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<Text-field style="Normal256" layout="Normal256">Superfractal forms</Text-field>
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<Text-field style="Normal258" layout="Normal258">Wieslaw Kotarski <Hyperlink linktarget="http://" hyperlink="true"></Hyperlink>(<Hyperlink linktarget="mailto:kotarski@math.us.edu.pl" hyperlink="true"><Font style="Hyperlink">kotarski@math.us.edu.pl</Font></Hyperlink>) &amp; Agnieszka Lisowska (<Hyperlink linktarget="mailto:alisow@ux2.math.us.edu.pl" hyperlink="true"><Font style="Hyperlink">alisow@ux2.math.us.edu.pl</Font></Hyperlink>)<Hyperlink linktarget="http://" hyperlink="true"></Hyperlink></Text-field>
<Text-field style="Normal258" layout="Normal258">Institute of Computer Science</Text-field>
<Text-field style="Normal258" layout="Normal258">Silesian University</Text-field>
<Text-field style="Normal258" layout="Normal258">Bedzinska 39</Text-field>
<Text-field style="Normal258" layout="Normal258">41-200 Sosnowiec, Poland</Text-field>
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<Text-field style="Normal257" layout="Normal257">(Abstract)</Text-field>
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<Text-field style="Normal" layout="Normal">The aim of this worksheet is to present Maple users some experiments related to superfractals which are a new kind of fractal objects recently discovered by M. Barnsley [4]. Classical fractals are self-similar objects that are uniquelly determined by their IFSs whereas in case of superfractals some probabilistic mechanism is implemented.</Text-field>
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<Text-field style="Heading 1" layout="Heading 1">1. Introduction</Text-field></Title>
<Text-field style="Normal" layout="Normal">The notion of superfractals can be explained on the example [3]  given below. Take two IFSs F={f1,f2} and G={g1,g2},</Text-field>
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<Text-field style="Normal" layout="Normal">Attractors generated by the above IFSs take the form presented in Fig. 1, where the upper part of the fractal is generated by F, whereas the lower part by G..</Text-field>
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<Text-field style="Normal257" layout="Normal257">Fig. 1. Two attractors (upper F and lower G).</Text-field>
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<Text-field style="Normal" layout="Normal">Let us carry out the following experiment. In two buffers the left L and the right R arbitrary initial sets are put. Next, the contents of both buffers are modified using the following probabilistic mechanism [3]:</Text-field>
<Text-field style="Normal" layout="Normal">1. Select an IFS (F or  G). Then select a buffer (L or  R) and draw its content changed by first transformation. Again, select a buffer (it may be the one used earlier) and draw it changed by second transformation. Finally, sum up the results (union of sets) of both transformations obtaining a new buffer L'. </Text-field>
<Text-field style="Normal" layout="Normal">2. Select an IFS (it may be the one used earlier). Then select a buffer (again L or R) and draw its content changed by first transformation. Select another buffer and draw it changed by second transformation. Finally, sum up the results of both transformations obtaining a new buffer R'. </Text-field>
<Text-field style="Normal" layout="Normal">3. Let L:=L' and  R:=R' Go to step 1.</Text-field>
<Text-field style="Normal" layout="Normal">After performing these steps new contents of buffers L and R are obtained. That process is repeated in such a way that probabilities of IFS and buffer choosing are equal to 1/2. After several iterations obtained contents of both buffers are independent on initial sets, at the very beginning stored in both buffers. Obtained attractors are probabilistic variations of input objects. They form a collection of fractal forms called superfractal. Some elements of superfractal collection, resembling necklases, are presented in Fig. 2. </Text-field>
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</Group>
<Text-field style="Normal257" layout="Normal257">Fig. 2. Some elements of superfractal collection obtained using F and G attractors.</Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field style="Normal" layout="Normal">The experiment described above can be performed  with  larger number of IFSs and buffers and with different probabilities of choosing IFSs and buffers.  Because of probabilistic character of  superfractal algorithm in every run of the program one can obtain different fascinating fractal forms. <Font bold="true" style="Text">So then, it is suggested to recalculate the worksheet several times!</Font></Text-field>
</Section>
<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 1" layout="Heading 1">2. Maple Program</Text-field></Title>
<Group labelreference="L4" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">restart;</Text-field>
</Input>
</Group>
<Group labelreference="L5" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(linalg):</Text-field>
</Input>
</Group>
<Group labelreference="L6" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(plots):</Text-field>
</Input>
</Group>
<Group labelreference="L7" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">with(plottools):</Text-field>
</Input>
</Group>
<Group labelreference="L8" drawlabel="true">
<Input>
<Text-field style="Text" bold="true" family="Monospaced" layout="Normal"><Font bold="true" family="Monospaced">Given two IFSs</Font></Text-field>
</Input>
</Group>
<Group labelreference="L53" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">f1:=transform((x, y)-&gt;evalf([1/2*x+3/8*y-1/16, 1/2*x-3/8*y+9/16])):</Text-field>
</Input>
</Group>
<Group labelreference="L55" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">f2:=transform((x, y)-&gt;evalf([1/2*x-3/8*y+9/16, -1/2*x-3/8*y+17/16])):</Text-field>
</Input>
</Group>
<Group labelreference="L57" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">g1:=transform((x, y)-&gt;evalf([1/2*x+3/8*y-1/16, -1/2*x+3/8*y+7/16])):</Text-field>
</Input>
</Group>
<Group labelreference="L42" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">g2:=transform((x, y)-&gt;evalf([1/2*x-3/8*y+9/16, 1/2*x+3/8*y-1/16])):</Text-field>
</Input>
</Group>
<Group labelreference="L15" drawlabel="true">
<Input>
<Text-field style="Text" bold="true" family="Monospaced" layout="Normal"><Font bold="true" family="Monospaced">Procedure n - number of iterations, p=0 for buffer L, p=1 for buffer R </Font></Text-field>
</Input>
</Group>
<Group labelreference="L50" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">supfrac:=proc(L,R,p,n)
  local L0,R0,d1,d2,d3,d4,h;
   
    d1:=rand(0..1):  ## IFS drawing

    if d1()=0 then 
       h:=[f1,f2];
    else 
       h:=[g1,g2];
    fi;

    d2:=rand(0..3); ## Buffer drawing and contents modification of L  
    
    if d2()=0 then 
    L0:=display({h[1](L),h[2](L)});
    elif d2()=1 then 
    L0:=display({h[1](L),h[2](R)});
    elif d2()=2 then 
    L0:=display({h[1](R),h[2](L)});
    else 
    L0:=display({h[1](R),h[2](R)});
    fi;

    d3:=rand(0..1); ## IFS drawing
        
    if d3()=0 then 
       h:=[f1,f2];
    else 
       h:=[g1,g2];
    fi;

    d4:=rand(0..3); ## Buffer drawing and contents modification of R 

    if d4()=0 then 
    R0:=display({h[1](L),h[2](L)});
    elif d4()=1 then 
    R0:=display({h[1](L),h[2](R)});
    elif d4()=2 then 
    R0:=display({h[1](R),h[2](L)});
    else 
    R0:=display({h[1](R),h[2](R)});
    fi;
## Buffer choice for displaying its contents (p=0 for L and p=1 for R)

    if n = 0 then 

    if p=0 then display(L);
    else display(R); 
    fi;

    else supfrac(L0,R0,p,n-1); ## Recurrence 
    
    fi:
end:</Text-field>
</Input>
</Group>
<Group labelreference="L16" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
<Text-field style="Normal" layout="Normal"></Text-field>
</Section>
<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 1" layout="Heading 1">3. Examples</Text-field></Title>
<Text-field style="Normal" layout="Normal"></Text-field>
<Group labelreference="L17" drawlabel="true">
<Input>
<Text-field style="Text" bold="true" family="Monospaced" layout="Normal"><Font bold="true" family="Monospaced">3.1. Contents of one buffer is displayed. </Font></Text-field>
</Input>
</Group>
<Group labelreference="L18" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display(supfrac(polygonplot([[0,0],[1,0],[1,1]]),polygonplot([[0,0],[1,0],[1,1]]),1,10),axes=none,color=yellow,scaling=constrained);</Text-field>
<Text-field style="Maple Plot" layout="Maple Plot"></Text-field>
</Input>
<Output>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">dis1:=display(seq(supfrac(polygonplot([[0,0],[1,0],[1,1]]),polygonplot([[0,0],[1,0],[1,1]]),0,i),i=0..8),scaling=constrained,insequence=true,axes=none,color=yellow):</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display(dis1);</Text-field>
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<Output>
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<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">dis2:=display(seq(supfrac(polygonplot([[0,0],[1,0],[1,1]]),polygonplot([[0,0],[1,0],[1,1]]),1,i),i=0..8),scaling=constrained,insequence=true,axes=none,color=yellow):</Text-field>
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<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display(dis2);</Text-field>
</Input>
<Output>
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<Group labelreference="L23" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">dis1:=display(seq(supfrac(polygonplot([[0,0],[1,0],[1,1]]),polygonplot([[0,0],[1,0],[1,1]]),0,8),i=0..8),scaling=constrained,insequence=true,axes=none,color=yellow):</Text-field>
</Input>
</Group>
<Group labelreference="L24" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">dis2:=display(seq(supfrac(polygonplot([[0,0],[1,0],[1,1]]),polygonplot([[0,0],[1,0],[1,1]]),1,8),i=0..8),scaling=constrained,insequence=true,axes=none,color=yellow):</Text-field>
</Input>
</Group>
<Group labelreference="L25" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display(dis1);</Text-field>
</Input>
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</Output>
</Group>
<Group labelreference="L26" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display(dis2);</Text-field>
</Input>
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<Text-field style="Text" bold="true" family="Monospaced" layout="Normal"><Font bold="true" family="Monospaced">3.2. Joined contents of both buffers is displayed. Observe, that one can obtain nice fractal forms. Some of them may even inspire motifs for jewellery! </Font></Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display(supfrac(polygonplot([[0,0],[1,0],[1,1]]),polygonplot([[0,0],[1,0],[1,1]]),1,10),supfrac(polygonplot([[0,0],[1,0],[1,1]]),polygonplot([[0,0],[1,0],[1,1]]),1,10),axes=none,color=yellow);</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">dis2:=display(seq(display(supfrac(polygonplot([[0,0],[1,0],[1,1]]),polygonplot([[0,0],[1,0],[1,1]]),0,i),supfrac(polygonplot([[0,0],[1,0],[1,1]]),polygonplot([[0,0],[1,0],[1,1]]),1,i)),i=0..8),scaling=constrained,insequence=true,axes=none,color=yellow):</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display(dis2);</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">dis1:=display(seq(display(supfrac(polygonplot([[0,0],[1,0],[1,1]]),polygonplot([[0,0],[1,0],[1,1]]),0,i),supfrac(polygonplot([[0,0],[1,0],[1,1]]),polygonplot([[0,0],[1,0],[1,1]]),1,i)),i=0..8),scaling=constrained,insequence=true,axes=none,color=yellow):</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">dis2:=display(seq(display(supfrac(polygonplot([[0,0],[1,0],[1,1]]),polygonplot([[0,0],[1,0],[1,1]]),0,i),supfrac(polygonplot([[0,0],[1,0],[1,1]]),polygonplot([[0,0],[1,0],[1,1]]),1,i)),i=0..8),scaling=constrained,insequence=true,axes=none,color=yellow):</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">display(dis1);</Text-field>
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<Text-field style="Heading 1" layout="Heading 1">4. Further experiments</Text-field></Title>
<Text-field style="Text" layout="Normal">It is suggested to replace IFSs from the begining of this worksheet by the following given below. IFS F={f1,f2} generates the upper part, whereas IFS G={g1,g2} generates the lower part of the fractals presented in figures below. Some of generated superfractal objects present nice aesthetic art forms that can be used in creation of ornaments or talismans.   </Text-field>
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<Text-field style="Heading 2" layout="Heading 2">4.1. Example 1 </Text-field></Title>
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</Section>
<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 2" layout="Heading 2">4.2. Example 2</Text-field></Title>
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<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">f1:=transform((x, y)-&gt;evalf([1/2*x+3/28*y, 1/2*x-11/20*y])):</Text-field>
</Input>
</Group>
<Group labelreference="L85" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">f2:=transform((x, y)-&gt;evalf([1/2*x-3/28*y+1/2, -1/2*x-11/20*y+1/2])):</Text-field>
</Input>
</Group>
<Group labelreference="L84" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">g1:=transform((x, y)-&gt;evalf([1/2*x-7/44*y, -1/2*x-11/20*y])):</Text-field>
</Input>
</Group>
<Group labelreference="L83" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">g2:=transform((x, y)-&gt;evalf([1/2*x+7/44*y+1/2, 1/2*x-11/20*y-1/2])):</Text-field>
</Input>
</Group>
<Group labelreference="L82" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRicvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=</Equation></Text-field>
</Input>
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<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRicvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=</Equation></Text-field>
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<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 2" layout="Heading 2">4.3. Example 3 </Text-field></Title>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal">f1:=transform((x, y)-&gt;evalf([1/2*x-3/28*y, 1/2*x+11/20*y])):</Text-field>
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<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">f2:=transform((x, y)-&gt;evalf([1/2*x+3/28*y+1/2, -1/2*x+11/20*y+1/2])):</Text-field>
</Input>
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<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">g1:=transform((x, y)-&gt;evalf([1/2*x+7/44*y, -1/2*x+11/20*y])):</Text-field>
</Input>
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<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">g2:=transform((x, y)-&gt;evalf([1/2*x-7/44*y+1/2, 1/2*x+11/20*y-1/2])):</Text-field>
</Input>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRicvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=</Equation></Text-field>
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</Section>
<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 2" layout="Heading 2">4.4. Example 4</Text-field></Title>
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<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">f1:=transform((x, y)-&gt;evalf([1/2*x+3/28*y, 1/2*x-11/20*y])):</Text-field>
</Input>
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<Group labelreference="L227" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">f2:=transform((x, y)-&gt;evalf([1/2*x-3/28*y+1/2, -1/2*x-11/20*y+1/2])):</Text-field>
</Input>
</Group>
<Group labelreference="L226" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">g1:=transform((x, y)-&gt;evalf([1/2*x+6/25*y, -1/2*x+11/20*y])):</Text-field>
</Input>
</Group>
<Group labelreference="L225" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">g2:=transform((x, y)-&gt;evalf([1/2*x-6/25*y+1/2, 1/2*x+11/20*y-1/2])):</Text-field>
</Input>
</Group>
<Group labelreference="L91" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRicvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=</Equation></Text-field>
</Input>
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<Text-field style="Heading 1" layout="Heading 1">5. Final Remarks</Text-field></Title>
<Text-field style="Normal" layout="Normal">Classical fractals [1], [2] , that are self-similar deterministic objects, are described by IFSs. Their shapes are entirely determined by coefficients of affine transformations. In Nature such ideal fractal objects cannot be found. Usually, natural fractal forms are not ideally self-similar and they are charaterized by some diversity in shapes. Superfractal mechanism implements randomless to deterministic fractals. So then,  any superfractal collection  resembles natural objects much better than classical deterministic fractals. Superfractal forms inherit feature shapes from parent fractals defined by their IFSs. It is worth to mention that probabilistic mechanism can be joined with deterministic factor inserted by control points of fractals. Such approach, presented in [5], assures further possibilities to generate a great variety of fractal forms.</Text-field>
<Text-field style="Normal" layout="Normal">It should be pointed out that all superfractal forms generated in this worksheet represent connected sets. Such connected forms can be obtained under some restrictions imposed on IFSs. Namely, IFSs must have comman fixed points to produce &quot;continuous&quot; superfractal forms.</Text-field>
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<Text-field style="Heading 1" layout="Heading 1">6. References</Text-field></Title>
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<Text-field style="Normal" layout="Normal">[1] Mandelbrot B., The Fractal Geometry of Nature, Freeman and Company, San Francisco, 1983.
[2] Barnsley M., Fractals Everywhere, Academic Press, New York, 1988.
<Font encoding="UTF-8">[3] Barnsley M., Hutchinson J., Stenflo \303\226., V-variable Fractals and Superfractals, 2003, </Font></Text-field>
<Text-field style="Normal" layout="Normal"><Hyperlink linktarget="http://wwwmaths.anu.edu au/~barnsley/pdfs/V-var_super_fractals.pdf" hyperlink="true"><Font style="Hyperlink">http://wwwmaths.anu.edu au/~barnsley/pdfs/V-var_super_fractals.pdf</Font></Hyperlink>. 
[4 ]Barnsley M., Superfractals, Cambridge University Press, New York, Melbourne, 2006.
<Font encoding="UTF-8">[5] Kotarski W., Lisowska A., Domider R., Superfractals in Graphical Object Modelling, Proceedings of Conference Computer Methods and Systems, Krak\303\263w, Poland,  2007, 329-334.
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<Text-field style="Normal" layout="Normal">Legal Notice: The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities. </Text-field>
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