1. Transformation of Classical Fractals applying
Fractional Operators
Dario Domínguez1, Karol Toro1, Mariela Marín2
Universidad Militar Nueva Granada, Bogotá, Colombia
1
Centro International de Física, Bogotá, Colombia
2
Email: fracumng@umng.edu.co
Abstract: In this we defined a new type of fractional operators for then
transform the geometry of classical fractals into a new geometry of such self,
obtaining new variants of the fractals without altering its basic form and geometry.
We defined the fractional operator and then we applied the transform to the
Mandelbrot’s and Sierpinski’s sets obtaining for each of them some particular results
on its new transformations.
Keywords: Mandelbrot’s set, Sierpinski’s set, fractional operator, fractals.
1. Introduction
The fractional operators were defined in the nineteenth century, but were
applied after the second half of the twentieth century. The work of
Kolwankar1 in 1997 indicates a rapprochement with the operators to fractal
sets. Some new ideas were taken from the Mandelbrot’s2 ‘‘Fractals and
Chaos (2004)’’.
In this paper, we used this approach to define a new fractional operator,
which is about fractal geometry of the Mandelbrot’s and Sierpinski’s set. The
definition which we used is a variation on the so-called fractional operator of
Riemann-Liouville as indicated by Hilfer3. This operator generates a new
fractal inside the Mandelbrot’s and Sierpinskis’s sets and constitutes a
transformation of it.
The paper shows the action of the operator using alpha values since α =-0.28;
α =-0.25, and α =-0.22, and α =-0.20 and then for some positive values.
We also defined the fractional operators and some of their properties and we
indicate that the algorithm leads to these changes and at the same time acts on
the all set. Likewise, we show the operator on the values of alpha non-
negative, indicating that it was similar to the classic operators of derivation
and integration, in the sense that the positive values correspond to the
derivative and not adverse to the integral. Although it is possible to give
some topological aspects we will not mention them at this point. It can be
applied to non-negative values generating other types of fractals.
This paper is the first of a series in which the fractional operators acts over
others known fractals like Julia’s set, Cantor’s set and others.

2. Domínguez D., Toro K., Marín M.
2. Fractional operator (Mandelbrot)
2.1. Riemann- Liouville definition
We define the fractional operator as usual Hilfer3, Carpinteri, Mainardi4 and
Oldham, Spanier5.
From this operator we define a new fractional operator as follows:
, where , and
Г(x) is the Euler’s gamma function.
If , we have the Mandelbrot’s classic set (Figure 1). From this we
generate a new fractal for α =-0.28 as shown in Figure 2.
Figure 1. Mandelbrot’s classic set. Figure 2. Transformation α=-0.28
Here we can appreciate the appendix augmentation and the diminution on the
left part of the fractal.
Then we apply the fractional operator defined for different values of alpha.
There is a change on Figure 1, but it maintains the basic structure of the
Mandelbrot’s set. When we change alpha to new values, we get new
geometries shown on Figures 3, 4 and 5 with values -0.25, -0.22 and -0.20
respectively.

3. Transformation of Classical Fractals applying Fractional Operators -
So, if we call the Mandelbrot’s set (Μ), the fractional operator is a
transformation given by:
.
where
On a detailed observation of the figures, we can see that the change on the
alpha values, transform the package into new fractal equivalents.
Figure 3. =-0.25 Figure 4. =-0.22 Figure 5. =-0.20
2.2. Properties of the Fractional Operator
The operator has the properties of the Riemann_Liouville fractional operator:
1. Linearity.
2. Semi-group Properties.
3. The function f leaves intact the properties of the Mandelbrot’s set.
3. Non negative case
The following case is when alpha is not negative; there is another kind of
transformation. In this case the values of alpha are 0.25, 0.22 and 0.20
respectively. In this case, the shape of the fractal changes as shown in figures
6 and 7 for α = 0.25, α = 0.22 .
Figure 6. =0.25 Figure 7. =0.22

4. Domínguez D., Toro K., Marín M.
4. Fractional operator (Sierpinski)
We define the derivative and the integral type for a polynomial fractional
function deduced from the definition of Riemann-Liouville Hilfer3,
Carpinteri, Mainardi4 and Oldham, Spanier5.
.
The equations are:
Γ( p + 1)
()
Dα x p = x p −α
Γ( p + 1 − α )
α > 0 FractionalDerivate
Si
α <0 we have the fractional integrate. By Mandelbrot2, Hilfer3 we
When
have:
⎛ x ⎞ ⎛ Dα x ⎞
D ⎜ ⎟=⎜ α ⎟
α
⎜ y⎟ ⎜D y⎟
⎝⎠ ⎝ ⎠
The Sierpinski’s matrix is:
⎛ .5 0 0 . 5 0 0 1 / 3 ⎞
⎜ ⎟
S = ⎜ .5 0 0 . 5 .5 0 1 / 3 ⎟
⎜ .5 0 0 .5 .25 .5 1 / 3 ⎟
⎝ ⎠
( )
D α A = D α ai , when we vary the values of alpha, we find
If we define
the Sierpinski’s fractal transformations as shown in the following figures:
Figure 8. alfa=-0.185 Figure 9. alfa=0 Figure 10. alfa=-0.5
Finally when we applied the fractional integral we appreciate the behavior of
the fractional operator on the Sierpinski’s set as following:

5. Transformation of Classical Fractals applying Fractional Operators -
Figure 11. alfa=0.5 Figure 12. alfa=0.185
5. Conclusions
In this paper we examined the properties of a fractional operator which acts
as a transformation on the Mandelbrot’s and Sierpinski’s sets and generate
new fractals without altering the original geometry of them, controlling only
the value of alfa for the transformations on a set between [-1, 1].The resulting
images of the applycation of the Fractal Operators over the Classical Fractals
give us an idea to understand the behavior of similar spaces which we can
find on the biology field. Finally, the resulting images also can have an
approach for new tendences of fractal geometry for applications on digital
and computer art.
Acknowledgment
This work was supported by Grant: 1123240520182 from the Programa
Ciencias Básicas, COLCIENCIAS. Universidad Militar Nueva Granada and
Centro Internacional de Física, Bogotá, Colombia. The authors would like to
specially thanks Mr. Daniel Domínguez engineer on mechatronics and
masters degree student from the Université des Sciences et Technologies
de Lille in France for his support and conseils during the realization of this
paper.
References
1 K. M. Kolwankar. Studies of fractal structures and processes using methods of
the fractional calculus. Tesis Ph.D University of Pune, 1997.
2 B.Mandelbrot. Fractals and Chaos: the Mandelbrot set and beyond.Springer, 2004.
3 R. Hilfer. Applications of Fractional calculus in Physics. World Scientific
IIIIIPublishing 2000.
4 A. Carpinteri and F. Mainardi. Fractals and fractional calculus in continuum
mechanics. Springer, 1997.
5 K. Oldham and J. Spanier. The fractional calculus. Dover Publications Inc, 1974.