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Lesson Euclidean geometry For a straight line one measures the distance by two end points. But, the two distance measurements of base and height are needed for the area of a triangle, and another measurement of height is required for the volume of a pyramid. Therefore, our intuitive definition is that dimension is the number of distance measurements needed to specify how much room an object takes up in space. Put it differently, a one-dimensional line segment has only the length, a two-dimensional triangle covers an area in plane, and a three-dimensional pyramid occupies a volume in space. Hence, the larger the dimension of an object, the more space it occupies in the Euclidean world, so that dimension is a measure of how an object takes up space to exist. But, what about the Cantor set, Sierpinski triangle and pyramid, Cantor gasket, and Menger sponge of Lesson 5? They are so fractured or broken up in space that it is not even intuitively clear how to define their dimension, let alone quantifying it. For being fractured, Bernoit Mandelbrot first coined the word fractal from the Latin fractus. He is the founder of fractal geometry, educated in France and did most of his work at the IBM laboratory in NY. For fractal objects such as, the Cantor set, Sierpinski triangle, etc., we cannot come up with dimension by simply counting the number of distance measurements. It is therefore necessary to have an alternate definition of dimension that can be applied to any objects, be they Euclidean or fractal. It is based on the notion of capacity, that is, how an object takes up space to fill in, which was first suggested by the Russian mathematician Andrei Kolmogorov (1903-1987). As a preliminary requisite, we must first show that the capacity definition gives correctly the known dimensions 1, 2, and 3 when we have a straight line, square, and cube, respectively. |