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On Some Regular Polygons in the Taxicab 3−Space KoG•19–2015 S. Yuksel,¨ M. Ozcan:¨ On Some Regular Polygons in the Taxicab 3−Space Original scientific paper SULEYMAN¨ YUKSEL¨ Accepted 17. 12. 2015. MUNEVVER¨ O¨ ZCAN On Some Regular Polygons in the Taxicab 3−Space On Some Regular Polygons in the Taxicab O nekim pravilnim mnogokutima u taxicab trodi- 3−Space menzionalnom prostoru ABSTRACT SAZETAKˇ In this study, it has been researched which Euclidean reg- U ovom se radu prouˇcavakoji su euklidski pravilni mno- ular polygons are also taxicab regular and which are not. gokuti ujedno i taxicab pravilni, a koji to nisu. Takod-er se The existence of non-Euclidean taxicab regular polygons istraˇzujepostojanje taxicab pravinih mnogokuta koji nisu in the taxicab 3-space has also been investigated. pravilni u euklidskom smislu. Key words: taxicab geometry, Euclidean geometry, regu- lar polygons Kljuˇcnerijeˇci: taxicab geometrija, euklidska geometrija, pravilni mnogokuti MSC2010: 51K05, 51K99, 51N25 1 Introduction which is centered at M = (x0;y0) point with radius r. In 3, the equation of taxicab sphere is 3 R The taxicab 3-dimensional space RT is almost the same 3 jx − x j + jy − y j + jz − z j = r as the Euclidean analytical 3-dimensional space R . The 0 0 0 (4) points, lines and planes are the same in Euclidean and taxi- which is centered at M = (x0;y0;z0) point with radius r cab geometry and the angles are measured in the same way, (see [7], [8], [9]). Then, a taxicab circle can be defined by but the distance function is different. The taxicab metric is a plane and a taxicab sphere. defined using the distance function as in [4], [6] In R3; the taxicab circle is the intersection set of taxicab sphere and ax + by + cz + d = 0, (a;b;c;d 2 ), plane dT (A;B) = jb1 − a1j + jb2 − a2j + jb3 − a3j: (1) R which passes through the center of taxicab sphere. −! 3 Also, for a vector V = (v1;v2;v3) 2 R , taxicab norm of Recently regular polygons have been studied in the taxicab −! −! plane (see [3]). Although there do not exist Euclidean and V is noted as V as in [5] and T taxicab regular triangles in the taxicab plane (see [3]), there −! exist Euclidean and taxicab regular triangles in the taxicab V = jv1j + jv2j + jv3j: (2) 3 T 3-dimensional space RT (see Example 1, 2). Therefore, it can be interesting to study regular polygons in the taxi- Since taxicab plane and 3-dimensional taxicab space have cab 3-dimensional space. On the x = k ;y = k ;z = k different distance function from that in the Euclidean plane 1 2 3 (k ;k ;k 2 ) planes in the 3 , Euclidean and taxicab and 3-dimensional space, it is interesting to study the taxi- 1 2 3 R RT regular polygons can be investigated in the same way as cab analogues of topics that include the distance concept in the taxicab plane 2 . Therefore, in this study, reg- in the Euclidean plane and 3-dimensional space. During RT ular polygons which are not on x = k ;y = k ;z = k the recent years, many such topics have been studied in the 1 2 3 (k ;k ;k 2 ) planes are researched and we answer the taxicab plane and 3-dimensional space (see [1], [2], [3], 1 2 3 R following question: Which Euclidean regular polygons are [5], [6], [7], [8], [9], [10]). also the taxicab regular, and which are not? In addition, In 2, the equation of taxicab circle is R we investigate the existence and nonexistence of taxicab 3 jx − x0j + jy − y0j = r (3) regular polygons in the taxicab 3-dimensional space RT . 32 KoG•19–2015 S. Yuksel,¨ M. Ozcan:¨ On Some Regular Polygons in the Taxicab 3−Space 2 Taxicab Regular Polygons and kwkT = kwikT (since there exist an infinite number of points on a circle as in Case 2, it can be said that i 2 N). As in the Euclidean 3-space, a polygon in the taxicab 3- Let k 2 R. space consists of three or more coplanar line segments, the If the direction vector of w is (±k;±k;±k), then kwkE = line segments (sides) intersect only at endpoints, each end- kwk p T : point (vertex) belongs to exactly two line segments and no 3 two line segments with a common endpoint are collinear. If the direction vector of w is a member of If the number of sides of a polygon is n for n ≥ 3 and f(±k;0;0);(0;±k;0);(0;0;±k)g, then kwkE = kwkT : If the direction vector of w is not a member of n 2 N, then the polygon is called an n−gon. The following definitions for polygons in the taxicab 3-space are given f(±k;0;0);(0;±k;0);(0;0;±k);(±k;±k;±k)g; then kwk p T < kwk < kwk : by means of the taxicab lengths instead of the Euclidean 3 E T lengths: To determine wi vectors, there exist three different cases depending on the Euclidean and taxicab lengths of the w Definition 1 A polygon in the plane is said to be taxicab vector. kwk equilateral if the taxicab lengths of its sides are equal. Case 1. For kwk = p T ; as it is shown in Figure 2, the E 3 members of the intersection set of the Euclidean and taxi- Definition 2 A polygon in the plane is said to be taxicab cab sphere are equiangular if the measures of its interior angles are equal. kwk kwk kwk w = (± T ;± T ;± T ): (5) Definition 3 A polygon in the plane is said to be taxicab i 3 3 3 regular if it is both taxicab equilateral and equiangular. Definition 2 does not give a new equiangular concept be- cause the taxicab and the Euclidean measure of an angle are the same. That is, every Euclidean equiangular polygon is also the taxicab equiangular, and vice versa. However, since the taxicab 3-space has a different distance function, Definition 1 and Definition 3 are new concepts [3]. An n-gon can be formed by n vectors with a total of zero on a plane. If the lengths of n vectors are the same and all the angles between two consecutive vectors are equal, the n-gon is regular (see Figure 1). Therefore, we need to find n vectors that allow these requirements to form a regular n-gon. Figure 2. kwk Case 2. For p T < kwk < kwk ; as it is shown in Fig- 3 E T ure 3, the intersection sets are 8 circles on the taxicab sphere. One of the circles, C; is on x + y + z = kwkT plane and points of C are intersection of x + y + z = kwkT (6) plane and kwk 2 kwk 2 kwk 2 3kwk2 −kwk2 x− T + y− T + z − T = E T 3 3 3 3 Figure 1. (7) Let W be a vector set and the members of W are sides of Euclidean sphere. All the circles, for 1 ≤ j ≤ 8 Cj; are taxicab regular polygon as in Figure 1. It is clear that all the intersection of vectors are on the same plane and all the angles between kwk 2 kwk 2 kwk 2 3kwk2 −kwk2 two consecutive vectors are equal. x± T + y± T + z± T = E T Let w be a vector in R3. We would like to find geo- 3 3 3 3 metric locations of the wi vectors where kwkE = kwikE (8) 33 KoG•19–2015 S. Yuksel,¨ M. Ozcan:¨ On Some Regular Polygons in the Taxicab 3−Space Euclidean spheres and If 0 ≤ y ≤ kwkT and −kwkT ≤ x;z ≤ 0 then C8 = S+;−;+ \ P−;+;−: ±x ± y ± z = kwkT (9) Assume kwkE = kwkT . C9 = f(±kwkT ;0;0) ;(0;±kwkT ;0);(0;0;±kwkT )g. planes. Thus, wi vectors where kwkE = kwikE and kwkT = 8 Corollary 1 Let W be a vector set of the edges of Eu- kwikT are members of [ Cj: j=1 clidean regular polygon and w 2 W. A Euclidean regular 9 polygon is also taxicab regular if only if W ⊆ [ Cj where j=1 9 [ Cj is the same as in Proposition 1: j=1 Proof. For n ≥ 3; if an n−gon is Euclidean and taxicab regular, then all the w side vectors of n−gon are members of the intersection set of an origin-centered taxicab sphere with a radius of kwkT and an origin-centered Euclidean sphere with a radius of kwkE : It is clear that the intersec- 9 tion set is [ Cj as in Proposition 1. j=1 3 Euclidean Regular Polygons in Taxicab 3- Space Figure 3. Euclidean regular polygons are Euclidean equiangular. Case 3. For kwk = kwk ; as it is shown in Figure 20, the E T Since the taxicab angles are measured in the same way members of the intersection set of Euclidean and taxicab as Euclidean (see [3], [6]), the polygons are also taxicab sphere are equiangular. Thus, we are just interested in which Eu- clidean regular polygons are taxicab equilateral and which wi = (±kwk ;0;0);(0;±kwk ;0);(0;0;±kwk ): (10) T T T are not (see Definitions 1, 2, 3). Let us introduce the following abbreviations for the fol- Theorem 1 None of Euclidean regular n−gon (n > 12) is lowing Proposition 1. −! taxicab regular. ±x±y±z = k w kT plane equation is shown as P±;±;± and 2 2 2 2 2 x ± kwkT + y ± kwkT + z ± kwkT = 3kwkE −kwkT Proof. All the regular polygons are planar and the inter- 3 3 3 3 section set of Euclidean and taxicab spheres forms eight sphere equation is shown as S .
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