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Fig. 1. Trees are an example of a natural fractal object. Although the patterns observed at different magnifications do not repeat exactly, analysis shows them to have the same statistical qualities (photograph by R.P. Taylor).

Figure 1 Trees are an example of a natural fractal object. Although the patterns observed at different magnifications do not repeat exactly, analysis shows them to have the same statistical qualities (photograph by R.P. Taylor).

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Fig. 2. Examples of natural forms exhibiting different D- values: 1.0 (top: horizon line); 1.3 (middle: clouds); and 1.9 (bottom: tree branches).
Fig. 2. Examples of natural forms exhibiting different D- values: 1.0 (top: horizon line); 1.3 (middle: clouds); and 1.9 (bottom: tree branches).
Fractal dimension (D) of several typical natural forms Table 1
Fractal dimension (D) of several typical natural forms Table 1
Fig. 3. Natural images and corresponding D-values used in the present study: (top row) cauliflower (D = 1.1), mountain (D = 1.2), stars (D = 1.23); (middle row) river (D = 1.3), lightning (D = 1.3), waves (D = 1.3), clouds (1.33); (bottom row) mud cracks (D = 1.7), tree branches (1.9).
Fig. 3. Natural images and corresponding D-values used in the present study: (top row) cauliflower (D = 1.1), mountain (D = 1.2), stars (D = 1.23); (middle row) river (D = 1.3), lightning (D = 1.3), waves (D = 1.3), clouds (1.33); (bottom row) mud cracks (D = 1.7), tree branches (1.9).
Fig. 4. Mathematical fractal images used in this study: simulated coastline images with D values of 1.33 (top row); 1.50 (middle row); and 1.86 (bottom row).
Fig. 4. Mathematical fractal images used in this study: simulated coastline images with D values of 1.33 (top row); 1.50 (middle row); and 1.86 (bottom row).
Fig. 5. Selection of cropped images of Pollock’s paintings used in this study in the category of human produced fractals: (first row) five cropped images with a fractal dimension of 1.12, extracted from “Untitled”, 1945 (private collection); (second row) five cropped images with a fractal dimension of 1.50, extracted from “Number 14’, 1948 (Yale University Art Gallery, USA); (third row) five cropped images with a fractal dimension of 1.66, extracted from ‘Number 32”, 1950 (Kunstsammlung Nordhein-Westfalen, Germany); (fourth row) five cropped images with a fractal dimension of 1.89, extracted from an unnamed work from 1950 that is no longer in existence (i.e. Pollock painted over this picture).
Fig. 5. Selection of cropped images of Pollock’s paintings used in this study in the category of human produced fractals: (first row) five cropped images with a fractal dimension of 1.12, extracted from “Untitled”, 1945 (private collection); (second row) five cropped images with a fractal dimension of 1.50, extracted from “Number 14’, 1948 (Yale University Art Gallery, USA); (third row) five cropped images with a fractal dimension of 1.66, extracted from ‘Number 32”, 1950 (Kunstsammlung Nordhein-Westfalen, Germany); (fourth row) five cropped images with a fractal dimension of 1.89, extracted from an unnamed work from 1950 that is no longer in existence (i.e. Pollock painted over this picture).
Fig. 6. Aesthetic preference for fractal images of different origin: average proportion by which the image was preferred among others as a function of fractal dimension for natural images (top panel); simulated coastlines (middle panel); and cropped images of Pollock’s paintings (bottom panel).
Fig. 6. Aesthetic preference for fractal images of different origin: average proportion by which the image was preferred among others as a function of fractal dimension for natural images (top panel); simulated coastlines (middle panel); and cropped images of Pollock’s paintings (bottom panel).
Fig. 7. Aesthetic preference for control random images: average proportion by which the image was preferred among others as a function of image density.
Fig. 7. Aesthetic preference for control random images: average proportion by which the image was preferred among others as a function of image density.
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Fractal Geometry
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