40:
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2591:
1580:
20:
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2617:, which states: "Fifteen schoolgirls walk each day in five groups of three. Arrange the girlsā walk for a week so that in that time, each pair of girls walks together in a group just once." There are 35 different combinations for the girls to walk together. There are also 7 days of the week, and 3 girls in each group. Two of the seven non-isomorphic solutions to this problem can be stated in terms of structures in the Fano 3-space, PG(3,2), known as
3925:
3913:
1600:
2564:
1505:
2. (The order of an affine plane is the number of points on any line, see below.) Since no three are collinear, any pair of points determines a unique line, and so this plane contains six lines. It corresponds to a tetrahedron where non-intersecting edges are considered "parallel", or a square where
1595:
for projective plane geometries, meaning that any true statement valid in all these geometries remains true if we exchange points for lines and lines for points. The smallest geometry satisfying all three axioms contains seven points. In this simplest of the projective planes, there are also seven
2629:
of its points into disjoint lines, and a packing is a partition of the lines into disjoint spreads. In PG(3,2), a spread would be a partition of the 15 points into 5 disjoint lines (with 3 points on each line), thus corresponding to the arrangement of schoolgirls on a particular day. A packing of
2246:
all finite division rings are fields. In this case, this construction produces a finite projective space. Furthermore, if the geometric dimension of a projective space is at least three then there is a division ring from which the space can be constructed in this manner. Consequently, all finite
2441:
2027:
The last axiom eliminates reducible cases that can be written as a disjoint union of projective spaces together with 2-point lines joining any two points in distinct projective spaces. More abstractly, it can be defined as an
1260:
to a projective space over a finite field (that is, the projectivization of a vector space over a finite field). However, dimension two has affine and projective planes that are not isomorphic to Galois geometries, namely the
2193:
23:
Finite affine plane of order 2, containing 4 "points" and 6 "lines". Lines of the same color are "parallel". The centre of the figure is not a "point" of this affine plane, hence the two green "lines" don't
1305:, by contrast, any two lines intersect at a unique point, so parallel lines do not exist. Both finite affine plane geometry and finite projective plane geometry may be described by fairly simple
1482:
3175:
1591:
An examination of the first two axioms shows that they are nearly identical, except that the roles of points and lines have been interchanged. This suggests the principle of
1422:
1205:
are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite
2598:
Every point is contained in 7 lines. Every pair of distinct points are contained in exactly one line and every pair of distinct planes intersects in exactly one line.
1876:
that he developed, he considered a finite three dimensional space with 15 points, 35 lines and 15 planes (see diagram), in which each line had only three points on it.
3311:
1616:. If any of the lines is removed from the plane, along with the points on that line, the resulting geometry is the affine plane of order 2. The Fano plane is called
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1357:
1765:
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or too simple to be of interest, such as a single line with an arbitrary number of points on it), while the first two specify the nature of the geometry.
2579:. It has 15 points, 35 lines, and 15 planes. Each plane contains 7 points and 7 lines. Each line contains 3 points. As geometries, these planes are
2247:
projective spaces of geometric dimension at least three are defined over finite fields. A finite projective space defined over such a finite field has
2975:
2813:, pgs. 6ā7). Pasch was concerned with real projective space and was attempting to introduce order, which is not a concern of the VeblenāYoung axiom.
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2121:
1201:
is not finite, because a
Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the
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3118:
3349:
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the 2-dimensional (two independent generators) subspaces (closed under vector addition) of this vector space. Incidence is containment. If
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1168:
1252:. Finite geometries can also be defined purely axiomatically. Most common finite geometries are Galois geometries, since any finite
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The smallest integer that is not a prime power and not covered by the BruckāRyser theorem is 10; 10 is of the form
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2614:
1854:
1841:
The next smallest number to consider is 12, for which neither a positive nor a negative result has been proved.
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1936:
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Collino, Alberto; Conte, Alberto; Verra, Alessandro (2013). "On the life and scientific work of Gino Fano".
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points on a line, so the two concepts of order coincide. Such a finite projective space is denoted by
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if that is the largest number for which there is a strictly ascending chain of subspaces of this form:
1932:
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746:
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Dimension 0 (no lines): The space is a single point and is so degenerate that it is usually ignored.
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1935:. For a discussion of higher-dimensional finite spaces in general, see, for instance, the works of
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because of their regularity and simplicity. Other significant types of finite geometry are finite
123:
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229:
86:
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214:
99:
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PG(3,2) consists of seven disjoint spreads and so corresponds to a full week of arrangements.
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2436:{\displaystyle {{n+1} \choose {k+1}}_{q}=\prod _{i=0}^{k}{\frac {q^{n+1-i}-1}{q^{i+1}-1}},}
1663:
1647:
1382:
1342:
1065:
988:
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72:
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Dimension 2: There are at least 2 lines, and any two lines meet. A projective space for
8:
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68:
63:
44:
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1912:
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1427:
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1226:
1198:
1029:
756:
596:
224:
148:
138:
109:
94:
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2 because it is unique (up to isomorphism). In general, the projective plane of order
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3608:
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2707:
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In any finite projective space, each line contains the same number of points and the
1100:
888:
866:
791:
650:
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305:
197:
143:
104:
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1692:+ 1 points (for a projective plane). One major open question in finite geometry is:
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816:
726:
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1807:
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For every two distinct points, there is exactly one line that contains both points.
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For every two distinct points, there is exactly one line that contains both points.
1302:
1253:
1245:
1206:
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1080:
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202:
187:
52:
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3411:
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3251:
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3010:
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2939:
2900:
1860:
The first axiomatic treatment of finite projective geometry was developed by the
1722:
1286:
1249:
503:
366:
209:
133:
39:
3085:(1999). "Yea why try her raw wet hat: A tour of the smallest projective space".
1214:
1075:
1044:
978:
826:
771:
706:
3840:
3767:
3474:
2645:
2488:. These are much harder to classify, as not all of them are isomorphic with a
2188:{\displaystyle \varnothing =X_{-1}\subset X_{0}\subset \cdots \subset X_{n}=P.}
1233:
1218:
1131:
1039:
983:
948:
856:
766:
736:
696:
601:
3274:
3155:
3022:
1105:
716:
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3512:
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2527:
states in the finite case that every projective space of geometric dimension
2470:
Dimension 1 (exactly one line): All points lie on the unique line, called a
2204:
1985:
1880:
1853:(1847) and the systematic development of finite projective geometry given by
1575:
There exists a set of four points, no three of which belong to the same line.
1486:
There exists a set of four points, no three of which belong to the same line.
1110:
1095:
1024:
841:
801:
751:
526:
489:
456:
294:
290:
3256:
2825:, p. 28, where the formula is given, in terms of vector space dimension, by
2203:
A standard algebraic construction of systems satisfies these axioms. For a
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3565:
3469:
2670:
2650:
2639:
2235:
1892:
1718:
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1579:
1241:
1237:
1210:
1049:
998:
811:
666:
581:
371:
3327:
3244:
2727:
Fano, G. (1892), "Sui postulati fondamentali della geometria proiettiva",
1596:
lines; each point is on three lines, and each line contains three points.
3772:
3436:
3359:
3321:
3229:
3180:, London Mathematical Society Student Texts, Cambridge University Press,
2858:
2680:
2580:
1714:
1257:
1085:
958:
776:
711:
639:
611:
586:
3126:, New York: Cambridge university Press, pp. 488ā499, archived from
2274:
is the size (order) of the finite field used to construct the geometry.
3757:
3636:
3431:
3098:
3048:
3006:
2584:
1635: + 1 points and the same number of lines; each line contains
1612:
1604:
1584:
1557:(whose elements are called "points"), along with a nonempty collection
1506:
not only opposite sides, but also diagonals are considered "parallel".
1321:(whose elements are called "points"), along with a nonempty collection
1222:
1190:
943:
922:
912:
902:
861:
806:
701:
691:
591:
442:
19:
1572:
The intersection of any two distinct lines contains exactly one point.
1501:
The simplest affine plane contains only four points; it is called the
1273:
3211:
2602:
1943:) has many important applications in advanced mathematical theories.
1865:
1741:
elements. Planes not derived from finite fields also exist (e.g. for
1495:
953:
671:
634:
498:
470:
3040:
2989:
2222:(vector space dimension is the number of elements in a basis). Let
1731:), by using affine and projective planes over the finite field with
3661:
3580:
3507:
1931:
geometry and the geometry of higher-dimensional finite spaces, see
1868:. In his work on proving the independence of the set of axioms for
1861:
1728:
1659:
1186:
1034:
993:
963:
851:
846:
796:
521:
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428:
322:
285:
31:
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2917:
1830:. The non-existence of a finite plane of order 10 was proven in a
3446:
2576:
2516:
and are projective planes over finite fields, but there are many
2462:
Classification of finite projective spaces by geometric dimension
1725:
1662:ā PSL(3,2), which in this special case is also isomorphic to the
1650:
points (points on the same line) to collinear points is called a
1277:
Finite affine plane of order 3, containing 9 points and 12 lines.
968:
681:
475:
419:
219:
917:
907:
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731:
606:
569:
557:
512:
465:
383:
48:
2571:
The smallest 3-dimensional projective space is over the field
2523:
Dimension at least 3: Two non-intersecting lines exist. The
1265:. Similar results hold for other kinds of finite geometries.
2572:
2563:
2104:). The full space and the empty space are always subspaces.
2085:
of the space is defined as one less than this common number.
1599:
1306:
1225:, and their higher-dimensional analogs such as higher finite
1202:
973:
897:
831:
676:
280:
275:
1646:
A permutation of the Fano plane's seven points that carries
564:
414:
1688:
points (for an affine plane), or such that each line has
1610:
This particular projective plane is sometimes called the
3201:
2550:-dimensional projective space over some finite field GF(
1922:
3293:
on canonical geometric properties of small finite sets.
3023:"The Search for a Finite Projective Plane of Order 10"
2880:
2810:
2789:
2558:
2226:
be the 1-dimensional (single generator) subspaces and
2883:
Projective geometry: from foundations to applications
2302:
2124:
1747:
1450:
1430:
1405:
1385:
1365:
1345:
1285:. There are two main kinds of finite plane geometry:
1767:), but all known examples have order a prime power.
1697:
Is the order of a finite plane always a prime power?
2700:Laywine, Charles F.; Mullen, Gary L. (1998-09-17).
2613:PG(3,2) arises as the background for a solution of
1970:(the set of lines), satisfying these axioms :
2927:
2605:was the first to consider such a finite geometry.
2435:
2187:
1759:
1587:: Each point corresponds to a line and vice versa.
1476:
1436:
1416:
1391:
1371:
1351:
2976:Transactions of the American Mathematical Society
2910:
2778:Transactions of the American Mathematical Society
2745:
2336:
2307:
1939:. The study of these higher-dimensional spaces (
3943:
2936:Ergebnisse der Mathematik und ihrer Grenzgebiete
2881:Beutelspacher, Albrecht; Rosenbaum, Ute (1998),
2642:ā a generalization of a finite projective plane.
1849:Individual examples can be found in the work of
1658:is of order 168 and is isomorphic to the group
1565:(whose elements are called "lines"), such that:
1490:The last axiom ensures that the geometry is not
1329:(whose elements are called "lines"), such that:
2608:
2270:is the geometric dimension of the geometry and
2088:A subspace of the projective space is a subset
1516:More generally, a finite affine plane of order
16:Geometric system with a finite number of points
3177:Finite Geometry and Combinatorial Applications
2092:, such that any line containing two points of
1927:For some important differences between finite
1814:does not occur as the order of a finite plane.
1553:A projective plane geometry is a nonempty set
1477:{\displaystyle \ell \cap \ell '=\varnothing .}
1221:, which are examples of a general type called
3343:
3235:Essay on Finite Geometry by Michael Greenberg
2983:(2), American Mathematical Society: 229ā277,
2699:
1162:
2973:Hall, Marshall (1943), "Projective planes",
1639: + 1 points, and each point is on
1509:The affine plane of order 3 is known as the
3357:
2266:, where PG stands for projective geometry,
1907:-dimensional finite geometry is denoted PG(
1548:
1317:An affine plane geometry is a nonempty set
1281:The following remarks apply only to finite
3350:
3336:
3312:āProblem 31: Kirkman's schoolgirl problemā
2069:projective space requires one more axiom:
2004:are distinct points and the lines through
1169:
1155:
38:
2988:
2925:
2916:
2822:
2760:Finite Geometries? an AMS Featured Column
2198:
2062:stating which points lie on which lines.
1962:(the set of points), together with a set
1232:Finite geometries may be constructed via
3280:
3275:Galois Geometry and Generalized Polygons
2703:Discrete Mathematics Using Latin Squares
2589:
2562:
1826:, but it is equal to the sum of squares
1598:
1578:
1272:
18:
3081:
3072:
2567:PG(3,2) but not all the lines are drawn
1946:
1806:is not equal to the sum of two integer
1770:The best general result to date is the
1312:
3944:
3263:Books by Hirschfeld on finite geometry
3056:
2857:
2757:
1958:can be defined axiomatically as a set
1705:Affine and projective planes of order
269:Straightedge and compass constructions
3331:
3202:
3145:
3113:
2023:Any line has at least 3 points on it.
1923:Finite spaces of 3 or more dimensions
1915:and has an associated transformation
3912:
3221:Incidence Geometry by Eric Moorhouse
3173:
2972:
2963:
2726:
3924:
3020:
2559:The smallest projective three-space
2012:meet, then so do the lines through
1835:
13:
3259:, researcher on finite geometries
2864:Combinatorics of Finite Geometries
2811:Beutelspacher & Rosenbaum 1998
2790:Beutelspacher & Rosenbaum 1998
2504:(those that are isomorphic with a
2311:
2100:(that is, completely contained in
1672:
14:
3968:
3195:
1468:
1256:of dimension three or greater is
235:Noncommutative algebraic geometry
3923:
3911:
3900:
3899:
3887:
3226:Algebraic Combinatorial Geometry
3075:Fundamental Concepts of Geometry
2966:A Survey of Geometry: Volume One
1702:This is conjectured to be true.
1399:, there exists exactly one line
1268:
3808:Computational complexity theory
3318: (archived August 17, 2010)
2746:Collino, Conte & Verra 2013
2218:-dimensional vector space over
1684:is one such that each line has
3281:Carnahan, Scott (2007-10-27),
3270:AMS Column: Finite Geometries?
3087:The Mathematical Intelligencer
3077:, New York: Dover Publications
2968:, Boston: Allyn and Bacon Inc.
2867:, Cambridge University Press,
2816:
2795:
2783:
2763:
2751:
2739:
2720:
2693:
1903:+ 1 coordinates are used, the
628:- / other-dimensional
1:
3028:American Mathematical Monthly
2938:, Band 44, Berlin, New York:
2851:
2448:Gaussian binomial coefficient
2234:is finite then it must be a
1834:that finished in 1989 ā see (
1538:points, and each point is on
3322:Projective Plane of Order 12
2774:Finite Projective Geometries
2615:Kirkman's schoolgirl problem
2609:Kirkman's schoolgirl problem
2594:Square model of Fano 3-space
7:
2633:
2625:of a projective space is a
2244:Wedderburn's little theorem
2111:of the space is said to be
1883:and W. H. Bussey described
10:
3973:
3858:Films about mathematicians
3303:: CS1 maint: postscript (
3277:, intensive course in 1998
3150:, Universitext, Springer,
3073:Meserve, Bruce E. (1983),
2887:Cambridge University Press
2281:-dimensional subspaces of
2277:In general, the number of
1933:axiomatic projective space
1844:
1534:lines; each line contains
1248:so constructed are called
1215:Mƶbius or inversive planes
3881:
3831:
3788:
3698:
3660:
3627:
3579:
3551:
3498:
3445:
3427:Philosophy of mathematics
3402:
3367:
3245:Finite Geometry Resources
3156:10.1007/978-3-642-15627-4
3146:Shult, Ernest E. (2011),
2926:Dembowski, Peter (1968),
2706:. John Wiley & Sons.
2293:is given by the product:
1974:Each two distinct points
1621:projective plane of order
3863:Recreational mathematics
3240:Finite geometry (Script)
2801:also referred to as the
2686:
1982:are in exactly one line.
1851:Thomas Penyngton Kirkman
1549:Finite projective planes
124:Non-Archimedean geometry
3748:Mathematical statistics
3738:Mathematical psychology
3708:Engineering mathematics
3642:Algebraic number theory
3287:Secret Blogging Seminar
2729:Giornale di Matematiche
2666:Linear space (geometry)
2518:non-Desarguesian planes
1889:homogeneous coordinates
1832:computer-assisted proof
1774:of 1949, which states:
1654:of the plane. The full
1263:non-Desarguesian planes
1189:system that has only a
230:Noncommutative geometry
3894:Mathematics portal
3743:Mathematical sociology
3723:Mathematical economics
3718:Mathematical chemistry
3647:Analytic number theory
3528:Differential equations
3021:Lam, C. W. H. (1991),
2805:and mistakenly as the
2595:
2568:
2437:
2371:
2199:Algebraic construction
2189:
1891:with entries from the
1761:
1643: + 1 lines.
1607:
1588:
1478:
1438:
1418:
1417:{\displaystyle \ell '}
1393:
1373:
1353:
1297:, the normal sense of
1278:
198:Discrete/Combinatorial
25:
3873:Mathematics education
3803:Theory of computation
3523:Hypercomplex analysis
3289:, notes on a talk by
3174:Ball, Simeon (2015),
2964:Eves, Howard (1963),
2859:Batten, Lynn Margaret
2593:
2566:
2534:is isomorphic with a
2438:
2351:
2190:
1762:
1602:
1582:
1503:affine plane of order
1479:
1439:
1419:
1394:
1392:{\displaystyle \ell }
1374:
1354:
1352:{\displaystyle \ell }
1276:
181:Discrete differential
22:
3853:Informal mathematics
3733:Mathematical physics
3728:Mathematical finance
3713:Mathematical biology
3652:Diophantine geometry
3120:On Galois Geometries
3059:"Finite Geometries?"
2525:VeblenāYoung theorem
2456:binomial coefficient
2300:
2122:
2047:consisting of a set
1947:Axiomatic definition
1745:
1664:general linear group
1448:
1428:
1403:
1383:
1363:
1343:
1313:Finite affine planes
1301:lines applies. In a
1227:inversive geometries
3868:Mathematics and art
3778:Operations research
3533:Functional analysis
3283:"Small finite sets"
3257:J. W. P. Hirschfeld
2656:Generalized polygon
2514:Desargues's theorem
2502:Desarguesian planes
2109:geometric dimension
2030:incidence structure
1885:projective geometry
1772:BruckāRyser theorem
1760:{\displaystyle n=9}
1511:Hesse configuration
448:Pythagorean theorem
3813:Numerical analysis
3422:Mathematical logic
3417:Information theory
3250:2011-09-27 at the
3204:Weisstein, Eric W.
3099:10.1007/BF03024845
2803:VeblenāYoung axiom
2661:Incidence geometry
2596:
2575:and is denoted by
2569:
2433:
2185:
2073:The set of points
2057:incidence relation
1913:synthetic geometry
1757:
1677:A finite plane of
1667:GL(3,2) ā PGL(3,2)
1656:collineation group
1608:
1589:
1474:
1434:
1414:
1389:
1369:
1349:
1279:
1199:Euclidean geometry
26:
3939:
3938:
3538:Harmonic analysis
3291:Jean-Pierre Serre
3207:"finite geometry"
3165:978-3-642-15626-7
3057:Malkevitch, Joe.
2930:Finite geometries
2896:978-0-521-48364-3
2428:
2334:
2055:of lines, and an
2051:of points, a set
1937:J.W.P. Hirschfeld
1437:{\displaystyle p}
1372:{\displaystyle p}
1250:Galois geometries
1246:projective planes
1244:; the affine and
1179:
1178:
1144:
1143:
867:List of geometers
550:Three-dimensional
539:
538:
3964:
3927:
3926:
3915:
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3903:
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3892:
3891:
3823:Computer algebra
3798:Computer science
3518:Complex analysis
3352:
3345:
3338:
3329:
3328:
3324:on MathOverflow.
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3148:Points and Lines
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3115:Segre, Beniamino
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2676:Partial geometry
2545:
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2511:
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2486:projective plane
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2142:
2077:is a finite set.
2046:
1953:projective space
1942:
1911:). It arises in
1829:
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1801:
1793:
1784:positive integer
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1337:Playfair's axiom
1303:projective plane
1254:projective space
1236:, starting from
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1164:
1157:
885:
884:
404:
403:
337:Zero-dimensional
42:
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3763:Systems science
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3690:Homotopy theory
3656:
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3441:
3412:Category theory
3398:
3363:
3356:
3316:Wayback Machine
3296:
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3252:Wayback Machine
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3041:10.2307/2323798
2990:10.2307/1990331
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2940:Springer-Verlag
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1838:) for details.
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1673:Order of planes
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1583:Duality in the
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1339:: Given a line
1315:
1271:
1219:Laguerre planes
1197:. The familiar
1183:finite geometry
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3196:External links
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3187:978-1107518438
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2807:axiom of Pasch
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1988:'s axiom: If
1983:
1966:of subsets of
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1864:mathematician
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1234:linear algebra
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3957:Combinatorics
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3629:Number theory
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3513:Real analysis
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3133:on 2015-03-30
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2770:Oswald Veblen
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1275:
1269:Finite planes
1266:
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1247:
1243:
1239:
1238:vector spaces
1235:
1230:
1228:
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1220:
1216:
1212:
1211:affine spaces
1208:
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1196:
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1167:
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955:
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571:
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566:
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562:
559:
556:
555:
551:
545:
544:
533:
530:
528:
527:Circumference
525:
523:
520:
519:
518:
517:
514:
511:
510:
505:
502:
500:
497:
496:
495:
494:
491:
490:Quadrilateral
488:
487:
482:
479:
477:
474:
472:
469:
467:
464:
463:
462:
461:
458:
457:Parallelogram
455:
454:
449:
446:
444:
441:
439:
436:
435:
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430:
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421:
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411:
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332:
331:
324:
321:
319:
316:
314:
311:
310:
307:
304:
302:
299:
296:
295:Perpendicular
292:
291:Orthogonality
289:
287:
284:
282:
279:
277:
274:
273:
270:
267:
266:
265:
255:
252:
251:
246:
245:
236:
233:
232:
231:
228:
226:
223:
221:
218:
216:
215:Computational
213:
211:
208:
204:
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200:
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196:
194:
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182:
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177:
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111:
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107:
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103:
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101:
100:Non-Euclidean
98:
96:
93:
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62:
61:
59:
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30:
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3928:
3916:
3904:
3885:
3818:Optimization
3680:Differential
3618:
3604:Differential
3571:Order theory
3566:Graph theory
3470:Group theory
3286:
3210:
3176:
3147:
3135:, retrieved
3128:the original
3119:
3093:(2): 38ā43.
3090:
3086:
3074:
3062:. Retrieved
3032:
3026:
2980:
2974:
2965:
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2839:
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2671:Near polygon
2651:Finite space
2640:Block design
2618:
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2249:
2242:), since by
2239:
2236:finite field
2231:
2227:
2223:
2219:
2213:
2207:
2202:
2112:
2108:
2106:
2101:
2097:
2093:
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2066:
2065:Obtaining a
2064:
2059:
2052:
2048:
2042:
2038:
2034:
2026:
2017:
2013:
2009:
2005:
2001:
1997:
1993:
1989:
1979:
1975:
1967:
1963:
1959:
1955:
1952:
1950:
1928:
1926:
1908:
1904:
1900:
1896:
1893:Galois field
1878:
1871:
1859:
1848:
1840:
1821:
1817:
1811:
1803:
1797:
1789:
1786:of the form
1779:
1769:
1737:
1733:
1721:raised to a
1719:prime number
1710:
1706:
1704:
1701:
1696:
1689:
1685:
1681:
1678:
1676:
1652:collineation
1645:
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1617:
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1562:
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1500:
1491:
1489:
1359:and a point
1326:
1322:
1318:
1316:
1295:affine plane
1282:
1280:
1242:finite field
1231:
1182:
1180:
999:Parameshvara
812:Parameshvara
582:Dodecahedron
192:
166:Differential
24:"intersect".
3930:WikiProject
3773:Game theory
3753:Probability
3490:Homological
3480:Multilinear
3460:Commutative
3437:Type theory
3404:Foundations
3360:mathematics
3230:Terence Tao
2681:Polar space
2446:which is a
1870:projective
1715:prime power
1524:points and
1424:containing
1223:Benz planes
1124:Present day
1071:Lobachevsky
1058:1700sā1900s
1015:Jyeį¹£į¹hadeva
1007:1400sā1700s
959:Brahmagupta
782:Lobachevsky
762:Jyeį¹£į¹hadeva
712:Brahmagupta
640:Hypersphere
612:Tetrahedron
587:Icosahedron
159:Diophantine
3946:Categories
3758:Statistics
3637:Arithmetic
3599:Arithmetic
3465:Elementary
3432:Set theory
3137:2015-07-02
2874:0521590140
2852:References
2758:Malkevitch
2585:Fano plane
2581:isomorphic
2512:) satisfy
1855:von Staudt
1613:Fano plane
1605:Fano plane
1585:Fano plane
1444:such that
1291:projective
1258:isomorphic
1207:projective
1193:number of
984:al-Yasamin
928:Apollonius
923:Archimedes
913:Pythagoras
903:Baudhayana
857:al-Yasamin
807:Pythagoras
702:Baudhayana
692:Archimedes
687:Apollonius
592:Octahedron
443:Hypotenuse
318:Similarity
313:Congruence
225:Incidence
176:Symplectic
171:Riemannian
154:Arithmetic
129:Projective
117:Hyperbolic
45:Projecting
3685:Geometric
3675:Algebraic
3614:Euclidean
3589:Algebraic
3485:Universal
3212:MathWorld
3107:122352568
2999:0002-9947
2918:1311.7177
2792:, pp. 6ā7
2780:7: 241ā59
2735:: 106ā132
2627:partition
2603:Gino Fano
2601:In 1892,
2422:−
2398:−
2390:−
2353:∏
2164:⊂
2161:⋯
2158:⊂
2145:⊂
2137:−
2126:∅
1866:Gino Fano
1648:collinear
1469:∅
1459:ℓ
1455:∩
1452:ℓ
1408:ℓ
1387:ℓ
1347:ℓ
1187:geometric
1101:Minkowski
1020:Descartes
954:Aryabhata
949:KÄtyÄyana
880:by period
792:Minkowski
767:KÄtyÄyana
727:Descartes
672:Aryabhata
651:Geometers
635:Tesseract
499:Trapezoid
471:Rectangle
264:Dimension
149:Algebraic
139:Synthetic
110:Spherical
95:Euclidean
3906:Category
3662:Topology
3609:Discrete
3594:Analytic
3581:Geometry
3553:Discrete
3508:Calculus
3500:Analysis
3455:Abstract
3394:Glossary
3377:Timeline
3299:citation
3248:Archived
3117:(1960),
2861:(1997),
2634:See also
2619:packings
1899:). When
1879:In 1906
1857:(1856).
1836:Lam 1991
1729:exponent
1723:positive
1660:PSL(2,7)
1494:(either
1462:′
1411:′
1299:parallel
1293:. In an
1091:PoincarƩ
1035:Minggatu
994:Yang Hui
964:Virasena
852:Yang Hui
847:Virasena
817:PoincarƩ
797:Minggatu
577:Cylinder
522:Diameter
481:Rhomboid
438:Altitude
429:Triangle
323:Symmetry
301:Parallel
286:Diagonal
256:Features
253:Concepts
144:Analytic
105:Elliptic
87:Branches
73:Timeline
32:Geometry
3918:Commons
3700:Applied
3670:General
3447:Algebra
3372:History
3314:at the
3049:2323798
3015:0008892
3007:1990331
2958:0233275
2905:1629468
2772:(1906)
2583:to the
2577:PG(3,2)
2500:. The
1862:Italian
1845:History
1810:, then
1808:squares
1726:integer
1593:duality
1545:lines.
1492:trivial
1379:not on
1240:over a
1185:is any
1116:Coxeter
1096:Hilbert
1081:Riemann
1030:Huygens
989:al-Tusi
979:KhayyƔm
969:Alhazen
936:1ā1400s
837:al-Tusi
822:Riemann
772:KhayyƔm
757:Huygens
752:Hilbert
722:Coxeter
682:Alhazen
660:by name
597:Pyramid
476:Rhombus
420:Polygon
372:segment
220:Fractal
203:Digital
188:Complex
69:History
64:Outline
3619:Finite
3475:Linear
3382:Future
3358:Major
3184:
3162:
3105:
3064:Dec 2,
3047:
3013:
3005:
2997:
2956:
2946:
2903:
2893:
2871:
2748:, p. 6
2710:
2623:spread
2546:, the
2506:PG(2,
2067:finite
1986:Veblen
1887:using
1874:-space
1307:axioms
1287:affine
1283:planes
1203:pixels
1195:points
1191:finite
1137:Gromov
1132:Atiyah
1111:Veblen
1106:Cartan
1076:Bolyai
1045:Sakabe
1025:Pascal
918:Euclid
908:Manava
842:Veblen
827:Sakabe
802:Pascal
787:Manava
747:Gromov
732:Euclid
717:Cartan
707:Bolyai
697:Atiyah
607:Sphere
570:cuboid
558:Volume
513:Circle
466:Square
384:Length
306:Vertex
210:Convex
193:Finite
134:Affine
49:sphere
3846:lists
3389:Lists
3362:areas
3131:(PDF)
3124:(PDF)
3103:S2CID
3045:JSTOR
3003:JSTOR
2913:arXiv
2838:+ 1,
2687:Notes
2573:GF(2)
2484:is a
2083:order
1941:n ā„ 3
1929:plane
1917:group
1828:1 + 3
1782:is a
1713:is a
1679:order
1496:empty
1086:Klein
1066:Gauss
1040:Euler
974:Sijzi
944:Zhang
898:Ahmes
862:Zhang
832:Sijzi
777:Klein
742:Gauss
737:Euler
677:Ahmes
410:Plane
345:Point
281:Curve
276:Angle
53:plane
51:to a
3305:link
3182:ISBN
3160:ISBN
3066:2013
2995:ISSN
2944:ISBN
2891:ISBN
2869:ISBN
2708:ISBN
2621:. A
2450:, a
2216:+ 1)
2107:The
2016:and
2008:and
1978:and
1909:n, q
1802:and
1627:has
1603:The
1520:has
1289:and
1217:and
1209:and
1050:Aida
667:Aida
626:Four
565:Cube
532:Area
504:Kite
415:Area
367:Line
3228:by
3152:doi
3095:doi
3037:doi
2985:doi
2536:PG(
2532:ā„ 3
2490:PG(
2482:= 2
2283:PG(
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2252:+ 1
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1800:+ 2
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2458:.
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2018:bd
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2010:cd
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2102:X
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2002:d
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1994:b
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1980:q
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1968:P
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1367:p
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1170:e
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293:(
75:)
71:(
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