7/29/2019 Koch Properties

http://slidepdf.com/reader/full/koch-properties 1/6

Koch Snowflake

 A fractal, also known as the Koch island, which was first described by Helge von Koch in 1904. It is built bystarting with an equilateral triangle, removing the inner third of each side, building another equilateraltriangle at the location where the side was removed, and then repeating the process indefinitely. The Kochsnowflake can be simply encoded as a Lindenmayer system with initial string "F--F--F", string

rewriting rule "F" -> "F+F--F+F", and angle . The zeroth through third iterations of the

construction are shown above. The fractal can also be constructed using a base curve and motif,illustrated below.

Let be the number of sides, be the length of a single side, be the length of the perimeter , and

the snowflake's area after the th iteration. Further, denote the area of the initial triangle , and the

length of an initial side 1. Then

Solving the recurrence equation with gives

so as ,

The capacity dimension is then

7/29/2019 Koch Properties

http://slidepdf.com/reader/full/koch-properties 2/6

(Sloane's A100831; Mandelbrot 1983, p. 43).

Some beautiful tilings, a few examples of which are illustrated above, can be made with iterations towardKoch snowflakes.

In addition, two sizes of Koch snowflakes in area ratio 1:3 tile the plane, as shown above.

7/29/2019 Koch Properties

http://slidepdf.com/reader/full/koch-properties 3/6

 Another beautiful modification of the Koch snowflake involves inscribing the constituent triangles with filled-

in triangles, possibly rotated at some angle. Some sample results are illustrated above for 3 and 4iterations.

SEE ALSO: Cesàro Fractal, Exterior Snowflake, Gosper Island, Koch Antisnowflake, Peano-Gosper Curve, Pentaflake, Sierpiński Sieve REFERENCES:

Capacity Dimension

 A dimension also called the fractal dimension, Hausdorff dimension, and Hausdorff-Besicovitch dimensionin which nonintegral values are permitted. Objects whose capacity dimension is different from their 

Lebesgue covering dimension are called fractals. The capacity dimension of a compact metric space is a

real number  such that if denotes the minimum number of open sets of diameter less than or 

equal to , then is proportional to as . Explicitly,

(if the limit exists), where is the number of elements forming a finite cover of the relevant metric space 

and is a bound on the diameter of the sets involved (informally, is the size of each element used to cover the set, which is taken to approach 0). If each element of a fractal is equally likely to be visited, then

, where is the information dimension.

The capacity dimension satisfies

7/29/2019 Koch Properties

http://slidepdf.com/reader/full/koch-properties 4/6

where is the correlation dimension (correcting the typo in Baker and Gollub 1996).

Define the "information function" to be

where is the natural measure, or probability that element is populated, normalized such that

The information dimension is then defined by

If every element is equally likely to be visited, then is independent of , and

so

and

where is the capacity dimension.

It satisfies

where is the capacity dimension and is the correlation dimension (correcting the typo inBaker and Gollub 1996).

SEE ALSO: Capacity Dimension, Correlation Dimension, Correlation Exponent 

7/29/2019 Koch Properties

http://slidepdf.com/reader/full/koch-properties 5/6

Natural Measure

, sometimes denoted , is the probability that element is populated, normalized such that

SEE ALSO: Information Dimension, q-Dimension 

q-Dimension

where

is the box size, and is the natural measure. 

The capacity dimension (a.k.a. box-counting dimension) is given by ,

If all s are equal, then the capacity dimension is obtained for any .

The information dimension corresponds to and is given by

But for the numerator,

and for the denominator, , so use l'Hospital's rule to obtain

7/29/2019 Koch Properties

http://slidepdf.com/reader/full/koch-properties 6/6

Therefore,

(Ott 1993, p. 79).

is called the correlation dimension.

If , then

(Ott 1993, p. 79).

SEE ALSO: Capacity Dimension, Correlation Dimension, Fractal Dimension, Information Dimension