Ivars Peterson's MathTrek

April 13, 1998

Curves on Baseballs

It's that time of year again (at least in North America), when the thoughts of many sports fans turn to baseball. Over the years, the game has also attracted the attention of mathematicians and statisticians intrigued by the rules, geometric considerations, and record-keeping endemic to the sport.

Consider the leather or vinyl cover of the baseball itself. The cover consists of two identical pieces, stitched together, then stretched to cover the cork, rubber, or yarn-wound core of a ball. Its design goes back to the 1860s and an inventor named C.H. Jackson.

Jackson's task was to find an acceptable shape for a piece of leather, called a flat, that could be sewn to an identical piece to cover the ball. The result had to be a baseball 9 1/8 inches in circumference, having the seam located in such a way that a pitcher's fingers could get a good grip. The latter requirement meant that, at certain locations on the ball, different sections of the seam had to be only about 1 3/16 inches apart. Jackson also wanted a flat that was symmetrical about its horizontal and vertical axes.

"The difficulty of getting close to an acceptable flat that met these criteria must have required many trials -- and a lot of error," mathematician Richard N. Thompson of the University of Arizona comments in the January College Mathematics Journal.

The shape that Jackson eventually came up with and patented is still in use today.

(Left) Jackson's symmetrical flat. (Right) Two flats ready for stitching.

Thompson contends that Jackson's design is not the only possible solution to the baseball cover problem. "We have considerable freedom in designing [flats]," he insists.

From a mathematical viewpoint, the seam can be thought of as a simple closed curve on the surface of a sphere. The curve serves as a boundary between two identical (congruent) regions. Hence, one can approach the problem of designing a cover by looking for a curve that separates the sphere into two congruent regions.

Once that curve is established, it's possible to open up the seam and unwrap the surface mathematically to reveal the pattern for a flat -- something that Jackson couldn't do with freehand pen-and-ink drawings but mathematicians can now readily accomplish by applying geometric insight and using calculus and mathematical software.

Thompson demonstrates that a cover designer has considerable freedom in selecting a suitable curve to define a flat's shape. The smooth curves that Jackson favored could just as easily have been irregular and still have divided the sphere's surface into two equal parts that showed the right sort of symmetry and met the minimum seam separation requirements.

Thompson also investigated the question of how closely the seam of a real baseball fits on a sphere's surface, given the geometry of the two pieces of leather that are sewn together and then stretched over the ball's core. In other words, are real baseballs made with acceptable flats?

He started with a freshly cut leather flat of the shape currently used by the Rawlings Sporting Goods Co. to manufacture National League baseballs. Measuring the flat's dimensions, Thompson worked out an equation to represent its shape. For the resulting seam to be correct, all points on the seam would have to be the same distance from the sphere's center.

The analysis showed that points along the seam vary from that ideal, ranging from 1.554 to 1.624 inches from the sphere's center. "Trial-and-error designing has done very well, but it has not produced an acceptable flat," Thompson says.

However, the deviation from the ideal isn't large. The flats currently used appear to be about 0.04 inch too narrow at the waist and 0.04 inch too long. "The fact that trial-and-error designing came this close to finding an acceptable flat testifies to the great persistence and patience of Mr. Jackson and his corporate heirs!" Thompson declares.

Interestingly, though the analysis points to a ball with an average radius of 1.584 inches, the measured radius of a real baseball is 1.452 inches. "It may be that manufacturers have found it desirable to make a cover that will pucker some at the seams, but will require less stretching of the leather in the middle of the flats," Thompson remarks.

"Mathematical analysis can be used to replace the early trial-and-error methods of Mr. Jackson," he concludes. "However, it is modern computational tools that allow the mathematics to be of real use in the design process."