2. Dimension

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

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

5. A cube has 3 dimensions. What are they? 1 1 1 2 2 2 Volume = 1 3 Volume = 2323 What is E, the Euclidean dimension of a cube?

6. 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

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

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

9. Euclidean Dimension = E Plane Line Point Solid & space

10. There Are Other Types of Dimensions

11. 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

12. 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,

13. 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

14. 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.

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

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

17. 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:

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

19. 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

20. 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)

21. 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