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Parabola Power: Field Geometry Helps the Cardinals Win

Note to readers: Every now and then, I am intrigued enough by one of the proposals presented to me for a guest column that I decide to publish it. This is one of those times. As a former (but long-time non-practicing) student of mathematics, I bring to you the thoughts of two very distinguished educators and authors, PhDs both, who also happen to be baseball fans. In the essay that follows, they consider the possibilities of the power of the parabola in the success of the St. Louis Cardinals.

By Sujata Bhatia, Harvard University and Patrick Chura, University of Akron

From the foul lines to the angles of the bases, one would assume that the game of baseball is about straight edges, squares and diamonds. But the parabolic arch on the field of Busch Stadium hints otherwise.

Probably we are introduced to parabolas during high school. In math class, we learn that the parabola is an open-ended “U” with arms that reach toward infinity. In physics class, we earn that parabolas are fundamental to projectile motion.  Throw a ball and it will trace a parabola. Bounce the ball along the ground, and a series of diminishing parabolas will result. A parabola is the shape comets make when they orbit the sun. NASA calls such a path an “escape orbit” because an object that achieves such a trajectory can travel forever.

We might even get to talk about parabolas during English class. In “Walking,” the last essay Henry David Thoreau ever wrote, Thoreau said he wanted his beloved daily walks in the woods to follow the contour of “a parabola… opening westward.” Thoreau chose the parabola for its philosophic properties, seeing it as a perfect middle way between opposing cultural forces. It symbolized not only the type of beauty he wished his walks to have, but the shaping and altering effects he wished his life to have.

The parabola also has a lot to do with the game of baseball. Though we’re used to thinking of the baseball diamond as the epitome of lines and angles, the game is really defined by curves. For the All-Star Game in 2009, the St. Louis Cardinals tapped into the deep science of the sport by combing into the outfield grass of the new Busch Stadium a beautiful replica of Eero Saarinen’s 630-foot parabolic masterpiece, the St. Louis Gateway Arch. Cardinal PR rep Mark Taylor said that the figure “was so well received by the media and fans that the team decided to keep it as a signature feature of the ballpark.”

The Cardinals have had great success on this field, so it’s worth considering whether the club’s innovative take on “field geometry” helps them win.  As many fans know, the flight path of a hit baseball is parabolic. And according to accepted research in environmental psychology, so is the ideal path taken by an outfielder to intercept the ball’s flight. Catching a fly ball requires the calculation of a subconscious geometric equation that relies on the ability to predict parabolic motion. This ability is what brings outfielder and fly ball together in one of the game’s key kinetic processes—a process that, in essence, illustrates the joining of a parabola with a parabola. Making the necessary split-second calculations on a playing surface that is inscribed with the parabolic shape could make the Cardinal players more efficient. A Cardinal outfielder, whether he is looking straight ahead or upward toward a batted ball, always has the arc of a parabola—a shape replicating the game’s dynamic geometry—in his field of vision.

Of course, the visiting team plays on the same field.  But playing at Busch Stadium for a handful of games a year is not the same as taking regular practice and playing 81 regular season games there.  Part of the home field advantage is the psychological reinforcement that comes with familiarity, allowing the environment to enhance the judgment of home players in comparison to opponents who do not internalize it to an equal degree.

Psychologists claim that humans are always responding to environmental variables, and the Busch Stadium parabola seems a major one. The long-term imprinting of this field design could work for the Cardinals in many ways.  Using the constant locational cues made possible by the parabola, Cardinal outfielders can position themselves more accurately for hitters and set themselves for throws to the bases with more confidence.  Being familiar with the arc of the parabola can help guide throwing lines and the ball’s trajectory.

A similar parabolic consciousness might even make Cardinal hitters more comfortable at the plate. Second base and home plate bisect the parabola and lie along a geometric sweet-spot called an axis of symmetry.  This could give hitters an added reference point for guiding the ball toward gaps. Painting a straight white line to halve the field from home plate to centerfield would violate the rules of the game, but the curved parabola accomplishes the same thing another way.

So we shouldn’t be deceived by the perpendicular lines of the baseball diamond; the game is all turns, arches, bends and bows.  The swing of the batter, the path of the fielder, the flight of the baseball, even the red seams on the ball are all curving arcs.  By making a parabola the signature feature of their field, the Cardinals have unleashed the real beauty of baseball.  And the more we think about it, the more we suspect that this eye-catching innovation actually helps the Cardinals win.

On Wednesday night, on the field where St. Louis had a 55-28 record this year, the Cardinals won the deciding game of their division series against Pittsburgh. If they continue to win, other teams might eventually re-think their own field geometry.  And we’ll be comparing the arms of the parabola to the arms of a triumphant athlete, reaching upward toward infinity.

12 Responses to “Parabola Power: Field Geometry Helps the Cardinals Win”

  1. blingboy says:

    Dang, those guys took the words right out of my mouth.

    I am glad to see that even Ivy League luminaries are wondering how the Cardinals do it.

  2. JumboShrimp says:

    “I’d rather entrust the government of the United States to the first 400 people listed in the Boston telephone directory than to the faculty of Harvard University.” – William F. Buckley, Jr.

    • Brian Walton says:

      Your political views are not welcome here. In case you missed it, this is a baseball site.

      • JumboShrimp says:

        The famed engineer, James B. Eads, designed a bridge across the Mississippi that employs arches. Photos can be found on the Web of the Eads bridge with Saarinen’s arch behind it. Eads would have been one of Saarinen’s sources of inspiration for the Gateway arch.
        Great rivers like the Mississippi, Ohio, and Missouri were the interstate highways of the 18th and 19th centuries, providing access within a continent. Transits on rivers and building bridges over them were important. Accordingly when a monument was conceived for St. Louis, an arch was fitting in the local context of bridging rivers. St Louis and the Louisiana Purchase were also once the western frontier of the US and a jumping off point for further exploration. In a larger sense, St Louis was a gateway to the rest of the continent, linking the first states to those to the west. Saarinen’s arch doubly serves this apt transcontinental metaphor for binding together a nation.

        In 2013, the Atlanta Braves went 56-25 in their home park, the Oakland As 52-29. Can we attribute these home records to nearby McDonald’s restaurants for their Golden Arches?

    • JumboShrimp says:

      In the essay above, it is stated “according to accepted research in environmental psychology, the ideal path for an outfielder to intercept the ball’s flight” is parabolic. This does not sound correct. A parabolic path on the plane of the field would be inefficient. An outfielder needs to make a quick estimate of where the ball will land and set out to that point by the shortest path.

  3. wacko413 says:

    Interesting read.

    It’s a shame that the above commenters were too fixated on where the authors work rather than the article’s content. They missed on out on a fun and unique article.

  4. kray66 says:

    That’s it! I’m cutting an arch in my front yard for good luck. Or maybe I’ll just roll up the garden hose in a big “U” shape and kill a patch of grass. That might be easier.

  5. WestCoastbirdWatcher says:

    Solid………………………….. you can just tell they’ve spent a lot of time at field level…………

    This is rich…………………………

    “And make sure that there are new ways of doing things and [b]new ways of getting information that we take advantage of [/b]and not just say ‘Oh, we’ve got it all figured out.’ Because no one in this game never really figures it out.”

    Bernie did a great job of teeing Bill up for me………………… Branch Rickey? Solid………………………

  6. Patrick C says:

    Hi Jumbo Shrimp,
    I am one of the authors of the Parabola article. We appreciate your healthy skepticism about the ideas my friend Sujata and I shared, so I ‘m writing with just a few points in response.

    About Atlanta and Oakland having arches in the vicinity: Sure they do but the arches aren’t down on the field where the game is played. This is a key difference because our article is about “field geometry.” You probably remember that old Busch Stadium had 96 concrete arches encircling the stadium, but there wasn’t an arch stretching across the entire outfield.

    About the ideal path of an outfielder to the ball: I’m not an environmental psychologist and neither is Sujata, but our reading in this field suggested that a parabola was the ideal path to the ball because this is the way perception and tracking of moving objects works. In order to “judge” a ball moving in a parabolic curve, one needs to achieve a path that makes it seem to our perceptions that the ball is NOT moving but stationary. The way to do this is to move in a shape that replicates the ball’s movement. I’ve read that dogs also do this naturally when they catch a ball or a Frisbee.

    And did you see Car-Bel’s amazing cannon throw from RF in last night’s game? Unbelievable!!

    If you have time, take a look again at the video of that “Must C Cannon” to the plate. I think what happened at that moment helps to support the case we made in our article. I think both CF Jay and RF Beltran approach along the type of curves our article talks about, with the centerfielder visibly following the curved (not straight) shape on the field.

    As I see the play, Beltran calls off Jay and hesitates so he can sprint into the catch to create momentum for his throw—-but before he does this he probably looks at the field in front of him to determine his path for the last few steps of his approach, steps that will direct his momentum to the target. He ends up using the leg of the field’s curved arch shape as the LAUNCHING PAD for his unbelievably accurate throw.

    It really seems that the shape on the field could have helped Beltran position himself and make his throw. Some might say the “help” was only slight. But as I’m sure you know well, there’s nothing “slight” in baseball especially when you are trying to do something so near-impossible as pegging a runner out from that far away. Right?

    Here’s what our article said about this:
    “Using the constant locational cues made possible by the parabola, Cardinal outfielders can position themselves more accurately for hitters and set themselves for throws to the bases [and to home of course] with more confidence. Being familiar with the arc of the parabola can help guide throwing lines and the ball’s trajectory.”

    By the way there are parabolas all over the video of this catch and throw, especially as the camera follows the ball to the plate after it leaves Beltran’s arm.

    Another point we wanted to make in our little article is simply that any shape that divides the outfield into two halves and offers points of purchase for positioning alters the game somewhat. Most outfields have checkerboard patterns cut into them that don’t really allow a player the type of knowledge of one’s position that the St. Louis shape allows. We just thought the Cardinals might have an advantage in this area because they are consciously and unconsciously “imprinted” with the shape to a greater degree than their opponents because they play there more often. I hope we were right!

    One last remark: Someone is likely to point out that the shape of the arch isn’t a “perfect” parabola. It isn’t. It’s a catenary, the shape made when a chain hangs upside down between two level points. But someone as smart as Galileo thought a catenary was a parabola, and Saarinen, who knew it wasn’t a perfect parabola, often referred to it as “parabolic,” even saying that the landscape and buildings in the arch’s vicinity were part of the same “parabolic family” (his words) as the arch itself. Even the flight of a baseball isn’t a “perfect” parabola because it’s affected by variables like wind and weather. Saarinen had his reasons for making it a catenary, reasons I’ve tried to explore in other articles. We think the real point is that the arch is what Saarinen called “a completely natural shape.” In the article, we were thinking that there might be more completely natural shapes in the game of baseball than there are straight lines.

    Have a great day JumboShrimp! And thanks for being intrigued enough by our article to comment on it. Sujata and I understand the skepticism some have expressed about the article’s ideas but it was a beginning, and we still think we might be onto something with “parabola power”. . . .

    Now, Go Cards!!

    Patrick Chura

    • JumboShrimp says:

      Greetings Patrick: Regarding Beltran, he has more HRs in post season than anyone else, 16. Three-quarters have involved the Cards. Carlos hit 4 against us in 2004, 3 more in 2006. Last year, he hit 3 for us and 2 more against the Pirates this year. His distinguished post-season career has often involved St Louis, against or for.

      The Wikipedia entry for Eads mentions his arches were catenary as well.

      A famous mis-play by an OF was a line drive hit right at Curt Flood in Game 7 during 1968. Flood was a terrific gloveman and played deep, so it was surprising to see a ball pass well above him. The reason for his misread of distance was perspective. When a ball is hit directly at an OF, he cannot gain valuable information (speed, rate of ascent and fall) that can be gleaned from a bit more side perspective. Maintaining a good read on a fly ball may offer a reasonable rationale for an OF sometimes being well served not to run directly to an estimated point of intersection, but to choose to stay a little off the projected path of the ball, so as to maintain some side perspective on its trajectory. (This would rely on having enough time to be able to take a more meandering path, because the ball’s hang time is sufficient to allow this.) I do not know if such a longer path for the OF could be called parabolic, but it might be prudently justified. An OF might also take a longer path to avoid losing a fly against a bright sun or other visual distraction.

      We played the Twins in 1987 and Cards players, unfamiliar with this AL park, seemed to have a terrible time seeing flies against the ceiling of this dome. In due course we lost all 4 games held in Minneapolis, while besting the Twins at a much more conventional park in St Lou. One felt the Twins had a tremendous home advantage. I do not doubt there are rational reasons for unusual characteristics at other parks translating into a few extra wins for a home team. It is plausible the biggest game impacts will be mis-reads by outfielders, such as in 2009 when Matt Holliday lost a fly during twilight in Chavez Ravine. The perspectives of defending OFers are important to processing and solving the geometry of the game.

      • JumboShrimp says:

        The first book (1965) by the writer John McPhee was a biography of college basketball player Bill Bradley entitled A Sense of Where You Are, if I recall correctly. A player in many sports will need to develop a keen intuitive geographic sense of the field or court, basket or goal.

        A key word in the essay above may have been “ideal” as in an ideal path of an OF is parabolic. Many times an OF should run on a straight line, since this is the shortest distance to travel. On Beltran’s two run double in Game 1, the Dodgers CF would have had a good side perspective on the ball’s flight, so should have been able to run straight and thus efficiently toward an accurately computed interception point. However, for a fly hit at a CF, but deeper or shallower, in these contexts a curved path may make sense, so as to move away a bit from the line of flight so as to obtain a side perspective. Perhaps a parabolic path for the OF would be ideal in instances of balls hit directly at an OF.

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