This book examines advancements in complexity science and fractal geometry. It is divided into two sections. The first section focuses on the theoretical aspects of complexity and fractality. The second section contains interdisciplinary applications. Chapter 1 presents excellent connections between fractal geometry and human and environmental biology. Chapter 2 deals with computational algorithms to study the dynamics of logistic equations. Chapter 3 applies a parametric iteration method (PIM) to solve nonlinear integral Fredholm differential equations (NFIDEs). Chapter 4 focuses on the orbits in some 3n+a Collatz sequences, observing a saturation point. Chapter 5 presents the butterfly effect in a chaotic, nonlinear, damped, and forced oscillator that transitions to chaos through cascades of periodic bifurcations. Chapter 6 shows how complexity and fractality can influence architecture. Chapter 7 introduces the telomere "funnel effect," which describes the convergence, partial independence, and equivalence of nine major factors (diet, physical activity, psychological factors, etc.) on telomeres. Chapter 8 describes urban organization on a small scale. In particular, garden design is discussed as a combination of architecture, esthetics, geometry, and fractality. Chapter 9 presents an intervention protocol supporting sports psychologists in mentally preparing for competitions and recovery after intensive training sessions. The target group of this study consists of professional and elite athletes. Chapter 10 shows how complexity economics criticizes conventional economic theory based on two results that undoubtedly characterize it: general economic equilibrium and the representation of the entire economic system by a structure corresponding to a microeconomic system. Chapter 11 describes the logarithmic decrement technique for determining the parameters of damped oscillation.
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