Product Description
Uniform polyhedron. Polyhedron, Regular polygon, Face (geometry), Vertex (geometry) , Isometry , Symmetry, Convex polytope , Star polyhedron , Prismatic uniform polyhedron , Nonconvex uniform polyhedron , Cube , Octahedron, Tetrahedral symmetry. ... Read more

Product Description
Chapters: Convex Uniform Honeycomb, Convex Uniform Honeycombs in Hyperbolic Space, Hypercubic Honeycomb, Icositetrachoric Honeycomb, Alternated Hypercubic Honeycomb, Weaire-phelan Structure, Uniform Tessellation, Tetrahedral-Octahedral Honeycomb, Bitruncated Cubic Honeycomb, Gyrated Tetrahedral-Octahedral Honeycomb, Order-4 Dodecahedral Honeycomb, Tesseractic Honeycomb, Quarter Cubic Honeycomb, Penteractic Honeycomb, Truncated Hexagonal Prismatic Honeycomb, Cantitruncated Cubic Honeycomb, Gyrated Triangular Prismatic Honeycomb, Demitesseractic Honeycomb, Gyroelongated Alternated Cubic Honeycomb, Octeractic Honeycomb, Rectified Cubic Honeycomb, Demipenteractic Honeycomb, Truncated Alternated Cubic Honeycomb, Order-5 Dodecahedral Honeycomb, Demiocteractic Honeycomb, Demihepteractic Honeycomb, Truncated Square Prismatic Honeycomb, Order-3 Icosahedral Honeycomb, Cantitruncated Alternated Cubic Honeycomb, Omnitruncated Cubic Honeycomb, Runcinated Alternated Cubic Honeycomb, Rhombitriangular-Hexagonal Prismatic Honeycomb, Omnitruncated Triangular-Hexagonal Prismatic Honeycomb, Snub Triangular-Hexagonal Prismatic Honeycomb, Gyroelongated Triangular Prismatic Honeycomb, Snub Square Prismatic Honeycomb, Disphenoid Tetrahedral Honeycomb, Demihexeractic Honeycomb, Cantellated Cubic Honeycomb, Runcitruncated Cubic Honeycomb, Order-5 Cubic Honeycomb, Cell, Rhombic Dodecahedral Honeycomb, Trapezo-Rhombic Dodecahedral Honeycomb, Voronoi Tessellation. Source: Wikipedia. Pages: 160. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In geometry, a convex uniform honeycomb is a tessellation of convex uniform polyhedron cells. In hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs in hyperbolic 3-space, generated as Wythoff constructions, and represented by ring permuta...More: http://booksllc.net/?id=22952807 ... Read more

Product Description
At the heart of this monograph is the Brunn-Minkowski theory. It can be used to great effect in studying such ideas as volume and surface area and the generalizations of these. In particular the notions of mixed volume and mixed area arise naturally and the fundamental inequalities that are satisfied by mixed volumes are considered in detail. The author presents a comprehensive introduction to convex bodies and gives full proofs for some deeper theorems that have never previously been brought together. Many hints and pointers to connections with other fields are given, and an exhaustive reference list is included. ... Read more

Product Description

New variational methods by Aubry, Mather, and Mane, discovered in the last twenty years, gave deep insight into the dynamics of convex Lagrangian systems. This book shows how this Principle of Least Action appears in a variety of settings (billiards, length spectrum, Hofer geometry, modern symplectic geometry). Thus, topics from modern dynamical systems and modern symplectic geometry are linked in a new and sometimes surprising way. The central object is Mather's minimal action functional. The level is for graduate students onwards, but also for researchers in any of the subjects touched in the book.

Product Description

This book constitutes the thoroughly refereed post-conference proceedings of the Kyoto Conference on Computational Geometry and Graph Theory, KyotoCGGT 2007, held in Kyoto, Japan, in June 2007, in honor of Jin Akiyama and Vašek Chvátal, on the occasion of their 60th birthdays.

The 19 revised full papers, presented together with 5 invited papers, were carefully selected during two rounds of reviewing and improvement from more than 60 talks at the conference. All aspects of Computational Geometry and Graph Theory are covered, including tilings, polygons, impossible objects, coloring of graphs, Hamilton cycles, and factors of graphs.

Product Description

Product Description

Projective geometry is one of the most fundamental and at the same time most beautiful branches of geometry. It can be considered the common foundation of many other geometric disciplines like Euclidean geometry, hyperbolic and elliptic geometry or even relativistic space-time geometry. This book offers a comprehensive introduction to this fascinating field and its applications. It in particular explains how metric concepts may be best understood in projective terms. One of the major themes that appears throughout this book is the beauty of the interplay of geometry, algebra and combinatorics. This book can especially be used as a guide that explains how geometric objects and operations may be most elegantly expressed in algebraic terms, making it a valuable resource for mathematicians as well as for computer scientists and physicists. The book is based on the author’s experience in implementing geometric software and includes hundreds of high quality illustrations.

Product Description
The theory of toric varieties (also called torus embeddings) describes a fascinating interplay between algebraic geometry and the geometry of convex figures in real affine spaces. This book is a unified up-to-date survey of the various results and interesting applications found since toric varieties were introduced in the early 1970's. It is an updated and corrected English edition of the author's book in Japanese published by Kinokuniya, Tokyo in 1985. Toric varieties are here treated as complex analytic spaces. Without assuming much prior knowledge of algebraic geometry, the author shows how elementary convex figures give rise to interesting complex analytic spaces. Easily visualized convex geometry is then used to describe algebraic geometry for these spaces, such as line bundles, projectivity, automorphism groups, birational transformations, differential forms and Mori's theory. Hence this book might serve as an accessible introduction to current algebraic geometry. Conversely, the algebraic geometry of toric varieties gives new insight into continued fractions as well as their higher-dimensional analogues, the isoperimetric problem and other questions on convex bodies. Relevant results on convex geometry are collected together in the appendix. ... Read more

Customer Reviews (1)

Advanced but readable
This book is an advanced overview of the theory of toric varieties written for individuals with a strong background in algebraic geometry, topology, and algebra. It is very formal and not for a beginning course.

The author moves right into the necessary convex geometry in the first section of Chapter 1 and then defines a toric variety in the next section. He does not hesitate to use diagrams to illustrate the examples, which is good given the level of abstraction he employs in the book. The fundamental group of a toric variety is given an explicit characterization, but the proof is omitted unfortunately. This is followed by a discussion of when a toric variety is compact and nonsingular, with detailed proofs given. The Hironaka resolution of singularities theorem is discussed for toric varieties, the proof being a lot simpler of course in this case. A concrete realization of singularity resolution using continued fractions is given in the next section. The chapter ends with a very detailed and superb discussion of the birational geometry of toric varieties.

The next chapter is very involved and deals with Cartier divisors on toric varieties and toric projective varieties. The latter are related to convex polytopes by means of moment maps. In particular, integral convex polytopes have many connections with toric projective varieties, and these are outlined in detail in this chapter. A toric version of Mori's theorem is also outlined. Toric varieties offer a nice, intuitive picture of Mori's program for rational curves on projective varieties.

Chapter 3 deals with differential forms on toric varieties. The author employs the sheaf of germs of holomorphic vector fields with logarithmic zeroes and the sheaf of germs of p-forms with logarithmic poles to study holomorphic differential forms over toric varieties. In addition, Ishida complexes are used to study complexes of coherent sheaves on toric varieties. A very interesting discussion on the automorphism groups of toric varieties is given in terms of Cremona groups.

The last chapter discusses applications, such as Mumford toroidal embeddings, quotients of toric varieties, semisimple algebraic groups, and Newton polyhedra. Unfortunately, the author does not expound on these, but refers the reader to the literature. Instead, the author explains how to construct complex manifolds in dimension two by taking the quotient of an open set of a toric variety with respect to an action of a discrete map. Some interesting examples of compact quotientsof toric varieties are given, including complex tori, Hopf surfaces, and Inoue surfaces. The latter, for the case of parabolic Inoue surfaces, use elliptic curves in their constructions, interestingly.

The book does contain a review of convex geometry for a reader not well-versed in this area. There is a lot in this book, even though it is short. The price is very high so only for individuals seriously interested in this topic. ... Read more

Product Description
This unique monograph discusses the interaction betweenRiemannian geometry, convex programming, numerical analysis, dynamicalsystems and mathematical modelling. The book is the first account ofthe development of this subject as it emerged at the beginning of the'seventies. A unified theory of convexity of functions, dynamicalsystems and optimization methods on Riemannian manifolds is alsopresented. Topics covered include geodesics and completeness ofRiemannian manifolds, variations of the p-energy of a curve andJacobi fields, convex programs on Riemannian manifolds, geometricalconstructions of convex functions, flows and energies, applications ofconvexity, descent algorithms on Riemannian manifolds, TC and TPprograms for calculations and plots, all allowing the user to exploreand experiment interactively with real life problems in the languageof Riemannian geometry. An appendix is devoted to convexity andcompleteness in Finsler manifolds. For students and researchers in such diverse fields as pure andapplied mathematics, and applied sciences like physics, chemistry,biology, and engineering. ... Read more

Product Description
High Quality Content by WIKIPEDIA articles! In geometry, the tesseract, also called an 8-cell or regular octachoron, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8 cubical cells. The tesseract is one of the six convex regular 4-polytopes.A generalization of the cube to dimensions greater than three is called a ?hypercube?, ?n-cube? or ?measure polytope?. The tesseract is the four-dimensional hypercube, or 4-cube. ... Read more