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# Fractals and Sequences

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by

## Jennifer Preissel

on 18 December 2015

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#### Transcript of Fractals and Sequences

Sequences and Fractals
They are all repeating visual
patterns
, called
fractals
...
Investigate a fractal. Determine which mathematical equations generate the patterns (or
sequences
) occurring in the fractal.

First. Some review. Remember that
arithmetic

sequences
are generated by the repeated addition of a
common difference
.
Some examples:
2, 4, 6, 8, 10...
10, 5, 0, -5, -10....

Geometric sequences
are generated by the repeated multiplication of a
common ratio
. Some examples:
2, 4, 8, 16, 32....
1, 3, 9, 27, 81...
-1, 1, -1, 1, -1...
4, 2, 1, 1/2. 1/4, 1/8...
And
recursive sequences
are generated by repeatedly applying the same pattern to each term to arrive at the subsequent term. The most famous example of this is the
Fibonnacci sequence
, whose formula is as follows:
t(0) = 1, t(1)=1
t(n+1) = t(n) + t(n-1)
1, 1, 2, 3, 5, 8, 13, 21...
Visit these links for more resources on fractals...

See what art you can create from fractals:
https://professortiz.wordpress.com/2013/07/22/geogebra-art/#jp-carousel-636

Play with this fractal machine to generate your own fractals: http://sciencevsmagic.net/fractal/#0103,0495,2,2,0,1,2

Learn about the fractal nature of rivers: http://fractalfoundation.org/resources/fractivities/fractal-rivers/

Here are some more "Fractivities" you can do on your own: http://fractalfoundation.org/resources/fractivities/

Now it's time for you to see how sequence equations relate to fractal patterns.

We are going to investigate a fractal called the Serpinski's Triangle, which you can see below.

Run several stages of the activity here:
http://www.shodor.org/interactivate/activities/SierpinskiTriangle/

http://goo.gl/forms/vxts5fvpmj
What do this shell...
...the Grand Canyon...
...Romanesco broccoli...
...a flash of lightning...
...and the Milky Way
galaxy have in common?

What makes a fractal a fractal?

https://georgemdallas.wordpress.com/2014/05/02/what-are-fractals-and-why-should-i-care/

The Fibonnaci Sequence occurs frequently in nature. Flower petals and spirals (like seashells) replicate outwardly in this pattern. Watch this video to find out more about how nature relis on this pattern.
Visit the
quiz
here to pratice your application of these concepts:
http://ed.ted.com/on/PJCeHRda#review