Sierpinski

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Waclaw Sierpinski

• Born in Warsaw, Poland in 1882.
• 1919- promoted to professor at Warsaw in
  mathematics and physics.
• 1920- founded math journal- Fundamenta
  Mathematica specialized in set theory.
• Retired in 1960 and was given many awards.
• Died in 1969.

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Sierpinskis Carpet

• Go to http//www.shodor.org/interactivate/activiti
  es/carpet/index.html

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Side Length Area of Removed Square

• After 1st iteration, what is the length of the
  removed square? What is the area of this square?
• After 2nd iteration, what is the length of
  removed square? What is the area?
• After 3rd,.?
• See any patterns? Write the formulas for side
  length area of a removed square.

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Number of Removed Squares

• After 1st iteration, how many squares have been
  removed?
• After 2nd iteration, how many squares have been
  removed?
• After 3rd iteration, how many squares have been
  removed?
• See any patterns? Write down the formula for
  determining the number of removed squares.

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Total Area Remaining

• With no iterations, what is the area remaining of
  the figure.
• After the 1st iteration, how would you find the
  area remaining?
• After the 2nd iteration, what is the area
  remaining of the figure?
• After the 3rd,?
• What patterns do you see? Write down the explicit
  formula for the area remaining.

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Area Removed

• How would we find the total area of the squares
  we removed?
• What relationship is there between the area
  remaining and the area removed?

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Limit of the Remaining Area

• Using the formula of total area remaining, what
  is the limit if we send n to infinity?

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The End