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