Abstract
Mixed projective-affine-hyperbolic (MPAH) planes belong to both the branches of NeutroGeometries and mixed or Smarandache geometries. This kind of plane is a geometric structure containing a finite number of points. Some lines of the MPAH satisfy the axioms of projective planes, other lines satisfy the axioms of affine planes, and others satisfy the axioms of hyperbolic planes. Therefore, each of the axioms of parallelism is partially satisfied. This chapter describes version 1.0 of a new software called “NeutroGeometry Laboratory” coded in Python by the author that is used for the calculation and visualization related to MPAH planes. This software is easy to use by users once they know the theory of finite mixed projective-affine-hyperbolic planes, and particularly, it supports the study of this topic and finite planes in general.
TopIntroduction
NeutroGeometries are geometric structures where some axioms or definitions are satisfied by a set of elements and simultaneously not satisfied by other elements of the space (Smarandache, 2019, 2020, 2021). The Mixed or Smarandache Geometries are geometric structures where the elements of the geometric space are divided into subspaces, each of which is identified with a specific type of geometry (Iseri, 2002; Mao, 2006; Smarandache, 1969, 2009). These new types of geometries correspond to the need to match geometric theory with the reality of physical space, where Euclidean or non-Euclidean spaces coexist fulfilling more than one geometry.
On the other hand, Finite Geometries are geometric structures based on axioms where there is a finite number of points (Dembowski, 1997; Dissett, 2000; Kiss & Szonyi, 2020). Within this theory there are three types of planes, the projective planes are the most important ones, where there is no parallelism. In the affine planes, for each point outside a line passes one and only one parallel. Additionally, in a Bolyai-Lobachevsky or hyperbolic plane, for each point outside a line passes at least two lines parallel to the aforementioned one (Korchmáros & Sonnino, 2012, 2014).
The Mixed Projective-Affine-Hyperbolic planes (MPAH planes) contain axioms of these three types of finite planes, especially those related to parallelism. That is to say, this type of plane can be divided into three types of lines, namely, some that satisfy the axiom of parallelism of projective planes, others that satisfy the axiom of parallelism of affine planes, and a third set that satisfies the axiom of parallelism of hyperbolic planes. These planes constitute a previous contribution of the author of this chapter to the theory of NeutroGeometries and Mixed Geometries.
It is well known that there are no projective planes of certain orders, where the order of the plane is determined by the number of points contained in each line. On the other hand, it is also known that an affine plane is obtained by eliminating a line and all its points within a projective plane. Much less studied is the theory of Bolyai-Lobachevsky planes and there are algorithms to obtain them from projective planes.
The drawbacks to obtaining projective planes of some orders may be due to their restrictive axiomatic. MPAH planes can be a solution to this problem because it relaxes the traditional axiomatic of finite planes. E.g., a consequence of the well-known theorem of Bruck-Ryser is that projective planes of order 6 do not exist, where the order plus 1 is the number of points contained in every line and also it is the number of lines passing for every point; however we could define an MPAH of order 6.
Another point in favor of finite geometries is their application in topics such as Code Theory, and Cryptography and they are associated with orthogonal Greco-Latin Squares (Bishnoi, Gijsiwijt, D'haesleer, & Potukuchi, 2022; A.L. Horlemann, 2022; A. Ilić, 2022; Neri, Santonastaso, & Zullo, 2023; Szabó, 2022). This ensures that the new theory of MPAH planes can be applied to these types of problems from a new perspective. Thus, now we have an open question that is how can this new theory contribute to the theoretical development of finite geometry and its applications?
Key Terms in this Chapter
Hyperbolic Plane: Also knows as the Bolyai-Lobachevsky plane , it is the geometric structure of dimension 2 such that given a line and an external point, there are at least two parallel lines to the aforementioned one, which contain the point.
Finite Geometry: Geometric system with only a finite number of points.
Tkinter: Is a graphical library package for the Python programming language. It is considered a standard for the Graphical User Interface (GUI) for Python.
Affine Plane: Geometric structure of dimension 2 such that given a line and an external point, there is a unique line parallel to the aforementioned one, which contains the point.
Mixed Geometry: Also called Smarandache Geometry , it is geometry where some elements satisfy the axiomatic of some type of geometry, and other elements satisfy other types of geometries, all in the same space. For example, there are geometric models which are partially Euclidean, partially hyperbolic, and partially elliptic.
Python: Is an interpreted programming language whose philosophy emphasizes a syntax that favors readable code. It is a popular multi-paradigm programming language.
Turtle Graphic: Is a term used in computer graphics as a method to program vector graphics using a relative cursor (the turtle) to Cartesian coordinates.
Projective Plane: Geometric structure of dimension 2 where there is non-parallelism between the lines.
Incidence Matrix: A matrix whose every row represents every line of the finite plane and every column represents every point, or vice versa, such that the element of the matrix is 1 if there is an incidence relation between the corresponding pair of line and point, and it is 0 otherwise.
NeutroGeometry: A geometric structure whose definition or at least one axiom is simultaneously satisfied by some elements and not satisfied by others.