Course:Harris, Fall 07: Diary Week 15

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 $k$, its overall size increases by a factor of $k^{2}$.
  • 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 $k$, its overal size increases by a factor of $k^{3}$.
  • Thus dimension can be expressed as a ratio of logarithms: $log(k^{2})/log(k)=2$ and $log(k^{3})/log(k)=3$ (because $log(k^{d})=d(log(k))$).
  • 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 $r$ while creating $N$ copies of the object (at the new size), then the "fractional dimension" is $log(N)/log(r$), 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).