The Side-Approximation Theorem, 1963

"I will now tell of an idea I got as a boy that helped me in later research. I was assigned the task of drawing a map. One way was to take a map, put it on the window, place a piece of paper over it and then trace. I had more difficulty, however, when I was asked to draw a map of a different size. However, I found that if I divided the page containing the original map into rectangles and then similarly divided the paper on which I was to draw the map, then I could make a reasonable copy of the map without resorting to the use of the window. In fact, even if the paper on which the drawing was made was of a different size from the original map and if one put the corresponding parts into the corresponding rectangles, then one got a reasonable copy of the original map, especially if the rectangles were small. A mosaic copy of a picture looks somewhat like the original.

I noted that a person could take a map whose boundary lines were curved (or even squiggly) and approximate it with a map whose boundary lines were polygonal; then if one did not examine the changed map too carefully, it resembled the original. This is the idea behind the fact that if one looks at a mosaic from a distance, it looks like an object which is not polygonal. Also, if one looks at a TV set, one does not see the dots, but a picture.

Let me discuss how I used this notion many years later. In the 1950’s, I learned that [fellow mathematician Edwin E. Moise] had proved that if h is a homeomorphism of a 3-cell into Euclidean 3-space, then this homeomorphism could be approximated by a piecewise linear homeomorphism. This result has many applications, and I felt that it was very important...

As I wrestled with Moise’s paper, I decided that there might be a different way of proving the same result and recalled the techniques I had learned for copying a map when I was a boy. Hence, I decided to triangulate R3 and use the 3-simplexes as I had used the rectangles...With this step I got the same results that Moise did and was able to make the same applications that he did; namely, that a homeomorphism of a 3-cell into Rcan be approximated by a piecewise linear homeomorphism and that any 3-manifold can be triangulated. Later I was able to prove the side approximation theorem. I suspect that my results on the approximation theorem and the side approximation theorem even have a larger impact on 3-dimensional topology than any other of the results I have obtained in this area."