Course:Harris, Fall 07: Diary Week 15

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Mon:

  • Groups finished up Iteration Exploration (doing part 3 first helps to understand parts 1 and 2).
  • Spoke about dimension and scaling:
    • If a two-dimensional object is doubled in its linear dimensions, keeping its shape (i.e., a similarity transformation), then it increases 4-fold in size.
      • Example: It takes 4 copies of a rectangle to build up to a new one with twice the linear dimensions.
    • Similarly, if a two-dimensional object has its linear dimensions increased by a factor of <math>k</math>, its overall size increases by a factor of <math>k^2</math>.
    • If a three-dimensional object is doubled in its linear dimenions, keeping its shape, it increases 8-fold in size.
      • Example: It takes 8 copies of a box to build up to a new one with twice the linear dimensions.
    • Similarly, if a three-dimensional object has its linear dimensions increased by a factor of <math>k</math>, its overal size increases by a factor of <math>k^3</math>.
    • Thus dimension can be expressed as a ratio of logarithms: <math>log(k^2)/log(k) = 2</math> and <math>log(k^3)/log(k) = 3</math> (because <math>log(k^d) = d(log(k))</math>).
    • That is why self-similar sets like fractals can have a dimension calculated by looking at the process of creating them:
      • If the similarity transformation defining it has change in linear dimension by a factor of <math>r</math> while creating <math>N</math> copies of the object (at the new size), then the "fractional dimension" is <math>log(N)/log(r</math>), as in the Fractal Dimension Exploration.
  • Groups worked on the Fractal Dimension Exploration, finishing at the end.


Wed:

  • Essays on the shape of the universe were returned with questions and comments put individually on each.
    • Further points can be earned by further thoughts exploring those questions and comments.
  • Groups worked on their choice of additional Explorations--mostly choosing the Impossible Exploration, but one choosing the Flat Exploration.
  • Friday groups will continue, doing another Exploration.


Fri:

  • Self-Similarity Exercises collected.
  • Groups worked further on explorations (at least one chose Dimensions Exploration).
  • Some comments made on final exam (reflected in the description on the schedule for next week).