Euclidean Dimension = E Solid & space Plane Line Point

There Are Other Types of Dimensions

Fractal Dimension Fractal dimension is the same as the self-similarity dimension which is Hausdorff Besicovitch dimension D. Fractal dimension is a fractional dimension.

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

Topological Dimension = DT Boundary Point where boundary meets line Arbitrarily small neighborhood l A point is a 0-dimension set Line Boundary Point of intersection [In the figure above, how many dimensions (Euclidean dimensions, E) does each figure have? E(line)=1 E(boundary, a simple closed curve) = 1 E(point of intersection) = 0 ] Robert L. Devaney, A First Course in Chaotic Dynamical Systems: Theory and Experiment, Addison-Wesley, 1992, ISBN 0-201-55406-2 “A set S has topological dimension 0 if every point has arbitrarily small neighborhoods whose boundaries do not intersect the set.” Cantor Set

Topological Dimension = DT A point is a 0-dimension set, DT=0 Let 0 = k-1 where k is the least non-negative integer for which this holds. l For the line, DT = k So DT=0+1 A line has DT = k = 1 A line has a topological dimension of 1

Topological Dimension = DT Robert L. Devaney, A First Course in Chaotic Dynamical Systems: Theory and Experiment, Addison-Wesley, 1992, ISBN 0-201-55406-2 Figure: Ellipse Boundary:Ellipse Intersection: 2 points [In the case above, how many dimensions (Euclidean dimensions, E) does each figure have? E(Ellipse)=1 E(boundary, a simple closed curve) = 1 E(point of intersection) = 0 ]

Topological Dimension = DT Robert L. Devaney, A First Course in Chaotic Dynamical Systems: Theory and Experiment, Addison-Wesley, 1992, ISBN 0-201-55406-2 Figure = rectangle Boundary = ellipse Intersection = ellipse For the parallelogram above, the Euclidean dimension E is 2 For the boundary ring above, the Euclidean dimension E is 1 The Euclidean dimension E of their intersection is 1 The topological dimension DT of the parallelogram is 1

Topological Dimension = DT Robert L. Devaney, A First Course in Chaotic Dynamical Systems: Theory and Experiment, Addison-Wesley, 1992, ISBN 0-201-55406-2 Figure = parallelogram Boundary = ellipse Intersection = ellipse For the parallelogram above, what is the Euclidean dimension E ? Parallelogram has E=2 For the boundary ring above, what is the Euclidean dimension E Boundary has E=1 What is the Euclidean dimension E of their intersection? In the figure above, what topological dimension DT does the parallelogram have? DT = 0

Topological Dimension = DT Robert L. Devaney, A First Course in Chaotic Dynamical Systems: Theory and Experiment, Addison-Wesley, 1992, ISBN 0-201-55406-2 “A set S has topological dimension 0 if every point has arbitrarily small neighborhoods whose boundaries do not intersect the set.” Discrete points in a set Boundary No point of intersection In the figure above, what topological dimension does the figure have? Answer: 0

Topological Dimension = DT Robert L. Devaney, A First Course in Chaotic Dynamical Systems: Theory and Experiment, Addison-Wesley, 1992, ISBN 0-201-55406-2 Line Boundary Point of intersection [In the figure above, how many dimensions (Euclidean dimensions, E) does each figure have? E(line)=1 E(boundary, a simple closed curve) = 1 E(point of intersection) = 0 ]