The Role of Sketches in High-School Geometry

The book I used as a high-school geometry student emphasized deductive proof — done in a step/reason format. In order to show the need for such an approach, the text began by showing the limitations of drawing inferences from pictures. Later, when proof had been well developed, the text suggested a step-by-step procedure that students could use to develop a proof of some proposition offered for proof. Step 1 was to draw and label a picture! It never struck me at the time, but what the book was suggesting as the first step was to create something that it had previously been at pains to show could not be trusted.

The thing that's going on here is that for some things in high-school geometry a sketch is to be trusted—and for some things it is not. This was never made specific—and I never noticed, at the time, the lack of an explanation. But years later the situation left an uneasy feeling. It should not be left as an apparent contradiction.

In short, pictures in high-school geometry may be used to draw topological conclusions, but not metric conclusions. That is the rule that is implicitly operative (but never, as far as I know, explicitly stated). Here are two examples to show what we mean by "topological".

(1) Draw a circle on a paper and put a dot (point) inside the circle. Now suppose that what you thought was paper was really a thin rubber sheet. By stretching the sheet in various ways you can deform the circle so that it is no longer a circle—but it is, and will remain (no matter how you stretch the sheet) a closed curve. Being a closed curve is a topological property. The point you put inside the circle will remain inside the closed curve. This property of the point is a topological property.

(2) Draw a line on a rubber sheet and label points A, B, C, and D in that order along the line. You cannot, by stretching the sheet change the order to A, C, B, D. The order of points on the line is a topological property. Topological properties are those that are invariant under stretching the sheet.

Metric properties are those that involve distance: the actual size of an angle or length of a line segment, for example. In high-school geometry, you are never allowed to assert two line segments to be of equal length without proof. That was the point made at the beginning of the book.

The fact that pictures in high-school geometry may be used to draw topological conclusions, but not metric conclusions has a use beyond merely fixing up an omission in a text. It provides the clue for us to see what will be wrong in those supposed proofs of silly propositions. For example, I have seen a supposed proof that all triangles are isosceles. I don't remember the proof at all, but I do know where the error is to be found. Somewhere, along the line of constructions and argument, we are conned into accepting a picture that is not topologically equivalent to the correct picture.

In fact the "right" picture may be difficult to determine: who could tell whether a point defined by the intersection of two lines in our construction fell inside or outside a circle of a certain radius with center at some other point in our picture?

In high-school geometry it is legitimate (and necessary) to draw topological inferences from sketches. It is legitimate to draw metric inferences only from axioms and propositions proven from the axioms. David Hilbert has provided axioms for the needed topological inferences. So all propositions follow from the axioms—and it is not legitimate, in Hilbert's geometry, to draw any inferences from sketches. (But when constructing a proof, you had better have a sketch—so you will know what is going on!)