Skip to main content
Mathematics LibreTexts

6.3.2: Exercises

  1. Last updated
  2. Save as PDF
  • Page ID
    74345

    • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Section 6.2.3 Exercises

    1. Using the initiator and generator shown, draw the next two stages of the iterated fractal.
      1.  

        An intiator of a horizontal line segment. The generator is the line segment cut in thirds with the middle third replaced with a square with no base.

      2.  

        The initiator is a horizontal line segment. The generator cuts the line segment in half and distances them from each other to form parallel lines connected by a diagonal line segment of the same length. Looks like a lightning bolt drawing.

      3.  

        The initiator is a diagonal line segment. The generator attaches three line segments diagonally on each side. One is half the length of the original segment, is attached one fourth of the way up on the left. One is one third of the original length and is attached one third of the way up on the right. The last is one fourth the original length and is attached three-fourths of the way up on the left.

      4.  

        An intiator of a horizontal line segment. The generator is the line segment cut in thirds with the middle third removed.

      5.  

        The initiator is a solid square. The generator is the square cut in to nine smaller squares with the middle one removed.

      6.  

        The initiator is an equilateral triangle. The generator adds to of the same triangle connected to the based of the original triangle so that the base vertices of the original triangle connects just below the top vertex of the two new triangle about one fourth of the way down. The left base vertex of the new triangle on the left and the right base vertex of the new triangle on the right connect in the middle.

    2. Create your own version of Sierpinski gasket with added randomness.
    3. Create a version of the branching tree fractal from example #3 with added randomness.
    4. Determine the fractal dimension of the Koch curve.
    5. Determine the fractal dimension of the curve generated in exercise #1
    6. Determine the fractal dimension of the Sierpinski carpet generated in exercise #5
    7. Determine the fractal dimension of the Cantor set generated in exercise #4

    This page titled 6.3.2: Exercises is shared under a CC BY-SA license and was authored, remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.