Classification of projective planes
The usual classification of projective planes in synthetic geometry is based on the operation of the respective group of their collineations . The Lenz-Barlotti classification classifies the levels by the properties of the operation of certain subgroups of their collineation group, it refines the Lenz classification . For this purpose, the richness of the subgroups of the collineation group, which consist of central-axial collineations (flat, projective perspectives ), each with a fixed axis and a fixed center, is regarded as a feature of both classifications .
It turns out that the coarser class division according to Lenz usually assigns a class of ternary bodies characteristic to each class of planes : The coordinate range of a "higher" Lenz class fulfills - with a suitable choice of the projective point base for the coordination - stronger algebraic axioms than that of a lower one.
The Lenz-Barlotti classification is not a classification “except for isomorphism”: Isomorphic projective planes always belong to the same class, but planes of a class do not need to be isomorphic to one another. The only exceptions are the Lenz-Barlotti classes IVa.3 and IVb.3: In these classes, all representatives are mutually isomorphic levels of order 9.
history
In the 1940s, Hanfried Lenz developed a classification for projective levels, the Lenz classification . As a characteristic feature, he defined the Lenz figure of a projective plane, which was later named after him , a set of pairs that are each formed by an axis ( fixed point line ) and a point (the center ) on this axis. Adriano Barlotti expanded and refined this classification in the 1950s by allowing off-axis centers for the characteristic figure. This turns the Lenz figure into the Lenz Barlotti figure.
In the 1960s, Günter Pickert developed a formal definition of the classic term closure sentence with which “specializations” of a closure sentence and related forms (specializations with an excellent, constant straight line, the distant line) can be grasped and compared. The closing theorem , which is important for the classification of projective planes, is Desargues's theorem , all “transitivity properties” that characterize a Lenz or Lenz-Barlotti class are equivalent to the validity of one specialization of the Desargues theorem together with the invalidity of another of its specializations. Pickert was also able to show that the validity of every inference sentence is equivalent to the validity of certain algebraic axioms in a suitably chosen coordinate ternary field. In this respect, the Lenz-Barlotti classification also provides a classification of the coordinate ternary bodies. However, while the classes of the algebraic axioms and the closing theorems are in principle unlimited, the Lenz-Barlotti classification provides a finite number of classes based on the automorphism group of the level.
Definitions
In the following, let a projective plane and at the same time the set of projective points lying on the plane and the set of straight lines of the plane be the group of collineations of the plane. A collineation is called -collineation if it has the axis and the center , that is, if the following applies:
- For each point is and
- For every straight line that goes through is .
The subgroup of collineations of the plane is designated with. The projective plane is called -transitive if the group operates with transitive on for every straight line .
characters
The amount
is called Lenz figure of .
The amount
is called the Lenz-Barlotti figure of .
Invariance of the figures
Both the Lenz figure and the Lenz-Barlotti figure are invariant under every collineation, i.e. specifically:
If there is a projective plane and any collineation of this plane then holds
- and
- .
Coordination of the projective plane
For levels of Lenz classes I to IV, the algebraic structure of the coordinate ternary bodies depends on the choice of the projective coordinate system and the definition of the ternary construction based on it. The Lenz-Barlotti figures described in the next section are specified using the coordinate reference system described here. For this purpose, a coordinate representation of the points with abbreviated coordinates on the long-distance line is introduced, compare the figure on the right, the coordination and the designations are based on Prieß-Crampe, they go back to Marshall Hall :
- A projective coordinate system is determined by a suitable choice of a complete quadrilateral on the Lenz-Barlotti figure .
- The point O becomes the origin of the affine coordinate system, is the distance line, the affine points on , so the points in form the ternary body.
- The point O becomes the neutral element of addition and is designated as the element of K with 0.
- The point E becomes the neutral element of the multiplication and is designated as 1 as the element of K.
- All elements of the affine line K have affine coordinates
- The point with the coordinates is determined as the intersection of the straight lines and . Conversely, the coordinates of an affine point are obtained as intersection points and .
- Special cases:
- The affine points B on the straight line , i.e. all points of this straight line except V , have the coordinates , whereby the ternary body element ( coordinate ) is determined as the point of intersection . This special case of coordinate construction is shown in black in the figure.
- The affine points Y on the straight line , i.e. all points on this straight line except V , have the coordinates , the coordinate being defined as the point of intersection . This special case of coordinate construction is shown in green in the figure.
- A point on the long line receives the coordinate representation, whereby it is determined that the intersection point has the affine coordinates . The point V , to which no coordinate can be assigned in this way, receives the coordinate representation .
Ternary connection and line display
The ternary link is now defined for as follows, compare the second figure on the right:
- The far point is constructed to.
- To the point is constructed.
- The result is determined as the y -coordinate of the section , which is shown in green in the figure.
This ternary connection is equivalent to the affine definition described in the article Ternary Field. In contrast to what is described there, the straight lines are represented here by equations (and two special forms) that are explicit in the second coordinate y :
- The straight lines have the explicit straight line equation for the affine points and the far point , as an example in the figure the connecting straight line shown in gray . These are all straight lines of the projective plane, except those through . The connecting lines with the affine equation and the far point are a special case .
- The straight lines through V that intersect at an affine point have the equation for the affine points and the far point . An example is the straight line in the figure.
- The distance line contains exactly the points with the coordinates .
In the affine section of the plane, straight lines are parallel if and only if their projective continuations pass through the same far point , therefore the following applies to the sets of parallels:
- Two straight lines are parallel if and only if is. This number is the common slope of the associated family of parallels.
- All straight lines of the second type are parallel to one another, but not to any straight line of the first type.
Construction of the two-digit links
On each projective plane, by choosing a coordinate reference system , two two-digit links, an addition and a multiplication on the affine point set are established. The ternary link described above can be represented on all levels except those of the Lenz-Barlotti class I.1, i.e. on all levels above a linear ternary body through these two-digit links .
These special cases of the ternary link are shown in the figures below.
Addition in a projective plane. | Multiplication in a projective plane. |
The classes and their properties
Lenz classes
Lenz assigns each Lenz figure an ordinal number in the form of a Roman number between I and VII. A class with a higher class number fulfills all the properties of the classes with lower numbers, but its Lenz figure is a real superset of Lenz figures of the lower classes. Class VI is omitted because it was shown that no projective level with the corresponding Lenz figure exists. Instead, Lenz has already divided class IV into two subclasses IVa and IVb, which are dual to one another.
A projective plane has exactly one of the Lenz figures listed below:
Lenz type | Lenz figure | Coordinate area |
---|---|---|
I. | Ternary bodies | |
II | There is an axis and a center with | Cartesian group |
III | There is a straight line and a point such that
applies. |
special Cartesian group (always infinite!) |
IVa | There is an axis so that is true. | Left quasi-body |
IVb | There is a center so that is true. | Right quasi-body |
V | There is an axis and a center so that
applies. |
Half body |
VII | Alternative body |
Levels that have at least Lenz type IVa, i.e. belong to one of the classes IVa, V or VII, are also referred to as projective translation levels . If one slits such a plane along an axis that belongs to the Lenz figure, an affine translation plane is created . At levels of Lenz classes IVb, V or VII, the dual level is a projective translation level in this sense. The algebraic structure of the coordinate area is only independent of the choice of the projective coordinate system for planes of Lenz class VII , here all coordinate areas are isomorphic alternative bodies, see Moufang plane . For a class V plane, the coordinate areas are half-bodies isotopic to one another . In classes I to IV, the coordinate bodies are isotopic ternary bodies to one another and only with a suitable choice of the coordinate system do they have the "strongest possible" algebraic structure mentioned in the table.
Lenz-Barlotti classes
The Lenz-Barlotti classification refines the Lenz classification by allowing the Lenz-Barlotti figure to have the center not on the axis. The Roman numerals according to Lenz are retained; Arabic numerals are added to them, separated by a period. Each Lenz-Barlotti class is a sub-class of the Lenz classification described above. For example, class I from Lenz at Barlotti was originally divided into 8 subclasses (I.1 to I.8), although it later became apparent that there were no representatives of classes I.5, I.7 and I.8. The Lenz class V is the only one in the Lenz-Barlotti classification that does not disintegrate any further . Otherwise, the Lenz-Barlotti figure for the first Lenz-Barlotti class corresponds to the Lenz figure of the corresponding superordinate Lenz class.
A projective plane has exactly one of the following Lenz-Barlotti figures:
Lenz-Barlotti type | Lenz Barlotti figure | Ternary body K with respect to |
---|---|---|
I.1 | no additional properties | |
I.2 | K is linear, multiplication is associative | |
I.3 | K is linear, multiplication is associative, | |
I.4 | K is linear, multiplication is associative, both distributive laws | |
I.6 | , where is a bijective mapping of the points of to the set of straight lines through V other than UV , in coordinates for example . | K is linear, multiplication is associative, both distributive laws, other special properties |
II.1 | Cartesian group | |
II.2 | Cartesian group, associative multiplication | |
III.1 | Cartesian group with special characteristics | |
III.2 | Cartesian group with special properties, associative multiplication | |
IVa.1 | , Translational plane | Left quasi-body: |
IVa.2 | (Left) almost body | |
IVa.3 | , whereby an involutive, bijective, fixed point-free mapping of the straight line UV is on itself | clearly defined left fast body with 9 elements. |
IVb.1 | The Lenz Barlotti figure is dual to that of class IVa.1. | Dual to IVa.1. |
IVb.2 | The Lenz Barlotti figure is dual to that of class IVa.2. | Dual to IVa.2. |
IVb.3 | The Lenz Barlotti figure is dual to that of class IVa.3. | Dual to IVa.3. |
V | Half body | |
VII.1 | Alternative body | |
VII.2 | Oblique body |
Models
This section provides an overview of examples known today (as of 2011) for levels that represent certain Lenz-Barlotti classes. In particular, very little is known about infinite levels that fall into Lenz Class I, finite models for these “weakest” Lenz-Barlotti classes are sought today with massive use of computers, or attempts are made to refute their existence in this way. The following tabular list does not claim to be complete!
Lenz class | Lenz Barlotti class | finite models | infinite models |
---|---|---|---|
I. | I.1 | “Hughes planes” : For odd prime numbers , a “real” near field of the (suitable) order is made into a ternary field. Also one of the projective planes of order 9, which is not a translation plane, is of the Hughes type. | Hilbert and Beltramian line systems |
I.2 | unknown. | An Archimedean , linear ternary body with the usual multiplication and a modified addition, which coordinates a flat I.2 plane, was specified by Spencer in 1960. | |
I.3 | Analogous to the construction of a real moulton plane with a buckling constant from the center of the almost body, a moulton plane multiplication is introduced on an arranged, real fast body. The resulting ternary body then coordinates a flat I.3 level. | ||
I.4 | An Archimedean, linear ternary body with the usual multiplication and a modified addition, which coordinates a flat I.4 plane, was specified by Salzmann in 1957. | ||
I.6 | Do not exist! | unknown | |
II | II.1 | Walker planes, these are levels of order , where q is a suitable, odd prime power. | For both Lenz-Barlotti classes II.1 and II.2 there are infinite models that even allow an arrangement. See the examples in the article Cartesian Group . |
II.2 | Some examples of the order are found in the article by Coulter and Mathews. | ||
III | III.1 | Do not exist! | Analogous to III.2, but the Moulton plane multiplication is defined on an arranged, non-commutative (therefore necessarily infinite) oblique body with a positive buckling constant that is not in the center . See also the examples in the article Cartesian Group . |
III.2 | Do not exist! | Moulton planes of the real type consisting of an (infinite) arranged body . | |
IVa | IVa.1 | Levels over finite quasi-bodies that are neither almost nor half-bodies. Such quasi-bodies are, for example, those described in Quasi-bodies # Quasi-bodies of finite Moulton planes , constructed from finite bodies of odd order of prime numbers , provided that the body automorphism defining the multiplication is not involutive. | Infinite André quasi-bodies. There is no model that allows an Archimedean arrangement. |
IVa.2 | Planes over finite almost-fields that are not half-fields. Such near-bodies are, for example, those described in Quasi-bodies # Quasi-bodies of finite Moulton planes , constructed from finite bodies of odd order of prime numbers , provided that the body automorphism defining the multiplication is involutor. | Infinite Andrésche almost bodies. There is no model that allows an Archimedean arrangement. | |
IVa.3 | Except for isomorphism, there is exactly one model: the projective, non-Desargue translation plane of order 9 | Do not exist! | |
IVb | IVb.1 | Dual to IVa.1 | Dual to IVa.1 |
VIb.2 | Dual to IVa.2 | Dual to IVa.2 | |
IVb.3 | Except for isomorphism, there is exactly one model: The dual level to the level of type IVa.3 | Do not exist! | |
V | V | Infinite number of models: For every prime power there is a “real” finite half-body, see half-body models . The projective plane above such a body always belongs to class V. | Infinite number of models of "real" infinite half-bodies. Examples of such half-bodies, which even allow an arrangement, can be obtained from generalized formal power series. No Class V level allows an Archimedean arrangement. |
VII | VII.1 | Do not exist! (→ See Moufang level !) | above the real octonions and levels via alternative bodies constructed accordingly from formally real bodies . There is no model that allows an arrangement. |
VII.2 | For every prime power order (except ) there is exactly one model except for isomorphism: The Pappus plane over the finite field | For every infinite oblique body there is exactly one model, except for isomorphism: The infinite Desargue's plane above |
literature
On the history of the term and review article
- Adriano Barlotti: Le possibili configurazioni del sistema delle coppie punto-retta (A, a) per cui un piano grafico risulta (A, a) -transitivo . In: Bolletino Unione Matematica Italiana . tape 12 , 1957, pp. 212-226 (Italian).
- Walter Benz: A Century of Mathematics, 1890-1990 . Festschrift for the anniversary of the DMV . Vieweg, Braunschweig 1990, ISBN 3-528-06326-2 .
- Marshall Hall: Projective Planes . In: Transactions of the American Mathematical Society . No. 54 , 1943, pp. 229-277 (English).
- A. Heyting: Axiomatic Projective Geometry . 2nd Edition. North Holland Publishing Company, Amsterdam 1980 (English, first edition: 1963).
- Hanfried Lenz: Small Desarguessian sentence and duality in projective levels . In: Annual report of the German Mathematicians Association . tape 57 . Teubner, 1955, p. 20–31 ( Permalink to the digitized full text [accessed on June 10, 2011]).
- Charles Weibel: Survey of Non-Desarguesian Planes . In: Notices of the American Mathematical Society . tape 54 . American Mathematical Society, November 2007, pp. 1294–1303 (English, full text [PDF; 702 kB ; accessed on December 25, 2011]).
Original articles in which models or their non-existence are proven
- Johannes André: On non-Desarguean levels with transitive translation group . In: Mathematical Journal . Volume 60, 1954, pp. 156-186 .
- Johannes André: About projective planes of the Lenz-Barlotti type III 2 . In: Mathematical Journal . Volume 84, No. 3 , 1964, pp. 316-328 , doi : 10.1007 / BF01112588 .
- Robert S. Coulter, Rex W. Mathews: Planar functions and planes of Lenz-Barlotti-Class 2 . In: Designs, Codes and Cryptography . tape 10 , no. 2 , February 1997, p. 167-184 , doi : 10.1023 / A: 1008292303803 (English).
- NL Johnson, FC Piper: On planes of Lenz-Barlotti class II-1 . In: Bull. London Math. Soc. tape 6 , 1974, p. 152–154 (English, oxfordjournals.org [accessed December 30, 2012]).
- Christoph H. Hering, William M. Kantor: On the Lenz-Barlotti classification of projective planes . In: Archives of Mathematics . tape 22 , no. 1 , 1971, p. 221–224 , doi : 10.1007 / BF01222566 (English).
- DR Hughes: A class of non-Desarguesian projective planes . In: Canadian Journal of Mathematics . Volume 9. American Mathematical Society, 1957, ISSN 0008-414X , pp. 378–388 , doi : 10.4153 / CJM-1957-045-0 (English, full text (PDF; 1 MB) [accessed on April 9, 2012]).
- H. Mohrmann: Hilbertsche and Beltramische Liniensysteme . In: Mathematical Annals . tape 85 , 1922, pp. 177-183 .
- Helmut Salzmann: Topological projective levels . In: Mathematical Journal . tape 71 , no. 1 . Springer, 1959, p. 408-413 , doi : 10.1007 / BF01181412 .
- Jill CD Spencer: On the Lenz-Barlotti Classification of Projective Planes . In: The Quarterly Journal of Mathematics . tape 11 , no. 1 . Oxford Journals, Oxford 1960, pp. 241-257 , doi : 10.1093 / qmath / 11.1.241 (English).
- JCDS Yaqub: The non-existence of finite projective planes of Lenz-Barlotti class III.2 . In: Archives of Mathematics . tape 18 , no. 3 . American Mathematical Society, 1967, pp. 308-312 , doi : 10.1007 / BF01900639 (English).
Textbooks
- Peter Dembowski : Finite geometries (= Classics in mathematics ). Springer, Berlin / Heidelberg / New York 1997, ISBN 3-540-61786-8 (English, first edition: 1968).
- Günter Pickert : Projective levels . 2nd Edition. Springer, Berlin / Heidelberg / New York 1975, ISBN 3-540-07280-2 .
- Sibylla Prieß-Crampe: Arranged structures . Groups, bodies, projective levels (= results of mathematics and its border areas . Volume 98 ). Springer, Berlin / Heidelberg / New York 1983, ISBN 3-540-11646-X .
References and comments
- ↑ Benz 1990
- ↑ ^{a } ^{b} Lenz (1954)
- ↑ Barlotti (1957)
- ↑ Pickert (1975)
- ↑ ^{a } ^{b } ^{c } ^{d } ^{e } ^{f } ^{g } ^{h } ^{i } ^{j } ^{k } ^{l} Prieß-Crampe V.5: Lenz-Barlotti classification of arranged projective planes
- ^ Hall (1943), Heyting (1963) (1980)
- ^ Hauke Klein: Lenz types. Geometry. University of Kiel, accessed on January 17, 2011 (English, tabular overview of the Lenz classes).
- ↑ If you start from the dual plane for a level of Lenz class IVb, i.e. when constructing the coordinate area instead of the set of points, first use the set of straight lines and a suitable complete quadrilateral as a basis, then you can choose this quadrilateral so that the coordinate area is again dualized level is a right quasi-body.
- ↑ ^{a } ^{b } ^{c } ^{d } ^{e} Hauke Klein: Lenz Barlotti. Geometry. University of Kiel, accessed on December 25, 2011 (English, tabular overview of the Lenz-Barlotti classes).
- ↑ Hughes (1957)
- ^ Mohrmann (1922)
- ↑ ^{a } ^{b} Up to 1975 no models were known. Pickert (1975), Appendix, 6: The Lenz-Barlotti Classification
- ^ Spencer (1960), p. 256
- ^ Salzmann (1957)
- ↑ ^{a } ^{b } ^{c} Hering and Kantor (1971), p. 221
- ↑ When André wrote his article on January 23, 1964, the existence of this class was unclear! See p. 316 there
- ↑ Johnson & Piper (1974)
- ^ Coulter & Mathews (1997)
- ↑ ^{a } ^{b} André (1964)
- ↑ Yaqub (1967)
- ↑ ^{a } ^{b} André (1954)