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the construction of almost all the geometric concepts known at the time. However, Euclid's postulation of points was neither complete nor definitive, and he occasionally assumed facts about points that did not follow directly from his axioms, such as the ordering of points on the line or the existence of specific points. In spite of this, modern expansions of the system serve to remove these assumptions.
133:
1748:
In addition to defining points and constructs related to points, Euclid also postulated a key idea about points, that any two points can be connected by a straight line. This is easily confirmed under modern extensions of
Euclidean geometry, and had lasting consequences at its introduction, allowing
1705:
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2464:
of one over the entire real line. The delta function is sometimes thought of as an infinitely high, infinitely thin spike at the origin, with total area one under the spike, and physically represents an idealized
2261:
1883:
1543:
2180:
2276:
2039:
1992:
1948:
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1354:, defined as "that which has no part". Points and other primitive notions are not defined in terms of other concepts, but only by certain formal properties, called
1849:), there is no linearly independent subset. The zero vector is not itself linearly independent, because there is a non-trivial linear combination making it zero:
2012:
1968:
1917:
3804:
2797:
1378:, or pen, whose pointed tip can mark a small dot or prick a small hole representing a point, or can be drawn across a surface to represent a curve.
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2379:
Although the notion of a point is generally considered fundamental in mainstream geometry and topology, there are some systems that forgo it, e.g.
2701:
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2209:
3134:
1284:
2441:
Often in physics and mathematics, it is useful to think of a point as having non-zero mass or charge (this is especially common in
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with respect to the covering dimension because every open cover of the space has a refinement consisting of a single open set.
2673:
384:
100:
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objects, points are usually taken to be the fundamental indivisible elements comprising the space, of which one-dimensional
2411:
in a way that the operation "take a value at this point" may not be defined. A further tradition starts from some books of
72:
1824:
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A point has
Hausdorff dimension 0 because it can be covered by a single ball of arbitrarily small radius.
3222:
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862:
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2020:
1973:
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1700:{\displaystyle L=\lbrace (a_{1},a_{2},...a_{n})\mid a_{1}c_{1}+a_{2}c_{2}+...a_{n}c_{n}=d\rbrace ,}
239:
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2362:{\displaystyle \operatorname {dim} _{\operatorname {H} }(X):=\inf\{d\geq 0:C_{H}^{d}(X)=0\}.}
2015:
513:
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93:
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1842:
1741:, and other related concepts. A line segment consisting of only a single point is called a
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8:
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1329:, and higher-dimensional objects consist; conversely, a point can be determined by the
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872:
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340:
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subset. In a vector space consisting of a single point (which must be the zero vector
1532:
collection of points that conform to certain axioms. This is usually represented by a
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respectively) which looks like a well-known function space on the set: an algebra of
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originally defined the point as "that which has no part". In the two-dimensional
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2995:. 2004 paperback, Prometheus Books. Being the 1919 Tarner Lectures delivered at
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function which is usually defined on a finite domain and takes values 0 and 1.
1402:
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respectively. More precisely, such structures generalize well-known spaces of
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in mathematics. In all of the common definitions, a point is 0-dimensional.
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is the dimension of the space. Similar constructions exist that define the
1478:) with the additional third number representing depth and often denoted by
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on the real number line that is zero everywhere except at zero, with an
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817:
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3192:
3143:
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2445:, where electrons are idealized as points with non-zero charge). The
1389:. In modern mathematics, a space of points is typically treated as a
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Mathematical
Methods For Physicists International Student Edition
2102:
1375:
1326:
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591:
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335:
2049:
exists, the space is said to be of infinite covering dimension.
1385:, points are often defined or represented in terms of numerical
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2577:
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947:
792:
396:
391:
3112:
2912:. Vol. 1 (2nd ed.). New York: Dover Publications.
2430:
3416:
680:
530:
2710:, p. 58, More specifically, see Ā§15. The Ī“ function;
2979:
An
Enquiry Concerning the Principles of Natural Knowledge
2896:
Handbook of
Incidence Geometry: Buildings and Foundations
2721:
2387:. A "pointless" or "pointfree" space is defined not as a
1841:
The dimension of a vector space is the maximum size of a
1528:
Many constructs within
Euclidean geometry consist of an
1482:. Further generalizations are represented by an ordered
132:
1462:. This idea is easily generalized to three-dimensional
1313:, or its generalization to other kinds of mathematical
2816:(1922). "Point, line and surface as sets of solids,".
1466:, where a point is represented by an ordered triplet (
1454:, and the second number conventionally represents the
1406:
is an element of some subset of points which has some
3054:
2601:
2279:
2212:
2121:
2023:
2000:
1976:
1956:
1932:
1905:
1855:
1546:
2613:
2625:
2589:
2419:is assumed as a primitive together with the one of
2256:{\displaystyle \sum _{i\in I}r_{i}^{d}<\delta .}
60:. Unsourced material may be challenged and removed.
2928:
2361:
2255:
2174:
2033:
2006:
1986:
1962:
1942:
1911:
1877:
1699:
1899:The topological dimension of a topological space
3817:
2869:Generalized Functions: Properties and Operations
2661:
2305:
2112:such that there is some (indexed) collection of
1878:{\displaystyle 1\cdot \mathbf {0} =\mathbf {0} }
1413:
3035:
2781:(3rd ed.). New York: McGraw-Hill Series.
1823:There are several inequivalent definitions of
3432:
3128:
2858:
2711:
2473:. It was introduced by theoretical physicist
1278:
2353:
2308:
2169:
2122:
1691:
1553:
1525:of the space in which the point is located.
2755:
2727:
2041:in which no point is included in more than
1791:. Unsourced material may be challenged and
1422:, are one of the most fundamental objects.
1418:Points, considered within the framework of
3800:
3773:
3439:
3425:
3135:
3121:
3006:Process and Reality: An Essay in Cosmology
2779:The Fourier transform and its applications
2485:(or function). Its discrete analog is the
2374:
1410:containing no other points of the subset.
1333:of two curves or three surfaces, called a
1285:
1271:
154:
3002:
2986:
2972:
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2812:
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2687:
2672:sfnp error: no target: CITEREFGerla1985 (
2619:
2607:
2431:Point masses and the Dirac delta function
2175:{\displaystyle \{B(x_{i},r_{i}):i\in I\}}
1830:
1811:Learn how and when to remove this message
1540:is an infinite set of points of the form
120:Learn how and when to remove this message
2947:
2854:(4th ed.). Oxford University Press.
2715:
1888:
1364:that passes through two distinct points"
1305:is an abstract idealization of an exact
131:
2909:The Thirteen Books of Euclid's Elements
2894:. In Buekenhout, F.; Kantor, W (eds.).
1752:
1358:, that they must satisfy; for example,
136:A finite set of points (in red) in the
14:
3818:
2795:
2059:
1919:is defined to be the minimum value of
385:Straightedge and compass constructions
3420:
3116:
3036:
2965:Modern Calculus and Analytic Geometry
2926:
2902:
2886:
2844:
2707:
2667:
2631:
2595:
2583:
1442:) of numbers, where the first number
1789:adding citations to reliable sources
1756:
58:adding citations to reliable sources
29:
2898:. North-Holland. p. 1015ā1031.
2851:The Principles of Quantum Mechanics
24:
2802:Notre Dame Journal of Formal Logic
2285:
2026:
1979:
1935:
25:
3837:
3015:
351:Noncommutative algebraic geometry
3799:
3772:
3762:
3752:
3741:
3731:
3730:
3524:
3201:
3100:
3088:
3076:
3064:
2931:Elementary Geometry for Teachers
2714:, pp. 1ā5, See Ā§Ā§1.1, 1.3;
2045:+1 elements. If no such minimal
1871:
1863:
1761:
34:
2872:. Vol. 1. Academic Press.
2769:(6th ed.). Academic Press.
2481:it is often referred to as the
1430:, a point is represented by an
45:needs additional citations for
2993:. Cambridge: University Press.
2982:. Cambridge: University Press.
2962:Silverman, Richard A. (1969).
2680:
2637:
2344:
2338:
2299:
2293:
2154:
2128:
2034:{\displaystyle {\mathcal {A}}}
1987:{\displaystyle {\mathcal {B}}}
1943:{\displaystyle {\mathcal {A}}}
1604:
1556:
1370:are made with tools such as a
744:- / other-dimensional
27:Fundamental object of geometry
13:
1:
3142:
2748:
2565:Whitehead point-free geometry
3446:
1536:of points; As an example, a
1414:Points in Euclidean geometry
7:
2935:. Reading: Addison-Wesley.
2712:Gelfand & Shilov (1964)
2492:
1970:admits a finite open cover
1895:Lebesgue covering dimension
10:
3842:
3693:Banach fixed-point theorem
2443:classical electromagnetism
2434:
2391:, but via some structure (
1892:
1834:
3726:
3683:
3647:
3533:
3522:
3454:
3403:
3382:
3318:
3256:
3210:
3199:
3150:
2987:Whitehead, A. N. (1920).
2957:(in French). Vol. 1.
2954:ThƩorie des distributions
2927:Ohmer, Merlin M. (1969).
2818:The Journal of Philosophy
2728:Arfken & Weber (2005)
2560:Singular point of a curve
1923:, such that every finite
3003:Whitehead, A. N (1929).
2798:"Individuals and Points"
2571:
1837:Dimension (vector space)
1458:and is often denoted by
1450:and is often denoted by
1366:. As physical diagrams,
240:Non-Archimedean geometry
2796:Clarke, Bowman (1985).
2525:Foundations of geometry
2415:in which the notion of
2381:noncommutative geometry
2375:Geometry without points
346:Noncommutative geometry
3748:Mathematics portal
3648:Metrics and properties
3634:Second-countable space
2889:"Pointless Geometries"
2586:, p. 34–37.
2550:Point set registration
2363:
2257:
2176:
2105:of the set of numbers
2035:
2008:
1988:
1964:
1944:
1913:
1879:
1831:Vector space dimension
1701:
1360:"there is exactly one
314:Discrete/Combinatorial
141:
2990:The Concept of Nature
2364:
2258:
2177:
2036:
2009:
1989:
1965:
1945:
1914:
1889:Topological dimension
1880:
1702:
297:Discrete differential
135:
69:"Point" geometry
3703:Invariance of domain
3655:Euler characteristic
3629:Bundle (mathematics)
3319:Dimensions by number
2775:Bracewell, Ronald N.
2477:. In the context of
2458:generalized function
2456:, is (informally) a
2447:Dirac delta function
2437:Dirac delta function
2401:continuous functions
2277:
2210:
2119:
2021:
1998:
1974:
1954:
1930:
1903:
1853:
1843:linearly independent
1785:improve this section
1753:Dimension of a point
1544:
1381:Since the advent of
54:improve this article
3713:Tychonoff's theorem
3708:PoincarƩ conjecture
3462:General (point-set)
2530:Position (geometry)
2483:unit impulse symbol
2337:
2267:Hausdorff dimension
2243:
2060:Hausdorff dimension
1309:, without size, in
564:Pythagorean theorem
3698:De Rham cohomology
3619:Polyhedral complex
3609:Simplicial complex
3248:Degrees of freedom
3151:Dimensional spaces
3038:Weisstein, Eric W.
2645:"Hilbert's axioms"
2500:Accumulation point
2385:pointless topology
2359:
2323:
2253:
2229:
2228:
2172:
2031:
2004:
1984:
1960:
1940:
1909:
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1729:are constants and
1697:
1420:Euclidean geometry
1348:Euclidean geometry
1325:, two-dimensional
142:
3813:
3812:
3602:fundamental group
3414:
3413:
3223:Lebesgue covering
3188:Algebraic variety
2949:Schwartz, Laurent
2887:Gerla, G (1995).
2757:Arfken, George B.
2535:Point at infinity
2479:signal processing
2213:
2095:Hausdorff content
2007:{\displaystyle X}
1963:{\displaystyle X}
1912:{\displaystyle X}
1821:
1820:
1813:
1383:analytic geometry
1368:geometric figures
1295:
1294:
1260:
1259:
983:List of geometers
666:Three-dimensional
655:
654:
130:
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16:(Redirected from
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3826:Point (geometry)
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3211:Other dimensions
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3173:Projective space
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2983:
2974:Whitehead, A. N.
2969:
2958:
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2934:
2923:
2904:Heath, Thomas L.
2899:
2893:
2883:
2855:
2841:
2809:
2792:
2770:
2743:
2740:Bracewell (1986)
2737:
2731:
2725:
2719:
2705:
2699:
2686:Whitehead (
2684:
2678:
2677:
2665:
2659:
2658:
2657:
2656:
2641:
2635:
2629:
2623:
2620:de Laguna (1922)
2617:
2611:
2608:Silverman (1969)
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2599:
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2587:
2581:
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2368:
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2195:
2181:
2179:
2178:
2173:
2153:
2152:
2140:
2139:
2111:
2088:
2081:
2054:zero-dimensional
2040:
2038:
2037:
2032:
2030:
2029:
2013:
2011:
2010:
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1990:
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1983:
1982:
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1961:
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1947:
1946:
1941:
1939:
1938:
1918:
1916:
1915:
1910:
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1802:
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1481:
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1473:
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1453:
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1437:
1352:primitive notion
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1001:
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453:Zero-dimensional
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125:
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111:
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103:
62:
38:
30:
21:
18:Point (topology)
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3835:
3834:
3832:
3831:
3830:
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3815:
3814:
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3740:
3722:
3718:Urysohn's lemma
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3529:
3520:
3492:low-dimensional
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3445:
3415:
3410:
3399:
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3314:
3252:
3206:
3197:
3163:Euclidean space
3146:
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3101:
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3095:Systems science
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3087:
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3055:
3021:
3018:
3013:
2997:Trinity College
2920:
2891:
2880:
2864:Shilov, Georgiy
2860:Gelfand, Israel
2830:10.2307/2939504
2824:(17): 449ā461.
2789:
2751:
2746:
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2734:
2726:
2722:
2716:Schwartz (1950)
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2590:
2582:
2578:
2574:
2569:
2495:
2487:Kronecker delta
2451:
2439:
2433:
2413:A. N. Whitehead
2405:algebra of sets
2377:
2332:
2327:
2284:
2280:
2278:
2275:
2274:
2238:
2233:
2217:
2211:
2208:
2207:
2206:that satisfies
2197:
2192:
2187:
2148:
2144:
2135:
2131:
2120:
2117:
2116:
2106:
2083:
2073:
2062:
2025:
2024:
2022:
2019:
2018:
1999:
1996:
1995:
1978:
1977:
1975:
1972:
1971:
1955:
1952:
1951:
1934:
1933:
1931:
1928:
1927:
1904:
1901:
1900:
1897:
1891:
1870:
1862:
1854:
1851:
1850:
1839:
1833:
1817:
1806:
1800:
1797:
1782:
1766:
1755:
1730:
1726:
1722:
1717:
1714:
1708:
1679:
1675:
1669:
1665:
1647:
1643:
1637:
1633:
1624:
1620:
1614:
1610:
1598:
1594:
1576:
1572:
1563:
1559:
1545:
1542:
1541:
1518:
1514:
1505:
1498:
1491:
1487:
1479:
1475:
1471:
1467:
1464:Euclidean space
1459:
1451:
1446:represents the
1439:
1435:
1428:Euclidean plane
1416:
1350:, a point is a
1291:
1262:
1261:
998:
997:
988:
987:
778:
777:
761:
760:
746:
745:
733:
732:
669:
668:
657:
656:
517:
516:
514:Two-dimensional
505:
504:
478:
477:
475:One-dimensional
466:
465:
456:
455:
444:
443:
377:
376:
375:
358:
357:
206:
205:
194:
171:
138:Euclidean plane
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
3839:
3829:
3828:
3811:
3810:
3808:
3807:
3797:
3796:
3795:
3790:
3785:
3770:
3760:
3750:
3738:
3727:
3724:
3723:
3721:
3720:
3715:
3710:
3705:
3700:
3695:
3689:
3687:
3681:
3680:
3678:
3677:
3672:
3667:
3665:Winding number
3662:
3657:
3651:
3649:
3645:
3644:
3642:
3641:
3636:
3631:
3626:
3621:
3616:
3611:
3606:
3605:
3604:
3599:
3597:homotopy group
3589:
3588:
3587:
3582:
3577:
3572:
3567:
3557:
3552:
3547:
3537:
3535:
3531:
3530:
3523:
3521:
3519:
3518:
3513:
3508:
3507:
3506:
3496:
3495:
3494:
3484:
3479:
3474:
3469:
3464:
3458:
3456:
3452:
3451:
3444:
3443:
3436:
3429:
3421:
3412:
3411:
3404:
3401:
3400:
3398:
3397:
3392:
3386:
3384:
3380:
3379:
3377:
3376:
3368:
3363:
3358:
3353:
3348:
3343:
3338:
3333:
3328:
3322:
3320:
3316:
3315:
3313:
3312:
3307:
3302:
3300:Cross-polytope
3297:
3292:
3287:
3285:Hyperrectangle
3282:
3277:
3272:
3266:
3264:
3254:
3253:
3251:
3250:
3245:
3240:
3235:
3230:
3225:
3220:
3214:
3212:
3208:
3207:
3200:
3198:
3196:
3195:
3190:
3185:
3180:
3175:
3170:
3165:
3160:
3154:
3152:
3148:
3147:
3140:
3139:
3132:
3125:
3117:
3110:
3109:
3097:
3085:
3073:
3053:
3052:
3033:
3017:
3016:External links
3014:
3012:
3011:
3000:
2984:
2970:
2959:
2945:
2924:
2918:
2900:
2884:
2878:
2856:
2842:
2810:
2793:
2787:
2771:
2761:Weber, Hans J.
2752:
2750:
2747:
2745:
2744:
2732:
2720:
2700:
2679:
2660:
2636:
2634:, p. 154.
2624:
2612:
2600:
2598:, p. 153.
2588:
2575:
2573:
2570:
2568:
2567:
2562:
2557:
2552:
2547:
2542:
2537:
2532:
2527:
2522:
2517:
2515:Critical point
2512:
2510:Boundary point
2507:
2502:
2496:
2494:
2491:
2435:Main article:
2432:
2429:
2376:
2373:
2358:
2355:
2352:
2349:
2346:
2343:
2340:
2335:
2330:
2326:
2322:
2319:
2316:
2313:
2310:
2307:
2304:
2301:
2298:
2295:
2292:
2287:
2283:
2273:is defined by
2252:
2249:
2246:
2241:
2236:
2232:
2226:
2223:
2220:
2216:
2190:
2171:
2168:
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2162:
2159:
2156:
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2147:
2143:
2138:
2134:
2130:
2127:
2124:
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2058:
2028:
2003:
1981:
1959:
1937:
1908:
1893:Main article:
1890:
1887:
1873:
1869:
1865:
1861:
1858:
1835:Main article:
1832:
1829:
1819:
1818:
1769:
1767:
1760:
1754:
1751:
1745:line segment.
1720:
1712:
1696:
1693:
1690:
1687:
1682:
1678:
1672:
1668:
1664:
1661:
1658:
1655:
1650:
1646:
1640:
1636:
1632:
1627:
1623:
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1613:
1609:
1606:
1601:
1597:
1593:
1590:
1587:
1584:
1579:
1575:
1571:
1566:
1562:
1558:
1555:
1552:
1549:
1510:
1503:
1496:
1444:conventionally
1415:
1412:
1403:isolated point
1311:physical space
1293:
1292:
1290:
1289:
1282:
1275:
1267:
1264:
1263:
1258:
1257:
1256:
1255:
1250:
1242:
1241:
1237:
1236:
1235:
1234:
1229:
1224:
1219:
1214:
1209:
1204:
1199:
1194:
1189:
1184:
1176:
1175:
1171:
1170:
1169:
1168:
1163:
1158:
1153:
1148:
1143:
1138:
1133:
1125:
1124:
1120:
1119:
1118:
1117:
1112:
1107:
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1082:
1077:
1072:
1067:
1062:
1054:
1053:
1049:
1048:
1047:
1046:
1041:
1036:
1031:
1026:
1021:
1016:
1008:
1007:
999:
995:
994:
993:
990:
989:
986:
985:
980:
975:
970:
965:
960:
955:
950:
945:
940:
935:
930:
925:
920:
915:
910:
905:
900:
895:
890:
885:
880:
875:
870:
865:
860:
855:
850:
845:
840:
835:
830:
825:
820:
815:
810:
805:
800:
795:
790:
785:
779:
775:
774:
773:
770:
769:
763:
762:
759:
758:
753:
747:
740:
739:
738:
735:
734:
731:
730:
725:
720:
718:Platonic Solid
715:
710:
705:
700:
695:
690:
689:
688:
677:
676:
670:
664:
663:
662:
659:
658:
653:
652:
651:
650:
645:
640:
632:
631:
625:
624:
623:
622:
617:
609:
608:
602:
601:
600:
599:
594:
589:
584:
576:
575:
569:
568:
567:
566:
561:
556:
548:
547:
541:
540:
539:
538:
533:
528:
518:
512:
511:
510:
507:
506:
503:
502:
497:
496:
495:
490:
479:
473:
472:
471:
468:
467:
464:
463:
457:
451:
450:
449:
446:
445:
442:
441:
436:
431:
425:
424:
419:
414:
404:
399:
394:
388:
387:
378:
374:
373:
370:
366:
365:
364:
363:
360:
359:
356:
355:
354:
353:
343:
338:
333:
328:
323:
322:
321:
311:
306:
301:
300:
299:
294:
289:
279:
278:
277:
272:
262:
257:
252:
247:
242:
237:
236:
235:
230:
229:
228:
213:
207:
201:
200:
199:
196:
195:
193:
192:
182:
176:
173:
172:
159:
151:
150:
128:
127:
42:
40:
33:
26:
9:
6:
4:
3:
2:
3838:
3827:
3824:
3823:
3821:
3806:
3798:
3794:
3791:
3789:
3786:
3784:
3781:
3780:
3779:
3771:
3769:
3765:
3761:
3759:
3755:
3751:
3749:
3744:
3739:
3737:
3729:
3728:
3725:
3719:
3716:
3714:
3711:
3709:
3706:
3704:
3701:
3699:
3696:
3694:
3691:
3690:
3688:
3686:
3682:
3676:
3675:Orientability
3673:
3671:
3668:
3666:
3663:
3661:
3658:
3656:
3653:
3652:
3650:
3646:
3640:
3637:
3635:
3632:
3630:
3627:
3625:
3622:
3620:
3617:
3615:
3612:
3610:
3607:
3603:
3600:
3598:
3595:
3594:
3593:
3590:
3586:
3583:
3581:
3578:
3576:
3573:
3571:
3568:
3566:
3563:
3562:
3561:
3558:
3556:
3553:
3551:
3548:
3546:
3542:
3539:
3538:
3536:
3532:
3527:
3517:
3514:
3512:
3511:Set-theoretic
3509:
3505:
3502:
3501:
3500:
3497:
3493:
3490:
3489:
3488:
3485:
3483:
3480:
3478:
3475:
3473:
3472:Combinatorial
3470:
3468:
3465:
3463:
3460:
3459:
3457:
3453:
3449:
3442:
3437:
3435:
3430:
3428:
3423:
3422:
3419:
3409:
3408:
3402:
3396:
3393:
3391:
3388:
3387:
3385:
3381:
3375:
3373:
3369:
3367:
3364:
3362:
3359:
3357:
3354:
3352:
3349:
3347:
3344:
3342:
3339:
3337:
3334:
3332:
3329:
3327:
3324:
3323:
3321:
3317:
3311:
3308:
3306:
3303:
3301:
3298:
3296:
3293:
3291:
3290:Demihypercube
3288:
3286:
3283:
3281:
3278:
3276:
3273:
3271:
3268:
3267:
3265:
3263:
3259:
3255:
3249:
3246:
3244:
3241:
3239:
3236:
3234:
3231:
3229:
3226:
3224:
3221:
3219:
3216:
3215:
3213:
3209:
3204:
3194:
3191:
3189:
3186:
3184:
3181:
3179:
3176:
3174:
3171:
3169:
3166:
3164:
3161:
3159:
3156:
3155:
3153:
3149:
3145:
3138:
3133:
3131:
3126:
3124:
3119:
3118:
3115:
3108:
3098:
3096:
3086:
3084:
3079:
3074:
3072:
3067:
3062:
3061:
3058:
3048:
3047:
3042:
3039:
3034:
3030:
3029:
3024:
3020:
3019:
3009:. Free Press.
3008:
3007:
3001:
2998:
2992:
2991:
2985:
2981:
2980:
2975:
2971:
2967:
2966:
2960:
2956:
2955:
2950:
2946:
2942:
2938:
2933:
2932:
2925:
2921:
2919:0-486-60088-2
2915:
2911:
2910:
2905:
2901:
2897:
2890:
2885:
2881:
2879:0-12-279501-6
2875:
2871:
2870:
2865:
2861:
2857:
2853:
2852:
2847:
2843:
2839:
2835:
2831:
2827:
2823:
2819:
2815:
2814:de Laguna, T.
2811:
2807:
2803:
2799:
2794:
2790:
2788:0-07-007015-6
2784:
2780:
2776:
2772:
2768:
2767:
2762:
2758:
2754:
2753:
2741:
2736:
2730:, p. 84.
2729:
2724:
2717:
2713:
2709:
2704:
2697:
2693:
2689:
2683:
2675:
2669:
2664:
2650:
2646:
2640:
2633:
2628:
2621:
2616:
2609:
2604:
2597:
2592:
2585:
2580:
2576:
2566:
2563:
2561:
2558:
2556:
2553:
2551:
2548:
2546:
2545:Point process
2543:
2541:
2538:
2536:
2533:
2531:
2528:
2526:
2523:
2521:
2518:
2516:
2513:
2511:
2508:
2506:
2503:
2501:
2498:
2497:
2490:
2488:
2484:
2480:
2476:
2472:
2468:
2463:
2459:
2455:
2448:
2444:
2438:
2428:
2426:
2422:
2418:
2414:
2410:
2406:
2402:
2398:
2394:
2390:
2386:
2382:
2372:
2369:
2356:
2350:
2347:
2341:
2333:
2328:
2324:
2320:
2317:
2314:
2311:
2302:
2296:
2290:
2281:
2272:
2268:
2263:
2250:
2247:
2244:
2239:
2234:
2230:
2224:
2221:
2218:
2214:
2204:
2200:
2193:
2185:
2166:
2163:
2160:
2157:
2149:
2145:
2141:
2136:
2132:
2125:
2115:
2109:
2104:
2100:
2096:
2093:-dimensional
2092:
2086:
2080:
2076:
2071:
2067:
2057:
2055:
2050:
2048:
2044:
2017:
2001:
1957:
1926:
1922:
1906:
1896:
1886:
1867:
1859:
1856:
1848:
1844:
1838:
1828:
1826:
1815:
1812:
1804:
1794:
1790:
1786:
1780:
1779:
1775:
1770:This section
1768:
1764:
1759:
1758:
1750:
1746:
1744:
1740:
1736:
1723:
1711:
1694:
1688:
1685:
1680:
1676:
1670:
1666:
1662:
1659:
1656:
1653:
1648:
1644:
1638:
1634:
1630:
1625:
1621:
1615:
1611:
1607:
1599:
1595:
1591:
1588:
1585:
1582:
1577:
1573:
1569:
1564:
1560:
1550:
1547:
1539:
1535:
1531:
1526:
1524:
1513:
1509:
1502:
1495:
1485:
1465:
1457:
1449:
1445:
1433:
1429:
1425:
1421:
1411:
1409:
1405:
1404:
1398:
1396:
1392:
1388:
1384:
1379:
1377:
1373:
1369:
1365:
1363:
1362:straight line
1357:
1353:
1349:
1346:In classical
1344:
1342:
1338:
1337:
1332:
1328:
1324:
1320:
1316:
1312:
1308:
1304:
1300:
1288:
1283:
1281:
1276:
1274:
1269:
1268:
1266:
1265:
1254:
1251:
1249:
1246:
1245:
1244:
1243:
1239:
1238:
1233:
1230:
1228:
1225:
1223:
1220:
1218:
1215:
1213:
1210:
1208:
1205:
1203:
1200:
1198:
1195:
1193:
1190:
1188:
1185:
1183:
1180:
1179:
1178:
1177:
1173:
1172:
1167:
1164:
1162:
1159:
1157:
1154:
1152:
1149:
1147:
1144:
1142:
1139:
1137:
1134:
1132:
1129:
1128:
1127:
1126:
1122:
1121:
1116:
1113:
1111:
1108:
1106:
1103:
1101:
1098:
1096:
1093:
1091:
1088:
1086:
1083:
1081:
1078:
1076:
1073:
1071:
1068:
1066:
1063:
1061:
1058:
1057:
1056:
1055:
1051:
1050:
1045:
1042:
1040:
1037:
1035:
1032:
1030:
1027:
1025:
1022:
1020:
1017:
1015:
1012:
1011:
1010:
1009:
1006:
1003:
1002:
992:
991:
984:
981:
979:
976:
974:
971:
969:
966:
964:
961:
959:
956:
954:
951:
949:
946:
944:
941:
939:
936:
934:
931:
929:
926:
924:
921:
919:
916:
914:
911:
909:
906:
904:
901:
899:
896:
894:
891:
889:
886:
884:
881:
879:
876:
874:
871:
869:
866:
864:
861:
859:
856:
854:
851:
849:
846:
844:
841:
839:
836:
834:
831:
829:
826:
824:
821:
819:
816:
814:
811:
809:
806:
804:
801:
799:
796:
794:
791:
789:
786:
784:
781:
780:
772:
771:
768:
765:
764:
757:
754:
752:
749:
748:
743:
737:
736:
729:
726:
724:
721:
719:
716:
714:
711:
709:
706:
704:
701:
699:
696:
694:
691:
687:
684:
683:
682:
679:
678:
675:
672:
671:
667:
661:
660:
649:
646:
644:
643:Circumference
641:
639:
636:
635:
634:
633:
630:
627:
626:
621:
618:
616:
613:
612:
611:
610:
607:
606:Quadrilateral
604:
603:
598:
595:
593:
590:
588:
585:
583:
580:
579:
578:
577:
574:
573:Parallelogram
571:
570:
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557:
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550:
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430:
427:
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412:
411:Perpendicular
408:
407:Orthogonality
405:
403:
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398:
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331:Computational
329:
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217:
216:Non-Euclidean
214:
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81:
78:
74:
71: ā
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
3805:Publications
3670:Chern number
3660:Betti number
3543: /
3534:Key concepts
3482:Differential
3405:
3371:
3310:Hyperpyramid
3275:Hypersurface
3168:Affine space
3158:Vector space
3044:
3026:
3005:
2989:
2978:
2968:. Macmillan.
2964:
2953:
2930:
2908:
2895:
2868:
2850:
2821:
2817:
2805:
2801:
2778:
2765:
2742:, Chapter 5.
2735:
2723:
2718:, p. 3.
2708:Dirac (1958)
2703:
2682:
2668:Gerla (1985)
2663:
2653:, retrieved
2651:, 2024-09-24
2648:
2639:
2632:Heath (1956)
2627:
2615:
2610:, p. 7.
2603:
2596:Heath (1956)
2591:
2584:Ohmer (1969)
2579:
2505:Affine space
2482:
2471:point charge
2450:
2446:
2440:
2424:
2420:
2378:
2370:
2270:
2266:
2264:
2202:
2198:
2188:
2183:
2107:
2098:
2094:
2090:
2084:
2078:
2074:
2070:metric space
2065:
2063:
2051:
2046:
2042:
1920:
1898:
1846:
1840:
1822:
1807:
1798:
1783:Please help
1771:
1747:
1739:line segment
1718:
1709:
1527:
1511:
1507:
1500:
1493:
1432:ordered pair
1417:
1408:neighborhood
1401:
1399:
1394:
1380:
1359:
1345:
1340:
1334:
1331:intersection
1302:
1296:
1115:Parameshvara
928:Parameshvara
698:Dodecahedron
460:
282:Differential
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
3768:Wikiversity
3685:Key results
3395:Codimension
3374:-dimensions
3295:Hypersphere
3178:Free module
3071:Mathematics
2846:Dirac, Paul
2808:(1): 61ā75.
2540:Point cloud
2052:A point is
1387:coordinates
1319:dimensional
1240:Present day
1187:Lobachevsky
1174:1700sā1900s
1131:Jyeį¹£į¹hadeva
1123:1400sā1700s
1075:Brahmagupta
898:Lobachevsky
878:Jyeį¹£į¹hadeva
828:Brahmagupta
756:Hypersphere
728:Tetrahedron
703:Icosahedron
275:Diophantine
3614:CW complex
3555:Continuity
3545:Closed set
3504:cohomology
3390:Hyperspace
3270:Hyperplane
3028:PlanetMath
2749:References
2655:2024-09-29
2475:Paul Dirac
2467:point mass
2425:connection
1925:open cover
1801:March 2022
1743:degenerate
1448:horizontal
1317:. As zero-
1100:al-Yasamin
1044:Apollonius
1039:Archimedes
1029:Pythagoras
1019:Baudhayana
973:al-Yasamin
923:Pythagoras
818:Baudhayana
808:Archimedes
803:Apollonius
708:Octahedron
559:Hypotenuse
434:Similarity
429:Congruence
341:Incidence
292:Symplectic
287:Riemannian
270:Arithmetic
245:Projective
233:Hyperbolic
161:Projecting
110:March 2022
80:newspapers
3793:geometric
3788:algebraic
3639:Cobordism
3575:Hausdorff
3570:connected
3487:Geometric
3477:Continuum
3467:Algebraic
3280:Hypercube
3258:Polytopes
3238:Minkowski
3233:Hausdorff
3228:Inductive
3193:Spacetime
3144:Dimension
3046:MathWorld
2649:Knowledge
2555:Pointwise
2421:inclusion
2409:functions
2393:algebraic
2315:≥
2291:
2248:δ
2222:∈
2215:∑
2196:for each
2182:covering
2164:∈
1860:⋅
1825:dimension
1772:does not
1608:∣
1523:dimension
1395:point set
1217:Minkowski
1136:Descartes
1070:Aryabhata
1065:KÄtyÄyana
996:by period
908:Minkowski
883:KÄtyÄyana
843:Descartes
788:Aryabhata
767:Geometers
751:Tesseract
615:Trapezoid
587:Rectangle
380:Dimension
265:Algebraic
255:Synthetic
226:Spherical
211:Euclidean
3820:Category
3758:Wikibook
3736:Category
3624:Manifold
3592:Homotopy
3550:Interior
3541:Open set
3499:Homology
3448:Topology
3407:Category
3383:See also
3183:Manifold
2976:(1919).
2951:(1950).
2941:00218666
2906:(1956).
2866:(1964).
2848:(1958).
2777:(1986).
2763:(2005).
2493:See also
2462:integral
2454:function
2087:ā [0, ā)
1716:through
1530:infinite
1456:vertical
1327:surfaces
1307:position
1299:geometry
1207:PoincarƩ
1151:Minggatu
1110:Yang Hui
1080:Virasena
968:Yang Hui
963:Virasena
933:PoincarƩ
913:Minggatu
693:Cylinder
638:Diameter
597:Rhomboid
554:Altitude
545:Triangle
439:Symmetry
417:Parallel
402:Diagonal
372:Features
369:Concepts
260:Analytic
221:Elliptic
203:Branches
189:Timeline
148:Geometry
3783:general
3585:uniform
3565:compact
3516:Digital
3305:Simplex
3243:Fractal
3083:Physics
3057:Portals
3041:"Point"
3023:"Point"
2838:2939504
2397:logical
2103:infimum
2101:is the
2016:refines
1793:removed
1778:sources
1521:is the
1490:terms,
1376:scriber
1372:compass
1232:Coxeter
1212:Hilbert
1197:Riemann
1146:Huygens
1105:al-Tusi
1095:KhayyƔm
1085:Alhazen
1052:1ā1400s
953:al-Tusi
938:Riemann
888:KhayyƔm
873:Huygens
868:Hilbert
838:Coxeter
798:Alhazen
776:by name
713:Pyramid
592:Rhombus
536:Polygon
488:segment
336:Fractal
319:Digital
304:Complex
185:History
180:Outline
94:scholar
3778:Topics
3580:metric
3455:Fields
3262:shapes
2939:
2916:
2876:
2836:
2785:
2417:region
2403:or an
2194:> 0
2089:, the
2014:which
1707:where
1517:where
1506:,āā¦ā,
1484:tuplet
1424:Euclid
1356:axioms
1341:corner
1336:vertex
1323:curves
1315:spaces
1253:Gromov
1248:Atiyah
1227:Veblen
1222:Cartan
1192:Bolyai
1161:Sakabe
1141:Pascal
1034:Euclid
1024:Manava
958:Veblen
943:Sakabe
918:Pascal
903:Manava
863:Gromov
848:Euclid
833:Cartan
823:Bolyai
813:Atiyah
723:Sphere
686:cuboid
674:Volume
629:Circle
582:Square
500:Length
422:Vertex
326:Convex
309:Finite
250:Affine
165:sphere
96:
89:
82:
75:
67:
3560:Space
3366:Eight
3361:Seven
3341:Three
3218:Krull
2892:(PDF)
2834:JSTOR
2572:Notes
2449:, or
2186:with
2114:balls
2072:. If
2068:be a
1735:plane
1303:point
1202:Klein
1182:Gauss
1156:Euler
1090:Sijzi
1060:Zhang
1014:Ahmes
978:Zhang
948:Sijzi
893:Klein
858:Gauss
853:Euler
793:Ahmes
526:Plane
461:Point
397:Curve
392:Angle
169:plane
167:to a
101:JSTOR
87:books
3351:Five
3346:Four
3326:Zero
3260:and
3107:Maps
2937:OCLC
2914:ISBN
2874:ISBN
2783:ISBN
2696:1929
2692:1920
2688:1919
2674:help
2520:Cusp
2383:and
2265:The
2245:<
2082:and
2064:Let
1776:any
1774:cite
1725:and
1538:line
1393:, a
1301:, a
1166:Aida
783:Aida
742:Four
681:Cube
648:Area
620:Kite
531:Area
483:Line
73:news
3356:Six
3336:Two
3331:One
2826:doi
2469:or
2423:or
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2389:set
2306:inf
2282:dim
2269:of
2110:ā„ 0
2097:of
1994:of
1950:of
1787:by
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1005:BCE
493:ray
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