In this book, we will think linearly. In two dimensions, this means a line. In three dimensions, we're talking about a plane. In higher dimensions, math helps us work with hyperplanes. A lot of the world isn't linear. The world, especially the natural world, often offers beautiful curves. Yet, like the horizon, curvature, if viewed in the right way, can look linear.

The ability to approximate curves with lines will be important to many portions of this book. To get a visual sense of modeling curved space with lines, consider sketching only with straight lines. How about drawing a portrait? I'll lay down a series of dots that approximate an image and then I'll draw one continuous line through all the points. I'll start and end at the same point. See Figure 1.1 for an example. Recognize the image in Figure 1.1? Such visual art, called TSP Art, was introduced and developed in [3, 4]. “TSP” stands for “traveling salesperson problem” since the underlying dots can be viewed as cities and the line segments between dots indicate the route the salesperson will make through the cities. Such problems help minimize travel.

Though the image is not the original portrait, the drawing is recognizable. The line drawing captures, in this case, visual components of the original image. Later, we will use linear phenomenon to model sports performance enabling us to predict future play.

But, let's not get too far ahead of ourselves. Most of this book will explore linear systems, essentially puzzles written as equations. Let's see an example that I'll pose in the form of a magic trick.

Think of a number between 1 and 20. Double the number. Now, add 8. Next, divide by 2. Subtract your original number and mark the spot on the number line where this computed number lies with an x.

I bet I know where x lies on the number line. It lies at the value 4.