The concept of DimensionsThe concept of Dimensions

Musafare RuswaMusafare Ruswa

Topological DimensionTopological Dimension The dimension of a space can be described as The dimension of a space can be described as

the number of independent parameters needed the number of independent parameters needed to specify different points in the space.to specify different points in the space.

We can think of a line as being one We can think of a line as being one dimensional, a plane 2 dimensional and so dimensional, a plane 2 dimensional and so forth.forth.

Recall that an open ball is a subset S of the form Recall that an open ball is a subset S of the form S(xS(x0, 0, εε)={x )={x ϵϵ X| d(x X| d(x00,x)<,x)<εε for any given x for any given x0 0 which which is an element of X and radius is an element of X and radius εε>0>0

A subset S of X is open if it is in an arbitrary A subset S of X is open if it is in an arbitrary union of open balls in X.union of open balls in X.

In a topological space we do not assume we know In a topological space we do not assume we know the distance but we assume that we know what the the distance but we assume that we know what the open subsets are.open subsets are.

A covering of a subset S is a collection C of open A covering of a subset S is a collection C of open subsets in X whose union contains all points of Xsubsets in X whose union contains all points of X

A refinement of a covering C is another covering C` A refinement of a covering C is another covering C` of S such that each set B in C` is contained in another of S such that each set B in C` is contained in another set A in C. The idea being that each set in C` is set A in C. The idea being that each set in C` is smaller than those in C and they provide a fuller smaller than those in C and they provide a fuller coverage than the ones in C.coverage than the ones in C.

In the figure below the covering C is shown in red In the figure below the covering C is shown in red

and the refinement C` is shown in blueand the refinement C` is shown in blue..

We therefore define the topological dimension of a We therefore define the topological dimension of a space to be space to be mm if every covering C of X has a if every covering C of X has a refinement C’ in which every point of X occurs in at refinement C’ in which every point of X occurs in at most most m+1m+1 sets of C`, and m is the smallest such sets of C`, and m is the smallest such integer.integer.

From this definition the curve below (Koch curve) From this definition the curve below (Koch curve) has topological dimension 1.has topological dimension 1.

There are some cases where the topological There are some cases where the topological dimension makes unexpected predictions, for dimension makes unexpected predictions, for example the Cantor set is observed to have a example the Cantor set is observed to have a dimension of 0.dimension of 0.

The The Hausdorff-BesicovitchHausdorff-Besicovitch dimension is useful dimension is useful in describing these objects and may take in describing these objects and may take fractional values.fractional values.

FractalsFractals

A Fractal is an object that possesses the A Fractal is an object that possesses the following propertiesfollowing properties

It is self similarIt is self similar It can possess non integer dimensionsIt can possess non integer dimensions An object is self similar if each magnified An object is self similar if each magnified

portion of itself is its direct replica.portion of itself is its direct replica. Cantor set is a fractal with Dimension 0.6309Cantor set is a fractal with Dimension 0.6309 Koch curve has dimension 1.26..Koch curve has dimension 1.26..

Koch SnowflakeKoch Snowflake

In constructing the Koch snowflake curve, a In constructing the Koch snowflake curve, a simple line segment is divided into thirds, and simple line segment is divided into thirds, and the middle piece is replaced by two equal lines the middle piece is replaced by two equal lines to form an equilateral triangle without a base.to form an equilateral triangle without a base.

The next stage involves replacing each of The next stage involves replacing each of these four segments with four new segments these four segments with four new segments each with length 1/3 of the parent according to each with length 1/3 of the parent according to the initial pattern, the procedure is continued the initial pattern, the procedure is continued to infinity to generate the Koch snowflake to infinity to generate the Koch snowflake curvecurve..

Koch Snowflake cont..Koch Snowflake cont.. Any portion of the snowflake is composed of 4 Any portion of the snowflake is composed of 4

segments each scaled down by a factor of 1/3segments each scaled down by a factor of 1/3

Fractal DimensionFractal Dimension

The term was coined by Mandelbrot in 1975, and the The term was coined by Mandelbrot in 1975, and the fractal dimension is a positive real number (could be non fractal dimension is a positive real number (could be non integral).integral).

Is a simplification of the Hausdorff Dimension concept, Is a simplification of the Hausdorff Dimension concept, also called capacity of a geometric figurealso called capacity of a geometric figure

A line segment, which is considered one dimensional, can A line segment, which is considered one dimensional, can be divided into N identical parts each of which is scaled be divided into N identical parts each of which is scaled down by the ratio r=1/N.down by the ratio r=1/N.

A two dimensional object such as a square can be A two dimensional object such as a square can be divided into N similar parts each of which is scaled down divided into N similar parts each of which is scaled down by a factor of r=1/√(N).by a factor of r=1/√(N).

With Fractal Dimensions a With Fractal Dimensions a DD--dimensional self similar object can be dimensional self similar object can be divided into N smaller copies each of divided into N smaller copies each of which is scaled down by a factor r where which is scaled down by a factor r where r = 1/(r = 1/(DD√N).√N).

Given then a self similar object of N parts Given then a self similar object of N parts scaled by a ratio scaled by a ratio r r from the whole, its from the whole, its fractal or similarity dimension can be fractal or similarity dimension can be given by D = log(N)/log(1/r).given by D = log(N)/log(1/r).

. .

For example in the Koch Snowflake, N=4,and r=1/3For example in the Koch Snowflake, N=4,and r=1/3 D=log(4)/log(3)=1.26…,then gives us the dimension of the D=log(4)/log(3)=1.26…,then gives us the dimension of the

Koch snowflake curve. Koch snowflake curve. The Koch curve has infinite length (but infinite area), at each The Koch curve has infinite length (but infinite area), at each

stage in its construction there are 4stage in its construction there are 4nn,line segments of length ,line segments of length 1/(31/(3nn),total length is (4/3)),total length is (4/3)nn,which approaches infinity as n ,which approaches infinity as n approaches infinityapproaches infinity

As D increases from 1 to 2 the resulting picture progresses As D increases from 1 to 2 the resulting picture progresses from being “line like” to space filling.from being “line like” to space filling.

A fractal can therefore described as an object whose A fractal can therefore described as an object whose Hausdorff Dimension strictly exceeds its Topological Hausdorff Dimension strictly exceeds its Topological Dimension.Dimension.

Cantor SetCantor Set

Start with a line segment of length 1,remove Start with a line segment of length 1,remove middle third, repeat iteration middle third, repeat iteration

In this case N=2,and r=1/3In this case N=2,and r=1/3 D=log2/log3=0.6309D=log2/log3=0.6309 The Sierpenski Triangle can be constructed in The Sierpenski Triangle can be constructed in

the same waythe same way Begin with a closed equilateral triangle and the Begin with a closed equilateral triangle and the

first stage remove a triangle of size ½,at the first stage remove a triangle of size ½,at the second stage remove 3 open triangles of size ¼ second stage remove 3 open triangles of size ¼ and continue in this manner.and continue in this manner.

The triangle consists of 3The triangle consists of 3nn triangles with triangles with magnification factor 2magnification factor 2n.n.

Dimension is therefore log3/log2=1.585Dimension is therefore log3/log2=1.585