& Tornado Alley
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:
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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