One of the ways mathematics is very useful is its great power in building models.
Weather forecasts are largely based on mathematical models. Just as oil field exploration programs. And many, many more, from finance to medicine.
I recently bumped into a model dealing with... flowers.
In addition to being very attractive to us humans (for reasons no one has been able to explain scientifically), flowers are beautiful mathematically-inclined subjects.
Even the humblest and most common, as daisies, have an immensely complex structure. This, when we observe an entirely developed flower. But to arrive at that point, the flower must have grown.
Imagine: from a bunch of tiny cells, so small no one might see at naked eye, a spiraling pattern of tiny flowers develops. A question spontaneously arises, on how much "information" does this process require.
In my modeling exercise (which you too may reproduce with the NetLogo program included) I was so lucky to screw things up a little bit. Did this lead to a "discovery"?
Anyone of you willing to test the idea (and, possibly, correct the error) is more than welcome at the resource page of the sister site, http://www.mathsupportgals.org/resources.html
(The point where to look is the second in the list, "Mathematics may also be fun" - here you find pointers to both my crank article, and the program I've used)
I feel the material may be useful for a laboratory, at mid- and high-school level. With some guidance, students may play with the program and look at what happens as models parameters change. They may also try to improve it (I left here and there some roughnesses not impairing the overall work - just be a little patient) (and the "biology", as I mentioned, might need some refinement).
In my opinion, the value of this exercise (and the many on the same line you may develop on your own) is in:
- Understanding what a "mathematical model" is, in the most direct way: by delving into one
- Having an opportunity to appreciate serendipity in research
- Touching how cute daisies are, beyond common sense
- Realizing that all problems in applied mathematics are open, even when they seem to be "fully solved".
Have a nice (Zen) travel.