A Constructive Introduction to Finite Mixed Projective-Affine-Hyperbolic Planes

Introduction

Finite geometry is a branch of geometry closely related to algebra. Several concepts of classical Euclidean geometry are modified since finite geometry deals with geometric structures where there is a finite set of points and therefore it contains a finite set of lines, planes, and so on (Dembowski, 1997; Dissett, 2000; Kiss & Szonyi, 2020).

On the other hand, these structures are dependent on the axioms on which they are based and these axioms determine the type of finite geometric structure in question. In dimension 2 there are three main types of structures, they are the projective planes where there is no parallelism between lines (Cameron, 2000), in the affine planes for each point external to the line passes one and only one parallel line (Zumbragel, 2016), while in the Bolyai-Lobachevsky planes or hyperbolic planes through a point outside a line pass at least two lines parallel to the first one (Korchmáros & Sonnino, 2012). The relationship of these geometric structures with algebra is that they are modeled with the support of finite vector spaces, which are vector spaces over finite fields (Hirschfeld, 1979).

Among the three types of finite planes, perhaps the projective planes are the most important of them, of which a large number of books and scientific papers can be found where they are studied exhaustively. These planes are applied in subjects like Code Theory, Cryptography, and are related to orthogonal Latin squares (Beutelspacher, 1990; Klein & Storme, 2011; Zangalho-Raposo, 2014). A useful theorem is the Bruck-Ryser theorem (Bruck & Ryser, 1949), which is a necessary condition for the existence of finite projective planes of certain order n, where n+1 is the number of points that each line contains. This theorem, because it is a necessary condition, allows us to rule out some orders with which there are no finite projective planes, for example, there is no projective plane with an order as small as n = 6. The projective plane of order n = 10 has fascinated some researchers since n = 10 satisfies the necessary conditions of the theorem, but this plane does not exist, which was demonstrated computationally, (Lam, 1991; Lam, Thiel, & Swiercz, 1989).

Less abundant in the literature is the study of affine planes (Zumbragel, 2016). It is well known that to obtain an affine plane from a projective plane we have to eliminate a fixed line and all the points contained in it. Conversely, from an affine plane, a projective plane can be obtained by adding a point for each class of parallel lines and connecting all these new points by a unique line. Affine planes can also be represented by finite vector spaces, and parallelism is an equivalence relation.

Compared to the other types of finite planes, Bolyai-Lobachevsky’s ones are found in a few pieces of literature, many of them were written in the sixties or seventies of the 20th century, with few pages dedicated to planes and almost non-existent literature dedicated to spaces of this type (Korchmáros & Sonnino, 2012). A relevant result is that from a projective plane of order greater than or equal to seven, a hyperbolic one can be obtained by eliminating three lines whose intersection is empty (Sandler, 1963). Other ways generalize this method (Olgun & Gunaltili, 2007). It is also possible to define hyperbolic planes from inversive planes of even order (Crowe, 1965, 1966). Inversive planes are defined from points and circles instead of lines, which satisfy certain axioms.

An idea that has inspired this chapter is to define Mixed or Smarandache Geometry (Iseri, 2002; Mao, 2006; Smarandache, 1969, 2009), wherein a finite geometric plane some elements satisfy the projective axioms, others the affine axioms, and a third set the hyperbolic axioms. We will call this the Mixed Projective-Affine-Hyperbolic plane (MPAH plane for short). This type of geometric structure is an example of NeutroGeometry, taking into account that each of these planes can be divided into three sets of lines satisfying different axioms of parallelism.