188:, pp. 124–125) and a table of the then known existence results (for both collineation groups and planes having such a collineation group) in both the finite and infinite cases appears on page 126. As of 2007, "36 of them exist as finite groups. Between 7 and 12 exist as finite projective planes, and either 14 or 15 exist as infinite projective planes."
239:
can be defined in several different ways that can be proved to be equivalent. In non-Desarguesian planes these proofs are no longer valid and the different definitions can give rise to non-equivalent objects. Theodore G. Ostrom had suggested the name
175:
Hanfried Lenz gave a classification scheme for projective planes in 1954, which was refined by
Adriano Barlotti in 1957. This classification scheme is based on the types of point–line transitivity permitted by the
108:
Regarding finite non-Desarguesian planes, every projective plane of order at most 8 is
Desarguesian, but there are three non-Desarguesian examples of order 9, each with 91 points and 91 lines. They are:
333:
Artzy has given an example of a
Steiner conic in a Moufang plane which is not a von Staudt conic. Garner gives an example of a von Staudt conic that is not an Ostrom conic in a finite semifield plane.
400:. According to the footnote on this page, the original "first" example appearing in earlier editions was replaced by Moulton's simpler example in later editions.
465:
Barlotti, Adriano (1957). "Le possibili configurazioni del sistema delle coppie punto-retta (A,a) per cui un piano grafico risulta (A,a)-transitivo".
655:
17:
155:. All known constructions of finite non-Desarguesian planes produce planes whose order is a proper prime power, that is, an integer of the form
270:
The set of points of intersection of corresponding lines of two pencils which are projectively, but not perspectively, related is known as a
389:, translated by Leo Unger from the 10th German edition (2nd English ed.), La Salle, IL: Open Court Publishing, p. 74,
541:
Ostrom, T.G. (1981), "Conicoids: Conic-like figures in Non-Pappian planes", in
Plaumann, Peter; Strambach, Karl (eds.),
70:
found that some projective planes do not satisfy it. The current state of knowledge of these examples is not complete.
804:
768:
750:
728:
710:
635:
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394:
244:
for these conic-like figures but did not provide a formal definition and the term does not seem to be widely used.
778:
82:
and infinite non-Desarguesian planes. Some of the known examples of infinite non-Desarguesian planes include:
788:
144:
133:
54:
of dimension not 2; in other words, the only projective spaces of dimension not equal to 2 are the classical
151:
Numerous other constructions of both finite and infinite non-Desarguesian planes are known, see for example
79:
839:
783:
97:
371:, English translation by E.J. Townsend (2nd ed.), La Salle, IL: Open Court Publishing, p. 48
204:
844:
583:
216:
212:
121:
260:
101:
277:
The set of points whose coordinates satisfy an irreducible homogeneous equation of degree two.
39:
694:
560:
528:
478:
451:
349:
Desargues' theorem is vacuously true in dimension 1; it is only problematic in dimension 2.
8:
438:
Lenz, Hanfried (1954). "Kleiner desarguesscher Satz und
Dualitat in projektiven Ebenen".
196:
192:
59:
55:
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624:
311:
177:
47:
800:
764:
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Other classification schemes exist. One of the simplest is based on special types of
129:
664:
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51:
43:
35:
690:
556:
524:
474:
447:
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137:
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195:(PTR) that can be used to coordinatize the projective plane. These types are
125:
93:
87:
67:
63:
114:
247:
There are several ways that conics can be defined in
Desarguesian planes:
96:
over alternative division algebras that are not associative, such as the
686:
596:
520:
224:
220:
200:
814:
208:
669:
274:. If the pencils are perspectively related, the conic is degenerate.
507:
Garner, Cyril W L. (1979), "Conics in Finite
Projective Planes",
132:, this plane was generalized to an infinite family of planes by
365:
100:. Since all finite alternative division rings are fields (
104:), the only non-Desarguesian Moufang planes are infinite.
251:
The set of absolute points of a polarity is known as a
182:
Lenz–Barlotti classification of projective planes
630:(2nd ed.), Boca Raton: Chapman & Hall/ CRC,
440:
Jahresbericht der
Deutschen Mathematiker-Vereinigung
494:, pg. 723 article on Finite Geometry by Leo Storme.
738:
623:
27:Projective plane not satisfying Desargues' theorem
656:Transactions of the American Mathematical Society
622:Colbourn, Charles J.; Dinitz, Jeffrey H. (2007),
136:. Hall planes are a subclass of the more general
831:
758:
428:for descriptions of all four planes of order 9.
425:
621:
491:
329:is based on a generalization of harmonic sets.
663:(2), American Mathematical Society: 229–277,
612:
281:Furthermore, in a finite Desarguesian plane:
653:Hall, Marshall (1943), "Projective planes",
502:
500:
799:, San Francisco: W.H. Freeman and Company,
700:
615:An Introduction to Finite Projective Planes
613:Albert, A. Adrian; Sandler, Reuben (1968),
46:), or in other words a plane that is not a
701:Hughes, Daniel R.; Piper, Fred C. (1973),
50:. The theorem of Desargues is true in all
794:
763:, Cambridge: Cambridge University Press,
668:
643:
497:
185:
152:
759:Room, T. G.; Kirkpatrick, P. B. (1971),
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408:
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776:
381:
358:
14:
832:
812:
617:, New York: Holt, Rinehart and Winston
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506:
412:
572:
543:Geometry – von Staudt's Point of View
403:
235:In a Desarguesian projective plane a
652:
437:
815:"Survey of Non-Desarguesian Planes"
230:
184:. The list of 53 types is given in
98:projective plane over the octonions
24:
25:
856:
721:Introduction to Finite Geometries
626:Handbook of Combinatorial Designs
255:. If the plane is defined over a
180:of the plane and is known as the
170:
18:Non-Desarguesian projective plane
795:Stevenson, Frederick W. (1972),
78:There are many examples of both
566:
545:, D. Reidel, pp. 175–196,
534:
485:
458:
431:
418:
375:
352:
343:
294:points, no three collinear in
167:is an integer greater than 1.
13:
1:
705:, New York: Springer Verlag,
606:
573:Artzy, R. (1971), "The Conic
322:is a conic, in sense 3 above.
147:of the Hall plane of order 9.
723:, Amsterdam: North-Holland,
7:
784:Encyclopedia of Mathematics
779:"Non-Desarguesian geometry"
745:, Berlin: Springer Verlag,
426:Room & Kirkpatrick 1971
73:
10:
861:
492:Colbourn & Dinitz 2007
205:alternative division rings
124:. Initially discovered by
648:, Berlin: Springer Verlag
644:Dembowski, Peter (1968),
813:Weibel, Charles (2007),
737:LĂĽneburg, Heinz (1980),
584:Aequationes Mathematicae
387:Foundations of Geometry
336:
777:Sidorov, L.A. (2001) ,
761:Miniquaternion Geometry
286:
38:that does not satisfy
32:non-Desarguesian plane
719:Kárteszi, F. (1976),
122:Hall plane of order 9
56:projective geometries
581:in Moufang Planes",
840:Projective geometry
509:Journal of Geometry
467:Boll. Un. Mat. Ital
193:planar ternary ring
819:Notices of the AMS
741:Translation Planes
597:10.1007/bf01833234
521:10.1007/bf01918221
178:collineation group
102:Artin–Zorn theorem
48:Desarguesian plane
40:Desargues' theorem
30:In mathematics, a
797:Projective Planes
703:Projective Planes
646:Finite Geometries
265:degenerate conics
225:right quasifields
52:projective spaces
16:(Redirected from
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253:von Staudt conic
231:Conics and ovals
217:right nearfields
153:Dembowski (1968)
44:Girard Desargues
36:projective plane
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845:Finite geometry
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312:Segre's theorem
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186:Dembowski (1968
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163:is a prime and
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5:
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383:Hilbert, David
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360:Hilbert, David
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261:characteristic
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171:Classification
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149:
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106:
105:
94:Moufang planes
91:
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9:
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752:0-387-09614-0
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712:0-387-90044-6
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302:is called an
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272:Steiner conic
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267:are obtained.
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88:Moulton plane
85:
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68:David Hilbert
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64:division ring
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45:
42:(named after
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327:Ostrom conic
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138:André planes
115:Hughes plane
107:
77:
66:). However,
31:
29:
473:: 212–226.
413:Weibel 2007
310:is odd, by
221:quasifields
117:of order 9.
834:Categories
607:References
263:two, only
213:nearfields
209:semifields
201:skewfields
130:Wedderburn
789:EMS Press
679:0002-9947
591:: 30–35,
446:: 20–31.
385:(1990) ,
362:(1950) ,
287:A set of
242:conicoid
159:, where
74:Examples
695:0008892
687:1990331
561:0621316
529:0525253
479:0089435
452:0061844
58:over a
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767:
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393:
316:PG(2,
296:PG(2,
197:fields
126:Veblen
80:finite
683:JSTOR
369:(PDF)
337:Notes
306:. If
257:field
237:conic
60:field
34:is a
801:ISBN
765:ISBN
747:ISBN
725:ISBN
707:ISBN
675:ISSN
632:ISBN
547:ISBN
424:see
391:ISBN
304:oval
223:and
145:dual
143:The
128:and
120:The
113:The
86:The
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665:doi
593:doi
517:doi
325:An
292:+ 1
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318:q
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298:q
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