Read articles, including Science News stories written for ages 9-14, on the SNK website.
Winning the World Series with math
A nearly circular path could be the fastest way to home plate.
A+ A- Text Size
Enlarge
Running the bases
Mathematicians computed that this path around the bases is, theoretically, the fastest. The red lines show the direction the runner is accelerating.
D. Carozza, S. Johnson, and F. Morgan

To run the bases faster, baseball players just need a bit of mathematics, according to research by an undergraduate math major and his professors. Their calculations show that the optimal path around the bases is one that perhaps no major-league ball player has ever run: It swings out a full 18.5 feet from the baseline.

The precise path the researchers calculated probably won’t turn out to be the very fastest in the real world, they acknowledge, because of physiological and practical complexities they couldn’t model. Still, the analysis suggests that runners might be able to improve their times by following much wider paths than they had ever considered.

“I would definitely experiment with it,” says former American Major League Baseball outfielder Doug Glanville, who last played with the Philadelphia Phillies. “There’s no question in my mind that runners could be more efficient.”

The issue is that turns slow runners down. The tighter the turn, the greater the slowdown, so while the straight-line path between the bases is the shortest, its sharp corners make it one of the slowest. Rounding the corner is faster, making the path a bit longer in favor of an efficient turn. And indeed, baseball players typically do this: They run straight along the baseline at the beginning and then, if they think they’ve hit a double or more, they bow out to make a “banana curve.”

But this can’t possibly be the quickest route, observes Davide Carozza, a math teacher at St. Albans School in Washington, D.C., who  studied the problem while an undergraduate at Williams College in Williamstown, Mass. It’d be faster, he reasons, to veer right from the beginning, running directly from the batter’s box to the widest portion of the curve. Of course, a runner is best off running straight toward first base until he’s certain he’s hit more than a single. But Carozza noticed that even when the ball heads straight for a pocket between fielders, making a double almost certain, runners almost never curve out right away.

To figure out just how critical the turns are, Carozza did a calculation comparing the straight-line path with a circle around the bases. A path that follows a circle turned out to be a whopping 25 percent faster.

When Carozza presented his calculation at a colloquium talk in the math department at Williams College, Stewart Johnson, one of the professors in the audience, got intrigued. The circular path is so long that it can hardly be the fastest, he figured. So what path is the fastest?

Johnson ran a simulation on his computer, tweaking the circular path in tiny ways to make it shorter and faster, until no more tweaks could improve it. The result was surprisingly close to a circle, both in its shape and its speed: It swung nearly as wide and was only 6 percent faster than Carozza’s circle. On this path, a runner would start running 25 degrees to the right of the baseline — toward the dugout rather than toward first base — and then swing wide around second and third base before running nearly straight to home. Johnson also computed the best path for a double, and it swings nearly as wide, venturing 14 feet from the baseline.

Carozza says he checked the rules of major league baseball, and although these routes are highly unconventional, they’re allowed. The only limits apply after a fielder has attempted to tag a runner with the ball. After that, the runner can veer no more than 3 feet from the straight line to the base.

“The math looks fine,” says Wayne Winston, a specialist in the mathematics of sports at the Kelley School of Business at Indiana University Bloomington, “but is it a good description of reality?”

The researchers acknowledge that in some ways it’s not. They assume that regardless of a runner’s speed, he can speed up, slow down or turn at the same rate (more precisely, that his maximum acceleration vector is constant), which isn’t strictly true. A particular concern is that a runner may not be able to speed up as quickly while running along a curved path as he can along a straight one. In that case, some form of a banana path might make sense, allowing a runner to go straight for the first few critical seconds.

“This cries out for an empirical test,” Winston says. “It would be easy to do. If it holds up, God, that goes in the New York Times sports section.”

Glanville points out another complication: Infielders might get in the way of such an unusual path. “They’re supposed to be out of your way, but that’s not usually what happens,” he says, “and if someone’s in your path, you’re going to end up breaking your stride.”

Still, the researchers say, baseball players should experiment with more exaggerated curves. “The fact of the matter is that based on what the optimal path looks like,” Carozza says, “I don’t think the way people do it now is the fastest path, no matter how you accelerate in real life.”

The payoff for such experimentation could be huge, says Frank Morgan of Williams College, a collaborator on the project. “I’d feel pretty bad if I were a coach and I’d seen this and not told my team and then lost the World Series by a fraction of a second.”

Comment

Carozza, D., Johnson, S. and Morgan, F. 2010. Baserunner’s Optimal Path. Mathematical Intelligencer 32 (1):10-15. [Go to].

Peterson, Ivars. 1998. Curves on Baseballs. Science News online.
[Go to]

Rehmeyer, Julie. 2009. The Noisy Game of Baseball. Science News.
[Go to]

Sanders, Laura. 2009. Baseball by the numbers. Science News. August 29. [Go to]

Please alert Science News to any inappropriate posts by clicking the REPORT SPAM link within the post. Comments will be reviewed before posting.

• I should expect that the best of the sluggers will already be used to practising this arcane artform!
PedroRoberto
Oct. 24, 2010 at 8:37am
• Unfortunately, current rules don't allow runners to fully experiment with this proposed route. There is a "box" from halfway down the first base line to the bag. A runner must be within this "box", or he may be called "out" by the umpire. Keep working.
Charles W Downing
Oct. 24, 2010 at 9:30am
• My intuition is that the optimal running pattern should change depending on whether the batter wants to go for a single, double, triple, or infield home run. This is trivially true for a single where the batter runs straight towards first. For a double, the batter would decrease his time by beginning to round his path, but would the path be as circular compared to the optimal path for running all the bases? What if the batter knows he has hit a likely single, possibly double? How much does he risk his chances of getting a single to improve his chances of getting a double?
thisiscrap@mailinator.com
Oct. 24, 2010 at 9:59am
• @charles downing... the batter is only at risk of being called out if he's outside the first base running lane and he's hit by a thrown ball or causes some interference with a fielder trying to catch a thrown ball.
pat ryan
Oct. 24, 2010 at 10:33am
• Don't mean to brag, but I played baseball since I was 8 till about 40 yo. I wasn't good at throwing or catching, but I could pitch, and boy, I could hit a bat and run. I knew then not to run a straight line and I 'felt' like I was 20% faster. Sometimes our bodies can feel the physics of what works, you know!
Harrrie
Oct. 24, 2010 at 11:53am
• As someone else mentioned, this seems to only be true for running ALL of the bases.

By simple math, at 18.5 degrees off center the batter has to turn further from batting position beffore getting to full speed, and would also be running a slightly longer distance, which equals longer time.

I certainly agree, though, that running all of the bases at once would certainly be faster if the runner could maintain the proper curve. If you imagine it in a much faster version, it's certainly faster for a racecar to make more rounded corners than have to slow down at each "corner".
Erik Johnson
Oct. 24, 2010 at 10:42pm
• This is silly. 99% of the time, if not more, the only time you would run all the way around the bases is if you hit a home run, in which case you could walk, for all it matters, since the ball is already out of play.
gemless_ear
Oct. 25, 2010 at 12:47am
• @thisis: Assuming constant maximum acceleration magnitude, and remembering that you can only run through first base (to the foul side) and home plate (in any direction), one can use symmetry arguments to show that the optimal route for a planned triple should have (reflectional) symmetry around an axis halfway between the home-first basepath and the second-third basepath, and that for a planned double should have symmetry around an axis pointing from first to third. For a single, one should of course run straight from home to first.

In actuality, I think you'd probably decelerate faster in a slide than you would accelerate from the batter's box; if so, the optimal path should bow out more from home to first than in the last leg.

As the probability increases of extending a single to a double, or a double to a triple, you'd expect the path to bow out more and more in the early legs. How much more sounds like a problem in the calculus of variations, though the existence of a maximum speed seems like it might complicate things.
Brian Tung
Nov. 17, 2010 at 6:14pm
Registered readers are invited to post a comment. To encourage fruitful discussion, please keep your comments relevant, brief and courteous. Offensive, irrelevant, nonsensical and commercial posts will not be published. (All links will be removed from comments.)

You must register with Science News to add a comment. To log-in click here. To register as a new user, follow this link.