That One Line - Where Does It Go?
The problem is to draw a cube in the easiest kind of perspective, called one-point perspective. (Don't worry about a technical definition; I'll describe the problem in a non-technical way, the only way I knew in art school!)
Following a standard procedure, I started by drawing the "one point," called a vanishing point, on a "horizon line," plus a square to serve as the front face of the cube (Figure 1). I also lightly drew blue lines from three corners of the square to the vanishing point. This is what I was taught, and although I didn't understand it at the time, so far everything is correct. Next (Figure 2), I need to draw a vertical line segment to represent the right rear edge of the cube, but where do I put it? I'm stuck already! (Do you know the answer? If not, read on!) At this point I did what I always did in art school: take a guess and plunge ahead.
The Solution: Including the Viewer in the Process
What is perspective, really? It's not just a bag of tricks with vanishing points, but rather a process that people can participate in. That's what we do in a Mathematics and Art class that Ayelet Lindenstrauss and I teach at Indiana University. Here's how we do it. First, we have "art directors" stand in front of windows, rooted to the spot, with one eye closed (Figure 5). Then our art directors direct their teams of "artists" (Figure 6) to outline on the window the view of the world they see with their one, immobilized eye.
Here's the resolution to my uncertainty about the cube. The box I drew can be considered a correct drawing of a cube, but it will only appear so if you look at it with just one eye, from the correct viewpoint! This is the same viewpoint an "art director" would have assumed if the drawing had been of a real cube outside a window. You can find that viewpoint and see for yourself as follows. First, your viewing eye should be directly in front of the vanishing point, but we also need to determine the correct viewing distance. To do this (Figure 9), we draw a dashed line connecting diagonally opposite corners of the top of the cube, until it intersects the horizon line at the point W. It turns out that the viewing distance is the distance between W and the vanishing point V.
- Close your right eye and place your left eye directly in front of the vanishing point V, at a distance equal to the indicated viewing distance - only about 3.5 inches (9 cm) from the computer screen.
- Stare directly at the point V from that close range for a moment, and then, keeping your right eye closed, and without moving your head, let your left eye roll to the left and downward to gaze at the box. It should have a much more cube-like appearance!
- Still keeping your right eye closed, move your head back from the computer screen; the "dumpster" distortion will reappear. Move your eye towards and away from the point V and watch how the apparent shape of the box changes!
In fact, it's possible to start over with Figure 1, choose a longer viewing distance first, and then draw the same kinds of lines and points (but in a different order!) to arrive at a cube that is more comfortable to view - a drawing done with awareness! Can you guess how to do it?
Viewpoints: What Happens in an Art Gallery?
The full explanation of this problem, and many others that crop up in perspective drawing, can be found in Lessons in Mathematics and Art, the text we use in our Math and Art course. It also adresses the question, "What happens in an art gallery?" My cube drawing was flawed in a sense, but perspective artwork in a gallery or museum usually doesn't have such obvious distortions. Is mathematical awareness of any value there?
The answer is yes! Viewers in a gallery or museum almost never assume the correct perspective viewpoint, because these locations tend to be rather low, off-center, rather close, etc. It's true that the artwork may appear perfectly acceptable from most locations in the room. However, when one assumes the correct viewpoint and looks with one eye, it is common for the illusion of depth to be astonishingly more believable. You can almost "feel" the space! (Of course the work must be of high quality and very precise.) While you can't draw dashed lines on the artwork as we did in Figure 9, you can mentally trace appropriate lines by holding up shish kebab skewers, such as the teachers are doing in Figure 11 at one of the VIEWPOINTS Mathematics and Art workshops in the Phillips Museum at Franklin & Marshall College (F&M). Once the viewpoint is determined, it's really fun to enjoy the painting from that spot, as one of my former classes did in the Indianapolis Museum of Art (Figure 12). Different tricks are needed for different pictures, but the underlying principles are always the same.
And There's More ...
Of course there's more to perspective than I've said here, and much more to the connections between Mathematics and Art, all of it enjoyable. I owe a lot to the little drawing glitch that jogged my mathematical awareness! Many teachers have also come to enjoy and use these connections, through the VIEWPOINTS Mathematics and Art workshops run by Annalisa Crannell and myself at Franklin & Marshall College. If you're a college or high school instructor, or a graduate student headed for a teaching career, you might consider attending VIEWPOINTS 2003.
The Indiana University Mathematics Throughout the Curriculum project and the National Science Foundation (NSF - DUE 9555408) funded the joint development of Math and Art courses by myself and Annalisa Crannell. Additional funding was provided by the Indiana University Strategic Directions Initiative. These sources also supported the development of Lessons in Mathematics and Art. MTC and the MAA Professional Enhancement Programs have supported and made possible the VIEWPOINTS Mathematics and Art workshops.