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This site features a nice textual introduction to site percolation theory on a square lattice, starting with brief introductions to random number generation on a computer and Monte Carlo algorithms and moving through basic site percolation concepts. Also included in the site are three applets. The first applet introduces random numbers obtained from a computer and histograms. This applet simulates a number of throws of a die and graphs a histogram of the outcome. The second applet illustrates the use of the Monte Carlo method. It shows how Monte Carlo methods can be used to estimate pi. The third applet very effectively guides users to the discovery of many basic concepts in percolation theory. Several exercises make the students participate in the discovery of these concepts. More advanced percolation concepts (L extrapolation, phase transitions, fractal nature of percolating clusters) are addressed in a more cursory fashion in the text, with some direction given as to how the third applet may be used to investigate them.

This site provides a more advanced treatment of percolation than the Forest Fire and Percolation site. It functions at about the same level as the Percolation Applet,
which would be a nice complement to it as it focuses on different aspects of the theory.

Learning Goals:

Users should gain an understanding of basic percolation theory and discover the existence of the percolation threshold.

Target Student Population:

College students in probability, discrete math, modeling, or physics courses.

Prerequisite Knowledge or Skills:

Students should be familiar with integral calculus and elementary probability theory.

Type of Material:

Tutorial and simulation.

Recommended Uses:

This site could be used as a supplementary text/student project in any of the courses mentioned above.

Technical Requirements:

For the applet, a Java-enabled browser.

Evaluation and Observation

Content Quality

Rating:

Strengths:

This site?s text and accompanying applet provide an excellent, in depth investigation of the square lattice site percolation threshold phenomenon, and a more cursory investigation of surrounding concepts such as extrapolation to infinite lattices, scaling laws, phase transitions, and the fractal nature of percolating clusters. It would be quite effective as a stand-alone learning tool if only the percolation threshold phenomenon were to be investigated.

Concerns:

For most students, more explanation than the text provides would be necessary if concepts beyond the percolation threshold phenomenon were to be investigated.

The explanation of the fractal nature of percolating clusters is in preparation. There will eventually be an applet which will demonstrate this concept.

Potential Effectiveness as a Teaching Tool

Rating:

Strengths:

This site provides the deepest, most complete approach to the percolation threshold of any of the three reviewed sites mentioned above and thus holds out the potential for developing the greatest degree of understanding. The presentation of the material through proposed exercises is quite effective and the explanations are clear. A course in percolation theory would benefit greatly from the use of the applets in this site. For the student with a fair degree of mathematical sophistication the site could stand on its own. For a typical student with the prerequisite courses mentioned above, some explanation of both the text and the applet may be necessary.

Concerns:

None

Ease of Use for Both Students and Faculty

Rating:

Strengths:

Because it attempts more than the other two sites reviewed, it is necessarily more complicated. The text provides a reasonably good guide to the use of the applets. The site skims through several sophisticated concepts in percolation theory. Further detailed mathematical explanations of these concepts should be done in the classroom.

Concerns:

Sometimes the applet does not take account of user changes to the probability parameter.