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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.
122:
1737:
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
1694:
2356:
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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
2250:
1872:
1532:
2169:
2265:
2028:
1981:
1937:
2725:
1343:, 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
1838:), 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:
2001:
1957:
1906:
3769:
2762:
1367:, 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.
2832:
2368:
Although the notion of a point is generally considered fundamental in mainstream geometry and topology, there are some systems that forgo it, e.g.
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2853:
2198:
3099:
1273:
2430:
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
2045:
with respect to the covering dimension because every open cover of the space has a refinement consisting of a single open set.
2638:
373:
89:
1310:
objects, points are usually taken to be the fundamental indivisible elements comprising the space, of which one-dimensional
2400:
in a way that the operation "take a value at this point" may not be defined. A further tradition starts from some books of
61:
1813:
3403:
1841:
68:
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339:
108:
42:
1781:
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75:
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1220:
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285:
46:
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2503:
57:
2360:
A point has
Hausdorff dimension 0 because it can be covered by a single ball of arbitrarily small radius.
3187:
1883:
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3657:
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2961:
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851:
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1259:
2009:
1962:
1918:
3456:
3122:
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1825:
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1731:
1689:{\displaystyle L=\lbrace (a_{1},a_{2},...a_{n})\mid a_{1}c_{1}+a_{2}c_{2}+...a_{n}c_{n}=d\rbrace ,}
228:
2942:
3742:
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3325:
3305:
2513:
2369:
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1511:
1319:
654:
334:
191:
35:
3677:
3598:
3475:
3463:
3436:
3396:
3315:
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3059:
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2538:
2397:
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1360:
730:
441:
319:
204:
3672:
3519:
3446:
3320:
3300:
3295:
2894:
2814:
2729:
2351:{\displaystyle \operatorname {dim} _{\operatorname {H} }(X):=\inf\{d\geq 0:C_{H}^{d}(X)=0\}.}
2004:
502:
463:
422:
417:
270:
2928:
82:
3667:
3619:
3593:
3441:
2953:
2446:
2425:
2405:
1831:
1730:, and other related concepts. A line segment consisting of only a single point is called a
1170:
1093:
941:
846:
368:
263:
177:
8:
3514:
3197:
3192:
2969:
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2518:
2389:
1303:
1295:
1175:
1119:
1032:
886:
866:
791:
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552:
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233:
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149:
3718:
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3431:
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3167:
3035:
2798:
2778:
2508:
2488:
2373:
1986:
1942:
1891:
1526:
1444:
1408:
1336:
1318:, and higher-dimensional objects consist; conversely, a point can be determined by the
1134:
861:
701:
329:
253:
243:
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199:
1834:
subset. In a vector space consisting of a single point (which must be the zero vector
1521:
collection of points that conform to certain axioms. This is usually represented by a
3707:
3700:
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3389:
3207:
3030:
3002:
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respectively) which looks like a well-known function space on the set: an algebra of
2377:
1522:
1432:
1379:
1375:
1371:
1324:
1205:
993:
971:
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302:
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209:
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831:
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3426:
3137:
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2739:
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1723:
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1340:
1185:
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514:
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157:
3539:
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originally defined the point as "that which has no part". In the two-dimensional
608:
471:
314:
297:
238:
144:
126:
3722:
2960:. 2004 paperback, Prometheus Books. Being the 1919 Tarner Lectures delivered at
1180:
1149:
1083:
931:
876:
811:
3629:
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3249:
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function which is usually defined on a finite domain and takes values 0 and 1.
1391:
1299:
1236:
1144:
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1053:
961:
871:
841:
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respectively. More precisely, such structures generalize well-known spaces of
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1200:
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in mathematics. In all of the common definitions, a point is 0-dimensional.
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3544:
3490:
3274:
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3132:
3005:
2493:
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2102:
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is the dimension of the space. Similar constructions exist that define the
1467:) with the additional third number representing depth and often denoted by
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1154:
1103:
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771:
686:
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3634:
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3142:
3071:
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1307:
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on the real number line that is zero everywhere except at zero, with an
3578:
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2992:
2810:
2802:
2463:
2455:
2381:
1913:
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1027:
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1007:
966:
911:
806:
796:
696:
547:
3603:
3335:
3244:
3157:
3108:
3010:
2543:
2434:, where electrons are idealized as points with non-zero charge). The
1378:. In modern mathematics, a space of points is typically treated as a
1058:
776:
739:
603:
575:
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1751:
24:
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956:
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427:
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136:
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2731:
Mathematical
Methods For Physicists International Student Edition
2091:
1364:
1315:
1073:
786:
580:
524:
324:
2038:
exists, the space is said to be of infinite covering dimension.
1374:, points are often defined or represented in terms of numerical
2987:
2566:
1412:
1022:
1012:
891:
836:
711:
674:
662:
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570:
488:
153:
2698:
3226:
1472:
1344:
1311:
1078:
1002:
936:
781:
385:
380:
3077:
2877:. Vol. 1 (2nd ed.). New York: Dover Publications.
2419:
3381:
669:
519:
2675:, p. 58, More specifically, see Ā§15. The Ī“ function;
2944:
An
Enquiry Concerning the Principles of Natural Knowledge
2861:
Handbook of
Incidence Geometry: Buildings and Foundations
2376:. A "pointless" or "pointfree" space is defined not as a
1830:
The dimension of a vector space is the maximum size of a
2686:
1517:
Many constructs within
Euclidean geometry consist of an
1471:. Further generalizations are represented by an ordered
121:
1451:. This idea is easily generalized to three-dimensional
1302:, or its generalization to other kinds of mathematical
2781:(1922). "Point, line and surface as sets of solids,".
1455:, where a point is represented by an ordered triplet (
1443:, and the second number conventionally represents the
1395:
is an element of some subset of points which has some
3019:
2590:
2268:
2201:
2110:
2012:
1989:
1965:
1945:
1921:
1894:
1844:
1535:
2602:
2614:
2578:
2408:is assumed as a primitive together with the one of
2245:{\displaystyle \sum _{i\in I}r_{i}^{d}<\delta .}
49:. Unsourced material may be challenged and removed.
2893:
2350:
2244:
2163:
2022:
1995:
1975:
1951:
1931:
1900:
1866:
1688:
1888:The topological dimension of a topological space
3782:
2834:Generalized Functions: Properties and Operations
2626:
2294:
2101:such that there is some (indexed) collection of
1867:{\displaystyle 1\cdot \mathbf {0} =\mathbf {0} }
1402:
3000:
2746:(3rd ed.). New York: McGraw-Hill Series.
1812:There are several inequivalent definitions of
3397:
3093:
2823:
2676:
2462:. It was introduced by theoretical physicist
1267:
2342:
2297:
2158:
2111:
1680:
1542:
1514:of the space in which the point is located.
2720:
2692:
2030:in which no point is included in more than
1780:. Unsourced material may be challenged and
1411:, are one of the most fundamental objects.
1407:Points, considered within the framework of
3765:
3738:
3404:
3390:
3100:
3086:
2971:Process and Reality: An Essay in Cosmology
2744:The Fourier transform and its applications
2474:(or function). Its discrete analog is the
2363:
1399:containing no other points of the subset.
1322:of two curves or three surfaces, called a
1274:
1260:
143:
2967:
2951:
2937:
2926:
2777:
2738:
2704:
2660:
2656:
2652:
2637:sfnp error: no target: CITEREFGerla1985 (
2608:
2596:
2420:Point masses and the Dirac delta function
2164:{\displaystyle \{B(x_{i},r_{i}):i\in I\}}
1819:
1800:Learn how and when to remove this message
1529:is an infinite set of points of the form
109:Learn how and when to remove this message
2912:
2819:(4th ed.). Oxford University Press.
2680:
1877:
1353:that passes through two distinct points"
1294:is an abstract idealization of an exact
120:
2874:The Thirteen Books of Euclid's Elements
2859:. In Buekenhout, F.; Kantor, W (eds.).
1741:
1347:, that they must satisfy; for example,
125:A finite set of points (in red) in the
3783:
2760:
2048:
1908:is defined to be the minimum value of
374:Straightedge and compass constructions
3385:
3081:
3001:
2930:Modern Calculus and Analytic Geometry
2891:
2867:
2851:
2809:
2672:
2632:
2620:
2584:
2572:
1431:) of numbers, where the first number
1778:adding citations to reliable sources
1745:
47:adding citations to reliable sources
18:
2863:. North-Holland. p. 1015ā1031.
2816:The Principles of Quantum Mechanics
13:
2767:Notre Dame Journal of Formal Logic
2274:
2015:
1968:
1924:
14:
3802:
2980:
340:Noncommutative algebraic geometry
3764:
3737:
3727:
3717:
3706:
3696:
3695:
3489:
3166:
3065:
3053:
3041:
3029:
2896:Elementary Geometry for Teachers
2679:, pp. 1ā5, See Ā§Ā§1.1, 1.3;
2034:+1 elements. If no such minimal
1860:
1852:
1750:
23:
2837:. Vol. 1. Academic Press.
2734:(6th ed.). Academic Press.
2470:it is often referred to as the
1419:, a point is represented by an
34:needs additional citations for
2958:. Cambridge: University Press.
2947:. Cambridge: University Press.
2927:Silverman, Richard A. (1969).
2645:
2333:
2327:
2288:
2282:
2143:
2117:
2023:{\displaystyle {\mathcal {A}}}
1976:{\displaystyle {\mathcal {B}}}
1932:{\displaystyle {\mathcal {A}}}
1593:
1545:
1359:are made with tools such as a
733:- / other-dimensional
16:Fundamental object of geometry
1:
3107:
2713:
2554:Whitehead point-free geometry
3411:
1525:of points; As an example, a
1403:Points in Euclidean geometry
7:
2900:. Reading: Addison-Wesley.
2677:Gelfand & Shilov (1964)
2481:
1959:admits a finite open cover
1884:Lebesgue covering dimension
10:
3807:
3658:Banach fixed-point theorem
2432:classical electromagnetism
2423:
2380:, but via some structure (
1881:
1823:
3691:
3648:
3612:
3498:
3487:
3419:
3368:
3347:
3283:
3221:
3175:
3164:
3115:
2952:Whitehead, A. N. (1920).
2922:(in French). Vol. 1.
2919:ThƩorie des distributions
2892:Ohmer, Merlin M. (1969).
2783:The Journal of Philosophy
2693:Arfken & Weber (2005)
2549:Singular point of a curve
1912:, such that every finite
2968:Whitehead, A. N (1929).
2763:"Individuals and Points"
2560:
1826:Dimension (vector space)
1447:and is often denoted by
1439:and is often denoted by
1355:. As physical diagrams,
229:Non-Archimedean geometry
2761:Clarke, Bowman (1985).
2514:Foundations of geometry
2404:in which the notion of
2370:noncommutative geometry
2364:Geometry without points
335:Noncommutative geometry
3713:Mathematics portal
3613:Metrics and properties
3599:Second-countable space
2854:"Pointless Geometries"
2575:, p. 34–37.
2539:Point set registration
2352:
2246:
2165:
2094:of the set of numbers
2024:
1997:
1977:
1953:
1933:
1902:
1868:
1820:Vector space dimension
1690:
1349:"there is exactly one
303:Discrete/Combinatorial
130:
2955:The Concept of Nature
2353:
2247:
2166:
2025:
1998:
1978:
1954:
1934:
1903:
1878:Topological dimension
1869:
1691:
286:Discrete differential
124:
58:"Point" geometry
3668:Invariance of domain
3620:Euler characteristic
3594:Bundle (mathematics)
3284:Dimensions by number
2740:Bracewell, Ronald N.
2466:. In the context of
2447:generalized function
2445:, is (informally) a
2436:Dirac delta function
2426:Dirac delta function
2390:continuous functions
2266:
2199:
2108:
2010:
1987:
1963:
1943:
1919:
1892:
1842:
1832:linearly independent
1774:improve this section
1742:Dimension of a point
1533:
1370:Since the advent of
43:improve this article
3678:Tychonoff's theorem
3673:PoincarƩ conjecture
3427:General (point-set)
2519:Position (geometry)
2472:unit impulse symbol
2326:
2256:Hausdorff dimension
2232:
2049:Hausdorff dimension
1298:, without size, in
553:Pythagorean theorem
3663:De Rham cohomology
3584:Polyhedral complex
3574:Simplicial complex
3213:Degrees of freedom
3116:Dimensional spaces
3003:Weisstein, Eric W.
2489:Accumulation point
2374:pointless topology
2348:
2312:
2242:
2218:
2217:
2161:
2020:
1993:
1973:
1949:
1929:
1898:
1864:
1718:are constants and
1686:
1409:Euclidean geometry
1337:Euclidean geometry
1314:, two-dimensional
131:
3778:
3777:
3567:fundamental group
3379:
3378:
3188:Lebesgue covering
3153:Algebraic variety
2914:Schwartz, Laurent
2852:Gerla, G (1995).
2722:Arfken, George B.
2524:Point at infinity
2468:signal processing
2202:
2084:Hausdorff content
1996:{\displaystyle X}
1952:{\displaystyle X}
1901:{\displaystyle X}
1810:
1809:
1802:
1372:analytic geometry
1357:geometric figures
1284:
1283:
1249:
1248:
972:List of geometers
655:Three-dimensional
644:
643:
119:
118:
111:
93:
3798:
3791:Point (geometry)
3768:
3767:
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3731:
3721:
3711:
3710:
3699:
3698:
3493:
3406:
3399:
3392:
3383:
3382:
3176:Other dimensions
3170:
3138:Projective space
3102:
3095:
3088:
3079:
3078:
3070:
3069:
3068:
3058:
3057:
3056:
3046:
3045:
3034:
3033:
3025:
3016:
3015:
2997:
2975:
2959:
2948:
2939:Whitehead, A. N.
2934:
2923:
2909:
2899:
2888:
2869:Heath, Thomas L.
2864:
2858:
2848:
2820:
2806:
2774:
2757:
2735:
2708:
2705:Bracewell (1986)
2702:
2696:
2690:
2684:
2670:
2664:
2651:Whitehead (
2649:
2643:
2642:
2630:
2624:
2618:
2612:
2609:de Laguna (1922)
2606:
2600:
2597:Silverman (1969)
2594:
2588:
2582:
2576:
2570:
2442:
2357:
2355:
2354:
2349:
2325:
2320:
2278:
2277:
2251:
2249:
2248:
2243:
2231:
2226:
2216:
2194:
2184:
2170:
2168:
2167:
2162:
2142:
2141:
2129:
2128:
2100:
2077:
2070:
2043:zero-dimensional
2029:
2027:
2026:
2021:
2019:
2018:
2002:
2000:
1999:
1994:
1982:
1980:
1979:
1974:
1972:
1971:
1958:
1956:
1955:
1950:
1938:
1936:
1935:
1930:
1928:
1927:
1907:
1905:
1904:
1899:
1873:
1871:
1870:
1865:
1863:
1855:
1805:
1798:
1794:
1791:
1785:
1754:
1746:
1721:
1717:
1713:
1704:
1695:
1693:
1692:
1687:
1673:
1672:
1663:
1662:
1641:
1640:
1631:
1630:
1618:
1617:
1608:
1607:
1592:
1591:
1570:
1569:
1557:
1556:
1509:
1505:
1478:
1470:
1466:
1462:
1458:
1450:
1442:
1430:
1426:
1341:primitive notion
1276:
1269:
1262:
990:
989:
509:
508:
442:Zero-dimensional
147:
133:
132:
114:
107:
103:
100:
94:
92:
51:
27:
19:
3806:
3805:
3801:
3800:
3799:
3797:
3796:
3795:
3781:
3780:
3779:
3774:
3705:
3687:
3683:Urysohn's lemma
3644:
3608:
3494:
3485:
3457:low-dimensional
3415:
3410:
3380:
3375:
3364:
3343:
3279:
3217:
3171:
3162:
3128:Euclidean space
3111:
3106:
3076:
3066:
3064:
3060:Systems science
3054:
3052:
3040:
3028:
3020:
2986:
2983:
2978:
2962:Trinity College
2885:
2856:
2845:
2829:Shilov, Georgiy
2825:Gelfand, Israel
2795:10.2307/2939504
2789:(17): 449ā461.
2754:
2716:
2711:
2703:
2699:
2691:
2687:
2681:Schwartz (1950)
2671:
2667:
2650:
2646:
2636:
2631:
2627:
2619:
2615:
2607:
2603:
2595:
2591:
2583:
2579:
2571:
2567:
2563:
2558:
2484:
2476:Kronecker delta
2440:
2428:
2422:
2402:A. N. Whitehead
2394:algebra of sets
2366:
2321:
2316:
2273:
2269:
2267:
2264:
2263:
2227:
2222:
2206:
2200:
2197:
2196:
2195:that satisfies
2186:
2181:
2176:
2137:
2133:
2124:
2120:
2109:
2106:
2105:
2095:
2072:
2062:
2051:
2014:
2013:
2011:
2008:
2007:
1988:
1985:
1984:
1967:
1966:
1964:
1961:
1960:
1944:
1941:
1940:
1923:
1922:
1920:
1917:
1916:
1893:
1890:
1889:
1886:
1880:
1859:
1851:
1843:
1840:
1839:
1828:
1822:
1806:
1795:
1789:
1786:
1771:
1755:
1744:
1719:
1715:
1711:
1706:
1703:
1697:
1668:
1664:
1658:
1654:
1636:
1632:
1626:
1622:
1613:
1609:
1603:
1599:
1587:
1583:
1565:
1561:
1552:
1548:
1534:
1531:
1530:
1507:
1503:
1494:
1487:
1480:
1476:
1468:
1464:
1460:
1456:
1453:Euclidean space
1448:
1440:
1435:represents the
1428:
1424:
1417:Euclidean plane
1405:
1339:, a point is a
1280:
1251:
1250:
987:
986:
977:
976:
767:
766:
750:
749:
735:
734:
722:
721:
658:
657:
646:
645:
506:
505:
503:Two-dimensional
494:
493:
467:
466:
464:One-dimensional
455:
454:
445:
444:
433:
432:
366:
365:
364:
347:
346:
195:
194:
183:
160:
127:Euclidean plane
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
3804:
3794:
3793:
3776:
3775:
3773:
3772:
3762:
3761:
3760:
3755:
3750:
3735:
3725:
3715:
3703:
3692:
3689:
3688:
3686:
3685:
3680:
3675:
3670:
3665:
3660:
3654:
3652:
3646:
3645:
3643:
3642:
3637:
3632:
3630:Winding number
3627:
3622:
3616:
3614:
3610:
3609:
3607:
3606:
3601:
3596:
3591:
3586:
3581:
3576:
3571:
3570:
3569:
3564:
3562:homotopy group
3554:
3553:
3552:
3547:
3542:
3537:
3532:
3522:
3517:
3512:
3502:
3500:
3496:
3495:
3488:
3486:
3484:
3483:
3478:
3473:
3472:
3471:
3461:
3460:
3459:
3449:
3444:
3439:
3434:
3429:
3423:
3421:
3417:
3416:
3409:
3408:
3401:
3394:
3386:
3377:
3376:
3369:
3366:
3365:
3363:
3362:
3357:
3351:
3349:
3345:
3344:
3342:
3341:
3333:
3328:
3323:
3318:
3313:
3308:
3303:
3298:
3293:
3287:
3285:
3281:
3280:
3278:
3277:
3272:
3267:
3265:Cross-polytope
3262:
3257:
3252:
3250:Hyperrectangle
3247:
3242:
3237:
3231:
3229:
3219:
3218:
3216:
3215:
3210:
3205:
3200:
3195:
3190:
3185:
3179:
3177:
3173:
3172:
3165:
3163:
3161:
3160:
3155:
3150:
3145:
3140:
3135:
3130:
3125:
3119:
3117:
3113:
3112:
3105:
3104:
3097:
3090:
3082:
3075:
3074:
3062:
3050:
3038:
3018:
3017:
2998:
2982:
2981:External links
2979:
2977:
2976:
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2935:
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2910:
2889:
2883:
2865:
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2843:
2821:
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2775:
2758:
2752:
2736:
2726:Weber, Hans J.
2717:
2715:
2712:
2710:
2709:
2697:
2685:
2665:
2644:
2625:
2623:, p. 154.
2613:
2601:
2589:
2587:, p. 153.
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2556:
2551:
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2516:
2511:
2506:
2504:Critical point
2501:
2499:Boundary point
2496:
2491:
2485:
2483:
2480:
2424:Main article:
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2418:
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2362:
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2281:
2276:
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2262:is defined by
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2017:
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1948:
1926:
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1882:Main article:
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1876:
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1854:
1850:
1847:
1824:Main article:
1821:
1818:
1808:
1807:
1758:
1756:
1749:
1743:
1740:
1734:line segment.
1709:
1701:
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1564:
1560:
1555:
1551:
1547:
1544:
1541:
1538:
1499:
1492:
1485:
1433:conventionally
1404:
1401:
1392:isolated point
1300:physical space
1282:
1281:
1279:
1278:
1271:
1264:
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829:
824:
819:
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804:
799:
794:
789:
784:
779:
774:
768:
764:
763:
762:
759:
758:
752:
751:
748:
747:
742:
736:
729:
728:
727:
724:
723:
720:
719:
714:
709:
707:Platonic Solid
704:
699:
694:
689:
684:
679:
678:
677:
666:
665:
659:
653:
652:
651:
648:
647:
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634:
629:
621:
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614:
613:
612:
611:
606:
598:
597:
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590:
589:
588:
583:
578:
573:
565:
564:
558:
557:
556:
555:
550:
545:
537:
536:
530:
529:
528:
527:
522:
517:
507:
501:
500:
499:
496:
495:
492:
491:
486:
485:
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479:
468:
462:
461:
460:
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453:
452:
446:
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438:
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434:
431:
430:
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420:
414:
413:
408:
403:
393:
388:
383:
377:
376:
367:
363:
362:
359:
355:
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348:
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344:
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342:
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327:
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290:
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278:
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251:
246:
241:
236:
231:
226:
225:
224:
219:
218:
217:
202:
196:
190:
189:
188:
185:
184:
182:
181:
171:
165:
162:
161:
148:
140:
139:
117:
116:
31:
29:
22:
15:
9:
6:
4:
3:
2:
3803:
3792:
3789:
3788:
3786:
3771:
3763:
3759:
3756:
3754:
3751:
3749:
3746:
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3736:
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3669:
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3664:
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3659:
3656:
3655:
3653:
3651:
3647:
3641:
3640:Orientability
3638:
3636:
3633:
3631:
3628:
3626:
3623:
3621:
3618:
3617:
3615:
3611:
3605:
3602:
3600:
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3595:
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3590:
3587:
3585:
3582:
3580:
3577:
3575:
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3568:
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3563:
3560:
3559:
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3555:
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3527:
3526:
3523:
3521:
3518:
3516:
3513:
3511:
3507:
3504:
3503:
3501:
3497:
3492:
3482:
3479:
3477:
3476:Set-theoretic
3474:
3470:
3467:
3466:
3465:
3462:
3458:
3455:
3454:
3453:
3450:
3448:
3445:
3443:
3440:
3438:
3437:Combinatorial
3435:
3433:
3430:
3428:
3425:
3424:
3422:
3418:
3414:
3407:
3402:
3400:
3395:
3393:
3388:
3387:
3384:
3374:
3373:
3367:
3361:
3358:
3356:
3353:
3352:
3350:
3346:
3340:
3338:
3334:
3332:
3329:
3327:
3324:
3322:
3319:
3317:
3314:
3312:
3309:
3307:
3304:
3302:
3299:
3297:
3294:
3292:
3289:
3288:
3286:
3282:
3276:
3273:
3271:
3268:
3266:
3263:
3261:
3258:
3256:
3255:Demihypercube
3253:
3251:
3248:
3246:
3243:
3241:
3238:
3236:
3233:
3232:
3230:
3228:
3224:
3220:
3214:
3211:
3209:
3206:
3204:
3201:
3199:
3196:
3194:
3191:
3189:
3186:
3184:
3181:
3180:
3178:
3174:
3169:
3159:
3156:
3154:
3151:
3149:
3146:
3144:
3141:
3139:
3136:
3134:
3131:
3129:
3126:
3124:
3121:
3120:
3118:
3114:
3110:
3103:
3098:
3096:
3091:
3089:
3084:
3083:
3080:
3073:
3063:
3061:
3051:
3049:
3044:
3039:
3037:
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3027:
3026:
3023:
3013:
3012:
3007:
3004:
2999:
2995:
2994:
2989:
2985:
2984:
2974:. Free Press.
2973:
2972:
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2957:
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2932:
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2925:
2921:
2920:
2915:
2911:
2907:
2903:
2898:
2897:
2890:
2886:
2884:0-486-60088-2
2880:
2876:
2875:
2870:
2866:
2862:
2855:
2850:
2846:
2844:0-12-279501-6
2840:
2836:
2835:
2830:
2826:
2822:
2818:
2817:
2812:
2808:
2804:
2800:
2796:
2792:
2788:
2784:
2780:
2779:de Laguna, T.
2776:
2772:
2768:
2764:
2759:
2755:
2753:0-07-007015-6
2749:
2745:
2741:
2737:
2733:
2732:
2727:
2723:
2719:
2718:
2706:
2701:
2695:, p. 84.
2694:
2689:
2682:
2678:
2674:
2669:
2662:
2658:
2654:
2648:
2640:
2634:
2629:
2622:
2617:
2610:
2605:
2598:
2593:
2586:
2581:
2574:
2569:
2565:
2555:
2552:
2550:
2547:
2545:
2542:
2540:
2537:
2535:
2534:Point process
2532:
2530:
2527:
2525:
2522:
2520:
2517:
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2512:
2510:
2507:
2505:
2502:
2500:
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2492:
2490:
2487:
2486:
2479:
2477:
2473:
2469:
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2457:
2452:
2448:
2444:
2437:
2433:
2427:
2417:
2415:
2411:
2407:
2403:
2399:
2395:
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2387:
2383:
2379:
2375:
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2358:
2345:
2339:
2336:
2330:
2322:
2317:
2313:
2309:
2306:
2303:
2300:
2291:
2285:
2279:
2270:
2261:
2257:
2252:
2239:
2236:
2233:
2228:
2223:
2219:
2213:
2210:
2207:
2203:
2193:
2189:
2182:
2174:
2155:
2152:
2149:
2146:
2138:
2134:
2130:
2125:
2121:
2114:
2104:
2098:
2093:
2089:
2085:
2082:-dimensional
2081:
2075:
2069:
2065:
2060:
2056:
2046:
2044:
2039:
2037:
2033:
2006:
1990:
1946:
1915:
1911:
1895:
1885:
1875:
1856:
1848:
1845:
1837:
1833:
1827:
1817:
1815:
1804:
1801:
1793:
1783:
1779:
1775:
1769:
1768:
1764:
1759:This section
1757:
1753:
1748:
1747:
1739:
1735:
1733:
1729:
1725:
1712:
1700:
1683:
1677:
1674:
1669:
1665:
1659:
1655:
1651:
1648:
1645:
1642:
1637:
1633:
1627:
1623:
1619:
1614:
1610:
1604:
1600:
1596:
1588:
1584:
1580:
1577:
1574:
1571:
1566:
1562:
1558:
1553:
1549:
1539:
1536:
1528:
1524:
1520:
1515:
1513:
1502:
1498:
1491:
1484:
1474:
1454:
1446:
1438:
1434:
1422:
1418:
1414:
1410:
1400:
1398:
1394:
1393:
1387:
1385:
1381:
1377:
1373:
1368:
1366:
1362:
1358:
1354:
1352:
1351:straight line
1346:
1342:
1338:
1335:In classical
1333:
1331:
1327:
1326:
1321:
1317:
1313:
1309:
1305:
1301:
1297:
1293:
1289:
1277:
1272:
1270:
1265:
1263:
1258:
1257:
1255:
1254:
1243:
1240:
1238:
1235:
1234:
1233:
1232:
1228:
1227:
1222:
1219:
1217:
1214:
1212:
1209:
1207:
1204:
1202:
1199:
1197:
1194:
1192:
1189:
1187:
1184:
1182:
1179:
1177:
1174:
1172:
1169:
1168:
1167:
1166:
1162:
1161:
1156:
1153:
1151:
1148:
1146:
1143:
1141:
1138:
1136:
1133:
1131:
1128:
1126:
1123:
1121:
1118:
1117:
1116:
1115:
1111:
1110:
1105:
1102:
1100:
1097:
1095:
1092:
1090:
1087:
1085:
1082:
1080:
1077:
1075:
1072:
1070:
1067:
1065:
1062:
1060:
1057:
1055:
1052:
1050:
1047:
1046:
1045:
1044:
1040:
1039:
1034:
1031:
1029:
1026:
1024:
1021:
1019:
1016:
1014:
1011:
1009:
1006:
1004:
1001:
1000:
999:
998:
995:
992:
991:
981:
980:
973:
970:
968:
965:
963:
960:
958:
955:
953:
950:
948:
945:
943:
940:
938:
935:
933:
930:
928:
925:
923:
920:
918:
915:
913:
910:
908:
905:
903:
900:
898:
895:
893:
890:
888:
885:
883:
880:
878:
875:
873:
870:
868:
865:
863:
860:
858:
855:
853:
850:
848:
845:
843:
840:
838:
835:
833:
830:
828:
825:
823:
820:
818:
815:
813:
810:
808:
805:
803:
800:
798:
795:
793:
790:
788:
785:
783:
780:
778:
775:
773:
770:
769:
761:
760:
757:
754:
753:
746:
743:
741:
738:
737:
732:
726:
725:
718:
715:
713:
710:
708:
705:
703:
700:
698:
695:
693:
690:
688:
685:
683:
680:
676:
673:
672:
671:
668:
667:
664:
661:
660:
656:
650:
649:
638:
635:
633:
632:Circumference
630:
628:
625:
624:
623:
622:
619:
616:
615:
610:
607:
605:
602:
601:
600:
599:
596:
595:Quadrilateral
593:
592:
587:
584:
582:
579:
577:
574:
572:
569:
568:
567:
566:
563:
562:Parallelogram
560:
559:
554:
551:
549:
546:
544:
541:
540:
539:
538:
535:
532:
531:
526:
523:
521:
518:
516:
513:
512:
511:
510:
504:
498:
497:
490:
487:
483:
480:
478:
475:
474:
473:
470:
469:
465:
459:
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400:Perpendicular
397:
396:Orthogonality
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320:Computational
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205:Non-Euclidean
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54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
3770:Publications
3635:Chern number
3625:Betti number
3508: /
3499:Key concepts
3447:Differential
3370:
3336:
3275:Hyperpyramid
3240:Hypersurface
3133:Affine space
3123:Vector space
3009:
2991:
2970:
2954:
2943:
2933:. Macmillan.
2929:
2918:
2895:
2873:
2860:
2833:
2815:
2786:
2782:
2770:
2766:
2743:
2730:
2707:, Chapter 5.
2700:
2688:
2683:, p. 3.
2673:Dirac (1958)
2668:
2647:
2633:Gerla (1985)
2628:
2621:Heath (1956)
2616:
2604:
2599:, p. 7.
2592:
2585:Heath (1956)
2580:
2573:Ohmer (1969)
2568:
2494:Affine space
2471:
2460:point charge
2439:
2435:
2429:
2413:
2409:
2367:
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2259:
2255:
2253:
2191:
2187:
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2063:
2059:metric space
2054:
2052:
2040:
2035:
2031:
1909:
1887:
1835:
1829:
1811:
1796:
1787:
1772:Please help
1760:
1736:
1728:line segment
1707:
1698:
1516:
1500:
1496:
1489:
1482:
1421:ordered pair
1406:
1397:neighborhood
1390:
1388:
1383:
1369:
1348:
1334:
1329:
1323:
1320:intersection
1291:
1285:
1104:Parameshvara
917:Parameshvara
687:Dodecahedron
449:
271:Differential
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
3733:Wikiversity
3650:Key results
3360:Codimension
3339:-dimensions
3260:Hypersphere
3143:Free module
3036:Mathematics
2811:Dirac, Paul
2773:(1): 61ā75.
2529:Point cloud
2041:A point is
1376:coordinates
1308:dimensional
1229:Present day
1176:Lobachevsky
1163:1700sā1900s
1120:Jyeį¹£į¹hadeva
1112:1400sā1700s
1064:Brahmagupta
887:Lobachevsky
867:Jyeį¹£į¹hadeva
817:Brahmagupta
745:Hypersphere
717:Tetrahedron
692:Icosahedron
264:Diophantine
3579:CW complex
3520:Continuity
3510:Closed set
3469:cohomology
3355:Hyperspace
3235:Hyperplane
2993:PlanetMath
2714:References
2464:Paul Dirac
2456:point mass
2414:connection
1914:open cover
1790:March 2022
1732:degenerate
1437:horizontal
1306:. As zero-
1089:al-Yasamin
1033:Apollonius
1028:Archimedes
1018:Pythagoras
1008:Baudhayana
962:al-Yasamin
912:Pythagoras
807:Baudhayana
797:Archimedes
792:Apollonius
697:Octahedron
548:Hypotenuse
423:Similarity
418:Congruence
330:Incidence
281:Symplectic
276:Riemannian
259:Arithmetic
234:Projective
222:Hyperbolic
150:Projecting
99:March 2022
69:newspapers
3758:geometric
3753:algebraic
3604:Cobordism
3540:Hausdorff
3535:connected
3452:Geometric
3442:Continuum
3432:Algebraic
3245:Hypercube
3223:Polytopes
3203:Minkowski
3198:Hausdorff
3193:Inductive
3158:Spacetime
3109:Dimension
3011:MathWorld
2544:Pointwise
2410:inclusion
2398:functions
2382:algebraic
2304:≥
2280:
2237:δ
2211:∈
2204:∑
2185:for each
2171:covering
2153:∈
1849:⋅
1814:dimension
1761:does not
1597:∣
1512:dimension
1384:point set
1206:Minkowski
1125:Descartes
1059:Aryabhata
1054:KÄtyÄyana
985:by period
897:Minkowski
872:KÄtyÄyana
832:Descartes
777:Aryabhata
756:Geometers
740:Tesseract
604:Trapezoid
576:Rectangle
369:Dimension
254:Algebraic
244:Synthetic
215:Spherical
200:Euclidean
3785:Category
3723:Wikibook
3701:Category
3589:Manifold
3557:Homotopy
3515:Interior
3506:Open set
3464:Homology
3413:Topology
3372:Category
3348:See also
3148:Manifold
2941:(1919).
2916:(1950).
2906:00218666
2871:(1956).
2831:(1964).
2813:(1958).
2742:(1986).
2728:(2005).
2482:See also
2451:integral
2443:function
2076:ā [0, ā)
1705:through
1519:infinite
1445:vertical
1316:surfaces
1296:position
1288:geometry
1196:PoincarƩ
1140:Minggatu
1099:Yang Hui
1069:Virasena
957:Yang Hui
952:Virasena
922:PoincarƩ
902:Minggatu
682:Cylinder
627:Diameter
586:Rhomboid
543:Altitude
534:Triangle
428:Symmetry
406:Parallel
391:Diagonal
361:Features
358:Concepts
249:Analytic
210:Elliptic
192:Branches
178:Timeline
137:Geometry
3748:general
3550:uniform
3530:compact
3481:Digital
3270:Simplex
3208:Fractal
3048:Physics
3022:Portals
3006:"Point"
2988:"Point"
2803:2939504
2386:logical
2092:infimum
2090:is the
2005:refines
1782:removed
1767:sources
1510:is the
1479:terms,
1365:scriber
1361:compass
1221:Coxeter
1201:Hilbert
1186:Riemann
1135:Huygens
1094:al-Tusi
1084:KhayyƔm
1074:Alhazen
1041:1ā1400s
942:al-Tusi
927:Riemann
877:KhayyƔm
862:Huygens
857:Hilbert
827:Coxeter
787:Alhazen
765:by name
702:Pyramid
581:Rhombus
525:Polygon
477:segment
325:Fractal
308:Digital
293:Complex
174:History
169:Outline
83:scholar
3743:Topics
3545:metric
3420:Fields
3227:shapes
2904:
2881:
2841:
2801:
2750:
2406:region
2392:or an
2183:> 0
2078:, the
2003:which
1696:where
1506:where
1495:,āā¦ā,
1473:tuplet
1413:Euclid
1345:axioms
1330:corner
1325:vertex
1312:curves
1304:spaces
1242:Gromov
1237:Atiyah
1216:Veblen
1211:Cartan
1181:Bolyai
1150:Sakabe
1130:Pascal
1023:Euclid
1013:Manava
947:Veblen
932:Sakabe
907:Pascal
892:Manava
852:Gromov
837:Euclid
822:Cartan
812:Bolyai
802:Atiyah
712:Sphere
675:cuboid
663:Volume
618:Circle
571:Square
489:Length
411:Vertex
315:Convex
298:Finite
239:Affine
154:sphere
85:
78:
71:
64:
56:
3525:Space
3331:Eight
3326:Seven
3306:Three
3183:Krull
2857:(PDF)
2799:JSTOR
2561:Notes
2438:, or
2175:with
2103:balls
2061:. If
2057:be a
1724:plane
1292:point
1191:Klein
1171:Gauss
1145:Euler
1079:Sijzi
1049:Zhang
1003:Ahmes
967:Zhang
937:Sijzi
882:Klein
847:Gauss
842:Euler
782:Ahmes
515:Plane
450:Point
386:Curve
381:Angle
158:plane
156:to a
90:JSTOR
76:books
3316:Five
3311:Four
3291:Zero
3225:and
3072:Maps
2902:OCLC
2879:ISBN
2839:ISBN
2748:ISBN
2661:1929
2657:1920
2653:1919
2639:help
2509:Cusp
2372:and
2254:The
2234:<
2071:and
2053:Let
1765:any
1763:cite
1714:and
1527:line
1382:, a
1290:, a
1155:Aida
772:Aida
731:Four
670:Cube
637:Area
609:Kite
520:Area
472:Line
62:news
3321:Six
3301:Two
3296:One
2791:doi
2458:or
2412:or
2384:or
2378:set
2295:inf
2271:dim
2258:of
2099:ā„ 0
2086:of
1983:of
1939:of
1776:by
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