This book is devoted to anyone who is in search of beauty in mathematics, and mathematics in the beauty around us. Attempting to combine mathematical rigor and magnificence of the visual perception, the author is presenting the mathematical study of phyllotaxis, the most beautiful phenomenon of the living nature. The distinctive feature of this book is an animation feature that explains the work of mathematical models and the transformation of 3D space. The analysis of the phyllotactic pattern as a system of discrete objects together with the mathematical tools of generalized sequences made it possible to find a universal algorithm for calculating the divergence angle. In addition, it is serving as a new proof of the fundamental theorem of phyllotaxis and analytically confirming well-known formulas obtained intuitively earlier as well as casting some doubts on a few stereotypes existing in mathematical phyllotaxis. The presentation of phyllotaxis morphogenesis as a recursive process allowed the author to formulate the hydraulic model of phyllotaxis morphogenesis and propose a method for its experimental verification. With the help of artificial intelligence, the author offered methodology for the digital measurement of phyllotaxis allowing a transition to a qualitatively new level in the study of plant morphogenesis. Due to the successful combination of mathematical constructions and their visual presentation, the materials of this study are comprehensible to readers with high school advanced mathematical levels.

Foreword by Stephen L Adler (Institute for Advanced Study, USA) Illustrations by Peggy Adler The term Phyllotaxis refers to the patterns on plants formed by the arrangement of repeated biological units. In nearly all cases, the Fibonacci Numbers and the Golden Ratio occur in these arrangements. This topic has long fascinated scientists. Over a period of more than two decades, Irving Adler wrote a number of papers that construct a rigorously derived mathematical model for Phyllotaxis, which are major and enduring contributions to the field. These papers are collected in this reprint volume to enable their access to a wider readership.

Biologists have long dismissed mathematics as being unable to meaningfully contribute to our understanding of living beings. Within the past ten years, however, mathematicians have proven that they hold the key to unlocking the mysteries of our world -- and ourselves. In The Mathematics of Life, Ian Stewart provides a fascinating overview of the vital but little-recognized role mathematics has played in pulling back the curtain on the hidden complexities of the natural world -- and how its contribution will be even more vital in the years ahead. In his characteristically clear and entertaining fashion, Stewart explains how mathematicians and biologists have come to work together on some of the most difficult scientific problems that the human race has ever tackled, including the nature and origin of life itself.

This book is the fifth volume of papers on advanced problems of phase transitions and critical phenomena, the first four volumes appeared in 2004, 2007, 2012, and 2015. It aims to compile reviews in those aspects of criticality and related subjects that are of current interest. The seven chapters discuss criticality of complex systems, where the new, emergent properties appear via collective behaviour of simple elements. Since all complex systems involve cooperative behaviour between many interconnected components, the field of phase transitions and critical phenomena provides a very natural conceptual and methodological framework for their study.As the first four volumes, this book is based on the review lectures that were given in Lviv (Ukraine) at the 'Ising lectures' — a traditional annual workshop on phase transitions and critical phenomena which aims to bring together scientists working in the field of phase transitions with university students and those who are interested in the subject.

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.

A breathtakingly illustrated look at botanical spirals and the scientists who puzzled over them Charles Darwin was driven to distraction by plant spirals, growing so exasperated that he once begged a friend to explain the mystery “if you wish to save me from a miserable death.” The legendary naturalist was hardly alone in feeling tormented by these patterns. Plant spirals captured the gaze of Leonardo da Vinci and became Alan Turing’s final obsession. This book tells the stories of the physicists, mathematicians, and biologists who found themselves magnetically drawn to Fibonacci spirals in plants, seeking an answer to why these beautiful and seductive patterns occur in botanical forms as diverse as pine cones, cabbages, and sunflowers. Do Plants Know Math? takes you down through the centuries to explore how great minds have been captivated and mystified by Fibonacci patterns in nature. It presents a powerful new geometrical solution, little known outside of scientific circles, that sheds light on why regular and irregular spiral patterns occur. Along the way, the book discusses related plant geometries such as fractals and the fascinating way that leaves are folded inside of buds. Your neurons will crackle as you begin to see the connections. This book will inspire you to look at botanical patterns—and the natural world itself—with new eyes. Featuring hundreds of gorgeous color images, Do Plants Know Math? includes a dozen creative hands-on activities and even spiral-plant recipes, encouraging readers to explore and celebrate these beguiling patterns for themselves.

Solving the Riddle of Microsoft and Your Computer provides easy to follow step by step instructions. Chapter one provides some useful facts about your computer. This Chapter describes such things as how to operate your mouse; how to modify your screen saver and printer settings; options for printing files; how to use different functions of the calculator; playing and storing music; using Snipping Tool to capture images; determining amount of Radom Access Memory (RAM) is on your computer; determining your computer name; how to use Microsoft Excel to keep track of financial transactions. Chapter two discusses how to use Microsoft File Explorer. I refer to File Explorer as the electronic filing cabinet. This chapter provides a description of the drives in your computer; how to create top level and sub-folders in your C-Drive also known as the hard drive; how to find files in File Explorer; copying files from the C-Drive and changing properties on files. Chapter three is the real meat of this book. It provides detailed information on the most useful Microsoft Word functions. You will learn how to use pull down menus to include (Find, Replace, Tracking Changes, Accepting or Rejecting Changes, saving files, Print Preview, Printing, inserting Headers and Footers); additional topics on cover pages, inserting blank pages, inserting page breaks, inserting tables and inserting pictures; next you will learn how to adjust margins, adjust orientation, adjusting size of paper, inserting columns, inserting page and section breaks, how to vertical line numbers; an in depth discussion concerning interactive table of contents; other useful tools include inserting symbols, inserting footnotes, inserting text boxes, inserting other files into Word, how to use Format Painter, description of formatting fonts and how to insert splits in Word.

In many physical problems several scales are present in space or time, caused by inhomogeneity of the medium or complexity of the mechanical process. A fundamental approach is to first construct micro-scale models, and then deduce the macro-scale laws and the constitutive relations by properly averaging over the micro-scale. The perturbation method of multiple scales can be used to derive averaged equations for a much larger scale from considerations of the small scales. In the mechanics of multiscale media, the analytical scheme of upscaling is known as the Theory of Homogenization.The authors share the view that the general methods of homogenization should be more widely understood and practiced by applied scientists and engineers. Hence this book is aimed at providing a less abstract treatment of the theory of homogenization for treating inhomogeneous media, and at illustrating its broad range of applications. Each chapter deals with a different class of physical problems. To tackle a new problem, the approach of first discussing the physically relevant scales, then identifying the small parameters and their roles in the normalized governing equations is adopted. The details of asymptotic analysis are only explained afterwards.

This enlightening and gorgeously illustrated book explores the beauty and mystery of the divine proportion in art, architecture, nature, and beyond. From the pyramids of Giza, to quasicrystals, to the proportions of the human face, the golden ratio has an infinite capacity to generate shapes with exquisite properties. Author Gary Meisner has spent decades researching the subject, investigating and collaborating with people across the globe in dozens of professions and walks of life. In The Golden Ratio, he shares his enlightening journey. Exploring the long history of this fascinating number, as well as new insights into its power and potential applications, The Golden Ratio invites you to take a new look at this timeless topic.