Malaya J. Mat. S(2)(2015) 366–373 Homeotoxal and Homeohedral Tiling Using Pasting Scheme S. Jebasingh, a ∗ Robinson Thamburaj b and Atulya K. Nagar c a Department of Mathematics, Karunya University, Coimbatore, India. b Department of Mathematics, Madras Christian College, Tambaram, Chennai - 600 059, India. c Department of Mathematics and Computer Science, Liverpool Hope University, Hope Park, Liverpool, L16 9JD, United Kingdom. Abstract The art of tiling has been in practice from the beginning of human civilization. Intricate tiling patterns were used to decorate and cover floors and walls [1]. Robinson Thamburaj introduced the notion of pasting rules [4] and studied the construction of kolam tiling. In this paper we study Pasting Scheme (PS) using special sets of tiles namely Wang tile, Escher tile and Truchet tile. The scheme is also used to construct the five types of Homeotoxal tiling [1] and the eleven types of Homeohedral tiling [1]. Generalizing Pasting Scheme we define Tabled Pasting Scheme (TPS) where pasting rules are grouped in tables and the tables of rules are used as many times as given in the scheme for the successive construction of a tiling. Keywords: Pasting rules, Tiling, Homeotoxal tiling, Homeohedral tiling. AMS 2010: 52C20. c 2012 MJM. All rights reserved. 1 Introduction Mathematical tiling theory provides an insight in geometry and computation. In the construction of complex tiling, the desired tiling rises spontaneously due to the various parameters defined in the system. Computational mechanisms play an important role in understanding how complex tiling is formed. The Pasting Scheme (PS) and Tabled Pasting Scheme (TPS) are syntactic methods, introduced by Robinson etc in [4], to generate plane tiling. These techniques use the notion of pasting rule [4] that allows the edges of the corresponding tiles to get glued or attached at the specified edges and thus generating interesting two dimensional tiling. In section 3, we study the construction of Wang tiling, Escher tiling and truchet tiling using Pasting Scheme. Homeotoxal tiling are tiling with the property that every edge is of the same type. That is every edge E of the tiling has the number of edges n and m for the two tiles which contain the edge E and the number of edges incident with (valences) the two vertices of E are p and q. In section 4 we study the construction of the five types of homeotoxal tiling using Pasting scheme. In section 5 we study the construction of the eleven types of homeohedral tiling using Pasting scheme. In section 6, we generalize the Pasting scheme and define Tabled Pasting Scheme. 2 Preliminaries A Tile is a two dimensional topological disk (region) whose boundary is a simple closed curve, whose ends join up to form a loop without crossing or branches. A plane tiling is a countable family of topological disks ∗ Corresponding author. E-mail address: jebasinghs@karunya.edu (S. Jebasingh), robinson@mcc.edu.in(Robinson Thamburaj), nagara@hope.ac.uk(Atulya K. Nagar).