How snowflakes pack



It is possible to use Maple to simulate how snowflakes arrange themselves when they fall. Of course, various assumptions need to made and not all these may prove to be realistic. Nevertheless, such exercises in simulation can give insight into physical processes which are difficult to study in nature.

Begin the process of modelling how snowflakes fall by considering the 2-dimensional version of the problem. Suppose that snowflakes are circles of uniform radius in the plane and that they fall onto the unit interval one after the other. Moreover, assume that they fall straight down and that their horizontal position is determined at random. The key assumption concerns how they arrive at their final resting place. For this project the simplifying assumption will be made that as soon as a falling flake touches either a flake on the ground or the ground itself it stops. This makes the mathematics of determining the final resting place of a falling flake quite simple.

Write a procedure which randomly drops some snowflakes and determines their final resting place using the above assumptions. Your procedure should produce as output a list of the centres of the rest positions of all the snow flakes. Using the plot command with the style=point, symbol=circle options it is possible to plot the centres of the snowflakes from your list. Can you get Maple to draw not only the centres but alos the circular flakes themselves in their rest positions? See what effect changing the radius of the flakes has on the model. Can you analyse your models to guess at what the density of this idealised snow is?

Try generalizing your methods to model 3-dimensional snow.


Instructor

Juris Steprans
email address: steprans@mathstat.yorku.ca
Department of Mathematics and Statistics.
Ross 536 North, ext. 33921
York University
Back to Department's Public Page