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

2. There Are Other Types of Dimensions

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

4. 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

5. 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
“A set S has topological dimension 0 if every point has arbitrarily small neighborhoods whose boundaries do not intersect the set.”
Cantor Set

6. 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

7. Topological Dimension = DT
Robert L. Devaney, A First Course in Chaotic Dynamical Systems: Theory and Experiment, Addison-Wesley, 1992, ISBN
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 ]

8. Topological Dimension = DT
Robert L. Devaney, A First Course in Chaotic Dynamical Systems: Theory and Experiment, Addison-Wesley, 1992, ISBN
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

9. Topological Dimension = DT
Robert L. Devaney, A First Course in Chaotic Dynamical Systems: Theory and Experiment, Addison-Wesley, 1992, ISBN
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

10. Topological Dimension = DT
Robert L. Devaney, A First Course in Chaotic Dynamical Systems: Theory and Experiment, Addison-Wesley, 1992, ISBN
“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

11. Topological Dimension = DT
Robert L. Devaney, A First Course in Chaotic Dynamical Systems: Theory and Experiment, Addison-Wesley, 1992, ISBN
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 ]