Ivars Peterson's MathTrek

March 25, 1996

## Pythagoras Plays Ball

Spring, baseball, statistics, math.

In many ways, baseball is a numbers game: Strikeouts, home runs, batting averages, innings pitched, complete games, earned run averages, number of players left on base in scoring position in the bottom of the ninth inning when a left-handed batter flies out. It's also a game played on a field of considerable geometric regularity. The baseball "diamond," for instance, is properly a square, 30 yards on a side.

As an assistant coach for his nine-year-old son's baseball team, Michael J. Bradley of Merrimack College in North Andover, Mass., spent a bit of time familiarizing himself with the official league rules. He came upon the following passage and accompanying diagram:

"Home base shall be marked by a five-sided slab of whitened rubber. It shall be a 12-inch square with two of the corners filled in so that one edge is 17 inches long, two are 8 1/2 inches and two are 12 inches."

There was something about these numbers that puzzled Bradley. "Wait a minute," he thought. "That 17 should be 12 times the square root of 2. It can't be an integer. It has to be an irrational number."

Indeed, the diagram implied the existence of a right triangle with sides 12, 12, and 17, and that's not quite a Pythagorean triple: 12^2 + 12^2 = 288 and 17^2 = 289.

At first, this discrepancy bothered Bradley. The dimensions for home plate cannot be mathematically correct. But baseball, like engineering and science, is a game of inches and feet. To the degree of accuracy required to construct a workable home plate, 17 is as good as (certainly more measurable than) the more exact value of 12 times the square root of 2.

"I soon realized that the problem did not reside in the realm of pure mathematics where I usually live, but rather in the world of applied mathematics where my engineer and scientist colleagues practice their professions," Bradley notes. There's a crucial difference between measured numbers (accurate to a certain number of significant digits) and purely mathematical numbers.

Bradley's observations made me curious about the origin of home plate as now defined. The playing field has been the same shape and size since the rules of baseball were first published nearly 140 years ago. But the size, placement, and shape of the bases, including home plate, have changed over the years.

Initially, the rules insisted that bases be 1 square foot in area (most simply, a 1 foot by 1 foot square). Home plate started as a circular iron plate, painted white, with a diameter not less than 9 inches. By the 1870s, however, home plate had become a square just like the other bases.

In 1877, the width of the bases was increased to 15 inches but home plate stayed at 12 inches. Finally, the year 1900 saw the introduction of the five-sided home plate, with a flat side rather than a point facing the pitcher. This extra rubber made it easier for both umpires and pitchers to judge when a ball "cut the corner," especially when dirt covered the corners of home plate.

Nowadays, all but one of the bases fit snugly inside the corners of the square that defines the infield. Until 1877, however, the bases had a slightly different position. The center of each base sat directly over a corner of the infield square. First and third base were moved to their present positions so that umpires could call foul balls more easily. Second base, however, still sticks out of the square.

Now, here's something else to ponder. Is the size of the piece of rubber that's home plate a little bit smaller in the cool days of spring training than during the hot days at the end of July? Does this expansion materially change the width of the strike zone, giving a pitcher the advantage on hot, sunny days?

Let the numbers fly and the games begin. Play ball!