Fractal Dimension

From Math Images

Revision as of 10:12, 16 June 2011 by Kderosier (Talk | contribs)
(diff) ←Older revision | Current revision (diff) | Newer revision→ (diff)

Contents

What is Fractal Dimension?

The fractal dimension, $D$, of a particular fractal is a measure of how the complexity of the figure increases as it scales. The dimension is the exponent that relates the scaling factor to the measure of the figure. That is, scale$^D$ = number of copies of the figure.

Or, $D = \frac{log(N)}{log(e)}$, where N is the number of copies of the figure and e is scale.

Example: Sierpinski's Triangle

For example, consider Sierpinski's triangle. If we double its size (as in the picture to the right), we create three copies of it. So its dimension can be calculated by $2^D=3\,$. So $D = \frac{log(3)}{log(2)} = 1.58$.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 5, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9455251097679138, "perplexity_flag": "middle"}