Logo ideas. Robert Lawler.
Some problems you want to put behind you--like having to do what you don't want to do, and not being able to do what you do want. Such problems should be resolved. Other kinds of problems have a friendlier face, and certain of them are worth solving and re-solving. Think about making a circle. Doing so is a classic Logo problem for beginners. Novice learners are typically asked to "do-it-yourself,' to walk through the problem by simulating the turtle. Their typical explanation of what they are doing as they walk in a circle is that they go forward a little and turn a little and do it again. This explanation translates directly into the Logo circle:
The Logo circle is very easy to make with a Logo computer, but it would be difficult to make such a circle by drawing on a piece of paper. The Logo circle is very perimeter-focussed because the turtle knows nothing at all about "centers.' (This leads to some interesting bugs and problems in turtle geometry procedures.) The Logo circle is natural in the sense that it is the path of an activity as familiar as walking is.
In plane geometry if you ask, "What's a circle?' the object, "the locus of all points in a plane equidistant from another point,' is easy to construct with a compass and not even hard to construct without one. The Euclidean circle is as "natural' as the Logo circle in the following sense: imagine a person sitting; the figure traced by the farthest reach of his arms is as circular as the path followed by any person imitating the Logo turtle. The Euclidean circle is center-focussed, and the circle is the boundary of the center's territory.
Can you get a computer to draw a Euclidean circle? There are several ways. If your computer speaks "polar,' you can specify the definition of a circle with the simplest of equations, radius = constant. Descriptions of circles in polar coordinates are simple, but they get complicated quickly if located away from the coordinate system origin.
While the description of a circle in polar coordinates still keeps in mind the relation of the circle to its center, and to a process a person could use unaided to make a circle, the description of a circle in a system of Cartesian coordinates becomes remote from the process of generating a circle:
X2 + Y2 = C2
This algebraic equation for an origin-centered circle (of radius C) specifies that the circle is the set of all point pairs (X, Y) in a Cartesian coordinate system which satisfy the equation. The primary relationship between the circle and "something else' is here between the circle and the Cartesian reference frame. This contrasts with the Logo circle (where the primary relation was between the circle and its process of creation) and the Euclidean circle (where the primary relation was between the circle and its center). The Cartesian description of the circle and other curved lines, although central to the development of modern mathematics and science, seems relatively unnatural as compared to the Logo and Euclidean circles because of the extent to which the person is removed from the description of the circle.
Scientists have recommended re-solving problems through the ages. Descartes recommends that whenever you encounter a new idea, you bring it into comparison with all the other ideas you hold as valuable and try to appreciate their interrelations. Feynman, a famous physicist of our time, relates that his practice as a student was typically one of solving a problem whatever way he could, then, with a worked out solution to guide him, to re-solve that same problem in as many different other formalisms or frames of reference as he could.
Sometimes You Need Another Idea
One of the most famous problems in the history of ideas puzzled the mathematicians of ancient Greece. They knew how to count very well (even though they used letters of their alphabet to represent numbers). They even knew about fractions, and this is where the puzzle came up. They knew about numbers like 1, and 1/2 and 1/4, but they wondered if there were any numbers that couldn't be represented by whole numbers or fractions made from whole numbers. The puzzle became a hot issue for them after the discovery of the Pythagorean theorem. They could prove that the areas of two squares constructed on the edges of a right triangle was equal to the area of a square constructed on the longer line (the hypotenuse), through the use of a technique such as shown below:
This proof helped make the problem more critical because it raised a specific question. If you start with a square one unit long on the side, and make a triangle by drawing the diagonal of the square, the sum of the areas of the two squares constructed on the side will be two units of area; but how long must be the hypotenuse, H, of the triangle made from half a unit square? H must be greater than 1 and less than 2. It must be more than 4/5 and less than 3/2.
Greek mathematicians suspected no fraction of whole numbers would result in the number 2 when multiplied by itself, and they began the attempt to prove there was no fraction of whole numbers equal to H. They tried to represent the number H as a fraction of two whole numbers, T (the top number) and B (the bottom number). They knew that H times H had to equal 2 and developed these equations:
first, H X H = 2
then, T/B X T/B = 2
or, T2/B2 = 2
finally, T2 = 2B2
Having reduced their relation of the possible whole numbers T and B to this simplest form, they were stuck. What else is there to do? Where can you go from here with this one idea? Think about it for a while. Can you go on from here?
Another idea is needed, another whole different way of looking at what T and B might be. The trick is to look "inside' T2 and B2. What must they be made of? No square can be a prime number (squares are made by multiplying at least two other numbers together). The factors of a square must be two (in number) if the roots are prime or some multiple of two if the roots are not prime, as in the example below:
Any number is either a prime number or can be decomposed into prime factors. Therefore every square must have an even number of factors. But think back about the equation: T2=2B2. Doesn't that imply there is at least one square, T2, which must have an odd number of factors? It surely does, and therefore it must be wrong. Consequently, there must exist numbers, like the square root of 2, which can not be expressed as the ratio of two whole numbers. That is, irrational numbers exist.
This mathematical proof was a difficult one for men to discover. Then someone realized that a new idea was needed, a new way of looking at the problem. Once a second way of describing the problem was brought to bear, its solution was relatively straightforward, almost obvious. When you have a very difficult problem, maybe you ought to think about whether some other description of the problem could help you with it. Finding the right description isn't always easy; it may, however, be necessary.
Photo: Hypotenuse Square
Photo: Two Side Squares