- Title Pages
- Dedication
- Preface
- Acknowledgments
- 0 Opening Remarks
- 1 Functions
- 2 Iterating Functions
- 3 Qualitative Dynamics: The Fate of the Orbit
- 4 Time Series Plots
- 5 Graphical Iteration
- 6 Iterating Linear Functions
- 7 Population Models
- 8 Newton, Laplace, and Determinism
- 9 Chaos and the Logistic Equation
- 10 The Butterfly Effect
- 11 The Bifurcation Diagram
- 12 Universality
- 13 Statistical Stability of Chaos
- 14 Determinism, Randomness, and Nonlinearity
- 15 Introducing Fractals
- 16 Dimensions
- 17 Random Fractals
- 18 The Box‐Counting Dimension
- 19 When do Averages Exist?
- 20 Power Laws and Long Tails
- 21 Infinities, Big and Small
- 22 Introducing Julia Sets
- 23 Complex Numbers
- 24 Julia Sets for the Quadratic Family
- 25 The Mandelbrot Set
- 26 Two‐Dimensional Discrete Dynamical Systems
- 27 Cellular Automata
- 28 Introduction to Differential Equations
- 29 One‐Dimensional Differential Equations
- 30 Two‐Dimensional Differential Equations
- 31 Chaotic Differential Equations and Strange Attractors
- 32 Conclusion
- A Review of Selected Topics from Algebra
- B Histograms and Distributions
- C Suggestions for Further Reading
- References
- Index

# The Box‐Counting Dimension

# The Box‐Counting Dimension

- Chapter:
- (p.187) 18 The Box‐Counting Dimension
- Source:
- Chaos and Fractals
- Author(s):
### David P. Feldman

- Publisher:
- Oxford University Press

There are several examples of fractals that are not exactly self-similar, as is the case with small parts of the random Koch curve which exhibit statistical self-similarity but not identicality. The dimension of a handful of fractal objects, including the Cantor set, the Sierpiński triangle and carpet, and the Koch curve, can be determined. This chapter considers the box-counting dimension, which extends the concept of dimension to objects that are not exactly self-similar. Instead of focusing on how many small copies of an object are contained in a large copy, it explains how the volume or size of the overall shape changes as measurement scales change.

*Keywords:*
fractals, self-similarity, box-counting dimension, Sierpiński triangle, Koch curve, Cantor set

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- Title Pages
- Dedication
- Preface
- Acknowledgments
- 0 Opening Remarks
- 1 Functions
- 2 Iterating Functions
- 3 Qualitative Dynamics: The Fate of the Orbit
- 4 Time Series Plots
- 5 Graphical Iteration
- 6 Iterating Linear Functions
- 7 Population Models
- 8 Newton, Laplace, and Determinism
- 9 Chaos and the Logistic Equation
- 10 The Butterfly Effect
- 11 The Bifurcation Diagram
- 12 Universality
- 13 Statistical Stability of Chaos
- 14 Determinism, Randomness, and Nonlinearity
- 15 Introducing Fractals
- 16 Dimensions
- 17 Random Fractals
- 18 The Box‐Counting Dimension
- 19 When do Averages Exist?
- 20 Power Laws and Long Tails
- 21 Infinities, Big and Small
- 22 Introducing Julia Sets
- 23 Complex Numbers
- 24 Julia Sets for the Quadratic Family
- 25 The Mandelbrot Set
- 26 Two‐Dimensional Discrete Dynamical Systems
- 27 Cellular Automata
- 28 Introduction to Differential Equations
- 29 One‐Dimensional Differential Equations
- 30 Two‐Dimensional Differential Equations
- 31 Chaotic Differential Equations and Strange Attractors
- 32 Conclusion
- A Review of Selected Topics from Algebra
- B Histograms and Distributions
- C Suggestions for Further Reading
- References
- Index