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This book, a translation of the German volume n-Ecke, presents an elegant geometric theory which, starting from quite elementary geometrical observations, exhibits an interesting connection between geometry and fundamental ideas of modern algebra in a form that is easily accessible to the student who lacks a sophisticated background in mathematics. It stimulates geometrical thought by applying the tools of linear algebra and the algebra of polynomials to a concrete geometrical situation to reveal some rather surprising insights into the geometry of n-gons. The twelve chapters treat n-gons, classes of n-gons, and mapping of the set of n-gons into itself. Exercises are included throughout, and two appendixes, by Henner Kinder and Eckart Schmidt, provide background material on lattices and cyclotomic polynomials.(Mathematical Expositions No. 18).

Polygons. --- Set theory. --- Aggregates --- Classes (Mathematics) --- Ensembles (Mathematics) --- Mathematical sets --- Sets (Mathematics) --- Theory of sets --- Logic, Symbolic and mathematical --- Mathematics --- Polygonal figures --- Geometry, Plane --- Shapes

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The main focus of this unique book is an in-depth examination of the polygonal technique; the primary method used by master artists of the past in creating Islamic geometric patterns. The author details the design methodology responsible for this all-but-lost art form and presents evidence for its use from the historical record, both of which are vital contributions to the understanding of this ornamental tradition. Additionally, the author examines the historical development of Islamic geometric patterns, the significance of geometric design within the broader context of Islamic ornament as a whole, the formative role that geometry plays throughout the Islamic ornamental arts (including calligraphy, the floral idiom, dome decoration, geometric patterns, and more), and the underexamined question of pattern classification. Featuring over 600 beautiful color images, Islamic Geometric Patterns: Their Historical Development and Traditional Methods of Construction is a valuable addition to the literature of Islamic art, architecture and geometric patterns. This book is ideal for students and scholars of geometry, the history of mathematics, and the history of Islamic art, architecture, and culture. In addition, artists, designers, craftspeople, and architects will all find this book an exceptionally informative and useful asset in their fields. Jay Bonner is an architectural ornamentalist and unaffiliated scholar of Islamic geometric design. He received his MDes from the Royal College of Art in London (1983). He has contributed ornamental designs for many international architectural projects, including the expansion of both the al-Masjid al-Haram (Grand Mosque) in Mecca, and the al-Masjid an Nawabi (Prophet’s Mosque) in Medina, as well the Tomb of Sheikh Hujwiri in Lahore, and the Ismaili Centre in London – to name but a few. He is committed to the revitalization of Islamic geometric design through the teaching of tradi tional methodological practices. To this end, in addition to publishing, Jay Bonner has lectured and taught design seminars at many universities and conferences in North America, Europe, North Africa and Asia.

Geometry in art. --- Polygons. --- Polygonal figures --- Mathematics. --- Architecture. --- Geometry. --- History. --- History of Mathematical Sciences. --- Architectural History and Theory. --- Geometry, Plane --- Shapes --- Mathematics --- Euclid's Elements --- Architecture, Western (Western countries) --- Building design --- Buildings --- Construction --- Western architecture (Western countries) --- Art --- Building --- Design and construction --- Annals --- Auxiliary sciences of history --- Math --- Science --- Architecture, Primitive

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This unique book gives a comprehensive account of new mathematical tools used to solve polygon problems. In the 20th and 21st centuries, many problems in mathematics, theoretical physics and theoretical chemistry – and more recently in molecular biology and bio-informatics – can be expressed as counting problems, in which specified graphs, or shapes, are counted. One very special class of shapes is that of polygons. These are closed, connected paths in space. We usually sketch them in two-dimensions, but they can exist in any dimension. The typical questions asked include "how many are there of a given perimeter?", "how big is the average polygon of given perimeter?", and corresponding questions about the area or volume enclosed. That is to say "how many enclosing a given area?" and "how large is an average polygon of given area?" Simple though these questions are to pose, they are extraordinarily difficult to answer. They are important questions because of the application of polygon, and the related problems of polyomino and polycube counting, to phenomena occurring in the natural world, and also because the study of these problems has been responsible for the development of powerful new techniques in mathematics and mathematical physics, as well as in computer science. These new techniques then find application more broadly. The book brings together chapters from many of the major contributors in the field. An introductory chapter giving the history of the problem is followed by fourteen further chapters describing particular aspects of the problem, and applications to biology, to surface phenomena and to computer enumeration methods.

Polygons --- Polyominoes --- Physics - General --- Geometry --- Physics --- Mathematics --- Physical Sciences & Mathematics --- Polygons. --- Polyominoes. --- Polygonal figures --- Physics. --- Chemometrics. --- Numerical analysis. --- Algorithms. --- Combinatorics. --- Statistical physics. --- Dynamical systems. --- Mathematical Methods in Physics. --- Numeric Computing. --- Statistical Physics, Dynamical Systems and Complexity. --- Math. Applications in Chemistry. --- Combinatorial designs and configurations --- Geometry, Plane --- Shapes --- Mathematical physics. --- Electronic data processing. --- Chemistry --- Complex Systems. --- Mathematics. --- Combinatorics --- Algebra --- Mathematical analysis --- Algorism --- Arithmetic --- Physical mathematics --- ADP (Data processing) --- Automatic data processing --- Data processing --- EDP (Data processing) --- IDP (Data processing) --- Integrated data processing --- Computers --- Office practice --- Foundations --- Automation --- Chemistry, Analytic --- Analytical chemistry --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Statics --- Mathematical statistics --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Measurement --- Statistical methods