Treks into Intuitive Geometry : The World of Polygons and Polyhedra This book is written in a style that uncovers the mathematical theories buried in our everyday lives such as examples from patterns that appear in nature art and traditional crafts and in mathematical mechanisms in techniques used by architects. The authors believe that through dialogues between students and mathematicians readers may discover the processes by which the founders of the theories came to their various conclusions their trials errors tribulations and triumphs. The goal is for readers to refine their mathematical sense of how to find good questions and how to grapple with these problems. Another aim is to provide enjoyment in the process of applying mathematical rules to beautiful art and design by examples that highlight the wonders and mysteries from our daily lives. To fulfill these aims this book deals with the latest unique and beautiful results in polygons and polyhedra and the dynamism of geometrical research history that can be found around us. The term intuitive geometry was coined by Lászlo Fejes Tóth to refer to the kind of geometry which in Hilberts words can be explained to and appeal to the man on the street. This book allows people to enjoy intuitive geometry informally and instinctively. It does not require more than a high school level of knowledge but calls for a sense of wonder intuition and mathematical maturity Chapter 1 Art From Tiling Patterns Chapter 2 The Tile Maker Theorem and Its Applications to Art and Designs Chapter 3 Patchwork Chapter 4 Reversible Pairs of Figures Chapter 5 Platonic Solids Chapter 6 Cross Sections of Polyhedra Chapter 7 Symmetry of Platonic Solids Chapter 8 Double Duty Solids Chapter 9 Nets of Small Solids with Minimum Perimeter Lengths Chapter 10 Tessellation Polyhedra Chapter 11 Universal Measuring Boxes Chapter 12 Wrapping a Box Chapter 13 Bees Pomegranates and Parallelohedra Chapter 14 Reversible Polyhedra Chapter 15 Elements of Polygons and Polyhedra Chapter 16 The Pentadron