Fractals
From GenerativeArt
(Difference between revisions)
(→Recursive Koch curve: improve pseudocode) |
(→Fractal or Box Counting dimension: Added content) |
||
<br> | <br> | ||
| - | ===Fractal or Box Counting | + | ===Fractal or Box Counting Dimension=== |
<br /> | <br /> | ||
| + | In traditional units a Koch curve will have infinite length but 0 area. This is not a useful measurement.<br /> | ||
A measurement of dimension filling. For example, D = 1.4 means it fills more than a line, but less than a plane.<br /> | A measurement of dimension filling. For example, D = 1.4 means it fills more than a line, but less than a plane.<br /> | ||
<br /> | <br /> | ||
| - | + | Where epsilon is the relative size of the division (1, 1/2, 1/3, etc.), and N is the number of copies or boxes.<br /> | |
| - | + | Using this formula for increasingly smaller divisions, if the object is a fractal the dimension will be about the same.<br /> | |
| - | + | If the dimension changes with scale the object is not a fractal.<br /> | |
| + | <br /> | ||
{{SingleImage|imageWidthPlusTen=510|imageURL=http://philipgalanter.com/generative_art/graphics/box_counting_formula.png|caption=}}<br /> | {{SingleImage|imageWidthPlusTen=510|imageURL=http://philipgalanter.com/generative_art/graphics/box_counting_formula.png|caption=}}<br /> | ||
| - | {{SingleImage|imageWidthPlusTen=510|imageURL=http://www.viz.tamu.edu/courses/viza626/10Fall/fractImg1.gif|caption=}}<br /> | + | {{SingleImage|imageWidthPlusTen=510|imageURL=http://www.viz.tamu.edu/courses/viza626/10Fall/fractImg1.gif|caption=Cantor dust breaks the segment in thirds and discards the middle one}}<br /> |
| - | {{SingleImage|imageWidthPlusTen=510|imageURL=http://www.viz.tamu.edu/courses/viza626/10Fall/fractImg2.gif|caption=}}<br /> | + | {{SingleImage|imageWidthPlusTen=510|imageURL=http://www.viz.tamu.edu/courses/viza626/10Fall/fractImg2.gif|caption=Koch curve breaks the segment into 4 segments}}<br /> |
| - | {{SingleImage|imageWidthPlusTen=510|imageURL=http://www.viz.tamu.edu/courses/viza626/10Fall/fractImg3.gif|caption=}}<br /> | + | {{SingleImage|imageWidthPlusTen=510|imageURL=http://www.viz.tamu.edu/courses/viza626/10Fall/fractImg3.gif|caption=Note that a line segment has a fractal dimension of 1}}<br /> |
| - | {{SingleImage|imageWidthPlusTen=510|imageURL=http://www.viz.tamu.edu/courses/viza626/10Fall/fractImg4.gif|caption=}}<br /> | + | {{SingleImage|imageWidthPlusTen=510|imageURL=http://www.viz.tamu.edu/courses/viza626/10Fall/fractImg4.gif|caption=Note that a square has a fractal dimension of 2}}<br /> |
| - | {{SingleImage|imageWidthPlusTen=510|imageURL=http://philipgalanter.com/generative_art/graphics/box_counting_dimension.png|caption=}}<br /> | + | {{SingleImage|imageWidthPlusTen=510|imageURL=http://philipgalanter.com/generative_art/graphics/box_counting_dimension.png|caption=Here is an illustration of the box counting method to measure the fractal dimension of actual objects}}<br /> |
