June 30, 2010

Pythagoras at the Plate

Baseball is played on a field of geometric regularity. The baseball "diamond," for instance, is properly a square, 30 yards on each side.


Official league rules also specify the size and shape of home plate: 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.


But something isn't quite right. The diagram implies the existence of a right triangle with sides 12, 12, and 17. If it were truly a right triangle, the Pythagorean theorem would hold, and 122 + 122 would be the same as 172. It's not: 122 + 122 = 288 and 172 = 289.

So, the dimensions of home plate (an irregular pentagon) are not mathematically correct.

But there's a difference between measured numbers (accurate to a certain number of significant digits) and purely mathematical numbers. To the degree of accuracy required to construct a workable home plate, 17 is as good as (and certainly more measurable than) the more exact value of 12 times the square root of 2.

The history of baseball sheds some light on how the dimensions of home plate came about.

The playing field has been the same shape and size since the rules of baseball were first published more than 140 years ago. The size, placement, and shape of the bases, however, 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). Out on the field, the center of each base sat directly over a corner of the infield 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. First and third base were moved to their present positions, where they fit snugly inside the corners of the square that defines the infield. This change was made so that umpires could call foul balls more easily. Second base, however, still stuck out of the square, where it remains to this day.

The year 1900 saw the introduction of the five-sided home plate, with a flat side rather than a point facing the pitcher. The extra rubber made it easier for both umpires and pitchers to judge when a ball "cut the corner," especially when dirt happened to cover the corners of home plate.

Original version posted March 25, 1996.
Updated July 12, 2004; June 30, 2010.

References:

Bradley, M.J. 1996. Building home plate: Field of dreams of reality. Mathematics Magazine 69 (February): 44-45.

Kreutzer, P., and T. Kerley. 1990. Little League's Official How-to-Play Baseball Book. New York: Doubleday.

Peterson, I. 2002. Pythagoras plays ball. In Mathematical Treks: From Surreal Numbers to Magic Circles. Mathematical Association of America.

Thorp, J., and P. Palmer, eds. 1995. Total Baseball, 4th ed. New York: Viking.

4 comments:

Anonymous said...

A baseball field is not a square as second base sits just outside the square that contains home plate, first, and third base.

The two sides of the square that run from first and third base respectively, bisect in the center of second base. This comes from the process of accurately laying out/measuring the bases. One must set second base before first and third base. To accurately center second base with home plate and the other two bases, one must set the POST of the second base bag (which lies in the exact center of the base) 127 feet, 3 3/8 inches (measured from the back of home plate through the middle of the pitching rubber).

This is unlike first and third base which are measured to the back of the actual base. Therefore, the four-sided figure drawn to include each of the four bases within its boundries is, in fact, a diamond.

SteveN said...

I enjoyed your posting, sorry I'm a few years behind schedule with my comment. I found your blog while looking into the use of "Diamond" in baseball and as a synonym for rhombus. You can view it here: http://mathonthemckenzie.blogspot.com/2013/02/square-vs-diamond.html
I will be looking into the home plate/Pythagorean Theorem dilemma in a future blog entry. Thanks

SteveN said...

I enjoyed your posting, sorry I'm a few years behind schedule with my comment. I found your blog while looking into the use of "Diamond" in baseball and as a synonym for rhombus. You can view it here: http://mathonthemckenzie.blogspot.com/2013/02/square-vs-diamond.html
I will be looking into the home plate/Pythagorean Theorem dilemma in a future blog entry. Thanks

Anonymous said...

The infield is a 90 foot rhombus, but not a square (see below). The vertices of the rhombus are:
1) the intersection of the first base line and the third base line
2) first base
3) second base
4) third base
First base, second base, and third base are points. The first base bag, the third base bag, and home plate are in the interior of the rhombus. The second base bag is centered on second base, partially in the infield and partially out. The infield cannot be a square because the diagonal of the infield from home plate to second base is 127 feet, 3 and 3/8 inches (see Rule 1.04). The ratio of the diagonal to a side is therefore127.28125/90 = 4,073/2,880, a rational number. But in a square, the ratio of diagonal to side is the square root of 2, an irrational number. So the infield cannot be a square, as Baseball Rule 1.04 mistakenly says.