Byungik Kahng and Jeremy Davis- Maximal Dimensions of Uniform Sierpinski Fractals

8/3/2019 Byungik Kahng and Jeremy Davis- Maximal Dimensions of Uniform Sierpinski Fractals

1/13

MAXIMAL DIMENSIONS OF UNIFORM SIERPINSKI FRACTALS

BYUNGIK KAHNG AND JEREMY DAVIS

Abstract. We study the invariant fractals in convex N-gons given by N identical pure

contractions at its vertices with non-overlapping images, which we will call uniform Sierpin-

ski fractals. We provide the explicit formulae for the maximal contraction ratio RN, and

the maximal Hausdorff dimension hN, for the uniform Sierpinski fractals. We use maximal

N-grams and principal crossing points as the main tool.

1. Introduction

It is well known that the iteration of similarity maps often produces an invariant fractal

with intricate self similarity structure that is aesthetically beautiful. See, for instance, [2, 3, 6]

for more about fractals and self similarities. One of the oldest and also the simplest classes

of such fractals is the class of Sierpinski fractals, which we define as follows.

Definition 1.1 (Sierpinski Fractals). LetPN be a convex N-gon in the Euclidean plane R2,

with the vertices v1, , vN R2. That is,

PN = N

i=1

ivi : 0 i 1,Ni=1

i = 1

.

LetP(PN) be its power set. Furthermore, let

f :P(PN) P(PN)

be a set function in PN given by

(1.1) f(S) = f1(S) fN(S),

where each fi : PN PN is a pure contraction at vi with the contraction ratio ri. Thatis, for each i {1, , N},

(1.2) fi(x) = ri(x vi) + vi, 0 < ri < 1.

Date: December 10, 2009.

2000 Mathematics Subject Classification. Primary: 51N05, 51N20, 28A80; Secondary: 51-01.Key words and phrases. Sierpinski fractal, self similarity, Hausdorff dimension, disturbed control dynam-

ical system, multiple valued iterative dynamics, maximal invariant set, global attractor.1


2/13

2 BYUNGIK KAHNG AND JEREMY DAVIS

Suppose further that the contractions, f1, , fN, have non-overlapping images. That

is, each intersection, fi(PN) fj(PN), i = j, has empty interior. Then, the set

(1.3) S(r1, ,rN)(PN) =

k=0

fk

(PN).

is called the Sierpinski fractal of PN with the contraction ratios r1, , rN. We say a

Sierpinski fractal is uniform if r1 = = rN = r, and we abbreviate it as,

Sr(PN) = S(r, ,r)(PN).

We say a uniform Sierpinski fractal Sr(PN) is maximal if the the common contraction ratio

r is the largest for given PN. Finally, a Sierpinski fractal is calledregular ifPN is a regular

polygon.

Figure 1.1 illustrates the uniform regular Sierpinski fractal Sr(PN) for selected N andr values. Our aim is to determine the condition to achieve the largest r value for given

N, which gives rise to the largest possible regular Sierpinski fractal in terms of Hausdorff

dimension The left hand side column of Figure 1.1 illustrates such maximal cases, where

adjacent images fi(PN) and fi+1(PN) apparently touch each other. In fact, we claim the

following.

Main Theorem (Theorem 3.2 and Corollary 3.3). Let Sr(PN) be a uniform Sierpinski

fractal of a regular N-gon, PN with vertices v1, , vN R2. Then, the largest possible

contraction ratio (r value in Definition 1.1), which we denote as RN, turns out to be,

(1.4) RN =1

2

1

tanN

N14

tan

N

+ N

N14

,

where x stands for the floor function. That is, x is the largest integer that does not

exceed x. As a direct consequence, we conclude that the maximal Hausdorff dimension hN is

attained when r = RN, and

(1.5) hN =ln N

ln(1/RN)=

ln N

ln RN.

The Main Theorem, which is a combination of Theorem 3.2 and Corollary 3.3, will be

proved in Section 3, along with the numerical illustrations. The interactive Mathematicaprograms to plot the regular Sierpinski fractals and to test the results and tools of this paper

will be posted in the first authors website [5].

2. Maximal N-grams and Principal Crossing Points

The main tool we use to prove the Main Theorem is the N-gram decomposition ofPN. In

general, an N-gram of a regular polygon PN is a union of selected diagonal line segments of

PN. Figure 2.1 Figure 2.4 exemplify some N-grams of various regular polygons. Clearly,


3/13

MAXIMAL DIMENSIONS OF UNIFORM SIERPINSKI FRACTALS 3

N = 3, r = 1/2.

N = 7, r 0.30798.

N = 6, r = 0.32.

N = 11, r = 0.21.

Figure 1.1. Selected Examples of Uniform Regular Sierpinski Fractals.

Figure 2.1. G5. Figure 2.2. G6.

an N-gram exists only if N 5 and there are more than one N-grams, if N 7. Among

them, we are particularly interested in one specific N-gram for each N 5, which we define

as follows.


4/13


Figure 2.3. G7 (thick). Figure 2.4. G9 (thick).

Definition 2.1 (Maximal N-gram). LetPN be a regular N-gon with vertices

vi = c0 +

cos

2i

N

, sin

2i

N

R2,

where c0 R2, > 0, i = 1, , N, and finally, N 5. Suppose further that

(2.1) m =

N 1

4

+ 1.

Then, we call the union of the diagonal line segments GN given by

(2.2) GN =

Nm

i=1 vi vi+m,

where p q stands for the line segment from p R2 to q R2, the maximal N-gram of PN.

The reason we chose the term maximalin Definition 2.1 is because of the maximal N-grams

are intimately related to the maximal regular Sierpinski fractals, or the regular Sierpinski

fractals given by the maximal contraction ratios. The following lemma justifies our choice.

It will play an important role in the proof of the Main Theorem.

Lemma 2.2 (Principal Crossing Points). Let Sr(PN) be a uniform Sierpinski fractal on a

regular N-gon, PN with vertices v1, , vN, N 5, introduced in Definition 2.1. Suppose further thatf1, , fN and f be as in Definition 1.1. Then, for each i {1, , N},

fi(vim) vi vim GN,(2.3)

fi(vim) vi vim GN,(2.4)

where and are the addition and the subtraction in {1, , N} under the (mod N) iden-

tification, and m is the number defined by the equality (2.1) in Definition 2.1. Furthermore,

the uniform contraction ratio r attains the maximum value, which we denote as RN, if and


5/13


Figure 2.5. N = 5, 0 < r < RN. Figure 2.6. N = 5, r = R5.

Figure 2.7. N = 7, 0 < r < RN. Figure 2.8. N = 7, r = R7.

only if

(2.5) vi vim vi1 vi(m1) = {fi(vim)},

for some i {1, , N}. Let us call the crossing points between the line segments vi vimand vi1 vi(m1), i {1, , N}, the principal crossing points of GN.

Figure 2.5 Figure 2.8 illustrate the first conclusion of Lemma 2.2, the set inequalities

(2.3) and (2.4). Note that each vertex fi(vim) stays on the diagonal line segment vi vim ofthe maximal N-gram GN, as r value increases from 0 to RN.

The second conclusion (the equivalence part), on the other hand, can be re-expressed

as follows. The uniform contraction ratio r reaches its maximum value if and only if each

fi(PN) touches the adjacent polygons, fi1(PN) andfi1(PN), at the principal crossing points.

Figure 2.6, Figure 2.8 and Figure 2.9 Figure 2.12 depict this claim. Figure 2.13 and Figure

2.14, on the other hand, illustrate what happens if we select wrong N-grams. Note that

the crossing points of the diagonal line segments do not match the touching points of the


6/13


Figure 2.9. N = 8, r = R8. Figure 2.10. N = 9, r = R9.

Figure 2.11. N = 12, r = R12. Figure 2.12. N = 13, r = R13.

Figure 2.13. N = 9, r = R9. Figure 2.14. N = 9, r 0.2835.

adjacent polygons, and compare them with Figure 2.10, which depict the correct maximal

N-gram for the same N value.

Proof of Lemma 2.2. Because fi is a pure contraction at vi, it preserves the line that runs

through vi. Hence, the equalities (2.3) and (2.4) follow. Figure 2.5, Figure 2.6 and Figure


7/13


2.7, Figure 2.8 illustrate this idea. All we have to prove now is the equivalence between r

reaching its maximum and the equality (2.5) holding for some i {1, , N}. Because of

the symmetry ofPN and the uniformity ofr, the equality (2.5) holds for every i {1, , N}

if it holds for any one i. That is, r attains the maximum if and only if (2.5) holds for everyi {1, , N}.

By symmetry, we conclude that the uniform contraction ratio r reaches its maximum when

and only when the farthest right hand side vertex of fi(PN) and the farthest left hand side

vertex of fi1(PN) coincide, as depicted in Figure 2.15 (the red point in the interior of PN).

Figure 2.16 is the expansion of the lower left hand side copy of the N-gon, fi(PN), of Figure

2.15. Note that the blue, red and green points of Figure 2.15 correspond to those of Figure

2.16. Let us abbreviate Vj = fi(vj) for convenience. Then clearly, Vi = vi, the blue point

in Figure 2.15 and Figure 2.16. Let be the distance between the adjacent vertices Vi and

Vi1, that is, = 1(Vi Vi), where 1() stands for the length.Because PN is a regular N-gon, the size of each outer-angle of PN, that is, the angle

between the pair of vectors vi vi1 = vi1 vi andvi1 vi2 = vi2 vi1, is 2/N. Note

that we used the notion p q to denote q p R2 for better understanding. Because fiis a pure contraction at vi = Vi, the same property holds also for fi(PN). For instance,

(Vi2Vi1M) = 2/N in Figure 2.16, where () means the radian measure of the

angle. Hence, we conclude that,

Vi Vij =

Vi Vi1 +

Vi1 Vi2 + +

Vi(j1) Vij

= (cos 0, sin 0) +

cos2

N, sin2

N

+ +

cos2(j 1)

N , sin2(j 1)

N

,

and that the x-coordinate ofVi Vij keeps on increasing until it reaches its maximum when

j 1 is the largest integer such that cos(2(j 1)/N) > 0. The reason is because the

x-coordinate ofVi Vij is no longer bigger than that of

Vi Vi(j1) from that point on and

keeps decreasing, as depicted as the gray arrow in the upper right hand side of Figure 2.16.

Therefore, we conclude that such j is the largest integer such that

(2.6) 2(j 1)/N < /2, or 4(j 1) < N .

Because j is an integer, we can re-express the condition (2.6) equivalently as 4(j 1) N1,or j 1 + (N 1)/4. Since j is the biggest such integer, we conclude

j =

N 1)

4

+ 1 = m,

as defined by the equality (2.1) of Definition 2.1. Hence, the farthest right hand side vertex

of fi(PN) and the farthest left hand side vertex of fi1(PN) coincide if and only if

vi vim vi1 vi(m1) = Vim = fi(vim),


8/13


vi vi 1

Figure 2.15

Vi

Vim

MVi 1

Vi 2

Figure 2.16

vi vi 1

Figure 3.1

vj

Figure 3.2

as claimed by the equality (2.5).

3. The Maximal Contraction Ratio

Let us make the following elementary observation before proving the Main Theorem.

Proposition 3.1. LetPN be a regular N-gon with vertices v1, , vN, as in Definition 2.1.

Then, (vivjvi1) = /N, for any i, j {1, , N} and j = i, i 1.

Proof. Because PN is a regular polygon, it can be inscribed into a circle as illustrated in

Figure 3.1. Consequently, (vivjvi) is all identical for each j {1, , N} such that

j = i, i 1. Let us denote the common quantity as . Then, from the symmetry, we must

have (vj1vjvj1) = (N 2) , as illustrated in Figure 3.2. Because the external angle of

a regular N-gon is 2/N, we must have (N 2) + 2/N = , as depicted in Figure 3.2.

Consequently, we must have = /N as claimed.

We are now ready to prove the first part of the Main Theorem, which we restate as follows.


9/13


Theorem 3.2 (Maximal Contraction Ratio). LetSr(PN) be a uniform Sierpinski fractal of

a regular N-gon, PN with vertices v1, , vN. Then, the largest possible contraction ratio,

which we denoted as RN in Lemma 2.2, turns out to be,

(3.1) RN =1

2

1

tanN

N14

tan

N

+ N

N14

.Proof. First, we make some trivial observations to reduce the problem. From symmetry, it is

easy to see that the uniform contraction ratio r must be the biggest when the polygon PN is

regular. For the remainder of the proof, therefore, let us assume PN is a regular N-gon with

the vertices v, , vN introduced in Definition 2.1. Now, observe that the equality (3.1) is

correct when N = 3 or N = 4, in which case, we get R3 = R4 =12 . Hence, we need only to

consider the case, N 5, in which case we can apply Lemma 2.2.

Lemma 2.2 tells us that we must concentrate our attention to the case where the adjacentpolygons fi(PN) and fi1(PN) share a common vertex at a principal crossing point ofGN, as

depicted in the lower region of Figure 2.15. For convenience, let us expand the region as in

Figure 3.3. As in the proof of Lemma 2.2, let Vj = fi(vj). Also, let fi1(vj) = Uj , and let M

be the midpoint of vi and vi1, or equivalently, that of Vi and Ui1. All these are illustrated

in Figure 3.3. Note that Vim is the principal crossing point between fi(PN) and fi1(PN),

as we proved in Lemma 2.2.

And then, let us name the common vertices vi and vi1 of PN as A (blue) and E (green),

respectively. Also, we name the vertices fi(vi1) and fi1(vi1) as B and D, respectively,

and let C be the midpoint of the line segment AE. Finally, let us label the principal crossing

point fi(vim) as F (red) and the midpoint of AE as C.

Now, let s be the side-lengh of PN, that is, s = 1(vi vi1) = 1(Vi Ui1). Then clearly,

(3.2) 1(Vi M) = 1(M Ui1) =s

2.

Also, from Lemma 2.2, we must have

(3.3) 1(Vi Vi1) = 1(Ui Ui1) = s RN, 1(Vi1 M) = 1(Ui D) =s

2 s RN.

On the other hand, using Proposition 3.1, we can express the size of the angle Ui1ViVimas

(Ui1ViVim) = (Vi1ViVim)(3.4)

= (Vi1ViVi2) + + (Vi(m1)ViVim)

=

N+ +

N=

N(m 1) =

N

N 1

4

.

Note that

The equality (3.4) tells us the size of internal angleVi1ViVim of the triangle Vi1ViVimis

N

N14

. Also, Proposition 3.1 tells us that the size of the angle ViVimVi1 is

N

. Hence,


10/13


Ui 1Vi

Vim

MVi 1 Ui

Figure 3.3

we conclude that

(VimVi1M) = (ViVi1Vim)(3.5)

= (ViVimVi1) + (Vi1ViVim)

=

N +

NN 1

4

.

Finally, we put everything together. From (3.2) and (3.4), we get

(3.6) 1(Vim M) = 1(Vi M)tan((Vi1ViVim)) =s

2tan

N

N 1

4

.

Also, from (3.3) and (3.5), we get

1(Vim M) = 1(Vi1 M)tan((VimVi1M))(3.7)

= s2

s RN tan

N+

NN 1

4 .

From (3.6) and (3.7), we get the equation,

(3.8)s

2tan

N

N 1

4

=s

2 s RN

tan

N+

N

N 1

4

,

where the positive constant s cancels itself out. Solving for RN through elementary algebra,

we get the equality (3.1). This completes the proof of Theorem 3.2, which is the first part

of the Main Theorem.


11/13


The second part of the Main Theorem follows immediately from Hutchinsons Theorem

[4], which can be found in most standard textbooks on fractals, say [2, 3, 6].

Corollary 3.3. Let Sr(PN) be a uniform Sierpinski fractal of a regular N-gon, PN with

vertices v1, , vN. Then, the maximal Hausdorff dimension hN of Sr(PN) is attained when

r = RN, given by the equality (3.1) of Theorem 3.2, and

(3.9) hN =ln N

ln(1/RN)=

ln N

ln RN.

Proof. From Hutchinsons Theorem on self similar fractals [4],

Ni=1

rhi = 1,

where ri is the contraction ratio of fi. Because r1 = = rN = r, we must have

Nrh = 1, rh =1

N, h = logr(1/N) =

ln N

ln r.

The maximum of h is attained, therefore, when r reaches the maximum, RN. Therefore,

hN = ln N

ln RN.

4. Discussion

In this paper, we set up our problem and the main result using convex polygons, primarily

for convenience. However, the equality (1.4) of the Main Theorem applies in more generalcircumstances as well. As we see from Figure 4.1 Figure 4.4, the Main Theorem applies

to a class of the self similar fractals given by N-grams, which we will call the regular N-

gram fractals. It is but a trivial exercise to set up the precise definition of the regular

N-gram fractals and the appropriate generalization of the Main Theorem, so we leave it to

the interested readers. It is not surprising at all that such generalization is possible. The

uniform contraction ratio r reaches the maximum when the outer vertices of the N-grams

touch each other (Figure 4.2), but the outer vertices of the N gram coincide with the vertices

ofPN (Figure 4.1), and thus the equality (1.4) applies. Indeed, as more iterations are taken,

the regular N-gram fractals gets more similarto the regular Sierpinski fractals, as illustratedin Figure 4.3 and Figure 4.4. Again, we leave the detail to the readers.

If we consider the inner vertices as well, however, the regular N-gram fractal problem

becomes even more interesting. As depicted in Figure 4.5 Figure 4.7, we need to consider

the contractions at the inner vertices as well, with different uniform contraction ratios. For

Figure 4.5 Figure 4.7, the contraction ratios ro = R7 0.308 and ri = 0.136 were selected,

for the contractions at the outer and the inner vertices, respectively. At this moment, we are

not sure what to expect in this case. In any case, it appears that the maximal contraction


12/13


Figure 4.1. N = 7, r = RN 0.30798. Figure 4.2. N = 7, first iteration.

Figure 4.3. N = 7, third iteration. Figure 4.4. N = 7, fourth iteration.

Figure 4.5. ro 0.308, ri = 0.136. Figure 4.6. N = 7, first iteration.

ratio formula (1.4) plays some role, but perhaps not in general. For the time being, therefore,

we leave this problem open for future research.


13/13


Figure 4.7. N = 7, second iteration. Figure 4.8. N = 7, third iteration.

Acknowledgments. This paper is partly based upon the undergraduate research project [1]

of the second author, supervised by the first author. It was supported in part by Morris Aca-

demic Partnership of University of Minnesota at Morris, and also in part my Mathematical

Association of America. The authors thank Mark Logan, who helped the authors as the sec-

ond reader of this undergraduate research project. They also thank the anonymous referees

of the journal, Fractals, for valuable criticisms and suggestions for the revision of this paper.

References

[1] J. Davis, Sierpinskii Fractals and their dimensionality, B.A. Thesis, University of Minnesota at Morris,

2009.

[2] G. Edgar, Measure, Topology, and Fractal Geometry, Undergraduate Texts in Mathematics, Springer,2007.

[3] K. Falconer, Fractal Geometry: Mathematical Foundations and Application, Wiley, 2nd ed., 2003.

[4] J. Hutchinson, Fractals and self-similarity, Indiana J. Math., 30 (1981), pp. 713747.

[5] B. Kahng, Sierpinski fractal plotter, http://cda.morris.umn.edu/kahng/Sierpinski.html, (2009).

[6] C. A. Rogers, Hausdorff Measures, Cambridge Mathematical Library, Cambridge Univ. Press, 2nd ed.,

1998.

Byungik Kahng, 600 East 4th Street, Division of Science and Mathematics, University

of Minnesota, Morris, MN 56267, USA

E-mail address: [email protected]

Jeremy Davis, 600 East 4th Street, UMM Mail 220, University of Minnesota, Morris, MN

56267, USA

E-mail address: [email protected]

Date post:	06-Apr-2018
Category:	Documents
Upload:	irokk
View:	225 times
Download:	0 times

Byungik Kahng and Jeremy Davis- Maximal Dimensions of Uniform Sierpinski Fractals

Documents