Learning and Computing ©2004, Robert W. Lawler

Re-solving Problems

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:

TO CIRCLE
FORWARD 1
RIGHT 1
CIRCLE
END

The Logo circle is very easy to make with a Logo capable 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 interesting bugs and problems in turtle geometry procedures.) The Logo circle is natural in the sense that it is no more than 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 un-natural as compared to the Logo and Euclidean circles, because of the extent to which the person is removed from the description of the circle.

SUMMARY

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.

Publication notes:

Written in 1981.
Published as MIT AI Memo 652 and Logo Memo 60, "Some Powerful Ideas,"1982, April.
Published as a series "Logo Ideas," in Creative Computing, 1982-83.