349:
38:
1009:
454:. This configuration describes two quadrilaterals that are simultaneously inscribed and circumscribed in each other. It cannot be constructed in Euclidean plane geometry but the equations defining it have nontrivial solutions in
661:) configuration in which the roles of "point" and "line" are exchanged. Types of configurations therefore come in dual pairs, except when taking the dual results in an isomorphic configuration. These exceptions are called
496:
and the twelve lines determined by pairs of these points. This configuration shares with the Fano plane the property that it contains every line through its points; configurations with this property are known as
161:, but with some additional restrictions: every two points of the incidence structure can be associated with at most one line, and every two lines can be associated with at most one point. That is, the
248:
1024:. In such cases, the restrictions that no two points belong to more than one line may be relaxed, because it is possible for two points to belong to more than one plane.
793:
There are several techniques for constructing configurations, generally starting from known configurations. Some of the simplest of these techniques construct symmetric (
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Another generalization of the concept of a configuration concerns configurations of points and circles, a notable example being the (8
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1348:
1318:
1472:
785:) configurations, and found 228 of them, but the 229th configuration, the Gropp configuration, was not discovered until 1988.
1409:
1306:
1371:
502:
1020:
The concept of a configuration may be generalized to higher dimensions, for instance to points and lines or planes in
781:) configuration. Gropp also points out a long-lasting error in this sequence: an 1895 paper attempted to list all (12
371:), the simplest possible configuration, consisting of a point incident to a line. Often excluded as being trivial.
600:
493:
278:
31:
1202:
131:
Configurations may be studied either as concrete sets of points and lines in a specific geometry, such as the
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1028:
513:
357:
1040:
1013:
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220:
1035:, consisting of twelve points and twelve planes, with six points per plane and six planes per point, the
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which are powers of primes, these constructions provide infinite families of symmetric configurations.
105:
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1052:
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These numbers count configurations as abstract incidence structures, regardless of realizability. As
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82:
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382:. Each of its three sides meets two of its three vertices, and vice versa. More generally any
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136:
78:
1435:(2006), "Configurations of points and lines", in Davis, Chandler; Ellers, Erich W. (eds.),
1358:
1059:) configurations, however, there exists a topological configuration with these parameters.
1043:, a configuration with 30 points, 12 lines, two lines per point, and five points per line.
1354:
1340:
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consisting of a 3Ă—3Ă—3 grid of 27 points and the 27 orthogonal lines through them, and the
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1329:
455:
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configurations and the notation is often condensed to avoid repetition. For example, (9
109:
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17:
1423:
1277:
1201:
This configuration would be a projective plane of order 6 which does not exist by the
1601:
1490:
547:
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113:
89:
to the same number of lines and each line is incident to the same number of points.
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732:
97:
92:
Although certain specific configurations had been studied earlier (for instance by
560:, formed by the 15 lines complementary to a double six and their 15 tangent planes
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1231:
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the number of points per line. These numbers necessarily satisfy the equation
166:
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1589:
1385:
Gropp, Harald (1990), "On the existence and non-existence of configurations
112:
wrote his dissertation on the subject in 1894, and they were popularized by
37:
379:
30:
This article is about points and lines. For incidences of polytopes, see
1535:
1519:(1986), "A resolution of the Sylvester–Gallai problem of J. P. Serre",
1008:
383:
765:) configurations, are realizable in the Euclidean plane, but for each
1051:
Configuration in the projective plane that is realized by points and
96:
in 1849), the formal study of configurations was first introduced by
1007:
347:
36:
1407:
Gropp, Harald (1997), "Configurations and their realization",
912:, then the construction results in a configuration of type ((
505:
that shows that they cannot be given real-number coordinates.
925:. Since projective planes are known to exist for all orders
1365:
Gévay, Gábor (2014), "Constructions for large point-line (n
1124:) are also used to describe configurations as defined here.
737:
934:
Not all configurations are realizable, for instance, a (43
363:
Notable projective configurations include the following:
863:(but not the points which lie on those lines except for
253:
as this product is the number of point-line incidences (
1396:
1309:(1999), "Self-dual configurations and regular graphs",
1189:
1134:
1132:
1130:
356:) configuration that is not incidence-isomorphic to a
223:
692:
The number of nonisomorphic configurations of type (
1031:, consisting of two mutually inscribed tetrahedra,
1328:
1166:
1113:
242:
125:
1439:, American Mathematical Society, pp. 179–225
1027:Notable three-dimensional configurations are the
942:has provided a construction which shows that for
147:. In the latter case they are closely related to
1475:, vol. 103, American Mathematical Society,
1337:Ergebnisse der Mathematik und ihrer Grenzgebiete
1437:The Coxeter Legacy: Reflections and Projections
906:is chosen to be a line which does pass through
285:. For instance, there exist three different (9
1263:(2000), "Counting symmetric configurations",
8:
1448:(2008), "Musing on an example of Danzer's",
1290:(2015), "Danzer's configuration revisited",
260:Configurations having the same symbol, say (
177:A configuration in the plane is denoted by (
169:of the configuration) must be at least six.
143:in that geometry), or as a type of abstract
435:. This configuration exists as an abstract
100:in 1876, in the second edition of his book
1501:(2nd ed.), Chelsea, pp. 94–170,
887:. The result is a configuration of type ((
165:of the corresponding bipartite graph (the
1554:Configurations from a Graphical Viewpoint
1534:
1422:
1276:
1121:
938:) configuration does not exist. However,
789:Constructions of symmetric configurations
239:
222:
77:in the plane consists of a finite set of
1162:
1138:
424:and complete quadrilateral respectively.
1247:The Electronic Journal of Combinatorics
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1101:
61:) (a complete quadrilateral, at right).
27:Points and lines with equal incidences
1214:
1190:Betten, Brinkmann & Pisanski 2000
1150:
939:
900:. If, in this construction, the line
772:there is at least one nonrealizable (
750:
735:, 2036, 21399, 245342, ... (sequence
392:sides forms a configuration of type (
297:and two less notable configurations.
7:
1118:tactical configuration of type (1,1)
1063:Configurations of points and circles
881:and all the points that are on line
757:) configurations, and all of the (11
546:, formed by 12 of the 27 lines on a
243:{\displaystyle p\gamma =\ell \pi \,}
104:, in the context of a discussion of
1522:Discrete and Computational Geometry
439:, but cannot be constructed in the
210:the number of lines per point, and
1469:Configurations of Points and Lines
25:
1451:European Journal of Combinatorics
665:configurations and in such cases
1167:Boben, GĂ©vay & Pisanski 2015
126:Hilbert & Cohn-Vossen (1952)
1473:Graduate Studies in Mathematics
958:) configuration exists for all
837:be a projective plane of order
499:Sylvester–Gallai configurations
1114:Hilbert & Cohn-Vossen 1952
753:discusses, nine of the ten (10
558:Cremona–Richmond configuration
1:
1424:10.1016/S0012-365X(96)00327-5
1372:Ars Mathematica Contemporanea
1339:, Band 44, Berlin, New York:
1278:10.1016/S0166-218X(99)00143-2
1108:In the literature, the terms
999:Unconventional configurations
1499:Geometry and the Imagination
1286:Boben, Marko; Gévay, Gábor;
1265:Discrete Applied Mathematics
985:is the length of an optimal
580:Grünbaum–Rigby configuration
708:, is given by the sequence
452:Möbius–Kantor configuration
1624:
1259:Betten, A; Brinkmann, G.;
1047:Topological configurations
124:, reprinted in English as
85:, such that each point is
29:
1608:Configurations (geometry)
1327:Dembowski, Peter (1968),
619:Duality of configurations
200:is the number of points,
1110:projective configuration
503:Sylvester–Gallai theorem
494:complex projective plane
336:) abbreviates to (9
300:In some configurations,
32:Configuration (polytope)
807:finite projective plane
514:Desargues configuration
358:Desargues configuration
18:Geometric configuration
1311:The Beauty of Geometry
1017:
601:Danzer's configuration
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293:) configurations: the
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139:(these are said to be
122:Anschauliche Geometrie
62:
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853:and all the lines of
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206:the number of lines,
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1495:Cohn-Vossen, Stephan
1410:Discrete Mathematics
1293:Advances in Geometry
1089:Perles configuration
1077:Miquel configuration
1033:Reye's configuration
1029:Möbius configuration
875:not passing through
869:) and remove a line
627:of a configuration (
569:Kummer configuration
467:Pappus configuration
295:Pappus configuration
283:incidence structures
221:
83:arrangement of lines
1549:Servatius, Brigitte
1369:) configurations",
1203:Bruck–Ryser theorem
1041:Schläfli double six
1014:Schläfli double six
857:which pass through
833:configuration. Let
612:Klein configuration
544:Schläfli double six
482:Hesse configuration
422:complete quadrangle
320:. These are called
71:projective geometry
51:complete quadrangle
1581:Weisstein, Eric W.
1536:10.1007/BF02187687
1037:Gray configuration
1018:
802:) configurations.
591:Gray configuration
529:Reye configuration
437:incidence geometry
361:
310:and consequently,
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145:incidence geometry
106:Desargues' theorem
102:Geometrie der Lage
63:
1482:978-0-8218-4308-6
1331:Finite geometries
1243:) configurations"
1004:Higher dimensions
486:inflection points
137:projective planes
53:, at left) and (6
41:Configurations (4
16:(Redirected from
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1458:: 1910-1918
1053:pseudolines
501:due to the
490:cubic curve
152:hypergraphs
118:Cohn-Vossen
67:mathematics
1236:"Movable (
1225:References
1215:GĂ©vay 2014
1151:Kelly 1986
433:Fano plane
279:isomorphic
167:Levi graph
141:realizable
1590:MathWorld
1379:: 175-199
1313:, Dover,
1253:(1): R104
989:of order
809:of order
761:) and (12
663:self-dual
322:symmetric
237:π
234:ℓ
228:γ
194:), where
156:biregular
133:Euclidean
1602:Category
1551:(2013),
1497:(1952),
1467:(2009),
1083:See also
974:, where
847:a point
815:is an ((
644:) is a (
527:), the
484:of nine
412:) and (6
380:triangle
344:Examples
326:balanced
173:Notation
87:incident
1402:: 34–48
1359:0233275
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