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Dron and Rhombic Triacontahedron KoG•19–2015 Z. Can, O.¨ Gelis¸gen, R. Kaya: On the Metrics Induced by Icosidodecahedron .... Original scientific paper ZEYNEP CAN Accepted 11. 5. 2015. O¨ ZCAN GELIS¸GEN RUSTEM¨ KAYA On the Metrics Induced by Icosidodecahe- dron and Rhombic Triacontahedron On the Metrics Induced by Icosidodecahedron O metrici induciranoj ikosadodekaedrom i and Rhombic Triacontahedron trijakontaedrom ABSTRACT SAZETAKˇ The theory of convex sets is a vibrant and classical field Teorija konveksnih skupova je vitalno i klasiˇcnopodruˇcje of modern mathematics with rich applications. If every moderne matematike s bogatom primjenom. Ako se sve points of a line segment that connects any two points toˇcke duˇzine,koja spaja bilo koje dvije toˇcke skupa, nalaze of the set are in the set, then it is convex. The more u tom skupu, tada je taj skup konveksan. Sve se viˇse geometric aspects of convex sets are developed introduc- geometrijskih aspekata o konveksnim skupovima razvija ing some notions, but primarily polyhedra. A polyhedra, uvode´ci neke pojmove, ponajprije poliedre. Konveksni n when it is convex, is an extremely important special solid poliedar je iznimno vaˇznoposebno tijelo u R . Neki primje- n in R . Some examples of convex subsets of Euclidean ri konveksnih podskupova euklidskog trodimenzionalnog 3-dimensional space are Platonic Solids, Archimedean prostora su Platonova tijela, Arhimedova tijela, tijela du- Solids and Archimedean Duals or Catalan Solids. In this alna Arhimedovim tijelima i Catalanova tijela. U ovom study, we give two new metrics to be their spheres an ˇclankuprikazujemo dvije metrike koje su sfere Arhime- archimedean solid icosidodecahedron and its archimedean dovom tijelu ikosadodekaedru i njemu dualnom tijelu, tri- dual rhombic triacontahedron. jakontaedru. Key words: Archimedean solids, Catalan solids, metric, Chinese Checkers metric, Icosidodecahedron, Rhombic tri- Kljuˇcne rijeˇci: Arhimedova tijela, Catalanova tijela, acontahedron metrika kineskog ˇsaha,ikosadodekaedar, trijakontaedar MSC2010: 51K05, 51K99, 51M20 1 Introduction by G. Chen for plane [1]. In [5], O.¨ Gelis¸gen, R. Kaya and M. Ozcan¨ extended Chinese-Checkers metric to Some mathematicians studied on metrics and improved 3−dimensional space. The CC−metric metric geometry (some of these are [2], [3], [6], [7], [8], 3 3 dCC : R × R ! [0;¥) is defined by [9]). Let P1 = (x1;y1;z1) and P2 = (x2;y2;z2) be two points 3 3 3 p in R . The maximum metric dM : R × R ! [0;¥) is de- dCC(P1;P2) = dL(P1;P2) + ( 2 − 1)dS(P1;P2) fined by where dL(P1;P2) = maxfjx1 − x2j;jy1 − y2j;jz1 − z2jg and d (P ;P ) = maxfjx − x j;jy − y j;jz − z jg: M 1 2 1 2 1 2 1 2 dS(P1;P2) = minfjx1 − x2j+jy1 − y2j;jx1 − x2j+jz1 − z2j; jy − y j + jz − z jg. Taxicab metric d : 3 × 3 ! [0;¥) is defined by 1 2 1 2 T R R Each of geometries induced by these metrics is a Minkowski geometry. Minkowski geometry is a non- dT (P1;P2) = jx1 − x2j + jy1 − y2j + jz1 − z2j: euclidean geometry in a finite number of dimensions that Then E. Krause asked the question of how to develop is different from elliptic and hyperbolic geometry (and a metric which would be similar to movement made by from the Minkowskian geometry of space-time). In a playing Chinese Checkers [11]. An answer was given Minkowski geometry, the linear structure is just like the 17 KoG•19–2015 Z. Can, O.¨ Gelis¸gen, R. Kaya: On the Metrics Induced by Icosidodecahedron .... Euclidean one but distance is not uniform in all directions. fore, there are some metrics in which unit spheres of space That is, the points, lines and planes are the same, and the furnished by them are convex polyhedra. That is, convex angles are measured in the same way, but the distance func- polyhedra are associated with some metrics. When a met- tion is different. Instead of the usual sphere in Euclidean ric is given we can find its unit sphere. On the contrary space, the unit ball is a certain symmetric closed convex a question can be asked; “Is it possible to find the metric set [13]. when a convex polyhedron is given?”. In this study we A polyhedron is a solid in three dimensions with flat faces, find the metrics of which unit spheres are an icosidodeca- straight edges and vertices. A regular polyhedron is a poly- hedron, one of the Archimedean Solids and a rhombic tri- hedron with congruent faces and identical vertices. There acontahedron which is Archimedean dual (a catalan solid) are only five regular convex polyhedra which are called of icosidodecahedron. platonic solids. Archimedes discovered the semiregular convex solids. However, several centuries passed before their rediscovery by the renaissance mathematicians. Fi- 2 Icosidodecahedron Metric nally, Kepler completed the work in 1620 by introducing One type of convex polyhedrons is the Archimedean prisms and antiprisms as well as four regular nonconvex solids. The fifth book of the “Synagoge” or “Collection” of polyhedra, now known as the Kepler–Poinsot polyhedra. the Greek mathematician Pappus of Alexandria, who lived Construction of the dual solids of the Archimedean solids in the beginning of the fourth century AD, gives the first was completed in 1865 by Catalan nearly two centuries known mention of the thirteen “Archimedean solids”. Al- after Kepler (see [10]). A convex polyhedron is said to though, Archimedes makes no mention of these solids in be semiregular if its faces have a similar configuration of any of his extant works, Pappus lists this solids and at- nonintersecting regular plane convex polygons of two or tributes to Archimedes in his book [16]. more different types about each vertex. These solids are commonly called the Archimedean solids. The duals are An Archimedean solid is a symmetric, semiregular con- known as the Catalan solids. The Catalan solids are all vex polyhedron composed of two or more types of regular convex. They are face-transitive when all its faces are the polygons meeting in identical vertices. A polyhedron is same but not vertex-transitive. Unlike Platonic solids and called semiregular if its faces are all regular polygons and Archimedean solids, the face of Catalan solids are not reg- its corners are alike. And, identical vertices are usually ular polygons. means that for two taken vertices there must be an isom- According to studies of mentioned researches unit spheres etry of the entire solid that transforms one vertex to the of Minkowski geometries which are furnished by these other. metrics are associated with convex solids. For example, One of the Archimedean solids is the icosidodecahedron. unit spheres of maximum space and taxicab space are An icosidodecahedron is a polyhedron which has 32 faces, cubes and octahedrons, respectively, which are Platonic 60 edges and 30 vertices. Twelve of its faces are regu- Solids [4], [6]. And unit sphere of CC-space is a del- lar pentagons and twenty of them are equilateral triangles toidal icositetrahedron which is a Catalan solid [5]. There- [14]. (b) (a) Figure 1: (a) Icosidodecahedron, (b) Net of icosidodecahedron 18 KoG•19–2015 Z. Can, O.¨ Gelis¸gen, R. Kaya: On the Metrics Induced by Icosidodecahedron .... To find the metrics of which unit spheres are convex poly- Thus icosidodecahedron distance between P1 and P2 is the hedrons, firstly, the related polyhedra are placed in the 3- sump of Euclidean lengths of these two line segments or dimensional space in such a way that they are symmetric 5−1 2 times the sum of Euclidean lengths of these three with respect to the origin. And then the coordinates of ver- line segments. tices are found. Later one can obtain metric which always Figure 2 illustrates icosidodecahedron wayp from P1 to supply plane equation related with solid’s surface. There- 5−1 P2 if maximum value is jx1 − x2j + ( )jy1 − y2j, fore we describe the metric which unit sphere is an icosi- p p 2 jx − x j + ( 5−1 )2 jz − z j ( 5−1 )(jx − x j + dodecahedron as following: 1 2 2 1 2 or 2 1 2 jy1 − y2j + jz1 − z2j). Definition 1 Let P1 = (x1;y1;z1) and P2 = (x2;y2;z2) be two points in 3. R Lemma 1 Let P1 = (x1;y1;z1) and P2 = (x2;y2;z2) be dis- The distance function d : 3 × 3 ! [0;¥) icosidodeca- 3 ID R R tinct two points in R . u; v; w denote jx1 − x2j, jy1 − y2j, hedron distance between P and P is defined by 1 2 jz1 − z2j, respectively. Then dID(P1;P2) = dID(P1;P2) ≥ u + (j−1)maxfv;(j−1)w;(1−j)u + v + wg; 8 u + (j − 1)maxfv;(j − 1)w;(1 − j)u + v + wg; 9 < = dID(P1;P2) ≥ v + (j−1)maxfw;(j−1)u;u + (1−j)v + wg; max v + (j − 1)maxfw;(j − 1)u;u + (1 − j)v + wg; dID(P1;P2) ≥ w + (j−1)maxfu;(j−1)v;u + v + (1−j)wg: : w + (j − 1)maxfu;(j − 1)v;u + v + (1 − j)wg ; Proof. Proof is trivial by the definition of maximum func- wherep u = jx1 − x2j, v = jy1 − y2j, w = jz1 − z2j and j = 1+ 5 tion. 2 the golden ratio. According to icosidodecahedron distance, there are three Theorem 1 The distance function dID is a metric. Also different paths from P1 to P2. These paths are according to dID, the unit sphere is an icosidodecahedron 3 i) union of two line segments which one is parallel top a in R . 5 coordinate axis and other line segment makes arctan( 2 ) 3 3 angle with another coordinate axis. Proof. Let dID : R × R ! [0;¥) be the icosidodec- ii) union of two line segments which one is parallel to a co- ahedron distance function and P1 = (x1;y1;z1), P2 = 1 ordinate axis and other line segment makes arctan( 2 ) angle (x2;y2;z2) and P3 = (x3;y3;z3) are distinct three points in 3 with another coordinate axis.
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