| 词条 | Non-Desarguesian plane |
| 释义 |
In mathematics, a non-Desarguesian plane, named after Girard Desargues, is a projective plane that does not satisfy Desargues' theorem, or in other words a plane that is not a Desarguesian plane. The theorem of Desargues is valid in all projective spaces of dimension not 2,[1] that is, all the classical projective geometries over a field (or division ring), but David Hilbert found that some projective planes do not satisfy it. Understanding of these examples is not complete, in the current state of knowledge. ExamplesSeveral examples are also finite. For a finite projective plane, the order is one less than the number of points on a line (a constant for every line). Some of the known examples of non-Desarguesian planes include:
ClassificationAccording to {{harvtxt|Weibel|2007|loc= pg. 1296}}, Hanfried Lenz gave a classification scheme for projective planes in 1954[3] and this was refined by Adriano Barlotti in 1957.[4] This classification scheme is based on the types of point–line transitivity permitted by the collineation group of the plane and is known as the Lenz–Barlotti classification of projective planes. The list of 53 types is given in {{harvtxt|Dembowski|1968|loc= pp.124–5}} 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. According to Charles Weibel, "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." Other classification schemes exist. One of the simplest is based on the type of planar ternary ring (PTR) which can be used to coordinatize the projective plane. The types are fields, skewfields, alternative division rings, semifields, nearfields, right nearfields, quasifields and right quasifields.[5] ConicsIn a Desarguesian projective plane a conic 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 In non-Desarguesian planes these proofs are no longer valid and the different definitions can give rise to non-equivalent objects.[6] Theodore G. Ostrom had suggested the name conicoid for these conic-like figures but did not provide a formal definition and the term does not seem to be widely used.[7] There are several ways that conics can be defined in Desarguesian planes:
Furthermore, in a finite Desarguesian plane:
Artzy has given an example of a Steiner conic in a Moufang plane which is not a von Staudt conic.[8] Garner gives an example of a von Staudt conic that is not an Ostrom conic in a finite semifield plane.[6] Notes1. ^Desargues' theorem is vacuously true in dimension 1; it is only problematic in dimension 2. 2. ^see {{harvnb|Room|Kirkpatrick|1971}} for descriptions of all four planes of order 9. 3. ^{{cite journal|last=Lenz|first=Hanfried|title=Kleiner desarguesscher Satz und Dualitat in projektiven Ebenen|journal=Jahresbericht der Deutschen Mathematiker-Vereinigung|year=1954|volume=57|pages=20–31|mr=0061844}} 4. ^{{cite journal|last=Barlotti|first=Adriano|title=Le possibili configurazioni del sistema delle coppie punto-retta (A,a) per cui un piano grafico risulta (A,a)-transitivo|journal=Boll. Un. Mat. Ital.|year=1957|volume=12|pages=212–226|mr=0089435}} 5. ^{{harvnb|Colbourn|Dinitz|2007|loc= pg. 723}} article on Finite Geometry by Leo Storme. 6. ^1 {{citation|first=Cyril W L.|last=Garner|title=Conics in Finite Projective Planes|journal=Journal of Geometry|year=1979|volume=12|issue=2|pages=132–138|doi=10.1007/bf01918221|mr=0525253}} 7. ^{{citation|first=T.G.|last=Ostrom|chapter=Conicoids: Conic-like figures in Non-Pappian planes|editor1-first=Peter|editor1-last=Plaumann|editor2-first=Karl|editor2-last=Strambach|title=Geometry - von Staudt's Point of View|publication-date=1981|publisher=D. Reidel|pages=175–196|isbn=90-277-1283-2|mr=0621316}} 8. ^{{citation|first=R.|last=Artzy|title=The Conic y = x2 in Moufang Planes|journal=Aequationes Mathematicae|year=1971|volume=6|pages=30–35|doi=10.1007/bf01833234}} References
2 : Projective geometry|Finite geometry |
| 随便看 |
|
开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。