I love models...

I was thinking about what I last wrote about modeling, and I realized that the area and circumference formulas I mentioned were also models. This time, the models are algebraic and represent some other physical model. Consider the variable r, which last time represented the radius of the circle. Physically, it represents the length of some line segment. This diagram is another way to model our generic line segment. Note that even though this diagram has some thickness, we are only measuring one dimension. Actual line segments don’t have thickness or width, only length.

Area cannot be measured the same way as length, since area is the “filling in” of some space. We cannot measure area in inches, or meters, or miles, or arm lengths; we cannot fill up some space with line segments, since they have no width. So we must measure in some units that fill up space, like squares, or triangles, or circles. Squares are convenient, since they can fill up a space without overlapping or leaving space between. (This is called “tessellating”, and it cannot be done with circles alone, and some triangles don’t tessellate as uniformly as squares.) So we measure in “square units”. This diagram shows a square with area equivalent to r-squared, since each side of the square has length r.

Here is a diagram showing four r-squared’s, or 4r^2. If we include the circle with radius r, we see that the area of the circle is less that 4r^2. I’m not going to go into the proof of why the area is pi*r^2, as I am feeling like this is rambling a bit too much, and I am suddenly feeling like I am trying to teach someone something about area, when I really want to talk about models.




One of the advantages of models is that if we play with the model – and see how far our tether can reach:) -- we can often come up with some further insights into the thing we are modeling. Let’s look at the circumference formula for example. We know that C=2*pi*r, and we can model that physically with pieces of string, each the same length as our radius. If we lay the pieces on the circle, marking where each piece stops, and continue as far as we can around the circle, we find that we can fit a little over six whole pieces of string. (See the model … er, diagram?...) The great thing about this is that it doesn’t matter what size circle we start with, we can always lay down a little more than six radius-lengthed strings. In fact, this 6.something or other is what the ancient Greeks figured was 2*pi. (They actually did this with a diameter-lengthed string, and were able to fit a little more than three of these around the circle – pi diameters around the circle.)

Here’s the further insight – If we animate our mental picture, sweeping the line segment radius around, from one end of a piece of string to the other, we get an angle that is the same no matter what size circle we have. And we can fit 2*pi of these angles into the circle. This special angle is called one radian. (Just like you can measure length using different units, like inches or miles, you can measure angles in different units.)

Okay, I’m still feeling all teachery, so I’m going to stop now. This whole idea of models intrigues me, and, since it is a different way of looking at math, I am exploring ground that is really familiar to me. I do want to get to some more unfamiliar topics like game theory and graphs on other kinds of grids, but I still have some ideas about models to explore (and I have not yet completely figured out how to get really good diagrams and algebraic notation on this blog).