Introductory Tiling Theory for Computer Graphics
Tiling theory is an elegant branch of mathematics that has applications in several areas of computer science. The most immediate application area is graphics, where tiling theory has been used in the contexts of texture generation, sampling theory, remeshing, and of course the generation of decorative patterns. The combination of a solid theoretical base (complete with tantalizing open problems), practical algorithmic techniques, and exciting applications make tiling theory a worthwhile area of study for practitioners and students in computer science. This synthesis lecture introduces the mathematical and algorithmic foundations of tiling theory to a computer graphics audience. The goal is primarily to introduce concepts and terminology, clear up common misconceptions, and state and apply important results. The book also describes some of the algorithms and data structures that allow several aspects of tiling theory to be used in practice. Table of Contents: Introduction / Tiling Basics / Symmetry / Tilings by Polygons / Isohedral Tilings / Nonperiodic and Aperiodic Tilings / Survey
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adjacent algorithm aperiodic tile aperiodic tile set apply Archimedean tiling aspect transforms Aspects 1 Rules chapter classiﬁcation computer graphics congruent construct corresponding deﬁne deﬁnition disc discrete symmetry group drawing edge shapes edge-to-edge Escher example ﬁgure ﬁnd ﬁnite ﬁrst frieze groups fundamental region geometric Grünbaum and Shephard Heesch implementation incidence symbol inﬁnite integer isohedral tilings isometry kite and dart Laves tilings matching conditions mathematical matrix monohedral tiling nonperiodic tilings orbifold overlap parameters patch of tiles Penrose tiles pentominoes perfect colourings period parallelogram permutation polyominoes prototile set reﬂection regular polygons rendering rhombs rigid motion rotation sample Section set of prototiles shown in Figure shows square subset substitution rules substitution system sufﬁcient symmetry group symmetry theory texture tile shape tile the plane tile’s tiling edges tiling polygon tiling theory tiling vertex parameterization tiling vertices tiling’s topological type translation vectors translational unit triangles vertex type wallpaper groups wallpaper patterns Wang tiles