A Guide to Infinity

Mathematics is not a dreary sequence of equations and calculations. Many ideas are brought to life through pictures. I hope that a casual bookstore browser might open this Guide to Infinity to the facing pages 60 and 61.

On page 60, we find two drawings of a rectangular box. On the left, the edges of the box are illustrated by parallel lines. On the right, the edges are drawn so that, when extended, they meet at points on an imaginary horizon; this gives a better sense of depth. That horizon is an embodiment of the line at infinity. Gazing across to page 61, the browser sees this idea in action in a Renaissance painting (A Man Weighing Gold by Cornelis de Man). We’ve extended parallel lines in the painting and see that they meet at common points on an implicit line at infinity. 

Inspired by this, I hope the reader grabs some art books to see how perspective is similarly rendered in other paintings. Now we’re ready to develop a new system of coordinates for the plane that allows us to understand that the line at infinity is not fundamentally different from any other line. 

There are many other illustrations for casual browsers to enjoy. Perhaps their bathrooms need retiling? Gosh, they might like to do that with pentagonal tiles as illustrated on page 96.