The fastest path between four points: Math and Baseball

No, I’m not exploring my creative side by doing one of those drawings where you get those little coloured plastic cogs, and a pen, and swirl them around a page. That diagram is the result of careful mathematical study of the geometry of baseball, it represents the fastest path around all four bases – useful only if you hit a deep ball that doesn’t go over the fence and you want to run home – it’ll shave milliseconds off your time.

If you are running to first, or between first and second (or second and third, or third and home), which I believe in baseball parlance is a single (what would I know, I’m from Australia, we play cricket) the straight line is no doubt still the best bet. This circuitous route shaves about 25% off the time taken for the run – because turning sharp angles slows runners down substantially.

Some quotes from the story:

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 was 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.”

One of Carozza’s colleagues, Stewart Johnson, optimised the path by computer (coming up with that diagram).

“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.”


Tim says:

As someone who has played baseball there are a few problems with this. Firstly the moment you hit the ball you go straight towards the base trying to get there as quick as possible. Only once momentum is reached and you are nearing the base do you look for the ball or look at your 1st or 3rd base coach. You are never positive if your hit has gone into the pocket and you run as fast as possible trusting your base coach. They tell you either to stop at the base, slide/dive at the base or to round it and keep going. Thus you only head out in the last five steps to start to round the base (go out around so you don’t have to do a right angle turn) keeping in mind you must step on the base.
Also if you stick to that large circle drawn there are fielders stationed exactly in that circle. they would slow you down or you would have to side step around them.
Finally If you kept to that circle you would probably run outside the baseline (direct path between bases) which is out. Also normally the direct line is dirt and outside that is grass- probably run quicker on dirt.

Nathan Campbell says:

The study actually seems to consider most of your objections (ie they acknowledge the interference of fielders and desire to get to first makes it implausible).

But the arcs are within legal parameters – they checked the dimensions of the curve against the legal area for running, and also checked what rules apply for deviations from a straight line (you can run curvy as long as a fielder isn’t trying to tag you).