Fractal Dimension

Smooth Curves: at any point on a smooth curve, the curve is well-approximated by its tangent line at that point. Zoom in on a smooth curve and al the complexity of the curve disappears into the distance, leaving only a straight line.

Koch Curve Zoom: zooming in on a fractal curve reveals similar structures at all scales. For example, little bits of the Koch curve don't look like straight lines. They keep looking like the whole curve.

Natural Fractal Zoom: natural fractals exhibit similar structures over a limited range of scales. his should be no surprise: the forces that sculpt or grow a particular shape dominate the formation of that shape over only a limited range of scales.

Scale Invariance: self-similarity has an interesting visual conswquence. Fractals do not have a preserred size. Distinguishing a small fractal viewed nearby from a large fractal seen far away is not so easy.

Coastlines: Coastlines are good examples of natural fractals. A consequence, obvious to anyone who has viewed a coastline from different altitudes, is that the length of a coastline depends on the scale at which it is measured. Smaller yardsticks will detect ever smaller features.

Measuring Dimensions: one common meaning of the word "dimension" can be extended to a measure of how many and what size smaller pieces make up a fractal.

Similarity Dimension: this notion of dimension is particularly well-suited for self-similar shapes. For applications, it can be extended to shapes exhibiting only approximate self-similarity over a limited range of scales. In another direction, it has deep mathematical generalizations.

Measuring Roughness: fractal dimension is the first quantitative measure of the roughness of natural objects.

A Family of Koch Curves: shapes between a straight line segment and a filled-in triangle can be interpolated by a family of Koch curves, of dimension increasing from 1 to 2.

Much more information about fractal dimensions can be found at http://classes.yale.edu/fractals/FracAndDim/welcome.html.

Smooth Curves: at any point on a smooth curve, the curve is well-approximated by its tangent line at that point. Zoom in on a smooth curve and al the complexity of the curve disappears into the distance, leaving only a straight line.
Koch Curve Zoom: zooming in on a fractal curve reveals similar structures at all scales. For example, little bits of the Koch curve don't look like straight lines. They keep looking like the whole curve.
Natural Fractal Zoom: natural fractals exhibit similar structures over a limited range of scales. his should be no surprise: the forces that sculpt or grow a particular shape dominate the formation of that shape over only a limited range of scales.
Scale Invariance: self-similarity has an interesting visual conswquence. Fractals do not have a preserred size. Distinguishing a small fractal viewed nearby from a large fractal seen far away is not so easy.
Coastlines: Coastlines are good examples of natural fractals. A consequence, obvious to anyone who has viewed a coastline from different altitudes, is that the length of a coastline depends on the scale at which it is measured. Smaller yardsticks will detect ever smaller features.
Measuring Dimensions: one common meaning of the word "dimension" can be extended to a measure of how many and what size smaller pieces make up a fractal.
Similarity Dimension: this notion of dimension is particularly well-suited for self-similar shapes. For applications, it can be extended to shapes exhibiting only approximate self-similarity over a limited range of scales. In another direction, it has deep mathematical generalizations.
Measuring Roughness: fractal dimension is the first quantitative measure of the roughness of natural objects.
A Family of Koch Curves: shapes between a straight line segment and a filled-in triangle can be interpolated by a family of Koch curves, of dimension increasing from 1 to 2.