Dynamic balances
- Author:
- Rafael Losada Liste
This activity belongs to the GeoGebra book Attractive projects.
There is a big difference between condition and calculation. For example, the condition for a real number to be a root of a function is that the numerical value of the function, for that number, must be zero. To calculate that root is another thing.
Usually, calculations require procedures whose learning is long and tedious. But the fact that we do not know how to perform those calculations is not an impediment to appropriate the mathematical idea that they approach. These ideas may seem much more attractive if we sacrifice some calculation ... or simply postpone it to a more advanced level.
2D project: create dynamic systems that stabilize by themselves.
Let's put two points inside a circle. Imagine that both the points and the edge of the circle are electrically charged, with the same charge. The two points repel each other, and are repelled by the circumference, with inversely proportional intensity to the square of the distance. Immediately , points will look for the balance, which will be reached when the two points are arranged symmetrically with respect to the center of the circle and a distance between them equal to one third of the diameter.
If we add more points, the equilibrium will be regular polygons, as you would expect. Here we see how 10 points are balanced forming the regular decagon (it could also be formed an enneagon with its center).
The result is not always as intuitive. In this case, apart from the five points in a square expected, four other possible distributions appear, symmetrical with each other.
You can even observe situations in which you can differentiate in the initial conditions the order of chaos. In this example, the points repel each other as before, but now they are also attracted, with double intensity, to the origin of the coordinates.
Author of the construction of GeoGebra: Rafael Losada