(Copyright: Ken Musgrave).
Barnsley's Fern
We can think of a fractal as a geometric object that has most or all of the
following features:
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has fine structure at arbitrarily small scales
is too irregular to be easily described in traditional Euclidean
geometric language.
is self-similar
has dimension greater than its topological dimension
has a simple and recursive definition.
Sierpinski gasket [1915]:
1. Draw an equilateral triangle and fill in the interior.
2. Put a hole in the triangle in the following way
 Find the midpoints of the triangle and connect these to make a
new triangle.
 Form a hole by removing the new triangle
3. Put a hole in each of the new triangles formed in step 2 using the
procedure in step 2.
4. Continue
Koch Snowflake [1904]
1. Draw an equilateral triangle.
2. In the middle of each side of the triangle attach another equilateral
triangle with sides 1/3 as long as the side it is attached to.
3. Continue.
Menger Sponge
LAB ASSIGNMENT #11
Fractals
DUE 11/20/2013
Name ____________________________
Grade ________________________
Mandelbrot set (60 points)
Look at the (x, y) coordinates in the area -2 < x < 0.7 and -1.2 < y < 1.2. Let z(0) = (x + iy) = C where i is
- 1 . The pattern for the Mandelbrot set is to see if z(n) < 2 or not where z(n+1) = z(n)2 + z(0). If z(n) < 2
as n →0, it belongs to the set. If z(n) > 2, color the (x, y) coordinate based on the value of n.
Find if the following points belong to the Mandelbrot set or not. If they do not, list the value of n when
z(n)>2.
Use Maple to color the Mandelbrot set. Use shift-enter for the procedure definition (proc)
> restart: with(plots):
> mset := proc(x,y)
local c, z, m;
c := evalf(x+y*I);
z := c;
for m from 0 to 40 while abs(z)<2 do
z := z^2+c
od;
m;
end:
> plot3d(0,-2..0.7,-1.2..1.2, orientation=[-90,0], grid=[250,
250],style=patchnogrid, color=mset);
Zoom in on an interesting part of the Mandelbrot set. Print or send me the image with the file named in a
word file named lastname_mset.docx. In the document, state the x and y range used.
Sierpinski gasket (40 points)
Draw your own Sierpinski gasket on white paper. The base of the
triangle should be at least 8”. Color each level differently.