimension
2248:inductive dimension
2132:, the dimension as
1828:algebraic varieties
1790:Poincaré conjecture
1698:algebraic dimension
1676:The dimension of a
1655:' discovery of the
1604:has four dimensions
1587:inductive dimension
1579:looks locally like
1571:Hausdorff dimension
1567:Minkowski dimension
1347:classical mechanics
1290:within it. Thus, a
529:Pythagorean theorem
4833:Tullio Levi-Civita
4776:Metric tensor (GR)
4690:Levi-Civita symbol
4543:Tensor contraction
4357:General relativity
4293:Euclidean geometry
4044:Degrees of freedom
3947:Dimensional spaces
3890:"Extra Dimensions"
3645:Vector Data Models
3631:2014-10-27 at the
3578:– via JSTOR.
3478:2006-10-27 at the
3462:Extract of page 24
3187:Higher dimensions
2927:Exterior dimension
2739:
2719:superstring theory
2707:fundamental forces
2655:general relativity
2651:special relativity
2597:temporal dimension
2423:linear combination
2405:Spatial dimensions
2370:fractal dimensions
2140:Topological spaces
2114:
1974:
1954:
1816:
1762:geometric topology
1709:length of a module
1641:, SchlÀfli's 1852
1475:degrees of freedom
1399:superstring theory
1395:general relativity
1334:. The inside of a
1276:mathematical space
107:
67:
4866:
4865:
4828:Hermann Grassmann
4784:
4783:
4736:Moment of inertia
4597:Differential form
4572:Affine connection
4387:Einstein notation
4370:
4369:
4298:Exterior calculus
4278:Coordinate system
4210:
4209:
4019:Lebesgue covering
3984:Algebraic variety
3873:978-0-670-03395-9
3849:978-0-19-286189-4
3822:978-0-486-77978-2
3798:978-0-19-992381-6
3770:978-0-7167-6015-3
3744:978-0-7867-2183-2
3720:Project Gutenberg
3691:978-981-4366-62-5
3591:Nuclear Physics B
3536:978-3-319-17045-9
3473:Fractal Dimension
3456:978-1-4008-7566-5
3396:978-0-465-02266-3
3209:Plane of rotation
2653:(and extended to
2588:
2587:
2584:
2583:
2535:
2534:
2508:(two-dimensional)
2483:
2482:
2386:orthonormal basis
2241:functionally open
2053:being a sequence
1883:algebraic variety
1861:spherical surface
1830:to work over the
1824:complex manifolds
1812:point at infinity
1802:Complex dimension
1796:Complex dimension
1637:. Riemann's 1854
1511:The dimension of
1411:quantum mechanics
1340:three-dimensional
1260:
1259:
1225:
1224:
948:List of geometers
631:Three-dimensional
620:
619:
18:Higher dimensions
16:(Redirected from
4906:
4889:Abstract algebra
4843:Bernhard Riemann
4675:
4518:Exterior product
4485:Two-point tensor
4470:Symmetric tensor
4352:Electromagnetism
4266:
4237:
4230:
4223:
4214:
4007:Other dimensions
4001:
3969:Projective space
3933:
3926:
3919:
3910:
3905:
3877:
3866:. Viking Press.
3853:
3826:
3802:
3774:
3748:
3723:
3709:
3700:Abbott, Edwin A.
3695:
3671:
3652:
3642:
3636:
3621:
3615:
3614:
3586:
3580:
3579:
3568:10.2307/20022840
3547:
3541:
3540:
3510:
3504:
3503:
3501:
3489:
3483:
3470:
3464:
3460:
3436:
3430:
3429:
3427:
3416:
3407:
3401:
3400:
3378:
3372:
3371:
3369:
3368:
3353:
3347:
3346:
3344:
3343:
3328:
3272:Dimension tables
3111:Axis of rotation
2978:
2967:
2942:Metric dimension
2780:
2760:electromagnetism
2717:. Most notably,
2711:extra dimensions
2677:as defined as a
2609:in one direction
2605:fourth dimension
2576:
2565:
2558:
2553:
2546:
2527:
2516:
2509:
2501:
2494:
2493:
2475:
2464:
2457:
2436:
2323:
2318:
2303:
2297:
2295:
2274:
2268:
2237:Tychonoff spaces
2234:
2228:
2221:
2215:
2208:
2202:
2196:
2190:
2180:
2162:
2152:
2123:
2121:
2120:
2115:
2113:
2112:
2107:
2106:
2090:
2089:
2084:
2083:
2073:
2072:
2067:
2066:
2043:commutative ring
1983:
1981:
1980:
1975:
1963:
1961:
1960:
1955:
1953:
1952:
1934:
1933:
1921:
1920:
1787:
1783:
1776:
1775:
1766:high-dimensional
1745:
1739:
1635:Bernhard Riemann
1594:
1584:
1578:
1564:
1558:
1552:
1542:
1537:
1531:
1525:
1518:
1439:parameter spaces
1437:or more general
1435:Euclidean spaces
1426:
1425:
1367:electromagnetism
1316:dimension of two
1296:dimension of one
1252:
1245:
1238:
966:
485:
418:Zero-dimensional
123:
109:
21:
4914:
4913:
4909:
4908:
4907:
4905:
4904:
4903:
4869:
4868:
4867:
4862:
4813:Albert Einstein
4780:
4761:Einstein tensor
4724:
4705:Ricci curvature
4685:Kronecker delta
4671:Notable tensors
4666:
4587:Connection form
4564:
4558:
4489:
4475:Tensor operator
4432:
4426:
4366:
4342:Computer vision
4335:
4317:
4313:Tensor calculus
4257:
4246:
4241:
4211:
4206:
4195:
4174:
4110:
4048:
4002:
3993:
3959:Euclidean space
3942:
3937:
3887:
3884:
3874:
3856:
3850:
3832:
3823:
3805:
3799:
3777:
3771:
3753:
3745:
3737:. Basic Books.
3726:
3712:
3698:
3692:
3669:
3664:
3661:
3659:Further reading
3656:
3655:
3643:
3639:
3633:Wayback Machine
3622:
3618:
3588:
3587:
3583:
3562:(11): 389â507.
3549:
3548:
3544:
3537:
3512:
3511:
3507:
3491:
3490:
3486:
3480:Wayback Machine
3471:
3467:
3457:
3438:
3437:
3433:
3425:
3414:
3409:
3408:
3404:
3397:
3380:
3379:
3375:
3366:
3364:
3355:
3354:
3350:
3341:
3339:
3330:
3329:
3325:
3320:
3315:
3262:
3257:
3190:in mathematics
2988:
2983:
2970:
2959:
2947:Order dimension
2899:
2897:More dimensions
2821:
2815:
2778:
2764:quantum gravity
2709:by introducing
2703:
2679:geometric point
2667:Minkowski space
2593:
2577:
2566:
2556:
2555:Cartesian
2554:
2528:
2517:
2507:
2502:
2476:
2465:
2443:
2441:
2407:
2402:
2394:Hamel dimension
2378:
2346:
2341:
2321:
2316:
2299:
2290:
2288:
2270:
2263:
2250:may be defined
2230:
2229:if and only if
2223:
2217:
2210:
2204:
2198:
2192:
2182:
2175:
2158:
2148:
2142:
2100:
2077:
2060:
2055:
2054:
2039:Krull dimension
2035:
2033:Krull dimension
2005:algebraic group
1989:algebraic stack
1966:
1965:
1944:
1925:
1912:
1907:
1906:
1879:
1873:
1832:complex numbers
1804:
1798:
1785:
1778:
1770:
1769:
1741:
1735:
1717:
1694:Hamel dimension
1684:for the space,
1674:
1668:
1631:Ludwig SchlÀfli
1608:has dimension 4
1590:
1580:
1574:
1560:
1554:
1548:
1538:
1533:
1527:
1521:
1514:
1506:Euclidean space
1467:
1455:abstract spaces
1423:
1422:
1383:Minkowski space
1256:
1227:
1226:
963:
962:
953:
952:
743:
742:
726:
725:
711:
710:
698:
697:
634:
633:
622:
621:
482:
481:
479:Two-dimensional
470:
469:
443:
442:
440:One-dimensional
431:
430:
421:
420:
409:
408:
342:
341:
340:
323:
322:
171:
170:
159:
136:
105:
43:
28:
23:
22:
15:
12:
11:
5:
4912:
4910:
4902:
4901:
4896:
4891:
4886:
4881:
4871:
4870:
4864:
4863:
4861:
4860:
4855:
4853:Woldemar Voigt
4850:
4845:
4840:
4835:
4830:
4825:
4820:
4818:Leonhard Euler
4815:
4810:
4805:
4800:
4794:
4792:
4790:Mathematicians
4786:
4785:
4782:
4781:
4779:
4778:
4773:
4768:
4763:
4758:
4753:
4748:
4743:
4738:
4732:
4730:
4726:
4725:
4723:
4722:
4717:
4715:Torsion tensor
4712:
4707:
4702:
4697:
4692:
4687:
4681:
4679:
4672:
4668:
4667:
4665:
4664:
4659:
4654:
4649:
4644:
4639:
4634:
4629:
4624:
4619:
4614:
4609:
4604:
4599:
4594:
4589:
4584:
4579:
4574:
4568:
4566:
4560:
4559:
4557:
4556:
4550:
4548:Tensor product
4545:
4540:
4538:Symmetrization
4535:
4530:
4528:Lie derivative
4525:
4520:
4515:
4510:
4505:
4499:
4497:
4491:
4490:
4488:
4487:
4482:
4477:
4472:
4467:
4462:
4457:
4452:
4450:Tensor density
4447:
4442:
4436:
4434:
4428:
4427:
4425:
4424:
4422:Voigt notation
4419:
4414:
4409:
4407:Ricci calculus
4404:
4399:
4394:
4392:Index notation
4389:
4384:
4378:
4376:
4372:
4371:
4368:
4367:
4365:
4364:
4359:
4354:
4349:
4344:
4338:
4336:
4334:
4333:
4328:
4322:
4319:
4318:
4316:
4315:
4310:
4308:Tensor algebra
4305:
4300:
4295:
4290:
4288:Dyadic algebra
4285:
4280:
4274:
4272:
4263:
4259:
4258:
4251:
4248:
4247:
4242:
4240:
4239:
4232:
4225:
4217:
4208:
4207:
4200:
4197:
4196:
4194:
4193:
4188:
4182:
4180:
4176:
4175:
4173:
4172:
4164:
4159:
4154:
4149:
4144:
4139:
4134:
4129:
4124:
4118:
4116:
4112:
4111:
4109:
4108:
4103:
4098:
4096:Cross-polytope
4093:
4088:
4083:
4081:Hyperrectangle
4078:
4073:
4068:
4062:
4060:
4050:
4049:
4047:
4046:
4041:
4036:
4031:
4026:
4021:
4016:
4010:
4008:
4004:
4003:
3996:
3994:
3992:
3991:
3986:
3981:
3976:
3971:
3966:
3961:
3956:
3950:
3948:
3944:
3943:
3938:
3936:
3935:
3928:
3921:
3913:
3907:
3906:
3883:
3882:External links
3880:
3879:
3878:
3872:
3854:
3848:
3830:
3828:Google preview
3821:
3803:
3797:
3775:
3769:
3751:
3750:
3749:
3743:
3724:
3696:
3690:
3660:
3657:
3654:
3653:
3637:
3623:Scott Watson,
3616:
3597:(2): 391â410.
3581:
3542:
3535:
3505:
3484:
3465:
3455:
3449:. p. 24.
3431:
3402:
3395:
3373:
3348:
3322:
3321:
3319:
3316:
3314:
3313:
3306:
3304:Mean dimension
3301:
3296:
3291:
3286:
3281:
3276:
3275:
3274:
3263:
3261:
3258:
3256:
3255:
3254:
3253:
3251:Function space
3248:
3240:
3239:
3238:
3237:
3236:
3231:
3226:
3218:
3217:
3216:
3211:
3206:
3201:
3196:
3185:
3184:
3183:
3178:
3173:
3168:
3163:
3158:
3153:
3148:
3143:
3135:
3134:
3133:
3128:
3123:
3118:
3113:
3108:
3103:
3098:
3093:
3091:Platonic solid
3085:
3084:
3083:
3078:
3073:
3068:
3066:Complex number
3063:
3058:
3053:
3048:
3040:
3039:
3038:
3033:
3028:
3023:
3018:
3010:
3009:
3008:
3003:
2998:
2989:
2987:
2984:
2982:
2981:
2980:
2979:
2968:
2949:
2944:
2939:
2934:
2932:Hurst exponent
2929:
2924:
2923:
2922:
2916:
2910:
2900:
2898:
2895:
2882:
2881:
2871:
2865:
2851:
2817:Main article:
2814:
2811:
2702:
2699:
2601:time dimension
2592:
2589:
2586:
2585:
2582:
2581:
2570:
2559:
2543:
2537:
2536:
2533:
2532:
2521:
2510:
2491:
2485:
2484:
2481:
2480:
2469:
2454:
2448:
2447:
2444:
2439:
2421:, moving in a
2406:
2403:
2401:
2398:
2377:
2376:Hilbert spaces
2374:
2345:
2342:
2319:for which the
2141:
2138:
2111:
2105:
2099:
2096:
2093:
2088:
2082:
2076:
2071:
2065:
2034:
2031:
2021:has dimension
2019:quotient stack
1973:
1951:
1947:
1943:
1940:
1937:
1932:
1928:
1924:
1919:
1915:
1875:Main article:
1872:
1869:
1865:Riemann sphere
1851:imaginary part
1800:Main article:
1797:
1794:
1757:at any point.
1749:For connected
1716:
1713:
1670:Main article:
1667:
1664:
1653:John T. Graves
1619:René Descartes
1466:
1465:In mathematics
1463:
1459:physical space
1433:. They may be
1415:function space
1302:, such as the
1258:
1257:
1255:
1254:
1247:
1240:
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1229:
1228:
1223:
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1136:
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1128:
1123:
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1113:
1108:
1103:
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1027:
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991:
986:
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973:
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945:
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935:
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870:
865:
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845:
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835:
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815:
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805:
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795:
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728:
727:
724:
723:
718:
712:
705:
704:
703:
700:
699:
696:
695:
690:
685:
683:Platonic Solid
680:
675:
670:
665:
660:
655:
654:
653:
642:
641:
635:
629:
628:
627:
624:
623:
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582:
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477:
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369:
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147:
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138:
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124:
116:
115:
104:
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96:
89:
82:
74:
26:
24:
14:
13:
10:
9:
6:
4:
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2:
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4721:
4718:
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4711:
4708:
4706:
4703:
4701:
4698:
4696:
4695:Metric tensor
4693:
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4683:
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4680:
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4608:
4607:Exterior form
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4480:Tensor bundle
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4458:
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4400:
4398:
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4380:
4379:
4377:
4373:
4363:
4360:
4358:
4355:
4353:
4350:
4348:
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4340:
4339:
4337:
4332:
4329:
4327:
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4320:
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4311:
4309:
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4299:
4296:
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4279:
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4271:
4267:
4264:
4260:
4256:
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4238:
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4226:
4224:
4219:
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4215:
4205:
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4198:
4192:
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4177:
4171:
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4148:
4145:
4143:
4140:
4138:
4135:
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4125:
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4120:
4119:
4117:
4113:
4107:
4104:
4102:
4099:
4097:
4094:
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4089:
4087:
4086:Demihypercube
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4000:
3990:
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3899:
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3894:Sixty Symbols
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3725:
3721:
3718:
3717:
3716:Flatland: ...
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3406:
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3398:
3392:
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3384:
3377:
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3362:
3358:
3352:
3349:
3338:on 2014-01-11
3337:
3333:
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3277:
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3264:
3259:
3252:
3249:
3247:
3246:Hilbert space
3244:
3243:
3241:
3235:
3232:
3230:
3229:String theory
3227:
3225:
3222:
3221:
3219:
3215:
3212:
3210:
3207:
3205:
3202:
3200:
3197:
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3186:
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3027:
3024:
3022:
3019:
3017:
3014:
3013:
3011:
3007:
3004:
3002:
2999:
2997:
2994:
2993:
2991:
2990:
2985:
2977:
2975:
2972:Correlation (
2969:
2966:
2964:
2958:
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2956:
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2872:
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2838:
2834:
2830:
2826:
2820:
2812:
2810:
2808:
2803:
2801:
2796:
2793:
2788:
2784:
2775:
2773:
2769:
2768:UV completion
2765:
2761:
2756:
2752:
2748:
2744:
2735:
2731:
2728:
2724:
2720:
2716:
2712:
2708:
2700:
2698:
2696:
2693:to change is
2692:
2688:
2684:
2680:
2676:
2672:
2668:
2664:
2660:
2656:
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2610:
2606:
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2598:
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2560:
2552:
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2511:
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2500:
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2479:
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2463:
2459:
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2428:
2424:
2420:
2416:
2412:
2404:
2399:
2397:
2395:
2391:
2387:
2383:
2382:Hilbert space
2375:
2373:
2371:
2367:
2363:
2362:box dimension
2359:
2358:metric spaces
2355:
2351:
2343:
2340:
2335:
2333:
2329:
2325:
2314:
2309:
2307:
2304:-dimensional
2302:
2296:)-dimensional
2293:
2286:
2282:
2278:
2273:
2266:
2261:
2260:new direction
2257:
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2238:
2233:
2226:
2220:
2213:
2207:
2201:
2195:
2189:
2185:
2178:
2173:
2169:
2166:
2161:
2156:
2151:
2147:
2139:
2137:
2135:
2131:
2126:
2109:
2097:
2094:
2091:
2086:
2074:
2069:
2052:
2048:
2044:
2040:
2032:
2030:
2028:
2025: â
2024:
2020:
2016:
2015:
2010:
2007:of dimension
2006:
2002:
1998:
1994:
1990:
1985:
1971:
1949:
1945:
1941:
1938:
1935:
1930:
1926:
1922:
1917:
1913:
1904:
1903:algebraic set
1899:
1896:
1892:
1888:
1887:tangent space
1884:
1878:
1870:
1868:
1866:
1862:
1857:
1855:
1852:
1848:
1845:
1841:
1837:
1833:
1829:
1825:
1821:
1813:
1808:
1803:
1795:
1793:
1791:
1781:
1773:
1767:
1763:
1758:
1756:
1752:
1747:
1744:
1738:
1734:to Euclidean
1733:
1730:
1726:
1722:
1714:
1712:
1710:
1706:
1701:
1699:
1695:
1691:
1687:
1683:
1679:
1673:
1666:Vector spaces
1665:
1663:
1660:
1658:
1654:
1650:
1646:
1645:
1640:
1636:
1632:
1628:
1624:
1623:Arthur Cayley
1620:
1615:
1613:
1609:
1605:
1601:
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1234:
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1038:
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1021:
1020:
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1010:
1007:
1005:
1002:
1000:
997:
995:
992:
990:
987:
985:
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980:
977:
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967:
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956:
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906:
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869:
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846:
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841:
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809:
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801:
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794:
791:
789:
786:
784:
781:
779:
776:
774:
771:
769:
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764:
761:
759:
756:
754:
751:
749:
746:
745:
737:
736:
733:
729:
722:
719:
717:
714:
713:
708:
702:
701:
694:
691:
689:
686:
684:
681:
679:
676:
674:
671:
669:
666:
664:
661:
659:
656:
652:
649:
648:
647:
644:
643:
640:
637:
636:
632:
626:
625:
614:
611:
609:
608:Circumference
606:
604:
601:
600:
599:
598:
595:
591:
586:
583:
581:
578:
577:
576:
575:
572:
571:Quadrilateral
568:
563:
560:
558:
555:
553:
550:
548:
545:
544:
543:
542:
539:
538:Parallelogram
535:
530:
527:
525:
522:
520:
517:
516:
515:
514:
511:
507:
502:
499:
497:
494:
492:
489:
488:
487:
486:
480:
474:
473:
466:
463:
459:
456:
454:
451:
450:
449:
446:
445:
441:
435:
434:
427:
424:
423:
419:
413:
412:
405:
402:
400:
397:
395:
392:
391:
388:
385:
383:
380:
377:
376:Perpendicular
373:
372:Orthogonality
370:
368:
365:
363:
360:
358:
355:
354:
351:
348:
347:
346:
336:
333:
332:
327:
326:
317:
314:
313:
312:
309:
307:
304:
302:
299:
297:
296:Computational
294:
292:
289:
285:
282:
281:
280:
277:
275:
272:
270:
267:
263:
260:
258:
255:
253:
250:
249:
248:
245:
241:
238:
236:
233:
232:
231:
228:
226:
223:
221:
218:
216:
213:
211:
208:
206:
203:
199:
196:
192:
189:
188:
187:
184:
183:
182:
181:Non-Euclidean
179:
177:
174:
173:
169:
163:
162:
155:
151:
148:
146:
143:
142:
140:
139:
135:
131:
127:
122:
118:
117:
114:
110:
101:
97:
94:
90:
87:
83:
80:
76:
75:
71:
64:
63:line segments
60:
56:
52:
47:
41:
37:
33:
19:
4858:Hermann Weyl
4662:Vector space
4647:Pseudotensor
4612:Fiber bundle
4601:
4565:abstractions
4460:Mixed tensor
4445:Tensor field
4252:
4201:
4167:
4166:
4106:Hyperpyramid
4071:Hypersurface
3964:Affine space
3954:Vector space
3939:
3893:
3862:
3838:
3834:Kaku, Michio
3811:
3807:Rucker, Rudy
3783:
3759:
3733:
3729:Stewart, Ian
3715:
3704:
3682:10.1142/8261
3673:
3648:
3640:
3619:
3594:
3590:
3584:
3559:
3555:
3545:
3518:
3508:
3499:math/0702552
3487:
3468:
3441:
3434:
3418:
3405:
3386:
3376:
3365:. Retrieved
3351:
3340:. Retrieved
3336:the original
3326:
3308:
3204:Vector space
3126:Skew polygon
2973:
2962:
2952:
2889:
2883:
2873:
2867:
2857:
2853:
2843:
2822:
2804:
2797:
2776:
2740:
2704:
2694:
2640:
2613:
2600:
2596:
2594:
2418:
2408:
2379:
2347:
2313:CW complexes
2310:
2300:
2291:
2284:
2280:
2276:
2271:
2264:
2259:
2256:discrete set
2245:
2240:
2231:
2224:
2218:
2216:. Moreover,
2211:
2205:
2199:
2193:
2187:
2183:
2176:
2167:
2159:
2149:
2143:
2134:vector space
2127:
2050:
2047:prime ideals
2036:
2026:
2022:
2013:
2008:
2000:
1996:
1992:
1986:
1900:
1880:
1858:
1853:
1846:
1839:
1835:
1820:real numbers
1817:
1779:
1771:
1765:
1759:
1748:
1742:
1736:
1732:homeomorphic
1723:topological
1718:
1703:For the non-
1702:
1697:
1693:
1678:vector space
1675:
1661:
1642:
1616:
1611:
1607:
1603:
1597:
1591:
1581:
1575:
1561:
1555:
1549:
1539:
1534:
1528:
1522:
1515:
1510:
1500:, such as a
1495:
1493:is two etc.
1468:
1453:; these are
1421:
1419:
1403:supergravity
1344:
1271:
1261:
1080:Parameshvara
893:Parameshvara
663:Dodecahedron
247:Differential
79:line segment
4798:Ălie Cartan
4746:Spin tensor
4720:Weyl tensor
4678:Mathematics
4642:Multivector
4433:definitions
4331:Engineering
4270:Mathematics
4191:Codimension
4170:-dimensions
4091:Hypersphere
3974:Free module
3898:Brady Haran
3220:in physics
3031:Real number
2886:Cartography
2862:interpolate
2689:moving any
2675:singularity
2661:, known as
2568:Cylindrical
2467:Number line
2390:cardinality
2298:balls have
2252:inductively
2017:, then the
1895:hyperplanes
1690:cardinality
1649:quaternions
1614:4" or: 4D.
1483:coordinates
1471:mathematics
1445:such as in
1373:consist of
1284:coordinates
1268:mathematics
1205:Present day
1152:Lobachevsky
1139:1700sâ1900s
1096:JyeáčŁáčhadeva
1088:1400sâ1700s
1040:Brahmagupta
863:Lobachevsky
843:JyeáčŁáčhadeva
793:Brahmagupta
721:Hypersphere
693:Tetrahedron
668:Icosahedron
240:Diophantine
4873:Categories
4627:Linear map
4495:Operations
4186:Hyperspace
4066:Hyperplane
3367:2014-03-03
3342:2014-03-03
3318:References
3166:Polychoron
3161:4-manifold
3156:Quaternion
3121:Skew lines
3106:3-manifold
3096:Polyhedron
2955:-dimension
2920:statistics
2878:polyhedron
2715:hyperspace
2442:dimensions
2440:Number of
2400:In physics
2384:admits an
2337:See also:
2306:boundaries
2172:open cover
2012:acting on
1513:Euclidean
1479:parameters
1447:Lagrangian
1065:al-Yasamin
1009:Apollonius
1004:Archimedes
994:Pythagoras
984:Baudhayana
938:al-Yasamin
888:Pythagoras
783:Baudhayana
773:Archimedes
768:Apollonius
673:Octahedron
524:Hypotenuse
399:Similarity
394:Congruence
306:Incidence
257:Symplectic
252:Riemannian
235:Arithmetic
210:Projective
198:Hyperbolic
126:Projecting
4879:Dimension
4766:EM tensor
4602:Dimension
4553:Transpose
4076:Hypercube
4054:Polytopes
4034:Minkowski
4029:Hausdorff
4024:Inductive
3989:Spacetime
3940:Dimension
3809:(2014) .
3727:—;
3713:—.
3242:Infinite
3141:Spacetime
2961:Fractal (
2908:mechanics
2807:universal
2800:cosmology
2741:In 1921,
2683:positions
2663:spacetime
2579:Spherical
2504:Cartesian
2324:-skeleton
2098:⊊
2095:⋯
2092:⊊
2075:⊊
1972:⊊
1942:⊊
1939:⋯
1936:⊊
1923:⊊
1871:Varieties
1844:real part
1721:connected
1715:Manifolds
1657:octonions
1600:tesseract
1371:spacetime
1324:longitude
1272:dimension
1182:Minkowski
1101:Descartes
1035:Aryabhata
1030:KÄtyÄyana
961:by period
873:Minkowski
848:KÄtyÄyana
808:Descartes
753:Aryabhata
732:Geometers
716:Tesseract
580:Trapezoid
552:Rectangle
345:Dimension
230:Algebraic
220:Synthetic
191:Spherical
176:Euclidean
100:tesseract
59:tesseract
4632:Manifold
4617:Geodesic
4375:Notation
4203:Category
4179:See also
3979:Manifold
3900:for the
3860:(2005).
3836:(1994).
3781:(2001).
3757:(1996).
3731:(2008).
3702:(1884).
3629:Archived
3576:20022840
3476:Archived
3423:archived
3361:Archived
3310:Flatland
3260:See also
3234:M-theory
3199:Sedenion
3194:Octonion
2858:Polyline
2787:D-branes
2723:M-theory
2659:manifold
2647:Einstein
2643:Poincaré
2354:fractals
2144:For any
1842:) has a
1725:manifold
1431:sciences
1407:M-theory
1379:observer
1320:latitude
1314:, has a
1308:cylinder
1304:boundary
1172:Poincaré
1116:Minggatu
1075:Yang Hui
1045:Virasena
933:Yang Hui
928:Virasena
898:Poincaré
878:Minggatu
658:Cylinder
603:Diameter
562:Rhomboid
519:Altitude
510:Triangle
404:Symmetry
382:Parallel
367:Diagonal
337:Features
334:Concepts
225:Analytic
186:Elliptic
168:Branches
154:Timeline
113:Geometry
4729:Physics
4563:Related
4326:Physics
4244:Tensors
4101:Simplex
4039:Fractal
3599:Bibcode
3056:Polygon
3051:Surface
3006:Integer
2874:Surface
2868:Polygon
2751:gravity
2636:entropy
2283:or the
2165:integer
2128:For an
1889:at any
1849:and an
1729:locally
1387:gravity
1300:surface
1264:physics
1197:Coxeter
1177:Hilbert
1162:Riemann
1111:Huygens
1070:al-Tusi
1060:KhayyĂĄm
1050:Alhazen
1017:1â1400s
918:al-Tusi
903:Riemann
853:KhayyĂĄm
838:Huygens
833:Hilbert
803:Coxeter
763:Alhazen
741:by name
678:Pyramid
557:Rhombus
501:Polygon
453:segment
301:Fractal
284:Digital
269:Complex
150:History
145:Outline
4657:Vector
4652:Spinor
4637:Matrix
4431:Tensor
4058:shapes
3870:
3846:
3819:
3795:
3767:
3741:
3688:
3635:(pdf).
3574:
3533:
3453:
3393:
3131:Volume
3087:Three
3036:Length
2831:, and
2691:object
2628:parity
2624:charge
2506:
2380:Every
2191:. For
2153:, the
2003:is an
1774:> 4
1768:cases
1519:-space
1502:circle
1389:; the
1375:events
1312:sphere
1294:has a
1280:object
1270:, the
1218:Gromov
1213:Atiyah
1192:Veblen
1187:Cartan
1157:Bolyai
1126:Sakabe
1106:Pascal
999:Euclid
989:Manava
923:Veblen
908:Sakabe
883:Pascal
868:Manava
828:Gromov
813:Euclid
798:Cartan
788:Bolyai
778:Atiyah
688:Sphere
651:cuboid
639:Volume
594:Circle
547:Square
465:Length
387:Vertex
291:Convex
274:Finite
215:Affine
130:sphere
86:square
57:and a
51:square
4577:Basis
4262:Scope
4162:Eight
4157:Seven
4137:Three
4014:Krull
3670:(PDF)
3572:JSTOR
3494:arXiv
3426:(PDF)
3415:(PDF)
3137:Four
3116:Knots
3046:Plane
3021:Curve
2996:Point
2992:Zero
2844:Point
2792:brane
2755:gauge
2753:with
2687:force
2599:, or
2519:Polar
2478:Angle
2427:Space
2415:space
2041:of a
1682:basis
1498:curve
1491:plane
1351:space
1332:plane
1306:of a
1288:point
1274:of a
1167:Klein
1147:Gauss
1121:Euler
1055:Sijzi
1025:Zhang
979:Ahmes
943:Zhang
913:Sijzi
858:Klein
823:Gauss
818:Euler
758:Ahmes
491:Plane
426:Point
362:Curve
357:Angle
134:plane
132:to a
4147:Five
4142:Four
4122:Zero
4056:and
3868:ISBN
3844:ISBN
3817:ISBN
3793:ISBN
3765:ISBN
3739:ISBN
3686:ISBN
3531:ISBN
3451:ISBN
3391:ISBN
3081:Area
3042:Two
3016:Line
3012:One
2976:= 2)
2965:= 1)
2854:Line
2695:time
2645:and
2626:and
2618:are
2591:Time
2429:and
2419:i.e.
2348:The
2227:= â1
2037:The
1999:and
1984:").
1826:and
1784:and
1705:free
1686:i.e.
1651:and
1633:and
1545:ball
1487:line
1405:and
1355:time
1353:and
1336:cube
1322:and
1292:line
1278:(or
1266:and
1131:Aida
748:Aida
707:Four
646:Cube
613:Area
585:Kite
496:Area
448:Line
93:cube
55:cube
53:, a
32:size
4152:Six
4132:Two
4127:One
3678:doi
3607:doi
3595:316
3564:doi
3523:doi
3061:Net
2918:in
2912:in
2906:in
2856:or
2649:'s
2638:).
2433:.)
2364:or
2294:+ 1
2277:new
2267:+ 1
2246:An
2243:".
2214:= â
2179:+ 1
2157:of
1901:An
1782:= 3
1760:In
1696:or
1547:in
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1469:In
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1262:In
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3647:,
3627:.
3605:.
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3385:.
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2785:.
2697:.
2611:.
2595:A
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1711:.
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1625:,
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1461:.
1417:.
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128:a
4236:e
4229:t
4222:v
4168:n
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2713:/
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2317:n
2301:n
2292:n
2289:(
2272:n
2265:n
2232:X
2225:X
2219:X
2212:X
2206:X
2200:n
2194:X
2188:n
2184:X
2177:n
2168:n
2160:X
2150:X
2110:n
2104:P
2087:1
2081:P
2070:0
2064:P
2051:n
2027:n
2023:m
2014:V
2009:n
2001:G
1997:m
1993:V
1950:d
1946:V
1931:1
1927:V
1918:0
1914:V
1854:y
1847:x
1836:x
1786:4
1780:n
1772:n
1743:n
1737:n
1592:E
1582:E
1576:E
1562:Δ
1556:Δ
1550:E
1540:n
1535:E
1529:n
1523:E
1516:n
1427:s
1251:e
1244:t
1237:v
378:)
374:(
156:)
152:(
102:.
95:.
88:.
81:.
42:.
20:)
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