In this paper, the Von Koch curve is generated in a general way from a vector approach without using the classic system of iterated functions. This approach also allows constructing diferent Koch islands based on any regular polygon; to do this, a Koch curve is drawn on each of the sides of the regular polygon, taking into account that if the vertices of the polygon are traversed counterclockwise, An island will be generated inside the polygon, but if the vertices are traversed clockwise, an island outside the polygon will be developed. Given the versatility of the method, an interior island and an exterior island are also united, giving origin to a new fractal. To visualize the results, Mathematica scientific software was used.
The system of iterated functions that generates the classical Von Koch curve is given by: ⎧ 1(, ) = ( 13 , 13 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪⎪ 2(, ) = ( 16 − ⎨ ⎪⎪ 3(, ) = ( 16 + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ 4(, ) = ( 13 + 23 , 13 ) √ 63 + 12 , − √ √ 63 + 13 , 63 + 16 ) √ 63 + 16 + √3 6 ) (d) Fourth iteration.
(e) Seventh iteration.
contractive operator with the Hausdorf metric and has a single fixed point [
As described above, numerically, it can be observed that efectively when evaluating the
transformations 1, 2, 3 and 4 starting at any pair of points in the plane, the values converge
to the points of the classical Von Koch curve defined in the interval [
The big question is how to generate the Von Koch curve by taking a segment with arbitrary
ends {, } of the plane. We want the Koch curve obtained to be located on the segment
. The interest in searching for diferent variants of the Von-Koch curve is because there
are exciting applications in the field of communications because, due to its shape, multiple
resonances can be obtained when the angle is modified from 600 to a diferent one [
The Von Koch curve classical is obtained from the following description algorithm [
The first transformation contracts the segment of length one to its third part; that is to say, 1(, ) = (︀ 13 , 1 )︀ . To obtain the second transformation rotate 1 an angle of 600 and we 3 move it to the point (︀ 31 , 0)︀ . Therefore:
⎛cos(600) − sin(600)⎞ ⎛ 13 ⎞ 2(, ) = ⎝ ⎠ . ⎝ sin(600) cos(600) ⎛ 1 ⎞
3 ⎠ + ⎝ ⎠ = ⎜ 0 ⎝ ⎛ 61 − √ 63 + 13 ⎞ ⎟ ⎠
Similarly, to determine the transformation 3 we must rotate 1 an angle of − 600 and translate it to the point ︁( 12 , √63 )︁ . Therefore:
⎛cos(− 600) − sin(− 600)⎞ ⎛ 13 ⎞ 3(, ) = ⎝ ⎠ . ⎝
Among the peculiarities of this curve is that it has infinite length because if we calculate its length in stages, it has that in the first stage, its length will be 1 = 4( 31 ), in the second stage is have 2 = 16( 19 ); and in the − th iteration it result = (︀ 4 )︀ . Therefore, the length of the 3
lim →∞ ︂( 4 )︂ 3 = ∞
On the other hand, it is a continuous curve everywhere but diferentiable in none of them [
log log (︀ 1 )︀
Where is the number of boxes with which the curve is covered and is the scale
that, in this case, will be 13 . Therefore the dimension of the Von Koch curve will be
dim = log 4 ︂) = 1 · 26186 [
3 Koch curve has a capacity of 1 · 26186 to occupy the space R2.
snowflake or Koch island can be constructed from it. This curve has acquired a lot of importance
because it is being used as an antenna in wireless communications [
In the same way, the snowflake can be generated starting with a square and, in general, from any polygon. This range of possibilities leads us to think of an algorithm that is simple to handle, given that if we use a system of iterated functions, twelve transformations would be needed when we start with a triangle. In the case of starting with a square sixteen and generally for a polygon with sides, 4 transformations with ≥ 3 are required. 2.2. Standard Mathematica tools for generating the Von Koch curve classical Mathematica incorporates the KochCurve function to generate the Von Koch curve classical.
which gives the line segments representing the (ℎ)-step Koch curve, which takes a series of steps of unit length at successive relative angles , and KochCurve[]
KochCurve[, { 1, 2, . . .}]
KochCurve[, {{1, 1}, {2, 2}, . . .}] which takes successive steps of lengths proportional to .
In[
In[
In[
It is known that the IFS that generates the Koch curve has four transformations, and if we start with a segment of extremes (1, 2) and (1, 2) when evaluating the IFS in the first iteration will be obtained 8 points; for the second 32 points; for the third 128 and so on, which would require more computational time.
This algorithm consists of three operations for which 5 points are obtained in the first iteration, 20 points for the second, 80 for the third, and so on. The tools that will be used to describe the algorithm are obtained from the following analysis.
Consider the scheme proposed in figure 7. The points and remain fixed, and the only thing we would need are the points 1, 2 and 3. Indeed: 1 is obtained through:
1 2 + 1 = + 3 ( − ) ⇒ 1 = 3
To get to 2 we use the complex transformation : R2 → R2 defined by (1, 2) = (− 2, 1) [10], given that the segment 2 is perpendicular to the segment , where is the midpoint of the segment . Therefore: • step 1: To consider two arbitrarily diferent points of the plane; • Step 2: To calculate the points {1, 2, 3} and add the points y , obtaining the list {, 1, 2, 3, }, which corresponds to the first iteration; • Step 3: Partition the previous list two by two, leaving:
{{, 1}, {1, 2}, {2, 3}, {3, }}
• Step 4: Re-partition the resulting list and apply step 2 again, as often as the desired iteration indicates; • Step 5: Draw the traverse from the points obtained in the required iteration; this traverse will be the Von Koch curve at the desired stage.
The coding of the above algorithm in the scientific software Mathematica is based on simple commands ofered by the software [11], which are detailed below:
The Flatten command removes the grouping signs corresponding to the braces from any list.
The Partition command partitions a list into a set of lists according to the user’s convenience.
In[
If in the previous entry after option 2, we add option 1; it results in a consecutive partition
increasing the number of lists. This is:
In[
In[
PuntP3[{P_,Q_}]:=(P+2Q)/3; {P,PuntP1[{P,Q}],PuntP2[{P,Q}],PuntP3[{P,Q}],Q}]
{, 1, 2, 3, } corresponding to the first iteration of the curve Von Koch. It should be noted that the command input corresponds to any two points of the plane {, } and the points {1, 2, 3} are calculated based on as mentioned in the description of the algorithm. Using the IniKoch command, the KochCurveGeneral command is defined that will generate the
In[
To increase the variants of the Koch curve, we can think about applying the previous commands to the sides of a regular polygon; however, determining the vertices of a regular polygon without computational help is very laborious. This dificulty is resolved by the command in Mathematica PoliRG2P detailed below. Let’s think of a point on the plane as the center of the polygon; then, from any other point on the plane, we can construct any regular polygon using a rotation matrix as indicated in the figure 8. From this idea, the command that will generate the vertices of the regular polygon is:
If we execute the previous command for the points (0, 2) and (3, 4) for = 6 is obtained: (c) Second iteration.
(d) Fifth iteration.
(a) First iteration.
Out[
(d) Sixth iteration.
the which are the vertices of a regular hexagon, see figure 9.
By combining commands 2 and 3, we can create a new command IslaKochGeneralInt that will draw the Koch curve internally on each side of the regular polygon. The command is defined as follows:
Out[
(d) Seventh iteration.
To draw the Koch curve externally on the sides of the polygon, we reverse the list of vertices using the Mathematica command Reverse; this will cause the command to traverse the vertices of the polygon clockwise; thus, command 4 is slightly modified by internally adding the option to invert the list, being defined as follows:
Out[8]:= IslaKochGeneralExt[Cen:{_,_},P:{_, _},nlados_,iter_,
Opts___]:= Module[{vv,ll},
vv=PoliRG2P[Cen,P,nlados];
vvr= Reverse[vv];
ll=Append[vvr,vvr[[
AxesLabel -> {X,Y}] ]
Another advantage of this method is that the input list of command 2 consists of only two points, namely and . However, it can be generalized so that the input is more than two (c) Fourth iteration.
(d) Seventh iteration. points so that Koch curves can be graphed in one polygonal. Having said this, the command will be:
(c) Fifth iteration.
(d) Seventh iteration.
Example 1. for the segment whose ends are {(0, 1), (2, 0)}, is obtain the results of figure 10.
Example 2. If we want the rotation to be downwards, we change the order of the points; that is, we take in the order {(2, 0), (0, 1)} and obtain the graphic of the figure 11.
Example 3. For the segment of extremes {(2, 1), (5, 5)}, is obtained (see figure 12): Example 4. If we apply command 4 to the points (0, 2) y (3, 4) for an equilateral triangle is obtain (see figure 13):
Example 5. If in the previous example we apply command 5 we obtain the island in the ifgure 14.
Example 6. If we apply commands 5 and 6 for the same center and vertex and then join them, we obtain a fascinating island, the same one formed by drawing the internal and external
Ejemplo 7. For the comannd 6, when entering points {{0, 0}, {2, 2}, {4, 2}, {5, 0}} is ob(c) Second iteration.
(d) Sixth iteration. tained the images of the figure 16:
Ejemplo 8. If we apply command 6 to the list of points {{0, 0}, {2, 2}, {4, 0}, {0, 0}}, That is, to the vertices of a triangle traveled counterclockwise, is obtained the images in the figure 17:
This article shows an algorithm described based on simple computational operations extracted from a vector point of view. The importance lies in the versatility of the algorithm since it is possible to construct a Koch curve simply by taking any initial segment without the need to build a system of iterated functions for each case. It was also shown that from this vector approach, it is possible to construct two Koch islands, one interior, and one exterior, from any regular polygon and generate a new fractal as the union of the two islands inherent to each polygon. In addition to this, a command was also implemented with the power to graph a Koch curve in each of the segments of a polygonal, giving rise to a new generalized Koch curve; it is recommended to computationally implement the approach proposed in this article for when the points belong to a sphere given that the world in which we live has a spherical shape.
Thanks to the National University of Piura authorities for purchasing the Mathematica scientific software license with which the diferent results were obtained. (c) Second iteration.
(d) Sixth iteration. [8] O. F. G. Palacios, R. E. D. Vargas, J. A. H. Perez, S. B. C. Erazo, S-band koch snowflake fractal antenna for cubesats, in: 2016 IEEE ANDESCON, IEEE, 2016, pp. 1–4. doi:10.1109/ andescon.2016.7836227.
[9] E. W. Weisstein, Koch snowflake, https://mathworld. wolfram. com/ (2008). [10] E. Abbena, S. Salamon, A. Gray, Modern diferential geometry of curves and surfaces with
[11] J. W. Gray, Mastering Mathematica®: Programming Methods and Applications, Academic Press, 2014.