Forms found in nature such as plants, clouds, mountains and water can be
described as graphs of mathematical functions. More complicated and realistic
looking models can be developed by adding more parameters and using
probabilities and random numbers to vary branching angles and lengths, sizes
and shapes of forms, and the hue, value and intensity of colors. Algorithms for
modeling such forms and incorporating them into a "mathscape" will be
described. Using geometry, trigonometry, elementary linear algebra and a little
probability, these algorithms are accessible to undergraduate math majors and
can be used for many interesting class projects.
From earliest times describing nature has been one of the goals of both artists
and scientists. Now computer graphics makes it possible to translate into a
visual experience a scientist's description of nature, which might be expressed
using mathematics. Curves and surfaces can be described as the graphs of
mathematical functions. Probabilities and random numbers can be employed to
simulate the effect of environment on natural forms by altering some of the
exact symmetry that may occur in a mathematical formula. These curves and
surfaces can be rendered as clouds, mountains, water and fields by imitating
the way a camera projects a scene on a flat surface. A light source can be
defined mathematically and used to create shadows and light to give the
illusion of three dimensions. Techniques from formal language theory, recursion
and stochastic matrices can be used to describe plant structures; complicated
objects can be generated from very simple ones with repeated application of a
small set of rules.
I would like to outline several algorithms for modeling some of the forms found
in nature: plants, clouds and surfaces such as mountains, deserts and water,
and describe how to put them together to form what one of my colleagues termed
"mathscapes". These algorithms and their implementation on a computer are
accessible to undergraduate mathematics majors who have taken a single
programming course such as Basic. They can even be programmed in Mathematica.
We use geometry, trigonometry, some elementary linear algebra and a little
probability. The importance of parameters in modeling is emphasized. Many of
the algorithms use recursion. Most students are fascinated when they see their
pictures on a computer screen and many interesting classroom projects using
computers can result depending on the interests and abilities of the students.
Each model or algorithm can be presented in a simple, easy to understand form.
Once it is mastered, more complicated and realistic looking models can be
developed by adding more parameters and using probabilities and random numbers
to vary branching lengths and angles, sizes and shapes of forms, hues, value
and intensity of colors and other properties.
If possible, I would like to illustrate some of these algorithms and the
resulting mathscapes interactively on a computer, but if that is not possible I
can use slides or transparencies.