PERFECT FOLDING OF THE PLANE

Transcription

1 SOOCHOW JOURNAL OF MATHEMATICS Volume 32, No. 4, pp , October 2006 PERFECT FOLDING OF THE PLANE BY E. EL-KHOLY, M. BASHER AND M. ZEEN EL-DEEN Abstract. In this paper we introduced the concept of perfect folding defined on E 2 equipped by perfect k-coloring of r-monohedral tilings. Then we studied the different cases of perfect foldings of 4-monohedral and 3-monohedral tilings of E 2. Also the permutations describing each folding are obtained. 1. Introduction A tiling of the plane is a family of sets called tiles that cover the plane without gaps or overlaps. Tilings are known as tessellations or pavings, they have appeared in human activities since prehistoric times. Their mathematical theory is mostly elementary, but nevertheless it contains a rich supply of interesting problems at various levels. The same is true for the special class of tiling called tiling by regular polygons [5]. The notions of tiling by regular polygons and perfect colorings of transitive tilings in the plane is introduced by B. Grunbaum and G. C. Shephard in [6]. For more details see [7, 8, 9, 10]. The notion of isometric folding is introduced by S. A. Robertson who studied the stratification determined by the folds or singularities ([11]). Then the theory of isometric foldings has been pushed and also different types of foldings are introduced by E. EL-Kholy and others [1, 2]. Definition 1.1. A tiling of the plane is a collection I = {T i : i I = {1, 2, 3,...}} of closed topological discs (tiles) which covers the Euclidean plane E 2 and is such that the interiors of the tiles are disjoint. Received November 6, 2004; revised March 6, AMS Subject Classification. 54C05, 05B45. Key words. folding, tiling. 521

2 522 E. EL-KHOLY, M. BASHER AND M. ZEEN EL-DEEN From now we will restrict attention to regular convex polygons as tiles. If such a polygon has n edges it is called an n-gon and will be denoted by {n}. Definition 1.2. A tiling is called edge-to-edge if the relation of any two tiles is one of the following three possibilities: (i) they are disjoint, or (ii) they have precisely one common point which is a vertex of each of the polygons, or (iii) they share a segment that is an edge of each of two polygons. Definition 1.3. The symmetry group S(I) of a tiling I is the group of isometries which leave I invariant, i.e., maps each tile of I onto a tile of I. Definition 1.4. Two tiles T 1 and T 2 of a tiling I are said to be equivalent if S(I) contains a transformation that maps T 1 onto T 2. Definition 1.5. A tiling I is called transitive if the symmetry group S(I) acts transitively on the tiles. Such tilings are called tile transitive or isohedral. Definition 1.6. A k-coloring of a tiling I is a partition of I into k-colorclasses I j, j = 1, 2,..., k, where I j = {T i : i I j } and I 1,..., I k is a partition of the index set I into non-empty sets. Each tile of I j is said to have color j. Definition 1.7. Let I be a transitive tiling and s (I) be a symmetry of I. A k-coloring of I is said to be compatible with s if s preserves the partition of I into the color-classes I 1,..., I k. In other words, the k-coloring is compatible if there exists a permutation σ of the colors 1, 2,..., k such that s maps each tile of color j into a tile of color σ j. Definition 1.8. A k-coloring of a transitive tiling I is called perfect if it is compatible with every symmetry s S(I), i.e., if I is colored in such a way that every symmetry of I can be extended to a colored symmetry. Theorem 1.1.([6]) The regular square tiling of the plane admits a perfect k- coloring if and only if k = n 2 or k = 2n 2 for some positive integer n. The regular hexagonal tiling admits a perfect k-coloring if and only if k = n 2, k = 3n 2. The regular triangular tiling admits a perfect k-coloring if and only if k = 2n 2, k = 6n 2, k = (3n 2) 2 or k = (3n 1) 2. In each case, for a given k, the perfect k-coloring is unique.

3 PERFECT FOLDING OF THE PLANE Perfect Folding The regular tiling I will be called monohedral if every tile in I is congruent to one fixed set T. The set T is called the prototile of I. Definition 2.1. Let E 2 perfectly k-colored. A map f : E 2 E 2 is said to be perfect folding if f maps {r} to {r} such that each {r} of color i is mapped to an {r} of color σ i, where σ is a permutation of colors {1, 2,..., k}. The set of points of E 2 at which f fails to be local homeomorphism is called the set of singularities and will be denoted by f. Thus f of perfect folding of E 2 will consists of union of vertices and edges. It is known ([11]) that for each x f the singularities of f near x form the image of an even number of lines emanating from x making alternate angles α 1,α 2,..., α n, β 1, β 2,..., β n where n α i = i=1 n β i = π. i=1 This condition on the angles will be called the angle folding relation. It should be noted that the valency of each vertex must be even [3]. Definition 2.2. By an r-monohedral tiling of E 2 we mean a planar monohedral edge-to-edge tiling by mutually congruent regular convex r-polygons whose vertices are of even valency and satisfy the angle folding relation. Theorem 2.1. Let I be a perfectly k-colorable r-monohedral tiling and f : E 2 E 2 be a perfect folding. Then either r = 3 or r = 4, i.e., the prototile of I is either a triangle or a square. First note that the only possible edge-to-edge tilings of E 2 by mutually congruent regular convex polygons are the three regular tilings by equilateral triangles, by squares or by regular hexagons. Thus the possible perfect folding of E 2 into itself will be valid if r = 3 or r = 4. The following two sections will give the precise way of perfect foldings of E 2, if E 2 is equipped with 3 or 4-monohedral tilings. Also in each case we describe the permutations of the perfect folding.

4 524 E. EL-KHOLY, M. BASHER AND M. ZEEN EL-DEEN 3. The Perfect Folding of 4-Monohedral Tiling of E 2 Lemma 3.1. Let E 2 perfectly k-colored of 4-monohedral tiling. Perfect foldings of E 2 into itself can be defined only by folding along the horizontal and vertical lines. The image f(e 2 ) in this case is a region containing k different colors which is a square for k = n 2 and a rectangle for k = 2n 2 for some positive integer n. Proof. We know that a regular square tiling of the plane E 2 admits a perfect k-coloring iff (i) k = n 2 or (ii) k = 2n 2 for some positive integer n, and for a given perfect coloring by squares we can see that each of the color classes I i must be congruent to every other color-class I j of I in the sense that I i can be brought into coincide with I j by a symmetry of I. Let n be the smallest positive integer such that a horizontal translation by n squares brings a square of color 1 into coincidence with another square of color 1. Thus every square obtained from a square of color j by a horizontal translation of n squares also has color j ([2]). Further, by considering rotation about the center of a tile through 90 we see that exactly the same assertion holds for vertical translation also. Thus, for each j, color j is assigned to, at least, all tiles that form a square lattice of mesh n. Consider now the color-class I 1. Two possibilities arise (a) either all the tiles of color 1 lie on a lattice of mesh n, or (b) some other tile has color 1. In case (a) we see that all the n 2 = k tiles in the mesh must have different colors, and that these same colors are repeated in the same way in every mesh. In case (b) consider reflections in the horizontal and vertical lines that pass through the centers of the tiles of color 1. contains two tiles of color 1. Hence each mesh of 2n 2 = k tiles In the first case, k = n 2. Thus in this case all the n 2 tiles in the mesh must have different colors, also the same colors are repeated in the same way in every mesh. Now let f : E 2 E 2 be a perfect folding. The only possibilities of perfect foldings, in this case, can be defined by folding either along the vertical lines or the horizontal lines that bound the meshes. If we first fold perfectly along the horizontal lines, we will obtain an infinite strip of repeated meshes, so we can fold again along the vertical lines that bounds the meshes. The image of these consequence foldings, f(e 2 ), will be a square mesh of different colors n 2. The set

5 PERFECT FOLDING OF THE PLANE 525 of singularities, f, in this case is the union of the horizontal and vertical lines that bounds the meshes. The permutations of the perfect folding f : E 2 E 2 are given by σ 1 (r + nl) = n 2 (l + 1)n + r where l = 0,..., n 1, r = 1,..., n and σ 2 (r + nl) = r + (l + 1)n + 1 where l = 0,..., n 1, r = 1,..., n. Figure 1 illustrates the case when n = 5, i.e. k = 25. Figure 1. A perfect folding of a perfect 25-coloring of the 4-monohedral tiling of E 2. In case n = 5, these permutations will take the following form: σ 1 = (1 21)(2 22)(3 23)(4 24 )(5 25 )(6 16 )(7 17 )(8 18)(9 19) (10 20)(11)(12)(13)(14)(15), σ 2 = (1 5)(2 4)(6 10)(7 9)(11 15 )(12 14 )(16 20)(21 25)(11 24) (17 19)(3)(8)(13)(18)(23).

6 526 E. EL-KHOLY, M. BASHER AND M. ZEEN EL-DEEN In the second case, k = 2n 2, and as mentioned before every color will be repeated twice in each mesh. Any perfect folding of E 2 into itself can be defined only either by folding along the vertical lines that pass through the centers of the meshes or by folding along the horizontal lines that bound the meshes. So if we first fold along the horizontal lines we will get an infinite strip, this strip can be folded again along the vertical lines that pass through the centers of the meshes. Then f(e 2 ) will be a rectangle which contains different colors 2n 2. The permutations of the perfect folding f : E 2 E 2 are given by σ 3 (r + nl) = 2n 2 (l +1)n+r where l = 0,..., 2n 1, r = 1,..., n and σ 4 (r +nl) = r +(l +1)n+1 where l = 0,..., n 1, r = 1,..., 2n. Figure 2 illustrates the case n = 3, i.e., k = 18. Figure 2. A perfect folding of a perfect 25-coloring of the 4-monohedral tiling of E 2.

7 PERFECT FOLDING OF THE PLANE 527 In case n = 3, these permutations will take the following forms: σ 3 = 7(1 16)(2 17)(3 18)(4 13)(5 14)(6 15) (7 10)(8 11)(9 12) σ 4 = (1 6)(2 5)(3 4)(7 12)(8 11)(9 10)(13 18)(14 17)(15 16). This completes the proof for the perfect folding of 4-monohedral tiling. 4. The Perfect Folding of 3-Monohedral Tiling of E 2 Lemma 4.1. Let E 2 perfectly k-colored of 3-monohedral tiling. A perfect folding of E 2 into itself can be defined only by foldings along the horizontal or the inclined lines. The image, g(e 2 ), of these folding is either (a) an infinite strip in the cases k = 2n 2 and k = (3n 2) 2 for some positive integer n and k = (3n 1) 2 for some positive even integer n; or (b) a triangle in the cases k = (3n 1) 2, where n is a positive odd integer and k = 6n 2, where n is a positive integer. Proof. The regular triangular tilings of the plane admits perfect k-color iff (i) k = 2n 2, (ii) k = 6n 2, (iii) k = (3n 2) 2 or (iv) k = (3n 1) 2 for some positive integer n ([2]). Considering only tiles of one particular aspect, (that is, translations of one another), we again define n to be the smallest positive integer such that a horizontal translation through a distance nb, (where b is the length of the side of a tile), brings a triangle of color 1 into coincidence with another tile of color 1. In case (i) we see that all the 2n 2 tiles in the mesh must have different colors, and that these colors are repeated in the same way in every mesh, where the mesh is a rhomb. If there is another tile of color 1 then the only position it can occupy is the center of one of the triangles of the mesh. In case (ii) we see that the triangle in the left half of the rhomb has a central tile of the same aspect as the tiles forming the mesh and each color repeats three times in the mesh. In case (iii) we see that the triangle in the left half of the rhomb has a central tile of the opposite aspect as the tiles forming the mesh and each color repeats twice in the mesh. Finally in case (iv) we see that the triangle in the right half of the rhomb has a central tile of the opposite aspect as the tiles forming the mesh and each color repeats twice in the mesh.

8 528 E. EL-KHOLY, M. BASHER AND M. ZEEN EL-DEEN In the case, k = 2n 2, all the 2n 2 tiles in the mesh have different colors, and the mesh takes the rhomb form. Any perfect folding g : E 2 E 2 can be done by folding along the lines that bound the rhomb meshes, and so g(e 2 ) will be an infinite strip, horizontal or inclined ones, which can not be folded any more, i.e. the image is unbounded. Figure 3 illustrates the case when n = 2, k = 8, and the folding lines are the horizontal lines. Figure 3. A perfect folding of a perfect 8-coloring of the 3-monohedral tiling of E 2. The permutations of this perfect folding are given by σ 5 = (1 6)(2 7)(3 8)(4 5). If k = (3n 2) 2, then as mentioned before each color will be repeated twice and the triangle in the left half of the rhomb has a central tile of the opposite aspect to the tiles forming the mesh. The set of singularities of any perfect folding of E 2 will consist of either the horizontal parallel lines, or the inclined parallel lines which bound the meshes. So we will have an infinite strip as an image which can not be folded any more because there is no permutation σ of colors {1, 2, 3,..., k} on this strip which can keep the colors perfect. Figure 4 illustrates the case for n = 2, i.e., k = 16.

9 PERFECT FOLDING OF THE PLANE 529 Figure 4. A perfect folding of a perfect 16-coloring of 3-monohedral tilings of E 2. The perfect folding permutations for n = 2 are σ 6 = (1 16)(2 6)(3 13)(4 8)(5 14)(7 15) (9) (10) (11). If k = (3n 1) 2, then each color will repeats again twice. The triangle in the right half of the rhomb has a central tile of the opposite aspect to the tiles forming the mesh. Now any perfect folding g : E 2 E 2 will depend on the number of colors in each mesh. In other words the image of the perfect folding for k even will be different from that of k odd. In the following we will discuss the cases separately: (1) Suppose k is odd Then we can perfectly fold E 2 along either the horizontal parallel lines, or the inclined parallel lines which bound the meshes to obtain infinite strip. But this strip can not be folded any more, i.e., the image in this case is unbounded. Figure 5 represents the case when n = 1, k = 25. The permutations of the folding are σ 7 = (1 5)(2 25)(3 17)(4 21)(5 19)(6 22)(7 11)(8 23)(9 13) (10 24)(18)(12)(14)(16).

10 530 E. EL-KHOLY, M. BASHER AND M. ZEEN EL-DEEN Figure 5. A perfect folding of a perfect 25-coloring of 3-monohedral tilings of E 2. (2) Suppose k is even In this case we can fold E 2 perfectly along either the horizontal parallel lines, or the inclined parallel lines which bound the meshes to get an infinite strip, then we can fold again along the diameters of the meshes to obtain a triangle. The image of these two perfect foldings is a triangle with k different colors. Figure 6 illustrates the case for n = 1, i.e. k = 4. Figure 6. A perfect folding of a perfect 4-coloring of 3-monohedral tilings of E 2.

11 PERFECT FOLDING OF THE PLANE 531 The perfect folding permutations are σ 8 = (1 3) (2) (4) and (3 4)(1)(2) if the two triangles lie in the same mesh σ 9 = (1 4)(2)(3) if the two triangle lie in adjacent meshes. Finally, in the case k = 6n 2, each color will be repeated three times in the rhomb mesh. Any perfect folding g : E 2 E 2 can be done firstly by folding along the parallel horizontal lines that bound the rhomb meshes, or the inclined lines and so we get an infinite strip. Secondly by folding along the inclined lines that bound the rhomb mesh or the horizontal lines, and then along the diameters of the rhomb meshes. Thus g(e 2 ) will be a triangle, i.e., will be bounded with k colors. Figure 7 illustrates the case when n = 1, k = 6. Figure 7. A perfect folding of a perfect 6-coloring of 3-monohedral tilings of E 2. The perfect folding permutations are σ 10 = (1 4)(2 5)(3 6) and (1 2)(3 4)(5 6) if the two triangles lie in the same mesh σ 11 = (1 6)(2 3)(4 5) if the two triangle lie in adjacent meshes.

12 532 E. EL-KHOLY, M. BASHER AND M. ZEEN EL-DEEN This completes the proof for the perfect folding of 3-monohedral tiling. References [1] E. EL-Kholy and M. EL-Ghoul, Simplicial foldings, Journal of the Faculty of Education, 18(1993), [2] E. EL-Kholy and R. M. Shahin, Cellular folding, Jour. of Inst. of Math & Comp. Sci., 3(1998), [3] H. R. Farran, E. EL-Kholy and S. A. Robertson, Folding a surface to a polygon, Geometriae Dedicata, 63(1996), [4] B. Grunbaum, J. C. P. Miller and G. C. Shephard, Tiling three-dimensional space with polyhedral tiles of a given isomorphism type, J. London Math. Soc., 29:2(1984), [5] B. Grunbaum and G. C. Shephard, Tilings by regular polygons, Mathematics Magazine Math., 50:5(1977), [6] B. Grunbaum and G. C. Shephard, Perfect colorings of transitive tilings and pattrens in the plane, Discrete Mathematics, 20(1977), [7] B. Grunbaum and G. C. Shephard, The eighty-one types of isohedral tilings in the plane, Math. Proc. Cambridge Phil. Soc., 82(1977), [8] B. Grunbaum and G. C. Shephard, The ninety-one types of isohedral tilings in the plane, Trans. Amer. Math. Soc., 242(1978), and 249(1979), 446. [9] B. Grunbaum and G. C. Shephard, Isohedral tilings, Pacific J. Math., 76(1978), [10] B. Grunbaum and G. C. Shephard, Isohedral tilings of the plane by polygons comment, Math. Helvet, 53(1978), [11] S. A. Robertson, Isometric folding of Riemannian manifolds, Proceedings of the Royal Society of Edinburgh, 79:3-4(1977), Department of Mathematics, Faculty of Science, Tanta University, Egypt. prof entesar@yahoo.com Department of Mathematics, Faculty of Education(Suez), Suez-Canal University, Egypt. m e basher@yahoo.com Department of Mathematics, Faculty of Education(Suez), Suez-Canal University, Egypt. mrzd 2@hotmail.com