Triangle and diamond shaped tiling using truchet tiles

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In this paper the pasting scheme model is used to study the construction of triangle shaped Truchet tiling and diamond shaped Truchet tiling using the Truchet tiles.

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Nội dung Text: Triangle and diamond shaped tiling using truchet tiles

International Journal of Mechanical Engineering and Technology (IJMET) Volume 10, Issue 03, March 2019, pp. 21–25, Article ID: IJMET_10_03_003 Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=3 ISSN Print: 0976-6340 and ISSN Online: 0976-6359 © IAEME Publication Scopus Indexed TRIANGLE AND DIAMOND SHAPED TILING USING TRUCHET TILES S. Jebasingh Department of Mathematics, Karunya Institute of Technlolgy and Sciences, Coiambatore, India G. Johnsy Arputhavalli Department of Physics, Karunya Institute of Technlolgy and Sciences, Coiambatore, India T. Robinson Department of Mathematics, Madras Christian College, Chennai, India ABSTRACT Pasting Scheme is a generating model for Tiling and Tessellation pattern. Truchet tiles are special type of tiles with continuous arc. In this paper the pasting scheme model is used to study the construction of triangle shaped Truchet tiling and diamond shaped Truchet tiling using the Truchet tiles. Key words: Pasting rule, Pasting Scheme, Tiling, Truchet tiles Cite this Article: S. Jebasingh, G. Johnsy Arputhavalli, T. Robinson, Triangle and Diamond Shaped Tiling Using Truchet Tiles, International Journal of Mechanical Engineering and Technology 10(3), 2019, pp. 21–25. http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=10&IType=3 1. INTRODUCTION Tiling consists of tiles made of stone and other similar material which fit together without holes to cover the plane [1]. Tiles made of recent materials such as plastic or metal sheet have applications in the modern engineering. These tiles have same constraints as that of tiles made of stone. Tiling, which is also referred as tessellation, had been studied extensively for their application in science and engineering. In the literature we find various methods of constructing tiling with regular polygons to cover the Euclidean plane [2, 3, 4, 5, 6]. In the beginning of 17 century Sebastian Truchet [7] described a simple half colored square tile that could be arranged to produce interesting tiling with geometric patterns. The Truchet tiling is obtained from a single square Truchet tile that was bisected along the diagonal into two uniquely colored isosceles right triangles. A variant of Truchet tile is constructed later where the triangles are replaced with quarter-circle arcs. http://www.iaeme.com/IJMET/index.asp 21 editor@iaeme.com
Triangle and Diamond Shaped Tiling Using Truchet Tiles The tiles are placed in a tiling in such a way that a continuous arc is produced. This tile produces tiling with sinuous structure. In section 2, we recall the principle of tiling by the method called Pasting scheme where tiles are glued with one another using pasting rule. In section 3, we use Truchet tiles to construct the triangle shaped tiling and diamond shaped tiling by pasting scheme. , Figure 1 A truchet tile set and the truchet tiling 2. PRELIMINARIES Definition: 2.1. A pasting rule (A, B), is an unordered pair of edge labels of tiles M and N which allows the edge to edge pasting of the two tiles. If A is the label of the upper edge of a truchet tile and G is the label of the lower horizontal edge of another truchet tile, then an application of the rule (A, G) pastes the tiles one over other ( joins edge to edge) so as form a continuous arc. When tiles are attached by pasting rules it results in the formation of Tiling. (A, G) By applying pasting rules to the edges of a tile or tiling a new tiling ti+1 is said to be  obtained from the tile ti. It is symbolically denoted as t i  t i 1 . Definition: 2.2. A Pasting Scheme (PS) is a 3 tuple G  (, P, t 0 ) where Σ is a finite non empty set of tiles with labeled edges, P is a finite set of pasting rules and t0 is the axiom tile or tiling. The patch of tiles constructed by applying the pasting rule n (n  1) number of time forms the language of tiling generated by the scheme G and it is denoted by L(G)  { / t 0    } . http://www.iaeme.com/IJMET/index.asp 22 editor@iaeme.com
S. Jebasingh, G. Johnsy Arputhavalli, T. Robinson 3. RESULTS In this section we construct triangle shaped tiling and diamond shaped tiling using the Truchet tiles. Theorem 3.1: The language of triangle tiling using Truchet tiles can be constructed by a Pasting Scheme. Proof: An array of rows of square tile in the shape of triangular region is referred as a triangle tiling. The number of rows of arrays in the triangular region is the size of the triangle. The language of triangle tiling is the collection of all triangle tiling of size n, where n is the number of rows. The construction of the triangle is done by pasting square tiles row by row starting from the axiom tile in the upward direction (bottom-up method). Consider the following pasting scheme to construct the language of triangle tiling, G  (, P, t 0 ) where, P = { (D, B) , (A, G), (F, H) } t0  Starting from the axiom tile, application of pasting rules of the scheme constructs the triangle tiling of size n, n  2 . Figure 2 A triangle tiling pattern of size 3 Theorem 3.2: The language of diamond tiling using Truchet tiles can be constructed by a Pasting Scheme. Proof: An array of rows of square tile in the shape of diamond region is referred as a diamond tiling. The number of rows of arrays in the region is the size of the diamond. The language of diamond tiling is the collection of all diamond tiling of size n, where n is the number of rows. The construction of the diamond tiling is done by adding square tiles row by row symmetrically starting from the axiom tile. The following pasting scheme constructs the class of diamond tiling G  (, P, t 0 ) , where http://www.iaeme.com/IJMET/index.asp 23 editor@iaeme.com
Triangle and Diamond Shaped Tiling Using Truchet Tiles P = {(D, B), (A, G), (F, H), (C, E) } t0  Figure 3 A diamond shaped tiling pattern. 4. CONCLUSIONS In this paper we have studied the construction of triangle shaped tiling and diamond shaped tiling using the Truchet tiles by the generative model pasting scheme. It is interesting to note that, the triangle shaped tiling was constructed with three pasting rules and the addition of one more pasting rule results in the construction of diamond shaped tiling. The model needs further investigation on the class of tiling it can generate using a special set of tiles. REFERENCES [1] Branko Gruenbaum, G.C. Shephard, Tilings and Patterns, W.H. Freeman Company, Newyork, 1986. [2] Thamburaj Robinson. Extended Pasting Scheme for Kolam Pattern Generation, Forma 22, 2007; 55 – 64 . [3] S. Jebasingh, T. Robinson, Atulya K. Nagar, A variant of Tile Pasting P System for Tiling Patterns, In: Proc. of IEEE Fifth International Conference on Bio-Inspired Computing: Theories and Applications, BIC-TA 2010 (UK), Vol. 2, 2010, pp. 1568-1576. [4] S. Jebasingh, T. Robinson, Atulya K. Nagar, Constructing Non-Periodic Tiling Patterns with P System, Romanian Journal of Information Science and Technology (Special issue), Volume 15, Number 1, 2012. http://www.iaeme.com/IJMET/index.asp 24 editor@iaeme.com
S. Jebasingh, G. Johnsy Arputhavalli, T. Robinson [5] N.G. de Bruijn, Updown generation of Penrose tilings, Indagationes Mathematicae, New Series 1(2), 1990, 201-219. [6] J. Kari, A small aperiodic set of Wang tiles, Discrete Mathematics 160, (1996), 259-264. [7] Cameron Browne, Duotone-Truchet like tilings, Journal of Mathematics and Arts, Vol 2, No 4, December 2008, 189-196. http://www.iaeme.com/IJMET/index.asp 25 editor@iaeme.com