In this invaluable book, the basic mathematical properties of the golden ratio and its occurrence in the dimensions of two- and three-dimensional figures with fivefold symmetry are discussed. In addition, the generation of the Fibonacci series and generalized Fibonacci series and their relationship to the golden ratio are presented. These concepts are applied to algorithms for searching and function minimization. The Fibonacci sequence is viewed as a one-dimensional aperiodic, lattice and these ideas are extended to two- and three-dimensional Penrose tilings and the concept of incommensurate projections. The structural properties of aperiodic crystals and the growth of certain biological organisms are described in terms of Fibonacci sequences. Contents: Basic Properties of the Golden Ratio; Geometric Problems in Two Dimensions; Geometric Problems in Three Dimensions; Fibonacci Numbers; Lucas Numbers and Generalized Fibonacci Numbers; Continued Fractions and Rational Approximants; Generalized Fibonacci Representation Theorems; Optimal Spacing and Search Algorithms; Commensurate and Incommensurate Projections; Penrose Tilings; Quasicrystallography; Biological Applications; Construction of the Regular Pentagon; The First 100 Fibonacci and Lucas Numbers; Relationships Involving the Golden Ratio and Generalized Fibonacci Numbers. Readership: Applied mathematicians.
Funded by the National Science Foundation and successfully field-tested in a variety of settings, the materials presented give teachers the opportunity to grow as learners for the classes they teach.
Containing more than 1,000 entries, the Dictionary of Classical and Theoretical Mathematics focuses on mathematical terms and definitions of critical importance to practicing mathematicians and scientists. This single-source reference provides working definitions, meanings of terms, related references, and a list of alternative terms and definitions. The dictionary is one of five constituent works that make up the casebound CRC Comprehensive Dictionary of Mathematics.
An alphabetically arranged handbook contains essays on two hundred key principles, from Kepler's laws of planetary motion and Mendel's laws of genetics, to lesser-known laws that explain everything from black holes to sunflower patterns.
With challenging new standards-based middle school mathematics curricula now in place, future teachers need college-level mathematics instruction that better prepares them for their professional careers. Addresses the importance of learning calculus in preparation for the teaching of middle school mathematics, focusing on concepts and applications to illuminate the connections that exist between college-level calculus and the mathematics taught in today's middle schools. Examines the unique needs of future teachers in comparison to general calculus books. Initiates new topics with engaging discussion rather than the standard formula-proof-example approach. Stresses the interplay between geometry and calculus, and demonstrates the essential power of calculus for computing areas, lengths, surface areas, and volumes. For current or future mathematics teachers, or anyone interested in learning more about calculus.
Presents maths using the rule of four by introducing concepts numerically, analytically, graphically and verbally. This text focuses on algebra as a pattern-seeking device and shows how maths is part of historical heritage.
This text places mathematics into a real-world setting. Concepts are developed through the Rule of Four - Numeric, Analytic, Graphic, and Verbal. Graphing utilities are integrated as a tool while the focus is on the mathematics.
A connection between abstract mathematical concepts and the real world settings is emphasized in this book. It provides analytical, graphical, numerical, and verbal approaches to major topics and includes regular use of graphing utilities where appropriate as well as discussions of the advantages and limitations of technology.
This book offers a general review of the voluminous theoretical and experimental literature pertaining to physical self-replicating systems. The principal focus here is on self-replicating machine systems. Most importantly, we are concerned with kinematic self-replicating machines: systems in which actual physical objects, not mere patterns of information, undertake their own replication. Following a brief burst of activity in the 1950s and 1980s, the field of kinematic replicating systems design received new interest in the 1990s with the emerging recognition of the feasibility of molecular nanotechnology. The field has experienced a renaissance of research activity since 1999 as researchers have come to recognize that replicating systems are simple enough to permit experimental laboratory demonstrations of working devices.
Starting with nothing more than basic high school algebra, this volume leads readers gradually from basic algebra to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. Features an informal writing style and includes many numerical examples. Emphasizes the methods used for proving theorems rather than specific results. Includes a new chapter on big-Oh notation and how it is used to describe the growth rate of number theoretic functions and to describe the complexity of algorithms. Provides a new chapter that introduces the theory of continued fractions. Includes a new chapter on “Continued Fractions, Square Roots and Pell's Equation.” Contains additional historical material, including material on Pell's equation and the Chinese Remainder Theorem. A useful reference for mathematics teachers.
What is discrete algorithmic mathematics. Mathematical preliminaries. Algorithms. Mathematical induction. Graphs and trees. Fundamental counting methods. Difference equations. Probability. An introduction to mathematical logic. Algorithmic linear algebra. Infinite processes in discrete mathematics. Sorting things out with sorting.
401 stories which sift through the implausible, the unlikely and the bizarre to reveal the bare facts. This book takes a look at the truths and the misconceptions surrounding the natural and the supernatural world, its past, present and future.
This series contains materials that cover Key Stage 3 National Curriculum science and the Scheme of Work for Science. This particular textbook includes learning objectives that are clearly defined at the start of each unit, key words, learning summary pages in each unit to support the development of study skills and a range of inspirational materials to challenge pupils.
One Of The Most Fascinating Math Theories In The World Of Nature Is The Fibonacci Sequence. Fibonacci Was A Man Who Calculated An Amazing Pattern That Is Followed By Many Things In Nature. Read About This Mind Boggling Theory And See For Yourself Where It Occurs In The Natural World.