بعد كسيري

11.5 x 200 = 2300 km

28 x 100 = 2800 km

70 x 50 = 3500 km

Figure 1. As the length of the measuring stick is scaled smaller and smaller, the total length of the coastline measured increases.

البعد الكسيري هي نسبة تعطي إشارة إحصائية حول التعقد. قد يكون البعد الكسيري عدداً غير صحيح.

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 مقدمة

Figure 2. A 32-segment quadric fractal scaled and viewed through boxes of different sizes. The pattern illustrates self similarity. The theoretical fractal dimension for this fractal is log32/log8 = 1.67; its empirical fractal dimension from box counting analysis is ±1% using fractal analysis software.

 دور المقياس

Figure 4. Traditional notions of geometry for defining scaling and dimension.

${\displaystyle{\displaystyle{N\propto\epsilon^{-D}}}}$

(1)

Figure 5. The first four iterations of the Koch snowflake, which has an approximate Hausdorff dimension of 1.2619.

${\displaystyle{\displaystyle{\log_{\epsilon}{N}={-D}={\frac{\log{N}}{\log{\epsilon}}}}}}$

(2)

 D هي ليست واصف منفرد

Figure 6. Two L-systems branching fractals that are made by producing 4 new parts for every 1/3 scaling so have the same theoretical ${\displaystyle{\displaystyle D}}$ as the Koch curve and for which the empirical box counting ${\displaystyle{\displaystyle D}}$ has been demonstrated with 2% accuracy.

 انظر أيضاً

• List of fractals by Hausdorff dimension
• Lacunarity
• Multifractal analysis
• Fractal derivative

 للاستزادة

• Mandelbrot, Benoit B., The (Mis)Behavior of Markets, A Fractal View of Risk, Ruin and Reward (Basic Books, 2004)

 وصلات خارجية

• TruSoft's Benoit, fractal analysis software product calculates fractal dimensions and hurst exponents.
• A Java Applet to Compute Fractal Dimensions
• Introduction to Fractal Analysis
• Bowley, Roger (2009). "Fractal Dimension". Sixty Symbols. Brady Haran for the University of Nottingham.