The Cantor Set: an Indepth Gander at Fractals

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The Cantor Set an In-depth Gander at Fractals

• Greg Blachut
• Keith Janson
• Kinan Hayani
• Jake Folkerts

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History

• The Cantor Set was first published in 1883.
• It is named after the German Mathematician Georg
  Cantor.
• It is probably the most important early
  mathematical set.

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An example of Cantors Dust

• You take a line (no depth) of x length.
• Divide it into 3 pieces, and take out the middle
  one.
• Repeat

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Formula for number of segments

• How many segments are there at step 1?

• How many segments are there at step 2?

• How many segments are there at step 3?

• How many segments are there at step 4?

• So you have 1, 2, 4, 8, . . .

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So the formula for number of segments is ? ? ?

• Two to the n minus one

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The length of a segment

• Okay, so remember that you divide it into three
  parts and take out the middle part.

• So, lets say the first parts length is X.

• So after dividing it into three parts, and taking
  out one segment, what is the length?

• The total length at step two is 2/3x.

• Remembering the last formula, how many segments
  are there at step two?

• So how long is each individual segment?

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Now the equation . . .

• So, the first segment is X long, then in the next
  iteration, each segment is 1/3x, and then the
  next one is 1/9x.

• So again, the lengths is a geometric sequence.
• The Length of an Individual Segment
• (x is the original length)

• What happens to the sequence as n goes to
  infinity? What does that mean?

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To Find the Total Length of the Segments . . .

• The total length (the length of one) (the
  total amount of segments)

• What happens to the sequence as n goes to
  infinity? What does that mean (again)?