Thayer Watkins
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Fractals and Fractional Dimensions

Consider first how to define the dimension of geometric objects such as lines, squares and cubes. Suppose we divide the edges of lines, squares and cubes into b equal parts. This creates smaller versions of the wholes and we can count the number of the smaller units it takes to cover the whole. For a line it takes b (=b1) copies of the smaller line segment to cover the original line segment. For a square it takes b2 smaller squares to cover the original square. And for the cube it takes b3 smaller cubes to cover the original cubes. An object is said to be n dimensional if it takes bn copies of the smaller version of the object created when its edges are divided into b parts. Thus the line segment is one dimensional, the square two dimensional and the cube three dimensional. The computation of the dimension of an object can be put into terms of a formula. Let b be the number of equal divisions made of the edges to create smaller similar objects. Let M be the number of copies of the smaller object required to exactly cover the larger object. If M= bn we could get the value of n by the formula:

log(bn)/log(b) =
nlog(b)/log(b) = n
or, in general,
n = log(M)/log(b).

For the objects considered so far the dimension has been an integral number. The surprising thing is that there are objects which are of a fractional dimension. Two such fractional dimension object (now called a fractal) are shown below. The fractal Cantor's Dust is created by taking a line segment and deleting the middle third. This creaes to two smaller line segments. The middle thirds of these two segments, thus creating six line segments. The middle thirds of these are also deleted and so on ad infinitum.

Sierpinski's Gasket is created by deleting the triangle formed from the midpoints of the sides of the triangles. This creates three sub-triangles. A middle triangle is deleted from each of thoose, thus creating nine sub-triangles, and again so on ad infinitum.

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