THE SNOWFLAKE CURVE AS AN ATTRACTOR OF AN ITERATED FUNCTION SYSTEM

Size: px

Start display at page:

Download "THE SNOWFLAKE CURVE AS AN ATTRACTOR OF AN ITERATED FUNCTION SYSTEM"

Cornelius Herbert Bryan
5 years ago
Views:

1 Commun. Korean Math. Soc. 8 (0), No., pp THE SNOWFLKE CURVE S N TTRCTOR OF N ITERTED FUNCTION SYSTEM Bünyamin Demir, Yunus Özdemir, and Mustafa Saltan bstract. lthough the snowflake curve, the boundary of the Koch snowflake, is one of the well-known fractals, there is no iterated function system (IFS) for it in the literature. In this study, we give an IFS for this familiar closed curve.. Introduction Many fractals can be obtained as attractors of iterated function systems (IFS) ([], [], []). The Koch curve is a well-known example. It is the attractor of the IFS consisting of the similarity contractions f,f,f,f 4 on the plane such that ( x f (x,y) =, y ( ) x y f (x,y) = 6 + ) x 6, + y 6 6 ( x y f (x,y) = x 6, + y ) ( x f 4 (x,y) = +, y ) (see Fig. ) [4]. It is known that there is another IFS for the Koch curve consisting of only two contractions with contractivity factor which map the triangle BC to the triangles DB and EC respectively as shown in Fig.. The Koch snowflake can also be obtained as the attractor of the IFS consisting of seven contractions which can be immediately written from Fig.. These contractions are also similarities. Since the images of an attractor under the contractions of an IFS are usually small copies of the attractor, the notion of self-similarity is geared to the notion of iterated function system. One can easily exhibit some IFS for the Koch curve Received December 6, 00; Revised pril 5, Mathematics Subject Classification. 880, 7C5. Key words and phrases. snowflake curve, Koch curve, iterated function system. 55 c 0 The Korean Mathematical Society

2 56 BÜNYMIN DEMİR, YUNUS ÖZDEMİR, ND MUSTF SLTN f (K) f (K) f (K) f 4 (K) Figure. The Koch curve K as the union of 4 similar copies of itself. B D E C Figure. The Koch curve K as the union of similar copies of itself. and the Koch snowflake. However, it is clear that there is no IFS consisting only of similarity contractions for the snowflake curve. In the next section, we give an IFS consisting of 8 contractions each of which is the composition of some similarities, symmetries and foldings.. n IFS for the snowflake curve Consider the contractions f,f : R R defined by f (x,y) = ( x, ) ( y + ), 6 ( f (x,y) = y, ) ( x + ),. Let f be the folding defined by f (x,y) ( = ( x,y) for all x,y R. Let f be the projective similarity with center C = 4, ) 4 and contractivity factor whichmapsthepolygonalpathbcdefgontothe path B C D E F G as shown in Fig. 4. Then the contraction f can be written as the composition of the folding f and the projective similarity f as shown in Fig. 5: f = f f.

THE SNOWFLKE CURVE 57 Figure. TheKochsnowflakeastheunionof7similarcopies of itself. ' C C' E' B' D' F' D G' B E F G Figure 4. f maps the path BCDEFG onto the path B C D E F G.

3 THE SNOWFLKE CURVE 57 Figure. TheKochsnowflakeastheunionof7similarcopies of itself. ' C C' E' B' D' F' D G' B E F G Figure 4. f maps the path BCDEFG onto the path B C D E F G. Let f and f be the foldings defined on the plane by f(x,y) = (x, y ) and f (x,y) ( = ( x,y) ) respectively. Let f be the projective similarity with centerc = and contractivityfactor which maps the polygonalpath 0, BCD onto the polygonal path B C D as shown in Fig. 6. Let f 4 be the

4 58 BÜNYMIN DEMİR, YUNUS ÖZDEMİR, ND MUSTF SLTN vertical folding f f f projective similarity Figure 5. The mapping f = f f. ' C' B' D' ' B'' C B D' C B C'' D D Figure 6. The projective similarity f (left) and the symmetry f 4 (right).

5 THE SNOWFLKE CURVE 59 horizontal folding f f vertical folding f 4 f symmetry f projective similarity Figure 7. The map f = f 4 f f f. symmetry with respect to the angle bisector between C and D as shown in Fig 6. Then the contraction f can be written as the composition of the foldings f,f and the projective similarity f and the symmetry f 4 as shown in Fig. 7: f = f 4 f f f. Note that f and f map the snowflake curve, S, onto its quadrant (see Fig. 8). The mappings f and f may be adapted as below (f to f 8 ) to map the snowflake curve onto its parts in the other three quadrants. f (x,y) = ( x, ) ( y + ), 6 ( ) ( f 4 (x,y) = y, x + ), f 5 (x,y) = ( x, y ) + (, ) 6

6 60 BÜNYMIN DEMİR, YUNUS ÖZDEMİR, ND MUSTF SLTN f ( S ) f 8 f ( S ) f 7 f f 5 f 4 f 6 Figure 8. The snowflake curve S as an attractor of the IFS. f 6 (x,y) = f 7 (x,y) = f 8 (x,y) = ( ) ( y, x +, ( x, ) ( ) y +, 6 ( y, ) x + (, ) ). The attractor of the IFS consisting of {f,f,f,f 4,f 5,f 6,f 7,f 8 } is the desired fractal, the snowflake curve (see Fig. 8). There are alternative iterated function systems for the snowflake curve each of which is formed by 6 contractions. Let g be the mapping that folds the right part of the line G onto the left part of it (as done by f before) and g be the mapping that folds the below part of the line EH onto the above part of it (see Fig. 9 (left)). g g maps the snowflake curve to the Koch curve. s done in the previous section (see Fig. ), the Koch curve can be obtained as the union of the images of itself under two similarities h and h. So that the Koch curve is the union of the images of the snowflake curve under the contractions f = h g g and f = h g g. One can apply similar procedure to obtain the other parts (the other two Koch curves) of the snowflake curve (see Fig. 9).

7 THE SNOWFLKE CURVE 6 f C B H D f ( S ) E G Figure 9. The lines G and EH (left) and the images of S under f and f (right). Note: In general, a folding map g that maps the half plane ax+by + c > 0 onto the other half plane can be defined by ( b ) x aby ac a ax+by+c g(x,y) = a +b, a y abx bc b ax+by+c a +b. Note that in our case a =, b = and c = 0. cknowledgements. uthors cordially thank the referees for their important contributions and creative reviews to the article. References [] M. F. Barnsley, Fractals Everywhere, cademic Press, Boston, 988. [] G. Helmberg, Getting cquainted with Fractals, Walter de Gruyter, Berlin, 007. [] J. E. Hutchinson. Fractals and self-similarity, Indiana Univ. Math. J. 0 (98), no. 5, [4] H. Jürgens, H. O. Peitgen, and D. Saupe. Chaos and Fractals: New Frontiers of Science, Springer-Verlag, New York, 99. Bünyamin Demir Deparments of Mathematics nadolu University 6470, Eskişehir, Turkey address: bdemir@anadolu.edu.tr

8 6 BÜNYMIN DEMİR, YUNUS ÖZDEMİR, ND MUSTF SLTN Yunus Özdemir Deparments of Mathematics nadolu University 6470, Eskişehir, Turkey address: Mustafa Saltan Deparments of Mathematics nadolu University 6470, Eskişehir, Turkey address:

THE GRAPH OF FRACTAL DIMENSIONS OF JULIA SETS Bünyamin Demir 1, Yunus Özdemir2, Mustafa Saltan 3. Anadolu University Eskişehir, TURKEY

International Journal of Pure and Applied Mathematics Volume 70 No. 3 2011, 401-409 THE GRAPH OF FRACTAL DIMENSIONS OF JULIA SETS Bünyamin Demir 1, Yunus Özdemir2, Mustafa Saltan 3 1,2,3 Department of