On Some Properties of Distance in TO-Space

The aim of this work is to investigate some properties of the truncated octahedron metric introduced in the space in further studies on metric geometry. With this metric, the 3dimensional analytical space is a Minkowski geometry which is a non-Euclidean geometry in a finite number of dimensions. In a Minkowski geometry, the unit ball is a certain symmetric closed convex set instead of the usual sphere in Euclidean space. The unit ball of the truncated octahedron geometry is a truncated octahedron which is an Archimedean solid. In this study, first, metric properties of truncated octahedron distance, dTO, in R 2 has been examined by metric approach. Then, by using synthetic approach some distance formulae in RTO 3 , 3dimensional analytical space furnished with the truncated octahedron metric has been found.


INTRODUCTION
The planar shape restricted with the line segments in the finite number is called a "polygon". A polyhedron is a three-dimensional figure consists of polygons. To define polyhedra the terms faces, edges and vertices are used. Polygonal parts of a polyhedron are called its faces. A line segment is called an edge, along which two faces come together. A point is called a vertex where several edges and faces come together. So, a polyhedron is a three dimensional solid with flat faces, straight edges and vertices.
Polyhedra have been studied by mathematicians and geometers during many years, because of their symmetries. There are many philosophers that worked on polyhedra among the ancient Greeks. The reason of Polyhedra attract people's interest is that polyhedral shapes are widely found in the nature. The kernels of some nuts and fruits contain many small seeds which grow in a restricted space. Pomegranates are one example. As each seed grows it presses up against its neighbours. The seeds prevent each other from expanding uniformly and they grow to fill the available space producing flat-faced seeds with sharp corners. If the seeds had a perfectly uniform distribution before they began to grow and were subjected to isotropic compression forces they would end up as rhombic dodecahedra. The principal of economy -maximising volume from given materialsleads to the construction of roughly spherical organisms. These sometimes have polyhedral substructures. Ernst Haeckel on his voyage on H.M.S. Challenger, in the 1880's, drew many pictures of microscopic single-celled creatures called radiolaria. A radiolarian has a spherical skeleton that is polyhedral in character. Haeckel named three of them circoporus octahedrus, circorrhegma dodecahedra and circogonia icosahedra because he thougt they resembled the Platonic solids. The recently discovered allotrope of carbon also forms polyhedral spheres, ellipsoids and tubes. In the smallest example, C60 , the sixty atoms are arranged in the same pattern as the vertices of a truncated icosahedron-familiar as a soccer ball.
Polyhedral molecules have been known for some time. Organic chemists have made carbonhydrogen structures such as cubane, C8H8, whose carbon atoms lie at the corners of a cube [1]. When we refer to polyhedron as a whole, that is, when we talk about a point or a polygon in a polyhedron, we can call this polyhedron a solid. Also, if the whole of the line segment connecting any two points remains on or in the surface of the polyhedron, this polyhedron is called convex, otherwise it is called concave. Geometrically, convexity of a polyhedron can be defined as a line connecting any two points of the polyhedron always lays in the interior of the polyhedron or on the surface of it.
A polyhedron with congruent faces and identical vertices is called a regular polyhedron.  Polyhedra, especially convex ones, have been studied by geometers for thousands of years because of their symmetries. Also metric space geometry is studied and improved by some mathematicians. In these studies it had been found that spheres of some metrics are certain convex solids. In taxicab space the unit sphere is an octahedron which is a Platonic solid; in maximum space the unit sphere is a cube which is another Platonic solid, and in CC-space the unit sphere is a deltoidal icositetrahedron which is a Catalan solid. Minkowski geometry is a non-Euclidean geometry in a finite number of dimensions. Here the linear structure is same as the Euclidean one, but distance is not uniform in all directions. That is, the points, lines and planes are the same, and the angles are measured in the same way, but the distance function is different. Instead of the usual sphere in Euclidean space, the unit ball is a general symmetric convex set [2]. The mentioned space geometries are examples of Minkowski geometries. It is easy to find unit sphere of a geometry when the metric is known. In Refs. [3][4][5][6][7][8][9], the authors have given some metrics which spheres are some of Platonic, Archimedean and Catalan solids by a reverse question; "If the sphere is known, then what is the metric of this geometry?". So there are some metrics which unit spheres are convex polyhedra.
In Ref [4], the truncated octahedron metric is introduced for 3-dimensional analytical space. By projection of truncated octahedron metric for 3-dimensional analytical space to 2-dimensional analytical plane, truncated octahedron distance, can be defined as for the points 1 , 2 ∈ ℝ 2 . If ℒ is the set of all lines in the Cartesian coordinate plane, and is the standard angle measure function in the Euclidean plane, then {ℝ 2 , ℒ , , } called TO-plane, is a model of protractor geometry. (This can be shown easily: the proof is similar to that of taxicab plane; refer to [10] or [11] to see that the taxicab plane is a model of protractor geometry.) TO-plane is also in the class of non-Euclidean geometries since it fails to satisfy the side-angle-side axiom. However, TO-plane is almost the same as Euclidean plane {ℝ 2 , ℒ , , } since the points are the same, the lines are the same and the angles are measured in the same way. Since the TO-plane (ℝ 2 ) geometry has a different distance function it seems interesting to study the TO-analogues of the topics that include the concepts of distance in the Euclidean geometry.
By these motivations, in this study, first it is shown that TO-plane geometry consisting of = ℝ 2 , ℒ and is a metric geometry. Then distance of a point to a line in the plane is found. Also in truncated octahedron space some other distance formulae are found such as distance of a point to a line, distance of a point to a plane and distance between two lines by a similiar process used in ref. [12] and which is different from refs. [13] and [14].

TO-Plane Geometry
The truncated octahedron metric is introduced in ref. [4] for 3-dimensional analytical space and for the plane this distance, can be defined as