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 √ 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) = max{|x1 − x2|,|y1 − y2|,|z1 − z2|} and d (P ,P ) = max{|x − x |,|y − y |,|z − z |}. M 1 2 1 2 1 2 1 2 dS(P1,P2) = min{|x1 − x2|+|y1 − y2|,|x1 − x2|+|z1 − z2|, |y − y | + |z − z |}. 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) = |x1 − x2| + |y1 − y2| + |z1 − z2|. 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 polyhedron, 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- sum√ 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 way√ from P1 to supply plane equation related with solid’s surface. There- 5−1 P2 if maximum value is |x1 − x2| + ( )|y1 − y2|, fore we describe the metric which unit sphere is an icosi- √ √ 2 |x − x | + ( 5−1 )2 |z − z | ( 5−1 )(|x − x | + dodecahedron as following: 1 2 2 1 2 or 2 1 2 |y1 − y2| + |z1 − z2|). 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 |x1 − x2|, |y1 − y2|, hedron distance between P and P is defined by 1 2 |z1 − z2|, respectively. Then dID(P1,P2) = dID(P1,P2) ≥ u + (ϕ−1)max{v,(ϕ−1)w,(1−ϕ)u + v + w},  u + (ϕ − 1)max{v,(ϕ − 1)w,(1 − ϕ)u + v + w},    dID(P1,P2) ≥ v + (ϕ−1)max{w,(ϕ−1)u,u + (1−ϕ)v + w}, max v + (ϕ − 1)max{w,(ϕ − 1)u,u + (1 − ϕ)v + w}, dID(P1,P2) ≥ w + (ϕ−1)max{u,(ϕ−1)v,u + v + (1−ϕ)w}.  w + (ϕ − 1)max{u,(ϕ − 1)v,u + v + (1 − ϕ)w}  Proof. Proof is trivial by the definition of maximum func- where√ u = |x1 − x2|, v = |y1 − y2|, w = |z1 − z2| and ϕ = 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 to√ 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. R . u, v, w denote |x1 − x2|, |y1 − y2|, |z1 − z2|, respec- 3 iii) union of three line segments each of which is parallel tively. To show that dID is a metric in R , the following 3 to a coordinate axis. axioms hold true for all P1, P2 and P3 ∈ R .

Figure 2: ID way from P1 to P2

19 KoG•19–2015 Z. Can, O.¨ Gelis¸gen, R. Kaya: On the Metrics Induced by Icosidodecahedron ....

M1) dID(P1,P2) ≥ 0 and dID(P1,P2) = 0 iff P1 = P2 SID = M2) dID(P1,P2) = dID(P2,P1)       |y|,(ϕ-1)|z|,   M3) dID(P1,P3) ≤ dID(P1,P2) + dID(P2,P3).   |x|+(ϕ-1)max ,    (1-ϕ)|x|+|y|+|z|   Since absolute values is always nonnegative value   |z|,(ϕ-1)|x|,   dID(P1,P2) ≥ 0. If dID(P1,P2) = 0 then there are possi- (x,y,z):max |y|+(ϕ-1)max , =1 .   |x|+(1-ϕ)|y|+|z|   ble three cases. These cases are   |x|,(ϕ-1)|y|,     |z|+(ϕ-1)max   Case I:   |x|+|y|+(1-ϕ)|z|   dID(P1,P2) = u+(ϕ−1)max{v,(ϕ−1)w,(1−ϕ)u+v+w} Case II: Thus the graph of SID is as in Figure 3. dID(P1,P2) = v+(ϕ−1)max{w,(ϕ−1)u,u+(1−ϕ)v+w} Case III: dID(P1,P2) = w+(ϕ−1)max{u,(ϕ−1)v,u+v+(1−ϕ)w}.

Case I: If dID(P1,P2) = u + (ϕ − 1)max{v,(ϕ − 1)w,(1 − ϕ)u + v + w}, then

u+(ϕ − 1)max{v,(ϕ − 1)w,(1 − ϕ)u + v + w}=0 ⇔ u = 0 and (ϕ −1)max{v,(ϕ −1)w,(1− ϕ)u + v + w} = 0 ⇔ x1 = x2, y1 = y2, z1 = z2 ⇔ (x1,y1,z1) = (x2,y2,z2) ⇔ P1 = P2

The other cases can be shown by similar way in Case I. Thus we get dID(P1,P2) = 0 iff P1 = P2. Figure 3: Icosidodecahedron Since |x − x | = |x − x |, |y − y | = |y − y | and 1 2 2 1 1 2 2 1 Corollary 1 The equation of the icosidodecahedron with |z1 − z2| = |z2 − z1|, obviously dID(P1,P2) = dID(P2,P1). center (x0,y0,z0) and radius r is That is, dID is symmetric.   |y − y |,(ϕ − 1)|z − z |,   Let P1 = (x1,y1,z1), P2 = (x2,y2,z2) and P3 = (x3,y3,z3)   0 0   3  |x − x | + (ϕ − 1)max (1 − ϕ)|x − x | ,  are distinct three points in R . u1, v1, w1, u2, v2, w2 denote  0 0    +|y − y | + |z − z |   |x − x |, |y − y |, |z − z |, |x − x |, |y − y |, |z − z |,  0 0  1 3 1 3 1 3 2 3 2 3 2 3   |z − z |,(ϕ − 1)|x − x |,   respectively.   0 0   max |y − y | + (ϕ − 1)max (1 − ϕ)|y − y | , Then by using the property |a − b + b − c| ≤ |a − b| + 0 0   +|x − x0| + |z − z0|   |b − c| we get      |x − x |,(ϕ − 1)|y − y |,    0 0    |z − z0| + (ϕ − 1)max (1 − ϕ)|z − z0|  dID(P1,P3)       +|x − x0| + |y − y0|    u1 + (ϕ − 1)max{v1, (ϕ − 1)w1,(1 − ϕ)u1 + v1 + w1}, =max v1 + (ϕ − 1)max{w1, (ϕ − 1)u1,u1 + (1 − ϕ)v1 + w1}, = r  w1 + (ϕ − 1)max{u1, (ϕ − 1)v1,u1 + v1 + (1 − ϕ)w1}  which is a polyhedron which has 32 faces with vertices;   v + v , (ϕ − 1)(w + w ),    1 2 1 2   such that all permutations of the three axis components  u + u + (ϕ − 1)max (1 − ϕ)(u + u ) ,  1 2 1 2  and all posible +/- sign changes of each axis√ component   +v + v + w + w   ϕ−1 ϕ  1 2 1 2  of (0,0,r), and ( r, 1 r, r), where ϕ = 1+ 5 the golden   w + w , (ϕ − 1)(u + u ),  2 2 2 2   1 2 1 2   ratio. ≤ max v1 + v2 + (ϕ − 1)max u1 + u2 + (1 − ϕ)(v1 + v2) ,   +w + w    1 2  Lemma 2 Let l be the line through the points P1 =       u1 + u2, (ϕ − 1)(v1 + v2),  (x ,y ,z ) and P = (x ,y ,z ) in the analytical 3-   1 1 1 2 2 2 2  w1 + w2 + (ϕ − 1)max u1 + u2 + v1 + v2  dimensional space and d denote the Euclidean metric. If     E +(1 − ϕ)(w1 + w2) l has direction vector(p,q,r), then = I. dID(P1,P2) = µ(P1P2)dE (P1,P2) Therefore one can easily ﬁnd that I ≤ dID(P1,P2) + where µ(P1P2) = dID(P2,P3) from Lemma 1. So, dID(P1,P3) ≤ dID(P1,P2)+ |p| + (ϕ − 1)max{|q|,(ϕ − 1)|r|,(1 − ϕ)|p| + |q| + |r|}, dID(P2,P3). Consequently, icosidodecahedron distance is a   metric in 3-dimensional analytical space. max |q| + (ϕ − 1)max{|r|,(ϕ − 1)|p|,|p| + (1 − ϕ)|q| + |r|},   Finally, the set of all points X = (x,y,z) ∈ 3 that icosido- |r| + (ϕ − 1)max{|p|,(ϕ − 1)|q|,|p| + |q| + (1 − ϕ)|r|} R p . decahedron distance is 1 from O = (0,0,0) is p2 + q2 + r2

20 KoG•19–2015 Z. Can, O.¨ Gelis¸gen, R. Kaya: On the Metrics Induced by Icosidodecahedron ....

Proof. Equation of l gives us x1 − x2 = λp, y1 − y2 = λq, 32 vertices and 60 edges. Its faces are rhombuses. The ra- z − z = λr, r ∈ . Thus, d (P ,P ) is equal to 1 2 R ID 1 2 tio of the long diagonal to√ the short diagonal of each face is 1+ 5   |q|,(ϕ − 1)|r|,  exactly equal to ϕ = 2 , which is the golden ratio [15].  |p| + (ϕ − 1)max ,    (1 − ϕ)|p| + |q| + |r|     Let P = (x ,y ,z ) and P = (x ,y ,z ) be   |r|,(ϕ − 1)|p|,  Deﬁnition 2 1 1 1 1 2 2 2 2 |λ|max |q| + (ϕ − 1)max ,  3 3 3  |p| + (1 − ϕ)|q| + |r|  two points in R . The distance function dRT : R × R →      |p|,(ϕ − 1)|q|,  [0,∞) Rhombic triacontahedron distance between P1 and  |r| + (ϕ − 1)max   |p| + |q| + (1 − ϕ)|r|  P2 is deﬁned by dRT (P1,P2) = p   and d (A,B) = |λ| p2 + q2 + r2 which implies the re- (ϕ + 1)|z1 − z2| + ϕ|y1 − y2|, E |x1 − x2| + (2ϕ − 3)max ,  |x1 − x2|  quired result.   ϕ  (ϕ + 1)|x1 − x2| + ϕ|z1 − z2|,  max |y1 − y2| + (2ϕ − 3)max , The lemma above says that dID−distance along any line 2 |y1 − y2|   is some positive constant multiple of Euclidean distance  (ϕ + 1)|y1 − y2| + ϕ|x1 − x2|,  |z1 − z2| + (2ϕ − 3)max  along same line. Thus, one can immediately state the fol-  |z1 − z2|  lowing corollaries. √ 1+ 5 where ϕ = 2 , the golden ratio. Corollary 2 If P1, P2 and X are any three collinear 3 points in R , then dE (P1,X) = dE (P2,X) if and only if According to the rhombic triacontahedron distance, there dID(P1,X) = dID(P2,X). are two types path from P1 to P2. These paths are:

Corollary 3 If P1,P2 and X are any three distinct i) union of three line segments which one is parallel to collinear points in the real 3-dimensional space, then a coordinate axis and other√ line segments are made arctan( 1 ) and arctan( 5 ) angle with other coordi- dID(X,P1) / dID(X,P2) = dE (X,P1) / dE (X,P2) . 2 2 nate axes. That is, the ratios of the Euclidean and dID−distances along a line are the same. ii) a line segment which is parallel to a coordinate axis.

Thus rhombic triacontahedron distance between P and 3 Rhombic Triacontahedron Metric 1 P2 is the Euclidean length of√ line segment which is par- 5+1 The duals of thirteen Archimedean solids are known as allel to a coordinate axis or 4 times the sum of Eu- Catalan solids. Unlike Platonic and Archimedean solids, clidean lengths of three line segments. Figure 5 shows faces of Catalan solids are not regular polygons. Rhombic that the path between√ P1 and √P2 in case of the maximum 3− 5 5−1 triacontahedron is one of the Catalan solids with 30 faces, is |y1 − y2| + 2 |z1 − z2| + 2 |x1 − x2|.

(a) (b)

Figure 4: (a) Rhombic triacontahedron, (b) Net of rhombic triacontahedron

21 KoG•19–2015 Z. Can, O.¨ Gelis¸gen, R. Kaya: On the Metrics Induced by Icosidodecahedron ....

Corollary 4 The equation of the rhombic triacontahedron with center (x0,y0,z0) and radius r is   (ϕ + 1)|z − z0| + ϕ|y − y0|, |x − x0| + (2ϕ − 3)max ,  |x − x0|    ϕ  (ϕ + 1)|x − x0| + ϕ|z − z0|,  max |y − y0| + (2ϕ − 3)max , 2 |y − y0|    (ϕ + 1)|y − y0| + ϕ|x − x0|,  |z − z0| + (2ϕ − 3)max   |z − z0|  =r. which is a polyhedron which has 30 faces with vertices; such that all permutations of the three axis components Figure 5: RT way from P1 to P2 in the case |y1 − y2| ≥ and all posible +/- sign changes of each axis component |x − x | ≥ |z − z | √ 1 2 1 2 of (µr,0,r), (0,δr,r) and (µr,µr,µr), where µ = 5−1 and √ 2 3− 5 δ = 2 . Lemma 3 Let P1 = (x1,y1,z1) and P2 = (x2,y2,z2) be distinct two points in R3. Then

dRT (P1,P2) ≥ ϕ 2 (|x1 − x2|+(2ϕ − 3)max{(ϕ+1)|z1 − z2|+ϕ|y1 − y2|,|x1 − x2|}) dRT (P1,P2) ≥ ϕ 2 (|y1 − y2|+(2ϕ − 3)max{(ϕ+1)|x1 − x2|+ϕ|z1 − z2|,|y1 − y2|}) dRT (P1,P2) ≥ ϕ 2 (|z1 − z2|+(2ϕ − 3)max{(ϕ+1)|y1 − y2|+ϕ|x1 − x2|,|z1 − z2|}). √ 1+ 5 where ϕ = 2 .

Proof. Proof is trivial by the deﬁnition of maximum function.

Theorem 2 The distance function dRT is a metric. Also according to dRT , unit sphere is a rhombic triacontahedron Figure 6: Rhombic Triacontahedron in R3. Lemma 4 Let l be the line through the points P1 = 3 3 Proof. Let dRT : R × R → [0,∞) be the rhombic tria- (x1,y1,z1) and P2 = (x2,y2,z2) in the analytical 3- contahedron distance function and P1 = (x1,y1,z1), P2 = dimensional space and dE denote the Euclidean metric. If (x2,y2,z2) and P3 = (x3,y3,z3) are distinct three points in l has direction vector (p,q,r), then 3 3 R . To show that dRT is a metric in R , the following ax- 3 dRT (P1,P2) = µ(P1P2)dE (P1,P2) ioms hold true for all P1, P2 and P3 ∈ R . M1) dRT (P1,P2) ≥ 0 and dRT (P1,P2) = 0 iff P1 = P2 where M2) dRT (P1,P2) = dRT (P2,P1)  |p| + (2ϕ − 3)max{(ϕ + 1)|r| + ϕ|q|,|p|}, ϕ   M3) dRT (P1,P3) ≤ dRT (P1,P2) + dRT (P2,P3). 2 max |q| + (2ϕ − 3)max{(ϕ + 1)|p| + ϕ|r|,|q|}, One can easily show that the rhombic triacontahedron dis-  |r| + (2ϕ − 3)max{(ϕ + 1)|q| + ϕ|p|,|r|}  µ(P1P2) = . tance function satisﬁes above axioms by similar way in pp2 + q2 + r2 Theorem 1. 3 Proof. Equation of l gives us x − x = λp, y − y = λq, Consequently, the set of all points X = (x,y,z) ∈ R that 1 2 1 2 rhombic triacontahedron distance is 1 from O = (0,0,0) is z1 − z2 = λr, r ∈ R. Thus, SRT = dRT (P1,P2) =   |p| + (2ϕ − 3)max{(ϕ + 1)|r| + ϕ|q|,|p|},  |x| + (2ϕ−3)max{(ϕ + 1)|z| + ϕ|y|,|x|},  ϕ    ϕ    |λ| max |q| + (2ϕ − 3)max{(ϕ + 1)|p| + ϕ|r|,|q|},  (x,y,z): max |y| + (2ϕ−3)max{(ϕ + 1)|x| + ϕ|z|,|y|}, =1 . 2 2  |r| + (2ϕ − 3)max{(ϕ + 1)|q| + ϕ|p|,|r|}   |z| + (2ϕ−3)max{(ϕ + 1)|y| + ϕ|x|,|z|}   p 2 2 2 and dE (A,B) = |λ| p + q + r which implies the re- Thus the graph of SRT is as in Figure 6. quired result.

22 KoG•19–2015 Z. Can, O.¨ Gelis¸gen, R. Kaya: On the Metrics Induced by Icosidodecahedron ....

The previous lemma says that dRT −distance along any line [9]R.K AYA,O.GELISGEN,S.EKMEKCI,A.BA- is some positive constant multiple of Euclidean distance YAR, On The Group of Isometries of The Plane with along same line. Thus, one can immediately state the fol- Generalized Absolute Value Metric, Rocky Mountain lowing corollaries: Journal of Mathematics 39(2) (2009), 591–603. Corollary 5 If P , P and X are any three collinear 1 2 [10]M.K OCA,N.KOCA,R.KOC¸, Catalan solids points in 3, then d (P ,X) = d (P ,X) if and only if R E 1 E 2 derived from three-dimensional-root systems and d (P ,X) = d (P ,X). RT 1 RT 2 quaternions, Journal of Mathematical Physics 51

Corollary 6 If P1,P2 and X are any three distinct (2010), 043501. collinear points in the real 3-dimensional space, then [11] E.F.KRAUSE, Taxicab Geometry, Addison-Wesley dRT (X,P1) / dRT (X,P2) = dE (X,P1) / dE (X,P2) . Publishing Company, Menlo Park, CA, 88p., 1975.

That is, the ratios of the Euclidean and dRT −distances along a line are the same. [12]R.S.M ILLMANN,G.D.PARKER, Geometry a Met- ric Approach with Models, Springer, 370p., 1991.

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