Volume 12 of the Math-Art series. This 3-part book is a visual exploration of knot geometry and ethnomathematics to celebrate the similarities between abstract geometry and unique cultures worldwide. Starting at latitude 0º, longitude 0º, the author set sail (virtually) westward at an average of 400 (nautical) knots a week to fully cover its circumference and explore 1 new knot each week for an entire year. Part I is the art portfolio extracted from the geometry models, part II is a detailed record of the original geometry used to create the artwork, and part III is the weekly wind map log showing the project’s positioning, actual winds, and currents in real-time. Each book includes 52 illustrations, notes, and references.

Volume 12 of the Math-Art series. This 3-part book is a visual exploration of knot geometry and ethnomathematics to celebrate the similarities between abstract geometry and unique cultures worldwide. Starting at latitude 0º, longitude 0º, the author set sail (virtually) westward at an average of 400 (nautical) knots a week to fully cover its circumference and explore 1 new knot each week for an entire year. Part I is the art portfolio extracted from the geometry models, part II is a detailed record of the original geometry used to create the artwork, and part III is the weekly wind map log showing the project’s positioning, actual winds, and currents in real-time. Each book includes 52 illustrations, notes, and references.

Volume 12 of the Math-Art series. This 3-part book is a visual exploration of knot geometry and ethnomathematics to celebrate the similarities between abstract geometry and unique cultures worldwide. Starting at latitude 0º, longitude 0º, the author set sail (virtually) westward at an average of 400 (nautical) knots a week to fully cover its circumference and explore 1 new knot each week for an entire year. Part I is the art portfolio extracted from the geometry models, part II is a detailed record of the original geometry used to create the artwork, and part III is the weekly wind map log showing the project’s positioning, actual winds, and currents in real-time. Each book includes 52 illustrations, notes, and references.

The 52 Illustration Prime Number series is a new chapter in the ongoing Math-Art collection exploring the world of mathematics and art. Inspired by the research of mathematicians from yesterday and today, this project aims to explore the visual aspect of numbers and highlight the unexpected connections between the challenging world of calculus, geometry, and art. Some will find references to ethnomathematics or a reflection on the universal cross-cultural appeal of mathematics; others will find a relation with the world we’re mapping for tomorrow, and hopefully, all will enjoy this unexpected interpretation of numbers from an artistic standpoint.

A 52 illustration two-part book on the exploration of minimal surfaces. Part 1 explores the surface from an artistic perspective, and part 2 visually reproduces the equations that stand in their own right as a beautiful expression of pure geometry. Each book includes notes from an informal work-in-progress diary and references directing the reader to the images’ original mathematical source. Both sides complement each other in helping us appreciate better these unrivaled expressions of our environment found in nature, from butterflies to black holes, and studied in statistics, material sciences, and architecture.

Knots are familiar objects. Yet the mathematical theory of knots quickly leads to deep results in topology and geometry. This work offers an introduction to this theory, starting with our understanding of knots. It presents the applications of knot theory to modern chemistry, biology and physics.

The study of 3-dimensional spaces brings together elements from several areas of mathematics. The most notable are topology and geometry, but elements of number theory and analysis also make appearances. In the past 30 years, there have been striking developments in the mathematics of 3-dimensional manifolds. This book aims to introduce undergraduate students to some of these important developments. Low-Dimensional Geometry starts at a relatively elementary level, and its early chapters can be used as a brief introduction to hyperbolic geometry. However, the ultimate goal is to describe the very recently completed geometrization program for 3-dimensional manifolds. The journey to reach this goal emphasizes examples and concrete constructions as an introduction to more general statements. This includes the tessellations associated to the process of gluing together the sides of a polygon. Bending some of these tessellations provides a natural introduction to 3-dimensional hyperbolic geometry and to the theory of kleinian groups, and it eventually leads to a discussion of the geometrization theorems for knot complements and 3-dimensional manifolds. This book is illustrated with many pictures, as the author intended to share his own enthusiasm for the beauty of some of the mathematical objects involved. However, it also emphasizes mathematical rigor and, with the exception of the most recent research breakthroughs, its constructions and statements are carefully justified.

A detailed mathematical derivation of space curves is presented that links the diverse fields of superfluids, quantum mechanics, Navier-Stokes hydrodynamics, and Maxwell electromagnetism by a common foundation. The basic mathematical building block is called the theory of quantum torus knots (QTK).

In this book the classical Greek construction problems are explored in a didactical, enquiry based fashion using Interactive Geometry Software (IGS). The book traces the history of these problems, stating them in modern terminology. By focusing on constructions and the use of IGS the reader is confronted with the same problems that ancient mathematicians once faced. The reader can step into the footsteps of Euclid, Viète and Cusanus amongst others and then by experimenting and discovering geometric relationships far exceed their accomplishments. Exploring these problems with the neusis-method lets him discover a class of interesting curves. By experimenting he will gain a deeper understanding of how mathematics is created. More than 100 exercises guide him through methods which were developed to try and solve the problems. The exercises are at the level of undergraduate students and only require knowledge of elementary Euclidean geometry and pre-calculus algebra. It is especially well-suited for those students who are thinking of becoming a mathematics teacher and for mathematics teachers.

For over 100 years the Poincare Conjecture, which proposes a topological characterization of the 3-sphere, has been the central question in topology. Since its formulation, it has been repeatedly attacked, without success, using various topological methods. Its importance and difficulty were highlighted when it was chosen as one of the Clay Mathematics Institute's seven Millennium Prize Problems. in 2002 and 2003 Grigory Perelman posted three preprints showing how to use geometric arguments, in particular the Ricci flow as introduced and studied by Hamilton, to establish the Poincare Conjecture in the affirmative. This book provides full details of a complete proof of the Poincare Conjecture following Perelman's three preprints. After a lengthy introduction that outlines the entire argument, the book is divided into four parts. The first part reviews necessary results from Riemannian geometry and Ricci flow, including much of Hamilton's work. The second part starts with Perelman's length function, which is used to establish crucial non-collapsing theorems. Then it discusses the classification of non-collapsed, ancient solutions to the Ricci flow equation. The third part concerns the existence of Ricci flow with surgery for all positive time and an analysis of the topological and geometric changes introduced by surgery. The last part follows Perelman's third preprint to prove that when the initial Riemannian 3-manifold has finite fundamental group, Ricci flow with surgery becomes extinct after finite time. The proofs of the Poincare Conjecture and the closely related 3-dimensional spherical space-form conjectu The existence of Ricci flow with surgery has application to 3-manifolds far beyond the Poincare Conjecture. It forms the heart of the proof via Ricci flow of Thurston's Geometrization Conjecture. Thurston's Geometrization Conjecture, which classifies all compact 3-manifolds, will be the subject of a follow-up article. The organization of the material in this book differs from that given by Perelman. From the beginning the authors present all analytic and geometric arguments in the context of Ricci flow with surgery. in addition, the fourth part is a much-expanded version of Perelman's third preprint; it gives the first complete and detailed proof of the finite-time extinction theorem. With the large amount of background material that is presented and the detailed versions of the central arguments, this book is suitable for all mathematicians from advanced graduate students to specialists in geometry and topology. Clay Mathematics Institute Monograph Series The Clay Mathematics Institute Monograph Series publishes selected expositions of recent developments, both in emerging areas and in older subjects transformed by new insights or unifying ideas. Information for our distributors: Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).