If you could attach little lights to the frog and the tip of your bow, darken the room, and play in front of a mirror, you’d see that bowing, and especially crossing, is a matter of geometry as well as artistry. If you were playing well, you’d see those lights make precise little circles and arcs, and every motion at the frog would be mimicked in reverse at the tip.

It’s what William J. Dick and Laurie P. Scott call the geometry of string crossing. Dick teaches at Southwestern University in Texas and Scott is a professor at the University of Texas at Austin. Their version of geometry doesn’t require you to brush up on Euclid or memorize the value of pi. But it does put bowing and crossing into the context of planes, lines, and arcs. And if your geometry is symmetrical, your string crossings should be clean.

Says Dick, “If you think about the violin, one and a half inches from the bridge to the fingerboard, that is our playing field. When the bow moves in that playing field, at the frog and the tip, it creates geometric patterns.” Those patterns differ, depending on what you’re up to. Sometimes the circles move clockwise, sometimes counterclockwise; sometimes the motion begins from the top, or 12 o’clock, point of the circle, and sometimes from 6 o’clock.

As Dick is fond of saying, “The shortest distance between two points is under construction.” For string players, that means that if you’re going to master the cleanest, most direct crossing methods, you’re going to have to put in some work.

Scott says, “String crossings are facilitated by the interrelationships of joints.” Dick echoes the sentiment: “Pulling the bow straight, parallel to the bridge and fingerboard, is a complex skill because all our joints move in arcs, yet the bow has to move straight. We have to figure out how two curves can make a straight line.”

First, you need to know the fulcrum of every bow pull. The bow hair balanced on a string, like a see-saw, is the fulcrum we usually think of first. But don’t forget that for bow strokes there’s also a fulcrum in your own body. You can pull the bow from your shoulder joint or from your elbow joint, both of which require little adjustments and good flexibility in your wrist. In fine string crossing techniques, the fulcrum may well be in your thumb. The control of your bow from some fulcrum in your body is what creates the geometry that you can see at the frog and tip.

Dick cites as an example the crossings in the Gigue from Bach’s Cello Suite No. 3. “There’s a figure-eight pattern that goes from the frog to the tip,” he says. “If the crossing is not even, neither is the geometric pattern. The eighths will be shallow, instead of nice and round.” See a shallow pattern, hear a messy crossing. “The ones with the most even geometric pattern are the most perfect crossings.”

Scott admits that thinking geometrically won’t work for everyone. “For some students, focusing on the shapes created by the movements interferes with the actual string crossing,” she says. “For those students, listening aurally to evenness of sound is the basis of their success. However, most students will become teachers themselves and do need to understand that every student learns differently, and while it may not be necessary for them personally to think about the ‘shape,’ they need to realize that a shape approach can facilitate success for some players.”

Let’s see how this works in practice. The starting point is “Twinkle, Twinkle, Little Star,” but with a complication: After playing each note, you cross to the next open string. So in this exercise, the first two notes turn into four.