The Fastest Way to First Base May Not be a Straight Line

Released: 5/27/2010 3:45 PM EDT
Source Newsroom: Williams College
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Newswise — Williams College math professor Frank Morgan is keeping an eye out for Major League hitters who veer toward the dugout on their way to first base. While at first glance this route might not seem the best way to start a sprint toward home plate, Morgan says his calculations prove it’s the fastest way around the diamond.

Although he says that the quickest path to first base is a straight line, for baseball players who hit a long ball, Morgan says the best way around the bases takes a more circular shape. By cutting off the corners, an average runner can round the bases 20 percent faster, saving approximately four seconds.

Work on the “Optimal Baserunner’s Path” was begun by Davide Carozza ’09, of Washington, D.C., for his senior colloquium project at the suggestion of Morgan, his advisor. “I’m a huge baseball fan. I picked it up and ran with it,” Carozza said.

Carozza literally did just that. He began his research by racing around Cole Field to get a feel for what a reasonable baseline acceleration speed might be, and to see which simple paths and shapes might work the best.
His presentation to faculty and fellow math majors generated a lot of discussion, leading math professor Stewart Johnson to join with Morgan to refine the optimal path.

The three Williams mathematicians’ calculations have not gone unnoticed. Their work was included in “The Mathematical Intelligencer” and in baseball author Rob Neyer’s blog, “Monday Mendozas.” According to Neyer, by taking the Optimal Baserunner’s Path, batters might hit more triples.

“If you hit that ball and you know you’re going to go farther than first base, you shouldn’t run straight for first base,” said Morgan, Atwell Professor of Mathematics at Williams College and vice president of the American Mathematical Society. “Right from the beginning, you should head to the right, 25 degrees off to toward the dugout, right from the base path to the outside. Don’t wait. Do it right away.”

As the runner rounds first, he should continue the curve as he heads to second base, and then bulge out even further between second and third so that he is lined up to make the straight sprint home, Morgan said.

In contrast, a base runner following the recommended “banana path” stays on the baseline halfway to first base before veering to the right to set up a better angle as he continues to second base. According to Johnson, even though he, Morgan and Carozza make a number of assumptions, it should be obvious that this is not the optimal path.

“By the time you’re done curving to the right and then curving back to the left with the banana path, you’ve wasted a lot of energy and time and distance. It’s much better to start out running angled off to the right and run around first base in one smooth arching curve,” Johnson explained.

In the weeks that followed Carozza’s colloquium presentation, Morgan and Johnson continued to discuss it.

“Frank Morgan does minimal surfaces and one of the areas I work in is optimal control theory,” said Johnson. “We had been going back and forth for years about how similar and different these two areas are, because they’re both trying to optimize things, but from a slightly different point of view.”

Using optimal control theory as it relates to minimal surfaces, the professors computed an optimal path calculated to take 16.7 seconds, compared to a time of 22.2 seconds for the banana path.

“It was very much a team effort,” said Johnson. “I would crunch some numbers and Frank would look at it from a much more abstract point of view and got some very beautiful curves.”

Morgan had the problem as part of a long list of interesting questions he had been compiling for more than two decades.

“I’d just thought of it,” he said. “That’s what mathematicians do all the time – we’re always thinking about things. After a while, you just see relationships everywhere. That’s sort of our job. You’re always thinking about things and saying, ‘Maybe that’s like something else. Maybe this concept of curvature that I’m teaching in my senior seminar, maybe this would be useful to baseball runners, because they have to decide how much to curve when they’re running around the field.”

Just as baseball is life to some, is math life? “No,” Morgan said, “it’s not that math is life. It’s that life is math. There’s math in everything. It’s a way of looking at the universe and seeing so much more of it. I’m just a math fan. When you like math, you like everything.”

While it’s up to individual organizations to decide how their hitters should run around the bases, should a baseball player choose Optimal Baserunner’s Path, it is legal, according to Carozza and Morgan, who researched the rules.
While Morgan and Johnson are optimistic the theory will catch on, Carozza is hesitant.

“I don’t think you’re ever going to convince a baseball player that if you hit the ball, his first couple of steps should be away from the field of play toward his own dugout,” said Carozza. “But it would be cool to see someone actually test it out.”

“It’s out there now, in a number of people’s awareness,” Johnson said. “I would be delighted this baseball season to see some runners actually running off to the right, instead.”

“We haven’t heard from the major leagues yet, but I don’t see why they shouldn’t give this more attention,” said Morgan. “We’ll see what happens. By the next World Series, maybe they’ll let me in the booth there with them. Sometimes, math takes thousands of years to catch on. I’m in no hurry.”

Founded in 1793, Williams College is the second oldest institution of higher learning in Massachusetts. The college’s 2,000 students are taught by a faculty noted for the quality of their teaching and research, and the achievement of academic goals includes active participation of students with faculty in their research. Students' educational experience is enriched by the residential campus environment in Williamstown, Mass., which provides a host of opportunities for interaction with one another and with faculty beyond the classroom. Admission decisions are made regardless of a student’s financial ability, and the college provides grants and other assistance to meet the demonstrated needs of all who are admitted.

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