691:
1188:. The configuration consisting of the three points of a triangle and the three lines joining pairs of these points is represented by a 6-cycle in the Heawood graph. A color-preserving automorphism of the Heawood graph that fixes each vertex of a 6-cycle must be the identity automorphism. This means that there are 168 labeled triangles fixed only by the identity collineation and only six collineations that stabilize an unlabeled triangle, one for each permutation of the points. These 28 triangles may be viewed as corresponding to the 28
361:
861:
818:
805:
797:
786:
778:
699:
1268:
317:
746:
1685:
24:
234:
The lines of the Fano plane may also be given homogeneous coordinates, again using non-zero triples of binary digits. With this system of coordinates, a point is incident to a line if the coordinate for the point and the coordinate for the line have an even number of positions at which they both have
1134:
of the quadrangle and a collineation fixes the quadrangle if and only if it fixes the diagonal line. Thus, there are 24 symmetries that fix any such quadrangle. The dual configuration is a quadrilateral consisting of four lines no three of which meet at a point and their six points of intersection,
1093:
The Fano plane contains the following numbers of configurations of points and lines of different types. For each type of configuration, the number of copies of configuration multiplied by the number of symmetries of the plane that keep the configuration unchanged is equal to 168, the size of the
538:
called the Cat's Cradle map. Color the seven lines of the Fano plane ROYGBIV, place your fingers into the two dimensional projective space in ambient 3-space, and stretch your fingers out like the children's game Cat's Cradle. You will obtain a complete graph on seven vertices with seven colored
732:
Dualities can be viewed in the context of the
Heawood graph as color reversing automorphisms. An example of a polarity is given by reflection through a vertical line that bisects the Heawood graph representation given on the right. The existence of this polarity shows that the Fano plane is
250:
On three of the lines, two of the positions in the binary triples of each point have the same value: in the line 110 (containing the points 001, 110, and 111) the first and second positions are always equal to each other, and the lines 101 and 011 are formed in the same
1060:
If this configuration lies in a projective plane and the three diagonal points are collinear, then the seven points and seven lines of the expanded configuration form a subplane of the projective plane that is isomorphic to the Fano plane and is called a
1104:
There are 7 points with 24 symmetries fixing any point and dually, there are 7 lines with 24 symmetries fixing any line. The number of symmetries follows from the 2-transitivity of the collineation group, which implies the group acts transitively on the
1680:
If you break one line apart into three 2-point lines you obtain the "non-Fano configuration", which can be embedded in the real plane. It is another important example in matroid theory, as it must be excluded for many theorems to hold.
1072:
states that if every complete quadrangle in a finite projective plane extends to a Fano subplane (that is, has collinear diagonal points) then the plane is
Desarguesian. Gleason called any projective plane satisfying this condition a
246:
On three of the lines the binary triples for the points have the 0 in a constant position: the line 100 (containing the points 001, 010, and 011) has 0 in the first position, and the lines 010 and 001 are formed in the same
1100:). Since the Fano plane is self-dual, these configurations come in dual pairs and it can be shown that the number of collineations fixing a configuration equals the number of collineations that fix its dual configuration.
114:
with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and lines cannot exist with this pattern of incidences in
196:, the seven points of the Fano plane may be labeled with the seven non-zero ordered triples of binary digits 001, 010, 011, 100, 101, 110, and 111. This can be done in such a way that for every two points
1192:. There are 84 ways of specifying a triangle together with one distinguished point on that triangle and two symmetries fixing this configuration. The dual of the triangle configuration is also a triangle.
235:
nonzero bits: for instance, the point 101 belongs to the line 111, because they have nonzero bits at two common positions. In terms of the underlying linear algebra, a point belongs to a line if the
737:. This is also an immediate consequence of the symmetry between points and lines in the definition of the incidence relation in terms of homogeneous coordinates, as detailed in an earlier section.
1011:
1123:
consisting of a line and a point on that line. Each flag corresponds to the unordered pair of the other two points on the same line. For each flag, 8 different symmetries keep it fixed.
1288:-design. The points of the design are the points of the plane, and the blocks of the design are the lines of the plane. As such it is a valuable example in (block) design theory.
1178:
694:
Bijection between the Fano plane as field with eight elements minus the origin and the projective line over the field with seven elements. Symmetries are made explicit.
139:", the first parameter is the geometric dimension (it is a plane, of dimension 2) and the second parameter is the order (the number of points per line, minus one).
1112:
of points, and each may be mapped by a symmetry onto any other ordered pair. For any ordered pair there are 4 symmetries fixing it. Correspondingly, there are 21
930:
1271:
The upper figure is an alternative representation of the Fano plane in grid layout – compare with one of the finite projective plane of order 3 below
216:
modulo 2 digit by digit (e.g., 010 and 111 resulting in 101). In other words, the points of the Fano plane correspond to the non-zero points of the finite
332:, the vertices of one part representing the points and the other representing the lines, with two vertices joined if the corresponding point and line are
2607:
2125:
1085:
collinear, a condition that holds in the
Euclidean and real projective planes. Thus, what Gleason called Fano planes do not satisfy Fano's axiom.
546:
will be at the center of the septagon inside. Now label this point as ∞, and pull it backwards to the origin. One can write down a bijection from
2415:
1264:
configurations of this type can be realized in the
Euclidean plane having at most one curved line (all other lines lying on Euclidean lines).
1116:
of points, each of which may be mapped by a symmetry onto any other unordered pair. For any unordered pair there are 8 symmetries fixing it.
395:
189:
with two elements. One can similarly construct projective planes over any other finite field, with the Fano plane being the smallest.
1020:
1037:
In any projective plane a set of four points, no three of which are collinear, and the six lines joining pairs of these points is a
905:
687:). The corresponding symmetries on the Fano plane are respectively swapping vertices, rotating the graph, and rotating triangles.
2358:
2307:
2282:
2201:
2089:
1130:
of four (unordered) points no three of which are collinear. These four points form the complement of a line, which is the
939:
2576:
2679:
2622:
2566:
2408:
2536:
1219:
form a triangle, and for every triangle there is a unique way of grouping the remaining four points into an anti-flag.
2340:
2257:
2137:
2107:
1199:), and six ways of permuting the Fano plane while keeping an anti-flag fixed. For every non-incident point-line pair
254:
In the remaining line 111 (containing the points 011, 101, and 110), each binary triple has exactly two nonzero bits.
1658:
is formed by taking the Fano plane's points as the ground set, and the three-element noncollinear subsets as bases.
2612:
320:
Bipartite
Heawood graph. Points are represented by vertices of one color and lines by vertices of the other color.
2643:
1788:
2185:
47:
2586:
2401:
1703:
The Fano plane can be extended in a third dimension to form a three-dimensional projective space, denoted by
712:
2664:
2617:
2602:
933:
162:
2674:
790:
453:
of points can be mapped by at least one collineation to any other ordered pair of points. (See below.)
442:
2042:
1602:
1142:
390:
of the Fano plane is a permutation of the 7 points that preserves collinearity: that is, it carries
2669:
2518:
1744:
1096:
1038:
812:
The 48 permutations with a complete 7-cycle form two distinct conjugacy classes with 24 elements:
649:
2684:
2551:
2193:
1226:
in which no three consecutive vertices lie on a line, and six symmetries fixing any such hexagon.
1189:
769:
411:
228:
193:
154:. Since it is a projective space, algebraic techniques can also be effective tools in its study.
111:
62:
2561:
2689:
2638:
2506:
1291:
With the points labelled 0, 1, 2, ..., 6 the lines (as point sets) are the translates of the
1260:)-configuration, is unique and is the smallest such configuration. According to a theorem by
1256:
lines with three points on each line and three lines through each point. The Fano plane, a (7
1233:
in which no three consecutive vertices lie on a line, and two symmetries fixing any pentagon.
1182:
triples of points, seven of which are collinear triples, leaving 28 non-collinear triples or
690:
2571:
2541:
2481:
2459:
2147:
1795:
1749:
1281:
429:
391:
341:
333:
1195:
There are 28 ways of selecting a point and a line that are not incident to each other (an
263:
Alternatively, the 7 points of the plane correspond to the 7 non-identity elements of the
8:
2581:
2546:
2526:
2424:
1042:
1032:
349:
264:
166:
143:
136:
1772:, but since the finite field of order 2 has no non-identity automorphisms, this becomes
2556:
2476:
2296:
2270:
2219:
2174:
2118:
2054:
915:
403:
224:
151:
116:
88:
2376:
2354:
2336:
2303:
2278:
2253:
2237:
2197:
2154:
2133:
2103:
2085:
1069:
762:
758:
446:
439:
2266:
123:
with two elements. The standard notation for this plane, as a member of a family of
2486:
2438:
2233:
2229:
2166:
2064:
1708:
1609:. This quasigroup coincides with the multiplicative structure defined by the unit
1328:
1094:
entire collineation group, provided each copy can be mapped to any other copy (see
182:
124:
2068:
1707:. It has 15 points, 35 lines, and 15 planes and is the smallest three-dimensional
2291:
2143:
2129:
1787:). Since the field has only one nonzero element, this group is isomorphic to the
1674:
1670:
1120:
1077:
thus creating some confusion with modern terminology. To compound the confusion,
860:
817:
804:
796:
785:
470:
360:
329:
169:
are always collinear. "The" Fano plane of 7 points and lines is "a" Fano plane.
99:
2469:
2380:
2316:
1646:
1596:
1296:
1261:
1113:
407:
285:. The lines of the plane correspond to the subgroups of order 4, isomorphic to
178:
932:
colors can be calculated by plugging the numbers of cycle structures into the
456:
Collineations may also be viewed as the color-preserving automorphisms of the
2658:
2241:
1666:
1127:
777:
619:
457:
345:
236:
147:
1669:
is necessary to characterize several important classes of matroids, such as
721:
between the point set and the line set that preserves incidence is called a
2464:
2073:
1661:
The Fano plane is one of the important examples in the structure theory of
1267:
1109:
702:
Duality in the Fano plane: Each point corresponds to a line and vice versa.
698:
635:
450:
399:
380:
374:
297:
217:
186:
120:
2210:
Manivel, L. (2006), "Configurations of lines and models of Lie algebras",
1734:
A line and a plane not containing the line intersect in exactly one point.
1725:
1049:
and pairs of sides that do not meet at one of the four points are called
337:
2496:
2443:
2393:
2178:
1606:
325:
2385:
2224:
2059:
2038:
1184:
745:
718:
365:
316:
107:
2170:
1684:
1610:
1230:
425:
386:
301:
1698:
1662:
1223:
227:, even though the plane is too small to contain a non-degenerate
1633:(omitting 1) if the signs of the octonion products are ignored (
750:
223:
Due to this construction, the Fano plane is considered to be a
428:
of order 168 = 2·3·7, the next non-abelian simple group after
1081:
states that the diagonal points of a complete quadrangle are
23:
912:
The number of inequivalent colorings of the Fano plane with
364:
A collineation of the Fano plane corresponding to the 3-bit
1015:
1026:
652:, and the basic transformations are reflections (order 2,
1718:
Each line is contained in 3 planes and contains 3 points.
480:, so the points of the Fano plane may be identified with
192:
Using the standard construction of projective spaces via
1858:
445:
on the 7 points of the plane, the collineation group is
239:
of the vectors representing the point and line is zero.
1894:
1846:
1836:
1834:
1819:
1147:
942:
518:. There is a relation between the underlying objects,
394:
points (on the same line) to collinear points. By the
2002:
1145:
1053:. The points at which opposite sides meet are called
918:
146:, so many of its properties can be established using
2100:
Introduction to the theory of groups of finite order
1952:
1006:{\textstyle {{n^{7}+21n^{5}+98n^{3}+48n} \over 168}}
539:
triangles (projective lines). The missing origin of
2374:
1831:
2295:
2117:
1906:
1731:Every pair of distinct planes intersect in a line.
1172:
1135:it is the complement of a point in the Fano plane.
1005:
924:
2265:
1969:
1946:
1840:
220:of dimension 3 over the finite field of order 2.
2656:
2126:Ergebnisse der Mathematik und ihrer Grenzgebiete
1715:Each point is contained in 7 lines and 7 planes.
308:) is that of the Fano plane, and has order 168.
1605:. As such, it can be given the structure of a
258:
2348:
1984:
242:The lines can be classified into three types.
119:, but they can be given coordinates using the
2409:
1980:
1978:
1163:
1150:
344:and each part contains 7 vertices. It is the
208:has the label formed by adding the labels of
800:42 permutations with a 4-cycle and a 2-cycle
2321:Über die construction der configurationen n
165:; in other words, the diagonal points of a
161:is a projective plane that never satisfies
2416:
2402:
2326:(Ph. D. thesis), Kgl. Universität, Breslau
2184:
2097:
1975:
1864:
418:
396:Fundamental theorem of projective geometry
172:
2330:
2275:Configurations from a Graphical Viewpoint
2223:
2115:
2058:
1923:
1900:
1825:
1721:Each plane contains 7 points and 7 lines.
1711:. It also has the following properties:
231:(which requires 10 points and 10 lines).
142:The Fano plane is an example of a finite
2423:
2315:
2190:Projective Geometries Over Finite Fields
2079:
1958:
1852:
1688:The Fano plane redrawn as a planar graph
1683:
1601:The Fano plane, as a block design, is a
1266:
744:
697:
689:
359:
315:
22:
2349:van Lint, J. H.; Wilson, R. M. (1992),
2290:
2247:
2209:
2153:
2020:
2008:
1996:
1935:
1912:
1888:
1876:
1211:, the three points that are unequal to
1027:Complete quadrangles and Fano subplanes
725:and a duality of order two is called a
336:. This particular graph is a connected
2657:
2080:Brown, Ezra; Guy, Richard K. (2021) ,
1275:
772:each define a single conjugacy class:
177:The Fano plane can be constructed via
2397:
2375:
324:As with any incidence structure, the
150:and other tools used in the study of
2037:
1634:
490:. The symmetry group may be written
1237:The Fano plane is an example of an
13:
1248:-configuration, that is, a set of
1229:There are 84 ways of specifying a
1222:There are 28 ways of specifying a
1154:
740:
18:Geometry with 7 points and 7 lines
14:
2701:
2368:
1665:. Excluding the Fano plane as a
1640:
1590:
1088:
808:56 permutations with two 3-cycles
2331:Stevenson, Frederick W. (1972),
2250:Fundamental Concepts of Geometry
1173:{\displaystyle {\tbinom {7}{3}}}
1126:There are 7 ways of selecting a
859:
816:
803:
795:
784:
776:
355:
2159:American Journal of Mathematics
2098:Carmichael, Robert D. (1956) ,
2014:
1990:
1963:
1940:
1929:
1917:
1789:projective special linear group
673:), and doubling (order 3 since
435:of order 60 (ordered by size).
2567:Cremona–Richmond configuration
2353:, Cambridge University Press,
2234:10.1016/j.jalgebra.2006.04.029
2157:(1956), "Finite Fano planes",
1882:
1870:
1762:
936:. This number of colorings is
638:of order 7. The symmetries of
1:
2128:, Band 44, Berlin, New York:
2069:10.1090/S0273-0979-01-00934-X
2030:
1970:Pisanski & Servatius 2013
1947:Pisanski & Servatius 2013
1841:Pisanski & Servatius 2013
1057:and there are three of them.
713:Duality (projective geometry)
311:
2644:Kirkman's schoolgirl problem
2577:Grünbaum–Rigby configuration
1812:
1581:
1569:
1563:
1550:
1547:
1535:
1516:
1513:
1501:
1488:
1482:
1479:
1454:
1448:
1445:
1420:
1414:
1411:
1386:
1380:
1377:
706:
259:Group-theoretic construction
7:
2537:Möbius–Kantor configuration
2248:Meserve, Bruce E. (1983) ,
1801:. It is also isomorphic to
1738:
1692:
753:numbering of the Fano plane
340:(regular of degree 3), has
10:
2706:
2623:Bruck–Ryser–Chowla theorem
2298:A Geometrical Picture Book
2082:The Unity of Combinatorics
1985:van Lint & Wilson 1992
1696:
1644:
1594:
1280:The Fano plane is a small
1215:and that do not belong to
1030:
710:
662:), translations (order 7,
204:, the third point on line
2680:Configurations (geometry)
2631:
2613:Szemerédi–Trotter theorem
2595:
2517:
2452:
2431:
2351:A Course in Combinatorics
2116:Dembowski, Peter (1968),
1313:. With the lines labeled
934:Pólya enumeration theorem
789:21 permutations with two
84:
76:
68:
56:
46:
38:
33:
2603:Sylvester–Gallai theorem
2335:, W.H. Freeman and Co.,
1755:
1745:Projective configuration
1097:Orbit-stabiliser theorem
1068:A famous result, due to
781:The identity permutation
148:combinatorial techniques
2608:De Bruijn–Erdős theorem
2552:Desargues configuration
2194:Oxford University Press
1190:bitangents of a quartic
1045:. The lines are called
891:is on the same line as
848:is on the same line as
611:labels the vertices of
412:projective linear group
328:of the Fano plane is a
229:Desargues configuration
194:homogeneous coordinates
173:Homogeneous coordinates
112:finite projective plane
2047:Bull. Amer. Math. Soc.
1689:
1677:, and cographic ones.
1282:symmetric block design
1272:
1174:
1007:
926:
761:of the 7 points has 6
754:
703:
695:
650:Möbius transformations
369:
321:
28:
2639:Design of experiments
1687:
1603:Steiner triple system
1331:(table) is given by:
1270:
1175:
1008:
927:
908:for a complete list.)
748:
701:
693:
363:
319:
157:In a separate usage,
26:
2572:Kummer configuration
2542:Pappus configuration
2425:Incidence structures
2186:Hirschfeld, J. W. P.
1853:Brown & Guy 2021
1796:general linear group
1750:Transylvania lottery
1143:
940:
916:
152:incidence geometries
2665:Projective geometry
2582:Klein configuration
2562:Schläfli double six
2547:Hesse configuration
2527:Complete quadrangle
2271:Servatius, Brigitte
2074:Online HTML version
1276:Block design theory
1043:complete quadrangle
1033:Complete quadrangle
570:and send the slope
167:complete quadrangle
144:incidence structure
137:projective geometry
69:Point orbit lengths
48:Lenz–Barlotti class
2675:Incidence geometry
2557:Reye configuration
2377:Weisstein, Eric W.
2212:Journal of Algebra
2155:Gleason, Andrew M.
1987:, pp. 196–197
1728:to the Fano plane.
1690:
1273:
1170:
1168:
1003:
922:
755:
704:
696:
469:is a degree-three
404:automorphism group
400:collineation group
370:
322:
225:Desarguesian plane
117:Euclidean geometry
77:Line orbit lengths
29:
2652:
2651:
2360:978-0-521-42260-4
2333:Projective Planes
2309:978-0-387-98437-7
2284:978-0-8176-8363-4
2203:978-0-19-850295-1
2120:Finite geometries
2091:978-1-4704-5667-2
1651:The Fano matroid
1586:
1585:
1284:, specifically a
1161:
1070:Andrew M. Gleason
1001:
925:{\displaystyle n}
909:
763:conjugacy classes
759:permutation group
449:meaning that any
447:doubly transitive
440:permutation group
125:projective spaces
96:
95:
2697:
2487:Projective plane
2439:Incidence matrix
2418:
2411:
2404:
2395:
2394:
2390:
2389:
2363:
2345:
2327:
2312:
2301:
2292:Polster, Burkard
2287:
2262:
2244:
2227:
2206:
2181:
2150:
2123:
2112:
2094:
2071:
2062:
2024:
2018:
2012:
2006:
2000:
1994:
1988:
1982:
1973:
1967:
1961:
1956:
1950:
1944:
1938:
1933:
1927:
1921:
1915:
1910:
1904:
1898:
1892:
1886:
1880:
1874:
1868:
1862:
1856:
1850:
1844:
1838:
1829:
1823:
1806:
1804:
1800:
1793:
1776:also denoted PGL
1775:
1771:
1766:
1709:projective space
1336:
1335:
1329:incidence matrix
1312:
1302:
1294:
1287:
1247:
1210:
1181:
1179:
1177:
1176:
1171:
1169:
1167:
1166:
1153:
1018:
1012:
1010:
1009:
1004:
1002:
997:
987:
986:
971:
970:
955:
954:
944:
931:
929:
928:
923:
903:
863:
820:
807:
799:
788:
780:
770:cycle structures
686:
676:
672:
661:
633:
632:
606:
569:
555:
517:
506:PSL(2, 7) = Aut(
503:
492:PGL(3, 2) = Aut(
489:
426:well-known group
416:
295:
284:
183:projective plane
134:
130:
31:
30:
2705:
2704:
2700:
2699:
2698:
2696:
2695:
2694:
2670:Finite geometry
2655:
2654:
2653:
2648:
2627:
2591:
2513:
2448:
2444:Incidence graph
2427:
2422:
2371:
2366:
2361:
2343:
2324:
2317:Steinitz, Ernst
2310:
2285:
2267:Pisanski, Tomaž
2260:
2204:
2171:10.2307/2372469
2140:
2130:Springer-Verlag
2110:
2092:
2043:"The Octonions"
2033:
2028:
2027:
2019:
2015:
2007:
2003:
1995:
1991:
1983:
1976:
1968:
1964:
1957:
1953:
1945:
1941:
1934:
1930:
1922:
1918:
1911:
1907:
1899:
1895:
1887:
1883:
1875:
1871:
1865:Carmichael 1956
1863:
1859:
1851:
1847:
1839:
1832:
1824:
1820:
1815:
1810:
1809:
1802:
1798:
1791:
1786:
1779:
1773:
1769:
1768:Actually it is
1767:
1763:
1758:
1741:
1701:
1695:
1657:
1649:
1643:
1632:
1625:
1618:
1599:
1593:
1561:
1530:
1499:
1468:
1437:
1406:
1375:
1344:
1341:
1326:
1319:
1304:
1300:
1292:
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1114:unordered pairs
1091:
1055:diagonal points
1035:
1029:
1014:
982:
978:
966:
962:
950:
946:
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743:
741:Cycle structure
715:
709:
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631:
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471:field extension
468:
433:
419:Hirschfeld 1979
414:
358:
330:bipartite graph
314:
307:
304:of the group (Z
294:
290:
286:
283:
279:
275:
271:
267:
261:
175:
132:
128:
100:finite geometry
91:
61:
19:
12:
11:
5:
2703:
2693:
2692:
2687:
2685:Matroid theory
2682:
2677:
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2649:
2647:
2646:
2641:
2635:
2633:
2629:
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2626:
2625:
2620:
2618:Beck's theorem
2615:
2610:
2605:
2599:
2597:
2593:
2592:
2590:
2589:
2584:
2579:
2574:
2569:
2564:
2559:
2554:
2549:
2544:
2539:
2534:
2529:
2523:
2521:
2519:Configurations
2515:
2514:
2512:
2511:
2510:
2509:
2501:
2500:
2499:
2491:
2490:
2489:
2484:
2474:
2473:
2472:
2470:Steiner system
2467:
2456:
2454:
2450:
2449:
2447:
2446:
2441:
2435:
2433:
2432:Representation
2429:
2428:
2421:
2420:
2413:
2406:
2398:
2392:
2391:
2370:
2369:External links
2367:
2365:
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2328:
2322:
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2308:
2288:
2283:
2277:, Birkhäuser,
2263:
2258:
2245:
2218:(1): 457–486,
2207:
2202:
2182:
2165:(4): 797–807,
2151:
2138:
2113:
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2090:
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2053:(2): 145–205,
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2013:
2001:
1989:
1974:
1962:
1951:
1939:
1928:
1924:Dembowski 1968
1916:
1905:
1901:Stevenson 1972
1893:
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1869:
1857:
1845:
1830:
1826:Stevenson 1972
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1732:
1729:
1724:Each plane is
1722:
1719:
1716:
1697:Main article:
1694:
1691:
1655:
1647:Matroid theory
1645:Main article:
1642:
1641:Matroid theory
1639:
1630:
1623:
1616:
1597:Steiner system
1595:Main article:
1592:
1591:Steiner system
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1234:
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1089:Configurations
1087:
1051:opposite sides
1031:Main article:
1028:
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711:Main article:
708:
705:
645:
629:
622:, noting that
615:
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543:
535:
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499:
485:
477:
466:
460:(see figure).
431:
421:, p. 131
408:symmetry group
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179:linear algebra
174:
171:
104:the Fano plane
94:
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27:The Fano plane
17:
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2493:Graph theory
2492:
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2485:
2483:
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2479:
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2463:
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2460:Combinatorics
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2259:0-486-63415-9
2255:
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2139:3-540-61786-8
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2111:
2109:0-486-60300-8
2105:
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2084:, MAA Press,
2083:
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2035:
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2017:
2010:
2005:
1998:
1993:
1986:
1981:
1979:
1972:, p. 221
1971:
1966:
1960:
1959:Steinitz 1894
1955:
1949:, p. 165
1948:
1943:
1937:
1932:
1926:, p. 168
1925:
1920:
1914:
1909:
1902:
1897:
1890:
1885:
1878:
1873:
1867:, p. 363
1866:
1861:
1855:, p. 177
1854:
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1843:, p. 171
1842:
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1710:
1706:
1705:PG(3, 2)
1700:
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1678:
1676:
1672:
1668:
1667:matroid minor
1664:
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1337:
1334:
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1332:
1330:
1323:
1316:
1311:
1307:
1303:in the group
1298:
1289:
1283:
1269:
1265:
1263:
1255:
1251:
1242:
1232:
1228:
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1221:
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1204:
1198:
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1191:
1187:
1186:
1158:
1155:
1137:
1133:
1132:diagonal line
1129:
1125:
1122:
1119:There are 21
1118:
1115:
1111:
1110:ordered pairs
1108:There are 42
1107:
1103:
1102:
1101:
1099:
1098:
1086:
1084:
1080:
1076:
1071:
1066:
1064:
1063:Fano subplane
1058:
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1039:configuration
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843:
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835:
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827:
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783:
779:
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774:
773:
771:
766:
764:
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747:
738:
736:
730:
728:
724:
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692:
688:
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666:
660:
656:
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641:
637:
625:
621:
620:edge coloring
614:
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586:
582:
578:
574:
567:
559:
550:
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534:
531:
524:
521:
512:
509:
504:. Similarly,
498:
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484:
476:
472:
465:
461:
459:
458:Heawood graph
454:
452:
448:
444:
441:
436:
434:
427:
422:
420:
413:
409:
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401:
397:
393:
389:
388:
383:
382:
377:
376:
367:
362:
356:Collineations
353:
351:
348:, the unique
347:
346:Heawood graph
343:
339:
335:
331:
327:
318:
309:
303:
302:GL(3, 2)
299:
266:
253:
249:
245:
244:
243:
240:
238:
237:inner product
232:
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155:
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126:
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118:
113:
109:
105:
101:
90:
87:
83:
79:
75:
71:
67:
64:
59:
57:Automorphisms
55:
51:
49:
45:
41:
37:
32:
25:
21:
16:
2690:Dot patterns
2632:Applications
2531:
2465:Block design
2384:
2381:"Fano Plane"
2350:
2332:
2320:
2302:, Springer,
2297:
2274:
2249:
2225:math/0507118
2215:
2211:
2189:
2162:
2158:
2119:
2099:
2081:
2060:math/0105155
2050:
2046:
2023:, p. 69
2021:Polster 1998
2016:
2011:, p. 29
2009:Meserve 1983
2004:
1999:, p. 23
1997:Polster 1998
1992:
1965:
1954:
1942:
1936:Manivel 2006
1931:
1919:
1913:Gleason 1956
1908:
1903:, p. 21
1896:
1891:, p. 15
1889:Polster 1998
1884:
1879:, p. 11
1877:Polster 1998
1872:
1860:
1848:
1828:, p. 34
1821:
1781:
1764:
1704:
1702:
1679:
1660:
1652:
1650:
1627:
1620:
1613:
1600:
1556:
1525:
1494:
1463:
1432:
1401:
1370:
1321:
1314:
1309:
1305:
1290:
1279:
1253:
1249:
1240:
1236:
1216:
1212:
1206:
1202:
1196:
1183:
1131:
1095:
1092:
1082:
1079:Fano's axiom
1078:
1074:
1067:
1062:
1059:
1054:
1050:
1046:
1036:
911:
902:
896:
892:
888:
884:
880:
876:
872:
868:
864:
853:
849:
845:
841:
837:
833:
829:
825:
821:
811:
767:
756:
734:
731:
726:
722:
716:
683:
679:
668:
664:
658:
654:
642:
639:
636:cyclic group
623:
612:
608:
607:, where now
602:
598:
591:
584:
580:
576:
572:
565:
557:
548:
540:
532:
529:
522:
519:
510:
507:
496:
493:
482:
474:
463:
462:
455:
451:ordered pair
437:
423:
385:
381:automorphism
379:
375:collineation
373:
371:
323:
298:automorphism
262:
241:
233:
222:
218:vector space
213:
209:
205:
201:
197:
191:
187:finite field
176:
163:Fano's axiom
159:a Fano plane
158:
156:
141:
135:stands for "
121:finite field
103:
97:
89:Desarguesian
20:
15:
2503:Statistics
1286:2-(7, 3, 1)
1252:points and
1041:known as a
768:These four
488:∖ {0}
398:, the full
368:permutation
338:cubic graph
2659:Categories
2532:Fano plane
2497:Hypergraph
2039:Baez, John
2031:References
1726:isomorphic
1607:quasigroup
1138:There are
1128:quadrangle
1075:Fano plane
1013:(sequence
424:This is a
326:Levi graph
312:Levi graph
85:Properties
34:Fano plane
2482:Incidence
2386:MathWorld
2252:, Dover,
2242:0021-8693
2102:, Dover,
1813:Citations
1803:PSL(2, 7)
1792:PSL(3, 2)
1774:PGL(3, 2)
1770:PΓL(3, 2)
1635:Baez 2002
1611:octonions
1301:{0, 1, 3}
1299:given by
1293:(7, 3, 1)
1197:anti-flag
1185:triangles
735:self-dual
719:bijection
707:Dualities
415:PGL(3, 2)
410:) is the
392:collinear
366:Gray code
185:over the
108:Gino Fano
92:Self-dual
63:PGL(3, 2)
60:2 × 3 × 7
2596:Theorems
2507:Blocking
2477:Geometry
2319:(1894),
2294:(1998),
2273:(2013),
2188:(1979),
2041:(2002),
1799:GL(3, 2)
1794:and the
1739:See also
1693:PG(3, 2)
1663:matroids
1262:Steinitz
1231:pentagon
867:maps to
824:maps to
791:2-cycles
727:polarity
387:symmetry
334:incident
131:. Here,
129:PG(2, 2)
2179:2372469
2148:0233275
1699:PG(3,2)
1675:graphic
1671:regular
1626:, ...,
1320:, ...,
1295:planar
1224:hexagon
1105:points.
1019:in the
1016:A241929
887:. Then
844:. Then
723:duality
342:girth 6
296:. The
181:as the
110:) is a
106:(after
2453:Fields
2357:
2339:
2306:
2281:
2256:
2240:
2200:
2177:
2146:
2136:
2106:
2088:
751:nimber
563:. Set
443:acting
350:6-cage
300:group
2220:arXiv
2175:JSTOR
2055:arXiv
1756:Notes
1340:Point
1121:flags
1083:never
1047:sides
904:(See
675:2 = 1
657:↦ −1/
634:is a
618:with
554:∪ {∞}
438:As a
406:, or
384:, or
272:) = Z
265:group
127:, is
52:VII.2
39:Order
2587:Dual
2355:ISBN
2337:ISBN
2304:ISBN
2279:ISBN
2254:ISBN
2238:ISSN
2198:ISBN
2134:ISBN
2104:ISBN
2086:ISBN
1343:Line
1327:the
1180:= 35
1021:OEIS
906:here
895:and
852:and
757:The
648:are
605:+ 1)
528:and
402:(or
251:way.
247:way.
212:and
200:and
2230:doi
2216:304
2167:doi
2065:doi
1637:).
1308:/ 7
1023:).
999:168
883:to
875:to
840:to
832:to
765:.
682:↦ 2
671:+ 1
597:/ (
568:= 0
556:to
473:of
372:A
291:× Z
280:× Z
276:× Z
98:In
2661::
2383:,
2379:,
2269:;
2236:,
2228:,
2214:,
2196:,
2192:,
2173:,
2163:78
2161:,
2144:MR
2142:,
2132:,
2124:,
2063:,
2051:39
2049:,
2045:,
1977:^
1833:^
1673:,
1619:,
1582:1
1551:1
1520:0
1489:1
1458:0
1427:0
1396:0
1365:6
1205:,
1065:.
992:48
976:98
960:21
879:,
871:,
836:,
828:,
749:A
729:.
717:A
677:,
667:↦
601:+
590:≅
583:∈
579:+
575:↦
378:,
352:.
268:(Z
206:pq
133:PG
102:,
2417:e
2410:t
2403:v
2323:3
2232::
2222::
2169::
2076:)
2072:(
2067::
2057::
1805:.
1785:2
1782:F
1780:(
1778:3
1656:7
1653:F
1631:7
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1621:e
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1614:e
1579:0
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