A Note On The Metrics Induced By Triakis Icosahedron And Disdyakis Triacontahedron

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1 urasian Academy of Sciences urasian Life Sciences Journal / Avrasya Fen Bilimleri Dergisi 01 Volume:1 S: 1 - Published Online May 01 ( A Note On The Metrics Induced By Triakis Icosahedron And Disdyakis Triacontahedron Zeynep Can* Zeynep Çolak** Öcan Gelişgen *** * Aksaray Üniversitesi -mail: eynepcan@aksaray.edu.tr ** Çanakkale Onseki Mart Üniversitesi olak.84@gmail.com *** skişehir Osmangai Üniversitesi gelisgen@ogu.edu.tr Copyright 01 Zeynep Can Zeynep Çolak Öcan Gelişgen. This is an open access article distributed under the urasian Academy of Sciences License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. ABSTRACT Polyhedrons have been studied by mathematicians and geometers during many years because of their symmetries. Geometricians who work in the field of polyhedra are aware of the origin of the Archimedean bodies. In geometry polyhedra are associated into pairs called duals where the vertices of one correspond to the faces of the other. The duals of the Archimedean solids are called the Catalan solids which are all convex and ırregular polyhedra. The number of Catalan solids is only thirteen. There are some relations between metrics and polyhedra. For example it has been shown that deltoidal icositetrahedron is Chinese Checker's unit sphere. In this study we introduce two new metrics that their spheres are Triakis Icosahedron and Disdyakis Triacontahedron which are catalan solids. Keywords: Catalan Solids Triakis Icosahedron Disdyakis Triacontahedron Metric Chinese Checker metric. MSC 000: 1K0 1K99 1M0 Triakis Icosahedron Ve Disdyakis Triacontahedrondan lde dilen Metrikler Üerine ÖZT Simetrilerinden dolayı matematikçiler ve geometriciler tarafından yıllardır çok yülüler üerinde çalışılmaktadır. Çok yülüler üerinde çalışan geometriciler Arşimedyan cisimlerin kökenini bilirler. Geometride çok yülüler birinin köşeleri diğerinin yülerine karşılık gelen ve dualler olarak isimlendirilen çiftlerden oluşmaktadırlar. Arşimedyan cisimlerin dualleri konveks ve irreguler olan Katalan cisimlerdir. Katalan cisimler tam olarak on üç tanedirler. Metrikler ve çok yülüler arasında baı ilişkiler bulunmaktadır. Örneğin deltoidal icositetrahedronun Çin Dama metriğinin birim küresi olduğu gösterilmiştir. Bi bu çalışmada birim küreleri Triakis Icosahedron ve Disdyakis Triacontahedron olan iki yeni metrik tanıtacağı.

2 A Note On The Metrics Induced By Triakis Icosahedron And Disdyakis Triacontahedron Anahtar Kelimeler: Katalan Cisimler Triakis Icosahedron Disdyakis Triacontahedron Metrik Çin Dama Metriği 1.INTRODUCTION The word polyhedron has slightly different meanings in geometry and algebraic topology. In geometry a polyhedron is simply a three-dimensional solid which consists of a collection of polygons usually joined at their edges. The term "polyhedron" is used somewhat differently in algebraic topology where it is defined as a space that can be built from such "building blocks" as line segments triangles tetrahedra and their higher dimensional analogs by "gluing them together" along their faces. 1 The word derives from the Greek poly (many) plus the Indo-uropean hedron (seat). A polyhedron is the three-dimensional version of the more general polytope which can be defined in arbitrary dimension. The plural of polyhedron is "polyhedra" (or sometimes "polyhedrons"). Polyhedra have worked by people since ancient time. arly civiliations worked out mathematics as problems and their solutions. Polyhedrons have been studied by mathematicians geometers during many years because of their symmetries. Recently polyhedra and their symmetries have been cast in a new light by mathematicians. A polyhedron is said to be regular if all its faces are equal regular polygons and same number of faces meet at every vertex. A polyhedra is convex if with every pair of points that belong to the shape the shape contains the whole straight line segment connecting the two points. Platonic solids are regular and convex polyhedra. Nowadays some mathematicians are working on platonic solid s metric. A polyhedron is called semi-regular if all its faces are regular polygons and all its vertices are equal. Archimedian solids are semi-regular and convex polyhedra. All the mathematicians and geometricians who work in the field of polyhedra are aware of the origin of the Archimedean bodies. The dual polyhedra of the Archimedean solids are Catalan solids. Catalan was the one who first described them in The Catalan solids are all convex and irregular polyhedra. The number of Catalan solids is thirteen. Minkowski geometry is non-uclidean geometry in a finite number of dimensions. Here the linear structure is the same as the uclidean one but distance is not uniform in all directions. Instead of the usual sphere in uclidean space the unit ball is a general symmetric convex set. The points lines and planes are the same and the angles are measured in the same way but the distance function is different. 4 Some mathematicians have been studied and improved metric space geometry. According to mentioned researches it is found that unit spheres of these metrics are associated with convex solids. For example unit sphere of maximum metric is a cube which is a Platonic Solid. Taxicab metric's unit sphere is an octahedron another Platonic Solid. And unit sphere of CC-metric is a deltoidal icositetrahedron a Catalan solid. 1 RMİŞ T. Dügün Çokyülülerin Metrik Geometriler ile İlişkileri Üerine Doktora Tei skişehir Osmangai Üniversitesi Fen Bilimleri nstitüsü THOMPSON A. C. Minkowski Geometry Cambridge University Press Cambridge 1996.

3 urasian conometrics Statistics & mprical conomics Journal 01 Volume: 1 3 So there are some metrics which unit spheres are convex polyhedrons. That is convex polyhedrons are associated with some metrics. This influence us to the question "Are there some metrics of which unit spheres are the Catalan Solids?". For this goal firstly the related polyhedra are placed as fully symmetric such that symmetry center of it is origin in the 3-dimensional space. And then the coordinates of vertices are found. Later one can obtain the metric which always supply plane equation related with solid's surface. In this work we introduce that new metrics of which spheres are Triakis Icosahedron and Disdyakis Triacontahedron.. Triakis Icosahedron The triakis icosahedron is an Archimedean dual solid or a Catalan solid. The Triakis Icosahedron has 3 vertices 60 faces and 90 edges. The Triakis Icosahedron contains 60 scale triangles. The dual of the Triakis Icosahedron is the truncated dodecahedron. Figure 1: Triakis Icosahedron We describe the metric that unit sphere is Triakis Icosahedron as following: Definition.1 : Let P1 = ( x1 y1 1) and P ( x y ) d R³ R³ [ 0 ) distance function TI : defined by = be distinct two points in R³. The Triakis Icosahedron distance between P 1 and P is ( 1 ) ( ) ( ϕ ) ( ϕ ) 1 1 x1 x + y1 y x1 x + max ϕ x1 x + + ϕ y1 y + ϕ 1 x1 x 1+ x1 x + 1 y1 y + 1 ( P1 P) = max y1 y + max ϕ x1 x ϕ y1 y + 1+ ϕ 1 y1 y x1 x + 1+ y1 y 1 + max ( 1 ϕ) x1 x ϕ y1 y ϕ where φφ = +1 the golden ratio.

4 4 A Note On The Metrics Induced By Triakis Icosahedron And Disdyakis Triacontahedron Triakis Icosahedron distance function may seem a bit complicated. In fact there is an orientation in d TI. Let aa = xx 1 xx bb = xx 1 xx cc = 1. This orientation is aa bb cc aa. According to orientation if one puts b c a instead of a b c respectively in the first term of distance function then obtains the second term. Similarly if one puts c a b instead of a b c respectively in the first term of distance function then obtains the third term. Lemma. : Let P = ( x y ) and P ( x y ) = be any distinct two points in R³. Then P P x x x x y y x x y y ( 1 ) + max { ( 1 ϕ) + + ( 1 + ϕ) ϕ + ( 1+ ϕ) + ϕ 1 } P P y y x x x x y y x x y y ( 1 ) + max { ( 1+ ϕ) + ( 1 ϕ) + ϕ ϕ + ( 1+ ϕ) 1 } P P y y x x y y x x y y ( 1 ) + max { + ( 1+ ϕ) + ( 1 ϕ) ( 1+ ϕ) + ϕ ϕ 1 } Proof: Proof is trivial by definition of maximum function. Theorem.3 : The distance function d TI is a metric of which unit sphere is a Triakis Icosahedron in R³. Proof: We have to show that da is positive definite and symmetric and da holds triangle inequality. Let P1 = ( x1 y1 1) P = ( x y ) and P3 = ( x3 y3 3) be three points in R³. Since absolute values is always nonnegative maximum of sums of absolute value is always nonnegative. Thus ( P1 P) 0. Obviously ( P1 P ) = 0if and only if P 1 = P. So d TI is positive definite. Clearly ( P1 P) = ( P P1) follows from xi xj = xj xi yi yj = yj yi i j = j i. That is d TI is symmetric. Now we try to prove that ( P1 P3) ( P1 P) + ( P P3) for all P1 ( x1 y1 1) P = ( x y ) and P = ( x y ) in R³. Then ( 1 ) ( ) ( ϕ ) ( ϕ ) = x1 x3 + y1 y x1 x3 + max ϕ x1 x3 + + ϕ y1 y3 + ϕ 1 3 x1 x3 1+ x1 x3 + 1 y1 y ( P1 P3) = max y1 y3 + max ϕ x1 x3 ϕ y1 y ϕ 1 3 y1 y3 x1 x y1 y max ( 1 ϕ) x1 x3 ϕ y1 y3 ϕ

5 urasian conometrics Statistics & mprical conomics Journal 01 Volume: 1 ( ) ( ϕ)( ) ( ) ( ϕ)( ) ϕ( x1 x + x x3 ) + ( 1+ ϕ)( y1 y + y y3 ) + ϕ( ) ( x1 x + x x3 ) ( ϕ)( ) ( ϕ)( ) ( + 3 ) ϕ( x1 x + x x3 ) ϕ( y1 y + y y3 ) + ( 1+ ϕ)( ) ( y1 y + y y3 ) ( ) ( ϕ)( ) ( ϕ)( ) ( 1+ ϕ)( x1 x + x x3 ) + ϕ( y1 y + y y3 ) ϕ( ) ( x1 x + x x3 ) + max 1 x1 x + x x3 + y1 y + y y max ( y1 y + y y3 ) + max 1+ x1 x + x x3 + 1 y1 y + y y3 + 1 ( ) + max x1 x + x x y1 y + y y = I One can easily find that I ( P1 P) ( P P3) d ( P P ) d ( P P ) d ( P P ) TI 1 3 TI 1 TI 3 inequality. Consequently the set + from Lemma.. So +. That is d TI distance function satisfies the triangle ϕ x + ( 1+ ϕ) y + ϕ ϕ x ϕ y + ( 1+ ϕ) ( + ϕ) x + ϕ y ϕ 1 x + y x + max x 1+ x + 1 y + STI = ( xy ) : ddt ( X0) = max y+ max = 1 y x + 1+ y max 1 is the set of all points X ( xy ) (See Figure ). = R³ triakis icosahedron distance is 1 from 0 = (000) Figure : Triakis Icosahedron in coordinate system Corollary.4 : The equation of the triakis icosahedron with center C ( x y ) r is = and radius 0 0 0

6 6 A Note On The Metrics Induced By Triakis Icosahedron And Disdyakis Triacontahedron ϕ x x0 + ( 1+ ϕ) y y0 + ϕ 0 ϕ x x0 ϕ y y0 + ( 1+ ϕ) 0 ( ϕ) x x ϕ y y ϕ 0 1 x x0 + y y x x0 + max x x 1+ x x + 1 y y + y y + = r y y0 x x y y max max 0 max Lemma. : Let l be the line through the points P = ( x y ) and P ( x y ) = in the analytical 3-dimensional space and d denote the uclidean metric. If l has direction vector ( pqr ) then d ( P P ) µ ( P P ) d ( P P ) TI = where ( P P ) µ is equal to p + max { r ( 1 ϕ) p + q + ( 1 + ϕ) r ϕ p + ( 1 + ϕ) q + ϕ r} max q + max { p ( 1+ ϕ) p + ( 1 ϕ) q + r ϕ p ϕ q + ( 1 + ϕ) r} r + max { q p + ( 1 + ϕ) q + ( 1 ϕ) r ( 1 + ϕ) p + ϕ q ϕ r}. p + q + r Proof: quation of l gives us x1 x = λ p y1 y = λq 1 = λr λ R. Thus p + max { r ( 1 ϕ) p + q + ( 1 + ϕ) r ϕ p + ( 1 + ϕ) q + ϕ r} ( P1 P) = λmax q + max { p ( 1+ ϕ) p + ( 1 ϕ) q + r ϕ p ϕ q + ( 1 + ϕ) r} r + max { q p + ( 1 + ϕ) q + ( 1 ϕ) r ( 1 + ϕ) p + ϕ q ϕ r} and ( ) 1 1 d P P = λ p + q + r which implies the required result. The above lemma says that d TI - distance along any line is some positive constant multiple of uclidean distance along same line. Thus one can immediately state the following corollaries: Corollary.6 : If P 1 P and X are any three collinear points in R³ then ( ) = ( ) if and only if d ( P X) d ( P X) d P X d P X 1 TI =. 1 TI Corollary.7 : If P 1 P and X are any three distinct collinear points in the real 3- dimensional space then That is the ratios of the uclidean and ( ) / ( ) ( ) / ( ) d X P d X P = d X P d X P. TI 1 TI 1 distances along a line are the same.

7 urasian conometrics Statistics & mprical conomics Journal 01 Volume: Disdyakis Triacontahedron Disdyakis Triacontahedron hexakis icosahedron or kisrhombic triacontahedron is a Catalan solid which is dual to the Archimedean truncated icosidodecahedron. It is composed of scalene triangles. It has 10 faces 180 edges and 6 vertices. 6 Figure - 3 : Disdyakis Triacontahedron We describe the metric which s unit sphere is Disdyakis Triacontahedron as following: Definition 3.1 : Let P1 = ( x1 y1 1) and P ( x y ) d R R [ 0 ) distance function : DT P is defined by = be distinct two points in R³. The Disdyakis Triacontahedron distance between P 1 and ϕ x1 x + ( + ϕ) y1 y + ϕ 1 ( 1 ϕ) x1 x + ( 1+ ϕ) y1 y + 3 ϕ 1 1 ( ϕ) x1 x + y1 y + ( ϕ+ 1) 1 1 ( ϕ) ( 1 ϕ) ϕ x1 x ϕ y1 y + ( ϕ+ ) 1 x1 x + ( 1 ϕ) y1 y + ( 1 + ϕ) 1 ( ϕ+ 1) x1 x + 1 ( ϕ) y1 y + 1 ( + ϕ) x1 x + ϕ y1 y ϕ 1 ( + ϕ) x1 x + ϕ y1 y + ( ϕ) 1 x x + ( ϕ+ 1) y y + 1 ( ϕ) y1 y x1 x + y1 y ϕ x1 x + max x1 x x1 x + y1 y + 1 4ϕ ddt ( P1 P) = max y1 y + max x1 x + y1 y x1 x + 1+ y1 y ϕ 1 + max where φφ = +1 the golden ratio. Disdyakis triacontahedrondistance function may be seem a bit complicated. In fact there is an orientation in d DT just like in d TI. As shown for d TI let be aa = xx 1 xx bb = xx 1 xx cc = 1. Here the orientation is aa bb cc aa. According to orientation if one puts b c a instead of a b c respectively in the first term of distance function then it is obtained the 6

8 8 A Note On The Metrics Induced By Triakis Icosahedron And Disdyakis Triacontahedron second term. Similarly if one puts c a b instead of a b c respectively in first term of distance function then it is obtained the third term. Lemma 3. : Let P = ( x y ) and P ( x y ) = be any distinct two points in R³. Then ( ) ( ) ( ) 1 ( ϕ) x x + y y + ( ϕ+ 1) y1 y x1 x + y1 y ϕ ddt ( P1 P) x1 x + max ϕ x1 x + + ϕ y1 y + ϕ 1 1 ϕ x1 x + 1+ ϕ y1 y + 3 ϕ ( ϕ) ( 1 ϕ) ( ) ( ) ( ) ( ϕ+ 1) x x + 1 ( ϕ) y y + x1 x x1 x + y1 y + 1 4ϕ ddt ( P1 P) y1 y + max ϕ x1 x ϕ y1 y + ϕ+ 1 x1 x + 1 ϕ y1 y ϕ x1 x y1 y x1 x ( 1 ϕ) y1 y ( 1 ϕ) 1 ( ϕ) ϕ ϕ ( ϕ) ϕ ( ϕ) x x + ( ϕ+ 1) y y + 1 ( ϕ) ϕ ddt ( P1 P) 1 + max + x1 x + y1 y 1 1+ x1 x + 3 y1 y Proof: Proof is trivial by definition of maximum function. Theorem 3.3 : The distance function d DT is a metric of which unit sphere is a Disdyakis Triacontahedron in R³. Proof: One can easily give the proof of theorem by similar way in Theorem.3. Consequently the set + ( ϕ) + + ( + ϕ) ϕ x + ( + ϕ) y + ϕ ( ϕ) x + ( 1+ ϕ) y + 3 ϕ 1 ( ϕ) x + y + ( ϕ + 1) + ( + ϕ) + ( ϕ) + ϕ x ϕ y + ( ϕ + ) x + ( ϕ) y + ( 1 + ϕ) ( ϕ + 1) x + 1 ( ϕ) y ( + ϕ) + ( ϕ) ( + ϕ) x + ϕ y ϕ ( 1+ ϕ) x + y + ( 1 ϕ) x + ( ϕ + 1) y + 1 ( ϕ) y 1 x y 1 4ϕ x + max x 1 x 1 y 4ϕ SDT = ( xy ) : ddt ( X0) = max y+ max = 1 x y x 1 y 1 4ϕ + max is the set of all points X ( xy ) 0 = ( 000) (See Figure 4). = R³ disdyakis triacontahedrondistance is 1 from

9 urasian conometrics Statistics & mprical conomics Journal 01 Volume: 1 9 Figure - 4 : Disdyakis Triacontahedron in coordinate system Corollary 3.4 : The equation of the disdyakis triacontahedron with center C ( x y ) radius r is ϕ x x0 + ( + ϕ) y y0 + ϕ 0 ( 1 ϕ) x x0 + ( 1+ ϕ) y y0 + 3 ϕ 0 1 ( ϕ) x x0 + y y0 + ( ϕ+ 1) 0 ϕ x x0 ϕ y y0 + ( ϕ+ ) 0 x x0 + ( 1 ϕ) y y0 + ( 1 + ϕ) 0 ( ϕ+ 1) x x0 + 1 ( ϕ) y y0 + 0 ( + ϕ) x x0 + ϕ y y0 ϕ 0 ( + ϕ) x x0 + ϕ y y0 + ( ϕ) 0 x x + ( ϕ+ 1) y y + 1 ( ϕ) = and y y x x0 + y y ϕ x x0 + max x x x x0 + 1 y y ϕ max y y0 + max = r x x0 + y y0 x x y y ϕ 0 + max Lemma 3. : Let l be the line through the points P = ( x y ) and P ( x y ) = in the analytical 3-dimensional space and d denote the uclidean metric. If l has direction vector d P P µ P P d P P µ P P is equal to ( pqr ) then ( ) ( ) ( ) DT = where ( )

10 10 A Note On The Metrics Induced By Triakis Icosahedron And Disdyakis Triacontahedron ( ) ( ) ( ) ( ϕ) + ( 1+ ϕ) + 3 ϕ ( 1 ϕ) + + ( ϕ+ 1) ϕ ϕ ( ϕ ) + ( ϕ) + ( 1 + ϕ) ( ϕ+ 1) + 1 ( ϕ) + p + q p + ( 1+ ) q + ( 1 ) r ( + ) p + q ( 1+ ϕ) + + ( ϕ) + ( ϕ+ 1) + 1 ( ϕ) 4ϕ q + r 1 ϕ p + q ϕ r ϕ p + + ϕ q + ϕ r p + max p q r p q r 4ϕ p + r 1+ p + 1 q + r p q + + r max q + max p q r p q r 4ϕ ϕ ϕ ϕ ϕ ϕ r r + max p q r p q r. p + q + r Proof : quation of l gives us x1 x = λ p y1 y = λq 1 = λr λ R. Thus ( ) ( ) ( ) ( ϕ) + ( 1+ ϕ) + 3 ϕ ( 1 ϕ) + + ( ϕ+ 1) ϕ ϕ ( ϕ ) + ( ϕ) + ( 1 + ϕ) ( ϕ+ 1) + 1 ( ϕ) + p + q p + ( 1+ ϕ) q + ( 1 ϕ) r ( ) ( 1+ ϕ) + + ( ϕ) + ( ϕ+ 1) + 1 ( ϕ) 4ϕ q + r 1 ϕ p + q ϕ r ϕ p + + ϕ q + ϕ r p + max p q r p q r 4ϕ p + r 1+ p + 1 q + r p q + + r ddt ( P1 P) = λ max q + max p q r p q r 4ϕ + ϕ p + ϕ q ϕ r r + max p q r p q r d P P = λ p + q + r which implies the required result. and ( ) 1 The above lemma says that ddt - distance along any line is some positive constant multiple of uclidean distance along same line. Thus one can immediately state the following corollaries: Corollary 3.6 : If P 1 P and X are any three collinear points in R³ then ( ) = ( ) if and only if d ( P X) d ( P X) d P X d P X 1 DT =. 1 DT Corollary 3.7 : If P 1 P and X are any three distinct collinear points in the real 3- dimensional space then ( ) / ( ) ( ) / ( ) d X P d X P = d X P d X P. DT 1 DT 1 That is the ratios of the uclidean and d TI distances along a line are the same.

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