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\begin{center}
{\large Math 331, Spring 2006, Problems}
\end{center}
\begin{enumerate}




\setcounter{enumi}{25}
  




\item Create a procedure that takes a polygon and a \textbf{linear} transformation 
yields as output the result of applying the transformation  to the  polygon. Find the 
coordinates of  the linear transformation that sends the 
blue (larger) square to the pink (smaller) square. Apply the transformation to the blue square. 
Test your result by plotting the two squares in the same graph. (Remark: In this exercise, you are asked to work with a \emph{linear} transformation, in class we worked with affine transformations.)

\includegraphics[scale=0.3]{twoSquares.eps}


\item Create a procedure that takes as input a positive integer and two real numbers
$a$ and $b$ 
and produces as output (the list of points corresponding to) 
an $n$-gon centered at $(a,b)$. Using this procedure, obtain a list of 21 decagons centered in 
$(0,0),(1,1),..(20,20)$ and plot them all in the same graph.


\item 
Create a procedure that takes a list of decagons and a linear transformation
gives as output the result of applying the linear transformation 
to all the decagons.
Using your procedure, rotate each of the decagons of your list by an angle of $\pi/4$. Plot 
the  decagons and their images in the same graph.




\item \begin{enumerate}

\item  Measure the length at each stage in constructing the Cantor set.  
Since the Cantor set is made by removing intervals, each of our 
measurements exceeds the length of the Cantor set, but we can look for 
a pattern and deduce the actual length. To begin, we cover the Cantor 
set with one interval of length 1. Next, we cover it with two intervals 
of length 1/3, so its length is less than 2/3. Continue this line of 
reasoning and make some conclusion about the length of the Cantor set.




\item Now measure the lengths of the intervals removed in forming the 
Cantor set and see what remains. For example, first we remove one 
interval of length 1/3, then two intervals of length 1/9, and so on. 
Using a calculator, add up the lengths removed (1/3 + 2/9 + ... ). What 
do you get for the length of the Cantor set?

\item  Do your answers for (a) and (b) agree?
\end{enumerate}

\item Compute IFS parameters and the similarity dimension of the following fractal.

\includegraphics[scale=0.3]{squares.eps}


\item Compute perimeter and area of the snowflake

\includegraphics[scale=0.2]{snowflake.eps}


\item \begin{enumerate}
\item Find the IFS parameters to generate atractor  of the Picture: a 
right gasket of side length 2 and compute the similarity dimension.

\includegraphics[scale=0.5]{ex1.eps}

\item Find the IFS parameters to generate atractor  of the Picture: a 
right gasket of side length 4 and compute the similarity dimension.

\includegraphics[scale=0.4]{ex2.eps}

\item Find the IFS parameters to generate atractor  of the Picture: a 
right gasket of side length 6 and compute the similarity dimension.

\includegraphics[scale=0.3]{ex3.eps}

\item Find the IFS parameters to generate  a 
right gasket of side length s (where s is any positive integer) and 
compute the similarity dimension.


\item  Do the e and f parameters (the "translation part " of the 
parameters of the similarity ) have any effect on the dimension of the 
resulting fractal?

\end{enumerate}


\item \begin{enumerate}
\item  Compute the similarity dimension of the Cantor set 
formed from 
the unit interval by removing the middle half of the interval (as 
opposed to removing the middle third, in the usual construction).

\item  Compute the similarity  dimension of the Cantor  set formed from the 
unit interval by removing the middle 2/3 of the interval.

\item  Compute the similarity dimension of the Cantor s set formed from the unit 
interval by removing the middle 1/5 of the interval.
\end{enumerate}


\item Compute the similarity dimension of the Cantor set formed from the unit 
interval by removing the middle segment of length t, where t is any 
number $0 < t < 1$.







\end{enumerate}

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