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SOME DISTANCE PROPERTIES of TWO SPACES INDUCED by DUAL CONVEX POLYHEDRA ZEYNEP ÇOLAK and ZEYNEP CAN

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SOME DISTANCE PROPERTIES of TWO SPACES INDUCED by DUAL CONVEX POLYHEDRA ZEYNEP ÇOLAK and ZEYNEP CAN INTERNATIONAL JOURNAL OF GEOMETRY Vol. 7 (2018), No. 2, 12 - 24 SOME DISTANCE PROPERTIES OF TWO SPACES INDUCED BY DUAL CONVEX POLYHEDRA ZEYNEP ÇOLAK and ZEYNEP CAN Abstract.Dual convex spaces have an important place in mathematics. In geometry, some polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. Arranging regular polyhedra into dual pairs needs some simple operations but it is not that easy to do this for semi-regular polyhedra. In further studies it has seen that there are some relations between polyhedra and metrics. For example, it has been shown that cube, octahedron and deltoidal icositetrahedron are maximum, taxicab and Chinese Checker’sunit spheres, respectively. In this paper some distance formulae of two spaces which are induced by two dual convex polyhedra has been researched by a new method di¤erent from [1] and [8], and by this method distance formulae in Minkowski geometries can be generalized. This polyhedra are cuboctahedron which is an Archimedean solid and rhombic dodecahedron a Catalan solid which is the dual of cuboc- tahedron. 1. Introduction Molecules, galaxies, sculpture, viruses, crystals, architecture and more, polyhedra are always in nature and art. It is at the same time hands- on, mind-turned-on introduction to one of the oldest and most fascinating branches of mathematics. Although it is so ancient it is not that easy to describe what a polyhedron is. To describe a polyhedron the terms faces, edges and vertices would be used. Polygonal parts of the polyhedron are called faces, line segments that along faces come together are called edges and points where several edges and faces come together are called vertex. So, simply, a polyhedron can be de…ned as a closed, three-dimensional …gure which faces are polygons. For example, any closed box is a polyhedron, for its faces are rectangles. ————————————– Keywords and phrases: Polyhedra, Dual polyhedra, Convex polyhe- dra, Metric geometry, Rhombic Dodecahedron, Cuboctahedron, Distance Formulae. (2010)Mathematics Subject Classi…cation: 51P99, 60A99 Received: 27.07.2018. In revised form: 30.09.2018. Accepted: 20.08.2018. Some distance properties of two spaces induced by dual convex polyhedra 13 A polyhedron is called convex if a line segment joining any two points in the interior of the …gure lies completely within the …gure. There are some classi…cations of convex polyhedra. The regular polyhedra (for each of these polyhedra, every face, every vertex and every edge is like every other) are also known as the "Platonic Solids" because the Greek philoso- pher Plato (427-347 B.C.E.) immortalized them in his dialogue Timaeus. Platonic solids are only …ve. An other class of polyhedra is Archimedean solids because they were …rst discovered by Archimedes (287-212 B.C.E.). Vertices of the Archimedean polyhedra are all alike, but their faces, which are regular polygons, are of two or more di¤erent kinds. For this reason they are often called semiregular [13]. And irregular polyhedra are de…ned by polygons that are composed of elements that are not all equal and they are called Catalan solids or Archimedean duals. There are thirteen Cata- lan solids just like Archimedean solids. They are named for the Belgian mathematician, Eugéne Catalan, who …rst described them in 1865. Two polyhedrons called dual if either of a pair of polyhedra in which the faces of one are equivalent to the vertices of the other. Starting with any regular polyhedra, the dual can be constructed in the following manner: (1) Place a point in the center of each face of the original polyhedron; (2) Connect each new point with the new points of its neighboring faces; (3) Erase the original polyhedron. This is an operation "of order 2" meaning that taking the dual of the dual of x gives back the original x. For example, take the dual of the octahedron and see that it is a cube and take the dual of the cube and see that is an octahedron. The relative sizes of the two dual polyhedra can be adjusted as shown here, so that their edges are the same distance from their common center, and so cross through each other. In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another with equivalent edges. So the regular polyhedra — the Platonic solids and Kepler-Poinsot polyhedra, — are arranged into dual pairs. But the same simple operations are not valid for semi-regular polyhedrons. Because of the centers of surfaces around a corner is not on the same plane. Thereby we have to put the semi-regular polyhedra into sphere and make a plane tangent to each hill. The dual polyhedra of a Platonic solid or Archimedean solid can be also drawn by constructing polyhedra edges tangent to the midsphere which are perpendicular to the original polyhedra edges. 14 ZeynepÇolakandZeynepCan Figure 1 is lightly rotated, showing the edges of rhombic dodecahedron (yellow), octahedron (green) and cube (blue). To measure the distance between two points in a space commonly Euclid- ean distance is used. Despite of it is so popular and ancient, sometimes it is not practical in the real world. For example taxicab distance is useful instead of Euclidean distance in an urban life where streets are planned par- allel or perpendicular to each other, as it is in Manhattan. Especially the maximum distance is a very useful model in the real world and its applica- tions are so much. For example, urban planning at the macro dimensions can be used for nearly calculations. Minkowski geometry is a non-Euclidean geometry in a …nite number of dimentions. 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 di¤erent. Instead of the usual sphere in Euclidean space, the unit ball is a general symmetric convex set [15]. In studies about metric space geometry it had seen that unit spheres of these geometries are associated with convex polyhedra. When a metric is given it is easy to …nd its unit sphere in the related space geometry. In [9 and 4] authors found cuboctahedron and rhombic dodecahedron metrics by a contrary question as if sphere is known how the metric would be found and by these metrics new geometries are induced. 3 For two points P1 = (x1; y1; z1) and P2 = (x2; y2; z2) in R , Cuboctahe- dron and Rhombic Dodecahedron metrics are de…ned as 1 dCO(P ;P ) = max x1 x2 ; y1 y2 ; z1 z2 ; ( x1 x2 + y1 y2 + z1 z2 ) 1 2 j j j j j j 2 j j j j j j and dRD(P1;P2) = max x1 x2 + y1 y2 ; x1 x2 + z1 z2 ; y1 y2 + z1 z2 fj j j j j j j j j j j jg 3 3 3 respectively. RCO , RRD and RE are denote analytic 3-spaces which are furnished by Cuboctahedron metric, Rhombic Dodecahedron metric and 3 3 Euclidean metric, respectively. Linear structure of the RCO , RRD are al- 3 most the same of RE, there is only one di¤erence, this di¤erence is that its distance function. It is so important to work on notions concerned to Some distance properties of two spaces induced by dual convex polyhedra 15 distance in geometric studies, because change of metric can bring out in- teresting results. In this study formulae of distance of a point to a plane, distance of a point to line and distance between two lines in two spaces which are induced by two dual convex polyhedra has been researched by a new method which is di¤erent from further calculations done in [1] and [8], and which enables generalizing distance formulae for Minkowski geometries. This convex polyhedra are cuboctahedron which is an Archimedean solid and rhombic dodecahedron a Catalan solid which is the dual of cuboctahe- dron. 2. DISTANCE FORMULAE OF CUBOCTAHEDRON AND RHOMBIC· DODECAHEDRON Cuboctahedron The cuboctahedron is a uniform polyhedron bounded by 6 squares and 8 triangles. A cuboctahedron has 14 faces, 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each sep- arating a triangle from a square. As such it is a quasiregular polyhedron, i.e. an Archimedean solid, being vertex-transitive and edge-transitive. Its dual polyhedron is the rhombic dodecahedron. It is edge-uniform, and its two kinds of faces alternate around each vertex, so it is also a quasi-regular polyhedron. It is, also called the heptaparallelohedron or dymaxion. The distance between the center of a cuboctahedron and each vertex is equal to the length of its edges. The regular hexagon and the cuboctahedron are the only uniform polytopes of their respective dimensions with this property. The cuboctahedron has the Oh octahedral group of symmetries. According to Heron, Archimedes ascribed the cuboctahedron to Plato (Heath 1981; Coxeter 1973, p. 30). The Cartesian coordinates of vertex of the cuboctahe- dron, centered on the origin and having edge length 2, are all permutations of coordinates and changes of sign of (0; 1; 1): Figure 2 Cuboctahedron De…nition 1. Let P1 = (x1; y1; z1) and P2 = (x2; y2; z2) be two points in 3 R . The cuboctahedron distance between P1 and P2 is 1 dCO(P1;P2) = max x1 x2 ; y1 y2 ; z1 z2 ; ( x1 x2 + y1 y2 + z1 z2 ) j j j j j j 2 j j j j j j In accordance with dCO-metric, the shortest way between points P1 and 1 P2 is a line segment that is parallel to a coordinate axis or 2 times of sum of three line segments each of them is parallel to a coordinate axis.
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