2. 
   Evolving Fractal (2 People): In order to draw the Sierpinski triangle, and some variations thereof, we are going to control a plotter through rotations and translations. This process needs to be self-similar, therefore we are going to use recursive functions to repeat the procedure over and over.

   a. Consider the cartesian axes as the reference ones in 2D. The basic op- erations are 3:

   i.  Draw a point at distance *l* from the origin *O* along the
       *x* axis (rotation angle 0)
   
   ii. Rotate by *α* = *π/*3 counter clockwise with respect to *x*
       and draw another point again at distance *l*
   
   iii. Rotate by *α* = *π/*3 clockwise with respect to *x* and
        draw another point again at distance *l*
   

Starting with O = (0*,* 0) and l = 1, this draws the first vertices of a triangle. To be fractal, every time you draw a point,

i. Set a new origin O^′^ at distance *l/*2 along the drawing direction.

This direction is going to be your new horizontal axis x^′^

ii. Now redo the 3 basic operations above using l^′^ = *l/*2

You can do this calling the same function recursively passing as argu- ments the l to use, the position of the origin O and an initial rotation angle with respect to an absolute reference system.

Remember to track the recursion depth and stop at some maximum value (7 recursions should be enough for a nice picture). Use a small dot ('.' for example) to have a less cluttered picture.

a. Now repeat the same procedure, but change the value of the angle α

for the side vertices.

Try a series of 4 configurations with α equally spaced between *π/*3 and

*π/*2, inclusive.

What do you obtain?

b. You can repeat all the above α configurations, but this time set the new origin O^′^ at a distance l(β − 1)/β and use a new distance l^′^ = l/β. You can try β = 2*.5,* 3*.*5.

What do you obtain?

c. Bonus: if instead of using the same α clock- and counterclock-wise you try different angles α1 and α2 of your choice, what do you see?