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Math 116 Fractals What is a fractal? A fractal is a geometric figure that is divided into smaller versions of itself. What does a fractal look like?
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All of these picture where generated by Suzanne Alejandre they can view at this website: http://mathforum.org/alejandre/workshops/fractal/fractal3.html How is a fractal created? The Sierpinski Triangle The Sierpinski Triangle is generated by draw a triangle and then dividing the triangle in four equal parts. All pictures are courtesy Cynthia Lanius. 1) Divide the equilateral triangle into four same equilateral triangles as shown:
2) Now, Remove the middle triangle.
3) Repeat the same pattern by taking the three triangles at the corners and divide those triangles up the as shown for the triangle in step 2
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4) Again, repeat the same pattern for the shaded triangles in step 3
By repeating the same processes over and over we can create an interesting fractal called the Sierpinski Triangle.
Example 2 The Koch Curve
Add the generator segment to each segment will give the following geometric figure
Repeat the same process by adding the same generator segment to new segment, will give the new geometric figure.
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Repeating this process one more time will give the following fractal
Graphics courtesy Vanderbilt University at: http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html The dimension of a fractal
( )
rs
objectoriginaltolengthtoobjectnewoflengthofratiorobjectsofNumberN
sNd
1
log)log(
=
==
=
Example 1
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( ) 5.147712.69897.
)3log()5log(
log)log(
3315
====
=
=
=
sdd
s
r
objectsnewN
Example 2
Find the dimension of the fractal
( ) 26.14771.6021.
)3log()4log(
log)log(
3314
====
=
=
=
sdd
s
r
objectsnewN
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Example 3
( ) 21.16990.8451.
)5log()7log(
log)log(
5517
====
=
=
=
sdd
s
r
objectsnewN
Interesting looking fractal
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Applications of Fractals Cellular Phone Engineer John Chenoweth discovered that fractal antennas are 25 percent more efficient than rubbery “stubby” antennas. In addition, these types of antenna are cheaper to manufacture and fractal antennas also can operate on multiple bands. Here are some examples of fractal antennas: Siepinski’s Carpet
Koch Curve
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Sierpenski’s Triangle
Other examples
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Solutions to the Fractals Activity Siepenski’s Triangle Iteration Picture Total Objects
(Triangles) 0
130 =
1
331 =
2
932 =
3
2733 =
4
8134 =
58.1)2log()3log(
)log()log(,
21,2,3 ======
sNdrsN
Koch Curve Iteration Picture Total Objects
Segments 0 140 =
1
441 =
2
1642 =
3
6443 =
4 25644 =
26.1)3log()4log(
)log()log(,
31,3,4 ======
sNdrsN
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Siepenski’s Carpet Use the space below to draw the first three iterations of Sierpenski’s carpet
Iteration Total Objects
0 180 =
1 881 =
2 6482 =
3 51283 =
4 409684 =
Find the dimension of Sierpenski’s carpet
89.14771.9031.
)3log()8log(
8
3,31
===
=
==
d
N
sr
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Using the pictures of the Menger Sponge, complete the following chart, and then find the dimension of the fractal
Iteration Total Objects
0 1200 =
1 20201 =
2 400202 =
3 8000203 =
4 160000204 =
Find the dimension
73.24771.
30.1)3log()20log(
20
3,31
===
=
==
d
N
sr