SAÜ Fen Bil Der 19. Cilt, 3. Sayı, s. 353-360, 2015

New metrics for deltoidal hexacontahedron and pentakis dodecahedron

Zeynep Çolak1*, Özcan Gelişgen2

26.02.2015Geliş/Received, 27.05.2015Kabul/Accepted ABSTRACT

There are only five regular convex polyhedra known as platonic solids. Semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices. These solids are called the Archimedian solids. Archimedean solids' s duals are known as the Catalan solids which are only thirteen. It has been shown that deltoidal icositetrahedron which is Chinese Checker' s unit sphere [1]. In this study, we introduce new metrics which their spheres are pentakis dodecahedron and deltoidal hexacontahedron

Keywords:pentakis dodecahedron, deltoidal hexacontahedron, metric, chinese checker metric, catalan solid.

Deltoidal hexacontahedron ve pentakis dodecahedron için yeni metrikler

ÖZ

Platonik cisimler olarak tanımlanan sadece beş tane düzgün konveks çokyüzlü vardır. Yarı-düzgün konveks çokyüzlülerin köşe noktalarında iki veya daha fazla tipten düzgün çokgen birleşir. Bu cisimlere Arşimet cisimleri adı verilir. Arşimet cisimlerinin dualleri Catalan cisimler olarak bilinirler ve sadece onüç tanedir. Son yıllardaki çalışmalarda Çin Dama metriğinin birim küresinin deltoidal icositetrahedron olduğu gösterildi [1]. Bu çalışmada birim küreleri deltoidal hexacontahedron ve pentakis dodecahedron olan metrikleri vereceğiz.

Anahtar Kelimeler: pentakis dodecahedron, deltoidal hexacontahedron, metrik, çin dama metriği, katalan cisim.

* Sorumlu Yazar / Corresponding Author 1 Çanakkale Onsekiz Mart Üniversitesi, Çanakkale - [email protected] 2 Osmangazi Üniversitesi,Fen-Edebiyat Fakültesi, Matematik Bölümü - [email protected] Z. Çolak, Ö. Gelişgen New metrics for deltoidal hexacontahedron and pentakis dodecahedron

1. INTRODUCTION This influence us to the question "Are there some metrics of which unit spheres are the Catalan Solids?". A polyhedron is a three dimensional solid which For this goal, firstly we put up the solid to coordinate consists of a collection of polygons, always joined at system to be its center the origin and some of solid's their edges. There are many thinkers that worked on surfaces distance from the origin are 1. And then, we polyhedra among the ancient Greeks. Early can have the metric which provide plane equation civilizations worked out mathematics as problems and related with solid's surface. In this work, we introduce their solutions. Polyhedrons have been studied by that new metrics of which spheres are Deltoidal mathematicians, scientists during many years, because Hexacontahedron and Pentakis Dodecahedron. of their symmetries [2-4]. 2. DELTOIDAL HEXACONTAHEDRON A polyhedron is called regular if all its faces are equal and regular polygons. It is called semi-regular if all its A deltoidal hexecontahedron (also sometimes called a faces are regular polygons and all its vertices are equal. trapezoidal hexecontahedron, a strombic An irregular polyhedron is defined by polygons that are hexecontahedron, or a tetragonal hexacontahedron) is a composed of elements that are not all equal. A regular catalan solid which looks a bit like either an polyhedron is called Platonic solid, a semi-regular overinflated dodecahedron or icosahedron. It is polyhedron is called Archimedean solid and an irregular sometimes also called the trapezoidal hexecontahedron polyhedron is called Catalan solid. or strombic hexecontahedron. Its dual polyhedron is the rhombicosidodecahedron. The 60 faces are deltoids or Platonic solids have been studied by mathematicians, kites (not trapezoidal). The short and long edges of geometers during many years. Nowadays some each kite are in the ratio 1.00/1.54. The Deltoidal mathematicians study new metrics of which spheres are Hexacontahedron has 60 faces, 120 edges and 62 = 12 + Platonic solids. The Archimedean solids take their name 20 + 30 vertices [8]. from Archimedes, who discussed them in a now-lost work. Pappus refers to it, stating that Archimedes listed 13 polyhedra. The Archimedean solids are distinguished by having very high symmetry. They are distinct from the Platonic which are composed of only one type of polygon meeting in identical vertices, whoseregular polygonal faces do not meet in identical vertices. The dual polyhedra of the Archimedean solids are called Catalan solids. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865. The Catalan solids are all Figure 1. Deltoidal hexacontahedron convex and ırregular polyhedra. The number of Catalan solids is thirteen.

Minkowski geometry is non-Euclidean geometry in a finite number of dimensions. Instead of the usual sphere in Euclidean space, the unit ball is symmetric closed convex set [7]. Some mathematicians have been studied and improved metric space geometry. The Chinese Checker metric plane and space geometry have been studied and developed by some mathematicians. O. Gelişgen, R. Kaya, M. Ozcan have defined CC- metric of which sphere is Deltoidal Icositetrahedron that is a Catalan solid (See [1]). In the 3-dimensional Figure 2. Net for deltoidal hexacontahedron analytical space the CC-metric is defined by We describe the metric that unit sphere is deltoidal hexacontahedron as following: dC (A,B)= max{|x₁-x₂|,|y₁-y₂|,|z₁-z₂|}+(√2-1)min {|x₁-x₂|+|y₁-y₂|,|x₁-x₂|+|z₁-z₂|,|y₁-y₂|+|z₁-z₂|} (1) Definition 2. 1:Let P₁=(x₁,y₁,z₁) and P₂=(x₂,y₂,z₂) be distinct two points in ℝ³ . The distance function where A=(x₁,y₁,z₁), B=(x₂,y₂,z₂) are two points in ℝ³. dDH :ℝ³x ℝ³→[0,∞) deltoidal hexacontahedron distance between P1 and P2 is defined by

354 SAÜ Fen Bil Der 19. Cilt, 3. Sayı, s. 353-360, 2015

New metrics for deltoidal hexacontahedron and pentakis Z. Çolak, Ö. Gelişgen dodecahedron

(, ) = where = 1 + √5⁄2is the golden ratio. | − | + (2 − 3) ⎧ 1 2 ⎫ Proof: Proof is trivial by definition of maximum ⎪ 1 − 2 + |1 − 2|, ⎪ ⎧ ⎫ function. ⎪ ⎪ (1 − )| − | + ⎪⎪ 1 2 ⎪ (1 + )| − |, ⎪ ⎪ 1 2 ⎪ Theorem 2. 3: The distance function dDH is a metric of ⎨−| − | + | − | +⎬ ⎪ ⎪ 1 2 1 2 ⎪⎪ which unit sphere is a deltoidal hexacontahedron in ℝ³. ⎪ ⎩ (1 + ) − ⎭⎪ ⎪ 1 2 ⎪ − + (2 − 3) Proof: Let dDH: ℝ³x ℝ³→ℝ and P₁=(x₁,y₁,z₁), ⎪ 1 2 ⎪ P₂=(x₂,y₂,z₂) and P₃=(x₃,y₃,z₃) distinct three points in ⎪ |1 − 2| + |1 − 2|, ⎪ ℝ³. To prove that d is a metric in ℝ³, the following ⎪ ⎧ ⎫⎪ DH ⎪ (1 − )1 − 2 + ⎪ axioms can be supplied for all P₁, P₂ and P₃∈ℝ³. (2) ⎨ (1 + )|1 − 2|, ⎬ ⎪ ⎨ ⎬⎪ M1) d(P₁,P₂)≥0 ve d(P₁,P₂)=0⇔P₁=P₂ −1 − 2 + |1 − 2| + ⎪ ⎪ ⎪⎪ M2) d(P₁,P₂)=d(P₂,P₁) ( )| | ⎪ ⎩ 1 + 1 − 2 ⎭⎪ M3) d(P₁,P₃)≤d(P₁,P₂)+d(P₂,P₃). ⎪ |1 − 2| + (2 − 3) ⎪ M1) Since absolute values is always nonnegative ⎪ − + |1 − 2|, ⎪ ⎪ ⎧ 1 2 ⎫⎪ maximum of sums of absolute value is always ⎪ (1 − )|1 − 2| + ⎪ ⎪ ⎪ ⎪⎪ nonnegative. Thus dDH(P₁,P₂)≥0. If dDH(P₁,P₂)=0 then ⎪ ⎪ (1 + )1 − 2, according to deltoidal hexacontahedron distance ⎪ ⎨ ⎬⎪ function three cases are possible. ⎪−|1 − 2| + − +⎪ ⎪ ⎪ 1 2 ⎪⎪ ( )| | ⎩ ⎩ 1 + 1 − 2 ⎭⎭ Case I: If where = 1 + √5⁄2is the golden ratio. dDH(P₁,P₂)=| − | + (2 − 3)

| − | + | − |, Deltoidal hexacontahedron distance function may seem (1 − )| − | + (1 + )| − |, a bit complicated. In fact there is an orientation in d . DH −| − | + | − | + (1 + )| − | Let = | − |, = | − |, = | − |. This orientation is a − b − c − a. According to orientation, the d (P₁,P₂)=0 ⇔ |x₁-x₂|=0,|y₁ - y₂|=0, |z₁ - z₂|=0 ⇔ if one can put b, c, a instead of a, b, c, respectively, in DH x₁=x₂, y₁=y₂, z₁=z₂⇔ P₁=P₂. first term of distance function, then it is obtained second term. Similarly, if one can put c,a, b instead of a, b, c, The other cases can be easily shown by similar way in respectively, in first term of distance function, then it is case i. Thus we get d (P₁,P₂)=0 iff P₁=P₂. obtained third term. DH M2) By the definition of absolute value

Lemma 2. 2: Let P₁=(x₁,y₁z₁) and P₂=(x₂,y₂,z₂) be any |x -x |=|x -x |, distinct two points in ℝ³. Then i j j i |y -y |=|y -y | , i j j i |z -z |=|z -z | ( , ) ≥ | − |+(2 -3) i j j i 3 for all xi, yi, zi,xj, yj, zj∈ℝ and i,j=1,2,3. Therefore one | − | + | − |, can get dDH(P₁,P₂) =dDH(P₂,P₁). (1 − )| − | + (1 + )| − |, −| − | + | − | + (1 + )| − | M3) Let P₁=(x₁,y₁,z₁) , P₂=(x₂,y₂,z₂) and P₃=(x₃,y₃,z₃) (, ) ≥ | − | + (2 − 3) be any distinct three points in ℝ³. Then | − | + | − |, (1 − )| − | + (1 + )| − |, −| − | + | − | + (1 + )| − | (, ) ≥ | − | + (2 − 3) | − | + | − |, (1 − )| − | + (1 + )| − |, −| − | + | − | + (1 + )| − |

SAÜ Fen Bil Der 19. Cilt, 3. Sayı, s. 353-360, 2015 355

Z. Çolak, Ö. Gelişgen New metrics for deltoidal hexacontahedron and pentakis dodecahedron

( , ) (, , ): (, ) = ⎧ ⎫ | − | + (2 − 3) || + (2 − 3) ⎧ ⎫ ⎪ ⎧ ⎫⎪ | − | + | − |, || + ||, (1 − )|| + (1 + )||, ⎪ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪⎪ (1 − )| − | + (1 + )| − |, −|| + || + (1 + )|| ⎪ ,⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎨ −| − | + | − | + ⎬ ⎪ ⎪ ⎪ || + (2 − 3) ⎪⎪ ⎪ ⎪ ⎩ (1 + )| − | ⎭ = || + ||, (1 − )|| + (1 + )||, ⎪ ⎪ ⎨ ⎨ ⎬⎬ ⎪ | − | + (2 − 3) ⎪ −|| + || + (1 + )|| ⎪ ⎪ ⎪⎪ ⎪ | − | + | − |, ⎪ || + (2 − 3) ⎧ ⎫ ⎪ ⎪ ⎪⎪ = (1 − )| − | + (1 + )| − |, ⎪ ⎪ || + ||, (1 − )|| + (1 + )||, ⎪⎪ ⎨ ,⎬ ⎨ −| − | + | − | + ⎬ ⎪ ⎩ −|| + || + (1 + )|| ⎭⎪ ⎪ ⎪ ⎩ ⎭ ⎪ ⎩ (1 + )| − | ⎭ ⎪ = 1 ⎪ | − | + (2 − 3) ⎪ ⎪ | − | + | − |, ⎪ is the set of all points X=(x,y,z)∈ℝ³ deltoidal ⎪ ⎧ ⎫ ⎪ (1 − )| − | + (1 + )| − |, hexacontahedron distance is 1 from O=(0,0,0). Thus the ⎪ ⎪ ⎪ ⎨ −| − | + | − | + ⎬ ⎪ graph of SDH is as in the figure ⎩ ⎩ (1 + )| − | ⎭ ⎭

| − + − | + (2 − 3) ⎧ ⎫ | − + − | + | − + − |, ⎪ ⎧ ⎫ ⎪ ⎪ ⎪ (1 − )| − + − | + ⎪ ⎪ ⎪ (1 + )| − + − |, ⎪ ⎨ ⎬ ⎪ ⎪−| − + − | + | − + − |⎪ ⎪ ⎪ ⎩ +(1 + )| − + − | ⎭ ⎪ ⎪ ⎪ | − + − | + (2 − 3) ⎪ | − + − | + | − + − |, ⎪ ⎪ ⎧ ⎫⎪ (1 − )| − + − | + = ⎪ ⎪ Figure 3. Deltoidal hexacontahedron ⎨ (1 + )| − + − |, ⎬ ⎨ ⎬ ⎪ ⎪−| − + − | + | − + − |⎪⎪ ⎪ ⎩ +(1 + )| − + − | ⎭⎪ Corollary 2. 4: The equation of the deltoidal ⎪ ⎪ | − + − | + (2 − 3) hexacontahedron with center C=(x₀,y₀,z₀) and radius r ⎪ | − + − | + | − + − |, ⎪ ⎪ ⎧ ⎫⎪ is ( )| | ⎪ ⎪ 1 − − + − + ⎪⎪ ⎪ (1 + )| − + − |, ⎪ ⎨ ⎬ | − | + (2 − 3) ⎪ ⎪−| − + − | + | − + − |⎪⎪ ⎧ ⎫ | − | + | − |, ⎩ ⎩ +(1 + )| − + − | ⎭⎭ ⎪ ⎧ ⎫⎪ ⎪ ⎪ (1 − )| − | + ⎪⎪ | − | + | − | + (2 − 3) ⎧ ⎫ ⎪ (1 + )| − |, ⎪ | − | + | − | + | − | + | − |, ⎪ ⎪ ⎨ ⎬ ⎧ ⎫ ⎪ −| − | + | − | ⎪ ⎪ ⎪ (1 − )(| − | + | − |) + ⎪ ⎪ ⎪ ⎪ ⎪ (1 + )(| − | + | − |), ⎪ ⎪ ⎩ +(1 + )| − | ⎭⎪ ⎨ ⎬ ⎪ ⎪−(| − | + | − |) + (| − | + | − |)⎪ ⎪ ⎪ ⎪ | − | + (2 − 3) ⎪ ⎩ +(1 + )(| − | + | − |) ⎭ ⎪ ⎪ ⎪ ⎪ ⎪ | − | + | − | + (2 − 3) | − | + | − |, ⎪ ⎪ ⎪ ⎧ ⎫⎪ | − | + | − | + | − | + | − |, ⎪ ⎧ ⎫⎪ ⎪ (1 − )| − | + ⎪ ⎪ (1 − )(| − | + | − |) + ⎪ = ≤ ( )| | ⎨ (1 + )(| − | + | − |), ⎬ ⎨ 1 + − , ⎬ ⎪ ⎨−(| − | + | − |) + (| − | + | − |)⎬⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪−| − | + | − |⎪⎪ ⎪ ⎩ +(1 + )(| − | + | − |) ⎭⎪ ⎪ ⎪ ⎪ ⎩ +(1 + )| − | ⎭⎪ | − | + | − | + (2 − 3) ⎪ ⎪ | − | + | − | + | − | + | − |, ⎪ | | ( ) ⎪ ⎪ ⎧ ⎫⎪ − + 2 − 3 ⎪ (1 − )(| − | + | − |) + ⎪ ⎪ ⎪ ⎪ ⎪ | − | + | − |, ⎪ (1 + )(| − | + | − |), ⎪ ⎪ ⎧ ⎫⎪ ⎨−(| − | + | − |) + (| − | + | − |)⎬ (1 − )| − | + ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎩ ⎩ +(1 + )(| − | + | − |) ⎭⎭ (1 + )| − |, = ⎪ ⎪ ⎨−| − | + | − |⎬ ⎪ ⎪ ⎪⎪ ⎩ ⎩ +(1 + )| − | ⎭⎭ One can easily find that I ≤ dDH(P₁,P₂) + dDH(P₂,P₃) from Lemma 2. 1. Lemma 2. 5: Let l be the line through the points P1 = (x1, y1, z1) and P2 = (x2, y2, z2) in theanalytical 3- So dDH(P₁,P₃)≤dDH(P₁,P₂)+dDH(P₂,P₃). That is, dDH distance function satisfies the triangle inequality. dimensional space and dE denotes the Euclidean metric. Consequently, the set If l has direction vector(p, q, r), then dDH(P1,P2) = µ(P1P2)dE(P1,P2), where µ(P1P2) is equal to

356 SAÜ Fen Bil Der 19. Cilt, 3. Sayı, s. 353-360, 2015

New metrics for deltoidal hexacontahedron and pentakis Z. Çolak, Ö. Gelişgen dodecahedron

|| + (2 − 3) dodecahedron. A pentakis dodecahedron has 60 faces, ⎧ ⎫ || + ||, (1 − )|| + (1 + )||, 90 edges and 62 = 12 + 20 + 32 vertices([9]). ⎪ ⎪ ⎪ −|| + || + (1 + )|| ⎪ ⎪ || + (2 − 3) ⎪ || + ||, (1 − )|| + (1 + )||, ⎨ −|| + || + (1 + )|| ⎬ ⎪ ⎪ ⎪ || + (2 − 3) ⎪ ⎪ || + ||, (1 − )|| + (1 + )||, ⎪ ⎩ −|| + || + (1 + )|| ⎭

+ +

Figure 4. Pentakis dodecahedron Proof: Equation of l gives us x1 - x2 =λp, y1 - y2 = λq, z1 - z2 = λr , λℝ. Thus,

(, ) = || + (2 − 3) ⎧ ⎫ || + ||, (1 − )|| + (1 + )||, ⎪ ⎪ ⎪ −|| + || + (1 + )|| ⎪ ⎪ || + (2 − 3) ⎪ || + ||, (1 − )|| + (1 + )||, ⎨ −|| + || + (1 + )|| ⎬ Figure 5. Net for pentakis dodecahedron ⎪ ⎪ || + (2 − 3) ⎪ ⎪ We describe the metric that unit sphere is Pentakis ⎪ || + ||, (1 − )|| + (1 + )||, ⎪ Dodecahedron metric as following: ⎩ −|| + || + (1 + )|| ⎭

Definition 3. 1:Let P₁=(x₁,y₁z₁) and P₂=(x₂,y₂,z₂) be and (, ) = + + which implies the distinct two points in ℝ³ . The distance function required result. dPD: ℝ³x ℝ³→[0,∞)Pentakis Dodecahedron metricdistance between P1 and P2 is defined by The above lemma says that dDH- distance along any line is some positive constant multiple of Euclidean distance (, ) = along same line. Thus, one can immediately state the following corollaries: | − | + ⎧ ⎫ ⎪ | − |, 2| − | + | − | + ⎪ Corollary 2. 6: If P1, P2 and X are any three collinear ⎧ ⎫ 3 ⎪ (| − | − | − |), ⎪ points in ℝ , then ⎪ ⎪ ⎨ | − | + 2| − | + ⎬ ⎪ ⎪ ⎩ 2(| − | − | − |) ⎭ dE(P1,X) = dE(P2,X) if and only if dDH(P1,X) = dDH(P2,X) ⎪ ⎪ | − | + ⎪ ⎪ ⎪ ⎪ Corollary 2. 7: If P , P and X are any three distinct | − |, 2| − | + | − | 1 2 ⎧ ⎫ collinear points in the real 3-dimensionalspace, then +(| − | − | − |), (3) ⎨ ⎬ | | | | ⎪ ⎨ − + 2 − + ⎬ ⎪ dDH(X, P1) / dDH(X, P2) = dE(X, P1) / dE(X, P2) . ⎪ ⎩ 2(| − | − | − |) ⎭ ⎪ ⎪ | − | + ⎪ That is, the ratios of the Euclidean and dDH distances ⎪ ⎪ | − |, 2| − | + | − | along a line are the same. ⎪ ⎧ ⎫ ⎪ ⎪ +(| − | − | − |), ⎪ ⎪ ⎨ | − | + 2| − | + ⎬ ⎪ 3. PENTAKIS DODECAHEDRON ⎩ ⎩ 2(| − | − | − |) ⎭ ⎭ where = √5 − 1⁄2. A pentakis dodecahedron is a Catalan solid. Its dual is the truncated icosahedron, an Archimedean solid. It can Pentakis dodecahedron distance function may seem a bit be seen as a dodecahedron with a pentagonal pyramid complicated. In fact there is an orientation in d just as covering each face; that is, it is the Kleetope of the PD in Deltoidal hexacontahedron metric. Let =

SAÜ Fen Bil Der 19. Cilt, 3. Sayı, s. 353-360, 2015 357

Z. Çolak, Ö. Gelişgen New metrics for deltoidal hexacontahedron and pentakis dodecahedron

| − |, = | − |, = | − |. This orientation dPD(P₁,P₂)=0⇔|x₁-x₂|=0,|y₁-y₂|=0, |z₁-z₂|=0⇔x₁=x₂, is a − b − c − a. According to orientation, if one can y₁=y₂, z₁=z₂⇔P₁=P₂. put b, c, a instead of a, b, c, respectively, in first term of distance function, then it is obtained second term. The other cases can be easily shown by similar way in Similarly, if one can put c,a, b instead of a, b, c, case i. Thus we get dPD(P₁,P₂)=0 iff P₁=P₂. respectively, in first term of distance function, then it is obtained third term. M2) By the definition of absolute value

Lemma 3. 2: Let P₁=(x₁,y₁z₁) and P₂=(x₂,y₂,z₂) be any |xi-xj|=|xj-xi|, distinct two points in ℝ³. Then |yi-yj|=|yj-yi| , |zi-zj|=|zj-zi| (, ) ≥ | − | + 3 3 for all xi, yi, zi,xj, yj, zj∈ℝ and i,j=1,2,3. Therefore one | − |, can get dPD(P₁,P₂) =dPD(P₂,P₁). 2| − | + | − | + (| − | − | − |), | − | + 2| − | + 2(| − | − | − |) M3) Let P₁=(x₁,y₁,z₁) , P₂=(x₂,y₂,z₂) and P₃=(x₃,y₃,z₃) ( , ) ≥ | − | + be any distinct three points in ℝ³. Then by using well 3 known property | + | ≤ || + || for all , ∈ ℝ,we | − |, have 2| − | + | − | + (| − | − | − |),

| − | + 2| − | + 2(| − | − | − |) (, ) = ⎧ | − | + ⎫ (, ) ≥ | − | + 3 3 ⎪ | − |, 2| − | + | − | ⎪ ⎪ ⎧ ⎫ ⎪ | − |, 2| − | + | − | +(| − | − | − |), ⎧ ⎫ ⎪ ,⎪ +(| − | − | − |), ⎪ ⎨ | − | + 2| − | ⎬ ⎪ ⎪ ⎪ | − | + 2| − | + ⎩ +2(| − | − | − |) ⎭ ⎨ ⎬ ⎪ ⎪ ⎩ (| | | |) ⎭ | − | + 2 − − − ⎪ 3 ⎪ ⎪ ⎪ where = √5 − 1⁄2. | − |, 2| − | + | − | ⎧ ⎫ +(| − | − | − |), Proof: Proof is trivial by definition of maximum ⎨ ,⎬ function. ⎪ ⎨ | − | + 2| − | ⎬ ⎪ ⎪ ⎩ +2(| − | − | − |) ⎭ ⎪ ⎪ ⎪ | − | + Theorem 3. 3: The distance function dPD is a metric of ⎪ 3 ⎪ which unit sphere is a deltoidal hexacontahedron in ℝ³. ⎪ | − |, 2| − | + | − | ⎪ ⎪ ⎧ ⎫ ⎪ +(| − | − | − |), Proof: Let dPD: ℝ³x ℝ³→ℝ and P₁=(x₁,y₁,z₁) , ⎪ ⎪ P₂=(x₂,y₂,z₂) and P₃=(x₃,y₃,z₃) distinct three points in ⎪ ⎨ | − | + 2| − | ⎬ ⎪ ⎩ ⎩ +2(| − | − | − |) ⎭ ⎭ ℝ³. To prove that dPD is a metric in ℝ³, the following axioms can be supplied for all P₁, P₂ and P₃∈ℝ³. ⎧ | − | + | − | + ⎫ 3 ⎪ | − | + | − |, ⎪ M1) d(P₁,P₂)≥0 ve d(P₁,P₂)=0⇔P₁=P₂ ⎪ ⎧ ⎫⎪ ⎪ ⎪ 2(| − | + | − |) + | − | ⎪⎪ M2) d(P₁,P₂)=d(P₂,P₁) ⎪ +(| − | + | − | − | − | − | − |), ⎪ ⎨ ⎬ ⎪ | − | + | − | + 2(| − | + | − |) + ⎪ M3) d(P₁,P₃)≤d(P₁,P₂)+d(P₂,P₃). ⎪ ⎪ ⎪⎪ ⎩ 2(| − | + | − | − | − | − | − |) ⎭ ⎪ ⎪ ⎪ | − | + | − | + ⎪ M1) Since absolute values is always nonnegative ⎪ 3 ⎪ ⎪ | − | + | − |, ⎪ maximum of sums of absolute value is always ⎧ ⎫ ≤ ⎪2(| − | + | − |) + | − | + | − | +⎪ ⎨ ⎬ nonnegative. Thus dPD(P₁,P₂)≥0. If dPD(P₁,P₂)=0 then (| − | − | − | − | − |), ⎪ ⎨| − | + | − | + 2(| − | + | − |) +⎬ ⎪ three cases are possible. ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2(| − | − | − | − | − |) ⎭ ⎪ ⎪ | − | + | − | + ⎪ Case I : If ⎪ 3 ⎪ ⎪ | − | + | − |, ⎪ d (P , P ) = | − | + ⎧ ⎫ 3 ⎪ 2(| − | + | − |) + | − | + | − | + ⎪ ⎪ ⎪ ⎪⎪ | − |, ⎪ (| − | + | − | − | − | − | − |), ⎪ ⎨ ( ) ⎬ 2| − | + | − | + (| − | − | − |), ⎪ ⎪| − | + | − | + 2 | − | + | − | +⎪⎪ ⎩ ⎩ 2(| − | + | − | − | − | − | − |) ⎭⎭ | − | + 2| − | + 2(| − | − | − |) then, = One can easily find that I ≤ dPD(P₁,P₂) + dPD(P₂,P₃) from Lemma 3. 2.

358 SAÜ Fen Bil Der 19. Cilt, 3. Sayı, s. 353-360, 2015

New metrics for deltoidal hexacontahedron and pentakis Z. Çolak, Ö. Gelişgen dodecahedron

⎧ | − | + ⎫ So dPD(P₁,P₃)≤dPD(P₁,P₂)+dPD(P₂,P₃). That is, dDH 3 ⎪ | − |, 2| − | + | − | ⎪ distance function satisfies the triangle inequality. ⎪ ⎧ ⎫ ⎪ +(| − | − | − |), ⎪ ,⎪ Consequently, the set ⎪ ⎨ | − | + 2| − | ⎬ ⎪ ⎪ ⎪ (, , ): (, ) = ⎩ +2(| − | − | − |) ⎭ ⎧ ⎫ ⎪ ⎪ | − | + ⎪⎧ || + ⎫⎪ ⎪ 3 ⎪ 3 ⎪ ⎪ ⎪⎪ ⎪⎪ | − |, 2| − | + | − | ⎪ ||, 2|| + || + (|| − ||), ⎪ ⎪ ⎪ ⎧ ⎫ = ⎪⎪ || + 2|| + 2(|| − ||) ⎪⎪ ⎨ +(| − | − | − |), ⎬ ⎪ ⎪ , ⎪ ⎪ ⎪ ⎨ | − | + 2| − | ⎬ ⎪ ⎪⎪ || + ⎪⎪ 3 ⎪ ⎩ +2(| − | − | − |) ⎭ ⎪ = ⎪ ⎪ ⎨ ||, 2|| + || + (|| − ||), ⎬ ⎨ ⎬ ⎪ | − | + ⎪ | | | | (| | | |) 3 ⎪⎪ + 2 + 2 − ⎪⎪ ⎪ ⎪ ⎪ ⎪ | − |, 2| − | + | − | ⎪ ⎪ ⎪ ⎧ ⎫⎪ ⎪⎪ || + ⎪⎪ +(| − | − | − |), ⎪⎪ 3 ⎪⎪ ⎪ ⎪ ⎪ ||, 2|| + || + (|| − ||), ⎪ ⎪ | − | + 2| − | ⎪ ⎪ ⎪ ⎨ ⎬ ⎪⎩ || + 2|| + 2(|| − ||) ⎭⎪ ⎩ ⎩ +2(| − | − | − |) ⎭⎭ ⎩ = 1 ⎭ is the set of all points X=(x,y,z)∈ℝ³ that pentakis Lemma 3. 5: Let l be the line through the points P1 = dodecahedron distance is 1 from O=(0,0,0). Thus the (x1, y1, z1) and P2 = (x2, y2, z2) in the graph of SPD is as in the figure 6: analytical 3-dimensional space and dE denotes the Euclidean metric. If l has direction vector

(p, q, r), then dPD(P1,P2) = µ(P1P2)dE(P1,P2), where µ(P1P2) is equal to

|| + ⎧ ⎫ ⎪ ||, 2|| + || + (|| − ||), ⎪ , ⎪ || + 2|| + 2(|| − ||) ⎪ ⎪ ⎪ ⎪ || + ⎪ ||, 2|| + || + (|| − ||), ⎨ ,⎬ || + 2|| + 2(|| − ||) ⎪ ⎪ ⎪ || + ⎪ Figure 6. Pentakis dodecahedron ⎪ ⎪ ⎪ ||, 2|| + || + (|| − ||), ⎪ Corollary 3. 4: The equation of the pentakis ⎩ || + 2|| + 2(|| − ||) ⎭ dodecahedron with center C=(x₀,y₀,z₀) and radius r is + +

Proof: Equation of l gives us x1 - x2 =λp, y1 - y2 = λq, z1 - z2 = λr , λℝ. Thus,

(, ) = || + ⎧ 3 ⎫ ⎪ ||, 2|| + || + (|| − ||), ⎪ ⎪ ,⎪ ⎪ || + 2|| + 2(|| − ||) ⎪ ⎪ ⎪ ⎪ || + ⎪ 3 ||, 2|| + || + (|| − ||), ⎨ ,⎬ ⎪ || + 2|| + 2(|| − ||) ⎪ ⎪ ⎪ ⎪ || + ⎪ ⎪ 3 ⎪ ⎪ ||, 2|| + || + (|| − ||), ⎪ ⎩ || + 2|| + 2(|| − ||) ⎭

SAÜ Fen Bil Der 19. Cilt, 3. Sayı, s. 353-360, 2015 359

Z. Çolak, Ö. Gelişgen New metrics for deltoidal hexacontahedron and pentakis dodecahedron

REFERENCES and (, ) = + + which implies the required result. [1] O. Gelişgen, R. Kaya and M. Ozcan, “Distance The above lemma says that dPD-distance along any line Formulae in The Chinese Checker Space”, Int. is some positive constant multiple of Euclidean distance J. Pure Appl. Math, 26(1),35-44, 2006. along same line. Thus, one can immediately state the [2] M. Atiyah, P. Sutcliffe, “Polyhedra in Physics, following corollaries: Chemistry and Geometry”, Milan Journal of Mathematics, 71(33-58), 2003. Corollary 3. 6: If P1, P2 and X are any three collinear points in ℝ3, then [3] T. Ermiş and R. Kaya, “On the Isometries the of 3- Dimensional Maximum Space”, Konuralp dE(P1,X) = dE(P2,X) if and only if dPD(P1,X) = dPD (P2,X) Journal of Mathematics, 3(1), 2015. Corollary 3. 7: If P1, P2 and X are any three distinct [4] O. Gelişgen and R. Kaya, “The Taxicab Space collinear points in the real 3-dimensional space, then Group”, Acta Mathematica Hungarica, DOI:10.1007/s10474-008-8006-9, 122(1-2), 187- dPD(X, P1) / dPD(X, P2) = dE(X, P1) / dE(X, P2) . 200, 2009. [5] T. Ermiş, “Düzgün Çokyüzlülerin Metrik That is, the ratios of the Euclidean and dPD distances Geometriler ile İlişkileri Üzerine”, Doktora Tezi, along a line are the same. Eskişehir Osmangazi Üniversitesi, Fen Bilimleri Enstitüsü 2014. ACKNOWLEDGMENTS [6] M. Koca, N. Koca and R. Koç, “Catalan solids This work was supported by the Scientific Research derived from three- dimensional-root systems Projects Commission of Eskişehir Osmangazi and quarternions”, Journal of Mathematical University under Project Number 201419A217 Physics, 51 (2010), 043501. [7] A. C. Thompson, Minkowski Geometry, Cambridge University Press, Cambridge, 1996. [8] https://en.wikipedia.org/wiki/Deltoidal_hexecont ahedron.

[9] https://en.wikipedia.org/wiki/Pentakis_dodecahe

dron.

360 SAÜ Fen Bil Der 19. Cilt, 3. Sayı, s. 353-360, 2015