Dimension A line segment has one dimension, namely length. length = 1 unit length = 2 units Euclidean Dimension = 1.

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Presentation transcript:

Dimension

A line segment has one dimension, namely length. length = 1 unit length = 2 units Euclidean Dimension = 1

A square has 2 dimensions, length & width. Euclidean Dimension = 2 length = 1 length = 2 width = 1 width = 2 Area = 1 = 1 2 Area = 4 =

A cube has 3 dimensions. What are they? Volume = 1 3 Volume = 2323 What is E, the Euclidean dimension of a cube?

A line A line has 1 dimension, length. It is infinitely long. It is also infinitely thin, but we give its drawing thickness to make it visible

A plane A plane is a flat surface that is infinitely long and infinitely wide. It has 2 dimensions.

Space Space has 3 dimensions: Infinite height (or depth) Infinite length Infinite width (or breadth)

Euclidean Dimension = E Plane Line Point Solid & space

There Are Other Types of Dimensions

Fractal Dimension What does it look like? It is a fractional dimension That exponent is a generally a fraction It is shown as an exponent

D = Fractal Dimension In 1977 Mandelbroit called fractional dimension (Hausdorff Besicovitch Dimension) a fractal dimension The Fractal Geometry of Nature (1977, 1983), p 15 B,

How do you find the fractal dimension? Because fractals are generally self-similar, we can use the self-similarity dimension. P. 37, The Fractal Geometry of Nature, 1977,1983

What does self-similar mean? Instead of comparing two separate shapes, Self-similar: The part is the same shape as the whole thing. we compare a part of a shape to the whole.

Let N = the number of rescaled objects in the generator that replace the initiator. N = Initiator: Generator:

Let N = the number of rescaled objects in the generator that replace the initiator. N = 2 Initiator: Generator:

Let m = how many times larger the figure in the initiator is than the the same figure in the generator. (Think m = magnification) Initiator: Generator:

Find the fractal dimension D N = m D N = 2 M = 3 2 = 3 D so 3 D = 2

Find the fractal dimension D 3 D = 2 We know 3 0 = 1 We know 3 1 = 3 D must be between 0 and 1

Using logs to find D Often our m is written as 1/r m = 1/r N = m D N = (1/r) D D = log N/log(1/r)

Mandelbrot’s Definition of a Fractal A fractal is by definition a set for which the Hausdorff Besicovitch dimension strictly exceeds the topological dimension. Mandelbrot, 1977,1983, p 15