WTSM 100L: Patterns in Nature

Study guide for second exam (Thursday, November 1, 2007)

Ground rules

You may bring a 3” x 5” index card on which you may write anything you want. You must turn this card in with your completed exam. You may bring a calculator (and may program it as you wish, but be sure you know how to use it and be sure the batteries are charged). Upon request, you may briefly use the computers to check answers. No other assistance will be permitted.

Full credit will not be given on any mathematical question unless you show your work (for example, make it clear what formula/procedure you’re using and what numbers you’re plugging in) and indicate your answer clearly. On questions requiring verbal answers, your answers should be clear, specific, and correct. You may supplement your words with clearly labeled sketches or illustrations.

The exam will be similar in length and format to the first exam.

Fractals

· Explain what scale invariance and self-similarity are. Be able to give examples and relate the terms to fractals.

· Sometimes fractals are said to have “symmetry under magnification.” Explain what this means, using the concepts of symmetry that we discussed earlier in the semester.

· Explain in words how a fractal such as the Koch curve, Sierpinski gasket, or Cantor set (for example) can be generated using the initiator/generator method.

· Given an initiator/generator, draw the first few stages of the corresponding fractal.

Fractal dimension; length and area

· Show (mathematically) what happens to the length of a fractal such as the Koch curve or Cantor middle-third set as the number of stages approaches infinity. Also show what happens to the area.

· Explain why the length of a coastline depends on the scale used to measure it.

· Given a regular mathematical fractal, identify the scale factor r and number of copies N, and be able to calculate its fractal dimension (similarity or Hausdorff dimension) from N and r.

· Given a fractal, be able to use the box-counting method to determine its fractal dimension. (Note: because of time and computer constraints, I won’t expect you to go through all the steps, but be able to describe/explain what should be done at each step, and given a graph with a trendline, be able to interpret it to determine the fractal dimension.)

· Explain what the fractal dimension tells you about an object.

IFS method

· Given a set of IFS transformations, describe in words what each of the transformations do (in terms of copies, scalings, reflections, rotations, translations).

· Find the IFS parameters needed to create a given fractal (similar to homework and in-class exercises).

· Given a (naturalistic) fractal and the IFS transformations that created it, identify which stage 1 pieces of the fractal correspond to which transformations.

· Explain why it doesn’t matter what the initial figure used in the IFS method is, just what the transformations are.

· Describe the strategy for winning the Chaos Game in a particular case.

· Explain how to find the “address” of a specific piece of a fractal in terms of the sequence of IFS transformations that generate that piece of the fractal. Show why this means that applying the IFS transformations repeatedly and randomly can generate a fractal.

Natural fractals

· Describe advantages to using fractals to represent natural forms.

· Explain why it is hard to generate realistic fractal images of plants such as trees.

· Explain why the image of a fractal on a computer screen (or a piece of paper) isn’t a true mathematical fractal.

· Give some examples of natural fractals and explain why they can be considered to be fractal.

· Explain how natural fractals are different from mathematical fractals.

· Explain what a random walk is. Show how a random walk can be used to create “fractal forgeries” of landscapes such as mountains and coastlines.

· Explain what a percolation cluster is. Give some examples of natural processes that can be modeled with a percolation cluster. For the Blaze Applet, describe what happens to the number of unburned trees as the fraction of “trees” growing increases.

· Explain what aggregation is. Describe the differences (in how they work and what kind of structures they produce) between diffusion-limited aggregation (DLA) and ballistic aggregation.

· Describe the Hele-Shaw cell experiment (viscous fingering) that was demonstrated in class.

· State the two factors that are common to the formation of many types of branching structures (electrical discharges, Hele-Shaw, aggregation, bacterial growth, etc.), and explain how these two factors act to make similar structures.

· In the context of snow crystal growth, explain what a dendrite is, and what affects the number of branches in a dendrite. Explain what faceting is, and what kinds of structures (including symmetries) are produced by faceting. Describe how the formation of snowflakes involves an interplay between randomness and symmetry (diffusion and faceting).

Mandelbrot set

· Be able to multiply and add complex numbers.

· Be able to plot a given complex number in its proper position in the complex plane.

· By multiplying and adding, show what happens to a specific complex number under the “square and add” procedure (for the complex number c to be tested, calculate z=0²+c, then calculate z²+c, etc.). Explain how the eventual behavior of this iterative procedure determines whether or not c is in the Mandelbrot set.

· Describe the procedure (algorithm) by which the Mandelbrot set is generated by a computer.

· Explain what the Mandelbrot set has to do with fractals.

· Point out some interesting features of the Mandelbrot set.