How do you draw a heptagon? What about a heptakaidecagon? How do you fit circles perfectly into triangles? And around them? If the computer is down - could you do it with ruler and compass? In this unique little book, Andrew Sutton guides you through the once treasured principles of ruler and compass constructions, used for centuries by architects, carpenters, stonemasons and master craftsmen. Designed to last until the lights go out, this is a timeless book. WOODEN BOOKS are small but packed with information. "Fascinating" FINANCIAL TIMES. "Beautiful" LONDON REVIEW OF BOOKS. "Rich and Artful" THE LANCET. "Genuinely mind-expanding" FORTEAN TIMES. "Excellent" NEW SCIENTIST. "Stunning" NEW YORK TIMES. Small books, big ideas.

In the study of the structure of substances in recent decades, phenomena in the higher dimension was discovered that was previously unknown. These include spontaneous zooming (scaling processes), discovery of crystals with the absence of translational symmetry in three-dimensional space, detection of the fractal nature of matter, hierarchical filling of space with polytopes of higher dimension, and the highest dimension of most molecules of chemical compounds. This forces research to expand the formulation of the question of constructing n-dimensional spaces, posed by David Hilbert in 1900, and to abandon the methods of considering the construction of spaces by geometric figures that do not take into account the accumulated discoveries in the physics of the structure of substances. There is a need for research that accounts for the new paradigm of the discrete world and provides a solution to Hilbert's 18th problem of constructing spaces of higher dimension using congruent figures. Normal Partitions and Hierarchical Fillings of N-Dimensional Spaces aims to consider the construction of spaces of various dimensions from two to any finite dimension n, taking into account the indicated conditions, including zooming in on shapes, properties of geometric figures of higher dimensions, which have no analogue in three-dimensional space. This book considers the conditions of existence of polytopes of higher dimension, clusters of chemical compounds as polytopes of the highest dimension, higher dimensions in the theory of heredity, the geometric structure of the product of polytopes, the products of polytopes on clusters and molecules, parallelohedron and stereohedron of Delaunay, parallelohedron of higher dimension and partition of n-dimensional spaces, hierarchical filling of n-dimensional spaces, joint normal partitions, and hierarchical fillings of n-dimensional spaces. In addition, it pays considerable attention to biological problems. This book is a valuable reference tool for practitioners, stakeholders, researchers, academicians, and students who are interested in learning more about the latest research on normal partitions and hierarchical fillings of n-dimensional spaces.

Classical Deformation Theory is used for determining the completions of the local rings of an eventual moduli space. When a moduli variety exists, a main result in the book is that the local ring in a closed point can be explicitly computed as an algebraization of the pro-representing hull (therefore, called the local formal moduli) of the deformation functor for the corresponding closed point.The book gives explicit computational methods and includes the most necessary prerequisites. It focuses on the meaning and the place of deformation theory, resulting in a complete theory applicable to moduli theory. It answers the question 'why moduli theory' and it give examples in mathematical physics by looking at the universe as a moduli of molecules. Thereby giving a meaning to most noncommutative theories.The book contains the first explicit definition of a noncommutative scheme, covered by not necessarily commutative rings. This definition does not contradict any of the previous abstract definitions of noncommutative algebraic geometry, but rather gives interesting relations to other theories which is left for further investigation.

One-Cocycles and Knot Invariants is about classical knots, i.e., smooth oriented knots in 3-space. It introduces discrete combinatorial analysis in knot theory in order to solve a global tetrahedron equation. This new technique is then used to construct combinatorial 1-cocycles in a certain moduli space of knot diagrams. The construction of the moduli space makes use of the meridian and the longitude of the knot. The combinatorial 1-cocycles are therefore lifts of the well-known Conway polynomial of knots, and they can be calculated in polynomial time. The 1-cocycles can distinguish loops consisting of knot diagrams in the moduli space up to homology. They give knot invariants when they are evaluated on canonical loops in the connected components of the moduli space. They are a first candidate for numerical knot invariants which can perhaps distinguish the orientation of knots.

In this volume, the authors present a collection of surveys on various aspects of the theory of bifurcations of differentiable dynamical systems and related topics. By selecting these subjects, they focus on those developments from which research will be active in the coming years. The surveys are intended to educate the reader on the recent literature on the following subjects: transversality and generic properties like the various forms of the so-called Kupka-Smale theorem, the Closing Lemma and generic local bifurcations of functions (so-called catastrophe theory) and generic local bifurcations in 1-parameter families of dynamical systems, and notions of structural stability and moduli.

Over the last number of years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important results. The first edition of Introduction to Symplectic Topology was published in 1995. The book was the first comprehensive introduction to the subject and became a key text in the area. A significantly revised second edition was published in 1998 introducing new sections and updates on the fast-developing area. This new third edition includes updates and new material to bring the book right up-to-date.

Theory and Computation of Tensors: Multi-Dimensional Arrays investigates theories and computations of tensors to broaden perspectives on matrices. Data in the Big Data Era is not only growing larger but also becoming much more complicated. Tensors (multi-dimensional arrays) arise naturally from many engineering or scientific disciplines because they can represent multi-relational data or nonlinear relationships.

Fractal Functions, Fractal Surfaces, and Wavelets, Second Edition, is the first systematic exposition of the theory of local iterated function systems, local fractal functions and fractal surfaces, and their connections to wavelets and wavelet sets. The book is based on Massopust's work on and contributions to the theory of fractal interpolation, and the author uses a number of tools-including analysis, topology, algebra, and probability theory-to introduce readers to this exciting subject. Though much of the material presented in this book is relatively current (developed in the past decades by the author and his colleagues) and fairly specialized, an informative background is provided for those entering the field. With its coherent and comprehensive presentation of the theory of univariate and multivariate fractal interpolation, this book will appeal to mathematicians as well as to applied scientists in the fields of physics, engineering, biomathematics, and computer science. In this second edition, Massopust includes pertinent application examples, further discusses local IFS and new fractal interpolation or fractal data, further develops the connections to wavelets and wavelet sets, and deepens and extends the pedagogical content.

This comprehensive reference begins with a review of the basics followed by a presentation of flag varieties and finite- and infinite-dimensional representations in classical types and subvarieties of flag varieties and their singularities. Associated varieties and characteristic cycles are covered as well and Kazhdan-Lusztig polynomials are treated. The coverage concludes with a discussion of pattern avoidance and singularities and some recent results on Springer fibers.

Geometry with Trigonometry Second Edition is a second course in plane Euclidean geometry, second in the sense that many of its basic concepts will have been dealt with at school, less precisely. It gets underway with a large section of pure geometry in Chapters 2 to 5 inclusive, in which many familiar results are efficiently proved, although the logical frame work is not traditional. In Chapter 6 there is a convenient introduction of coordinate geometry in which the only use of angles is to handle the perpendicularity or parallelism of lines. Cartesian equations and parametric equations of a line are developed and there are several applications. In Chapter 7 basic properties of circles are developed, the mid-line of an angle-support, and sensed distances. In the short Chaper 8 there is a treatment of translations, axial symmetries and more generally isometries. In Chapter 9 trigonometry is dealt with in an original way which e.g. allows concepts such as clockwise and anticlockwise to be handled in a way which is not purely visual. By the stage of Chapter 9 we have a context in which calculus can be developed. In Chapter 10 the use of complex numbers as coordinates is introduced and the great conveniences this notation allows are systematically exploited. Many and varied topics are dealt with , including sensed angles, sensed area of a triangle, angles between lines as opposed to angles between co-initial half-lines (duo-angles). In Chapter 11 various convenient methods of proving geometrical results are established, position vectors, areal coordinates, an original concept mobile coordinates. In Chapter 12 trigonometric functions in the context of calculus are treated. New to this edition: The second edition has been comprehensively revised over three years Errors have been corrected and some proofs marginally improved The substantial difference is that Chapter 11 has been significantly extended, particularly the role of mobile coordinates, and a more thorough account of the material is given

MESH ist ein mathematisches Video ber vielfl chige Netzwerke und ihre Rolle in der Geometrie, der Numerik und der Computergraphik. Der unter Anwendung der neuesten Technologie vollst ndig computergenierte Film spannt einen Bogen von der antiken griechischen Mathematik zum Gebiet der heutigen geometrischen Modellierung. MESH hat zahlreiche wissenschaftliche Preise weltweit gewonnen. Die Autoren sind Konrad Polthier, ein Professor der Mathematik, und Beau Janzen, ein professioneller Filmdirektor.

Der Film ist ein ausgezeichnetes Lehrmittel f r Kurse in Geometrie, Visualisierung, wissenschaftlichem Rechnen und geometrischer Modellierung an Universit ten, Zentren f r wissenschaftliches Rechnen, kann jedoch auch an Schulen genutzt werden.

The term "stereotype space" was introduced in 1995 and denotes a category of locally convex spaces with surprisingly elegant properties. Its study gives an unexpected point of view on functional analysis that brings this fi eld closer to other main branches of mathematics, namely, to algebra and geometry. This volume contains the foundations of the theory of stereotype spaces, with accurate definitions, formulations, proofs, and numerous examples illustrating the interaction of this discipline with the category theory, the theory of Hopf algebras, and the four big geometric disciplines: topology, differential geometry, complex geometry, and algebraic geometry.

This monograph provides a coherent development of operads, infinity operads, and monoidal categories, equipped with equivariant structures encoded by an action operad. A group operad is a planar operad with an action operad equivariant structure. In the first three parts of this monograph, we establish a foundation for group operads and for their higher coherent analogues called infinity group operads. Examples include planar, symmetric, braided, ribbon, and cactus operads, and their infinity analogues. For example, with the tools developed here, we observe that the coherent ribbon nerve of the universal cover of the framed little 2-disc operad is an infinity ribbon operad.In Part 4 we define general monoidal categories equipped with an action operad equivariant structure and provide a unifying treatment of coherence and strictification for them. Examples of such monoidal categories include symmetric, braided, ribbon, and coboundary monoidal categories, which naturally arise in the representation theory of quantum groups and of coboundary Hopf algebras and in the theory of crystals of finite dimensional complex reductive Lie algebras.