document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1012663>Question 1012663</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How could i make an equation for Von Koch curve?\r<BR>\n" );
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document.write( "I attempted it but no luck :/ \r<BR>\n" );
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document.write( "I do not know where to start, so a help would be very nice :D\r<BR>\n" );
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document.write( "I need to make an equation for perimeter and the area, please write each steps if possible.\r<BR>\n" );
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document.write( "Im so dumb T.T </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #628805 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=628805\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The Van Koch curve is a famous fractal construction<BR>\n" );
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document.write( "First, a straight line(unit length) is divided into three segments.  The middle segment is removed and replaced by two segments of equal length that form an equilateral triangle.<BR>\n" );
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document.write( "The above process is repeated for the four segments in the first step.<BR>\n" );
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document.write( "If we start with one line segment for iteration one and look at the the next two iterations we notice the following:<BR>\n" );
document.write( "iteration------segment length-----segment number------curve length<BR>\n" );
document.write( ":---0-----------------1------------------1------------------1<BR>\n" );
document.write( ":---1----------------1/3-----------------4------------------1.33<BR>\n" );
document.write( ":---2----------------1/9----------------16------------------1.77<BR>\n" );
document.write( ":---3----------------1/27---------------64------------------2.37<BR>\n" );
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document.write( "The perimeter increases by 4/3 for each iteration, therefore<BR>\n" );
document.write( "perimeter = (4/3)^n<BR>\n" );
document.write( "This shows that as the number of iterations increases, the curve length(perimeter) tends to infinity while it encloses a finite area<BR>\n" );
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document.write( "Now take a look at the area<BR>\n" );
document.write( "We take the first triangle formed on iteration 2 of unit area, then on the next iteration we have four small triangles added.  Each triangle is 1/9 the area of the unit triangle.<BR>\n" );
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document.write( "The next iteration produces 16 triangles but each trianlge is 81 times smaller than the unit triangle.<BR>\n" );
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document.write( "iteration-------triangle number-------triangle area-------curve area<BR>\n" );
document.write( ":---0------------------0---------------------0----------------0.00<BR>\n" );
document.write( ":---1------------------1---------------------1----------------1.00<BR>\n" );
document.write( ":---2------------------4---------------------0.44444----------1.44444<BR>\n" );
document.write( ":---3------------------16--------------------0.19753----------1.64197<BR>\n" );
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document.write( "Area = 1 + (4/9)^n where n is number of iteration after the iteration with 1 triangle.<BR>\n" );
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document.write( "Note that this equation converges to 0.8 as n goes to infinity, that is<BR>\n" );
document.write( "(4/9) / (1 - (4/9)) = 0.8<BR>\n" );
document.write( "   <BR>\n" );
document.write( " </SPAN>\n" );
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