page_title: Fractal dimension
section_title: Role of scaling
hierarchical_section_title: Fractal dimension / Role of scaling
caption_attribution_description: English: Aid to defining fractal dimensions. Traditional notions of geometry for defining scaling and dimension.
context_page_description: In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in; a fractal dimension does not have to be an integer.
  The essential idea of "fractured" dimensions has a long history in mathematics, but the term itself was brought to the fore by Benoit Mandelbrot based on his 1967 paper on self-similarity in which he discussed fractional dimensions. In that paper, Mandelbrot cited previous work by Lewis Fry Richardson describing the counter-intuitive notion that a coastline's measured length changes with the length of the measuring stick used. In terms of that notion, the fractal dimension of a coastline quantifies how the number of scaled measuring sticks required to measure the coastline changes with the scale applied to the stick. There are several formal mathematical definitions of fractal dimension that build on this basic concept of change in detail with change in scale.
context_section_description: The concept of a fractal dimension rests in unconventional views of scaling and dimension. As Fig. 4 illustrates, traditional notions of geometry dictate that shapes scale predictably according to intuitive and familiar ideas about the space they are contained within, such that, for instance, measuring a line using first one measuring stick then another 1/3 its size, will give for the second stick a total length 3 times as many sticks long as with the first. This holds in 2 dimensions, as well. If one measures the area of a square then measures again with a box of side length 1/3 the size of the original, one will find 9 times as many squares as with the first measure. Such familiar scaling relationships can be defined mathematically by the general scaling rule in Equation 1, where the variable  stands for the number of sticks,  for the scaling factor, and  for the fractal dimension:
  This scaling rule typifies conventional rules about geometry and dimension – for lines, it quantifies that, because  when  as in the example above,  and for squares, because  when 
  The same rule applies to fractal geometry but less intuitively. To elaborate, a fractal line measured at first to be one length, when remeasured using a new stick scaled by 1/3 of the old may not be the expected 3 but instead 4 times as many scaled sticks long. In this case,  when  and the value of  can be found by rearranging Equation 1:
  That is, for a fractal described by  when  a non-integer dimension that suggests the fractal has a dimension not equal to the space it resides in. The scaling used in this example is the same scaling of the Koch curve and snowflake. Of note, the images shown are not true fractals because the scaling described by the value of  cannot continue infinitely for the simple reason that the images only exist to the point of their smallest component, a pixel. The theoretical pattern that the digital images represent, however, has no discrete pixel-like pieces, but rather is composed of an infinite number of infinitely scaled segments joined at different angles and does indeed have a fractal dimension of 1.2619.
caption: Figure 4. Traditional notions of geometry for defining scaling and dimension.
   
    
      
        
          1
        
      
      {\displaystyle 1}
    
  , 
    
      
        
          
            1
            
              2
            
          
          
            =
          
          1
        
      
      {\displaystyle 1^{2}{=}1}
    
  , 
    
      
        
          
            1
            
              3
            
          
          
            =
          
          1
        
      
      {\displaystyle 1^{3}{=}1}
    
  
   
    
      
        
          2
        
      
      {\displaystyle 2}
    
  , 
    
      
        
          
            2
            
              2
            
          
          
            =
          
          4
        
      
      {\displaystyle 2^{2}{=}4}
    
  , 
    
      
        
          
            2
            
              3
            
          
          
            =
          
          8
        
      
      {\displaystyle 2^{3}{=}8}
    
  
   
    
      
        
          3
        
      
      {\displaystyle 3}
    
  , 
    
      
        
          
            3
            
              2
            
          
          
            =
          
          9
        
      
      {\displaystyle 3^{2}{=}9}
    
  , 
    
      
        
          
            3
            
              3
            
          
          
            =
          
          27
        
      
      {\displaystyle 3^{3}{=}27}
    
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