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Dave Baldwin is unique among former big-league pitchers. After a 16-year professional baseball career, including stints with the Senators (1966-1969), Brewers (1970), and White Sox (1973), Baldwin was a geneticist, engineer, and artist. He is now retired and living in Yachats, Oregon. His "Baseball Paradoxes" can be found at http://www.snakejazz.com.

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DL: Camilo Pascual had a great curveball.

DB: “Drilling for oil,” is the way that some players described it. Or “falling off the table.” Pascual (a righty) and Sandy Koufax (a southpaw) threw similar curveballs—the best I ever saw. Camilo came nearly directly overhand with it and the spin he put on the ball was amazing. When we were just playing catch and he’d tell me he was going to spin one, I’d brace myself because even though he was only tossing the ball, it would be coming at me like a buzz saw.

A spinning ball has two sources of deflection—gravity and the Magnus force, with gravity being the strongest of these forces. The Magnus force creates a spin-induced deflection which results from the combination of spin and forward movement of the ball. The angle of the spin axis determines the direction of deflection, and the forward velocity and spin velocity determine the magnitude of the deflection—the faster these velocities, the greater the magnitude.

Terry Bahill, an engineering professor at the University of Arizona, and I have used two right-hand rules to describe the direction of deflection from the Magnus force (see Bahill & Baldwin, “Describing baseball pitch movement with right-hand rules,” Computers in Biology and Medicine, 37 (2007) 1001–1008.). The first rule is applied to find the spin axis of a pitch. If you curl the fingers of your right hand in the spin direction, your extended thumb will point in the direction of the spin axis (fig. 1).

Fig. 1 Right-hand rule showing fingers pointing in direction of the ball’s spin (arrow) and thumb pointing in direction of spin axis. Copyright © 2004, A.T. Bahill, from http://www/sie.arizona.edu/sysengr/slides/ (used with permission).

A coordinate right-hand rule can then be used to determine the direction of deflection. Point the thumb of your right hand in the direction of the spin axis (as determined from the first right-hand rule), and point your index finger in the direction of forward motion of the pitch. Bend your middle finger so that it is perpendicular to your index finger. Your middle finger will be pointing in the direction of the spin-induced deflection (fig. 2). The right-hand rules apply to pitches of both right- and left-handed pitchers.

Fig. 2 Right-hand rule showing thumb pointing in direction of spin axis, index finger pointing in forward direction, and second finger pointing in direction of spin-induced deflection. Copyright © 2004, A.T. Bahill, from http://www/sie.arizona.edu/sysengr/slides/ (used with permission).

In throwing an overhand curveball, the pitcher imparts topspin to the ball. This sets the spin axis angle in a horizontal position, causing the Magnus force to deflect the pitch downward. On an overhand curve, then, the Magnus force and gravity are working in concert, thus creating a pitch that appears to be breaking downward very sharply.

Ballplayers speak of the “break” of a curveball, but this is misleading. The spin of a curve (and of a fastball, too) stabilizes the pitch, just as a gyroscope is stabilized by rotation. Therefore, the trajectory of a spinning pitch is smooth, with no sudden “breaks” as batters claim. The rate of curvature of a curve isn’t constant, however. From application of the coordinate right-hand rule, note that the direction of forward motion (indicated by your index finger) is continuously redirected by the Magnus force (second finger) and that the rate of curvature increases continuously.

Of course, a good curveball can be thrown from any arm angle, with the spin axis angle determined by the position of the fingers on the ball at the moment of release. A pitcher throwing with a 3/4 arm angle (e.g., a right-hander throwing out of one o’clock or two o’clock) will create a diagonal deflection, sweeping from upper right to lower left for a right-handed pitcher. A sidearm pitcher will create a sweeping horizontal deflection with the Magnus force, and gravity will cause the curve to dip. A slower forward motion of the curve allows gravity to work longer on the ball, resulting in a greater vertical deflection.

Some extremely fast pitchers were able to “go up the ladder” with the batter—throwing progressively faster fastballs, each a little higher than the previous pitch. Batters would swing at each one, and each swing would be a little too slow and a little under the ball. Camilo, however, would use his curve to “go down the ladder.” Each curve would be slower and lower than the previous curve. And the batter would swing over the top of each one. Once, I saw Camilo do this against one of the Yankee hitters. On the third nose-diving curve, the ball bounced just in front of the plate. The batter judged the trajectory of the pitch perfectly and clobbered the earth.

DL: Frank “Hondo” Howard’s brain is hard at work as the ball comes plateward.

DB: The part of Hondo’s brain that is at work during the pitch’s flight is somewhat surprising, however. And Hondo isn’t unique in that regard. All batters are forced into a particular mode of processing information and making decisions by the very short time frame in which these operations must be accomplished. A 90 miles per hour (mph) fastball, for example, takes about four-tenths of a second or 400 milliseconds (msec) to get from pitcher’s hand to catcher’s mitt. Because it will take Hondo about 130 msec to swing the bat, he has only about 270 msec to decide whether or not to swing, and if he is going to swing, where the barrel of the bat needs to be in order to best hit the ball.

But that doesn’t give Hondo time enough to consciously figure all that stuff out. About forty years ago, neurophysiologist Benjamin Libetconducted a series of experiments to determine the relationship between the conscious intention to carry out an action and the initial brain activity that must precede the action. It turns out that the brain lights up about 350 msec before the conscious mind even knows the action is to be taken. Since Hondo’s brain has only 270 msec to make its decisions, the conscious mind will be about 80 msec tardy.

Fortunately for Hondo, the unconscious mind is on the job from the get-go. As the pitcher releases the ball, Hondo’s conscious mind stays focused on picking up the pitch. That keeps it occupied and out of the way of the unconscious mind which is busy evaluating the appearance of the pitch (direction and speed of the spin, forward velocity, etc.) so that it can make the necessary calculations and decisions without Hondo’s awareness.

Immediately after the pitcher releases the ball the pitch seems to be coming very nearly directly at Hondo’s eyes, but as the ball gets closer to the contact point, the eyes have increasing difficulty keeping the image on the foveae. In fact, the eyes aren’t able to follow the pitch all the way to the ball/bat contact—the image “outruns” the foveae when the ball is about five feet from the contact point. This doesn’t matter since Hondo can do nothing at that point to alter the trajectory of the bat. During the last five feet of the pitch’s flight, Hondo would do just as well if he had his eyes closed.

Note that the ball “appears” to approach the contact point more rapidly in the later stages of its flight, even though the 90 mph pitch actually slows by about 11.5 mph overall because of drag force. Hondo’s mind makes adjustments for this phenomenon. To experience this illusion, watch the median stripes on a highway as you travel at a constant speed of 60 mph. A stripe that is quite distant down the highway will seem to creep toward you, while a stripe very near the car will seem to whiz by, even though the car is moving at the same speed relative to the two stripes, of course.

Besides checking the velocity and angle of the pitch, Hondo’s unconscious mind might also check out the ball’s spin pattern for indication of pitch behavior. Some batters report seeing a pattern of stripes (and maybe a dot) made by the red seam as it whirls around the axis of the ball, but others say they see only a gray blur. If a batter’s unconscious mind recognizes the spin of the pitch, it can anticipate the ball’s spin-induced deflection.

I surveyed 15 former major-league position players and found that only eight could recall seeing the spin pattern. This could be due to visual differences or to differences in pattern processing in the brain. Coaches generally assume that the ability to see the spin pattern will make for a better hitter, but the success of a batter doesn’t seem to be related to his ability to recall seeing this pattern. For example, Frank Robinson has reported he was able to see the seam, but another Hall of Famer, Mike Schmidt, has said he was never able to see it (see Schmidt & Ellis, The Mike Schmidt Study: Hitting Theory, Skills and Technique. McGriff and Bell, Atlanta, 1994).

DL: Ted Williams hit .406 and Joe Mauer can only dream.

DB: For decades, players, sportswriters, and fans have been theorizing why we will never see a .400 hitter again (e.g., see M. Durslag, “Why the .400 hitter is extinct,” Baseball Digest, Aug. 1975). Each season reinforces the basic assumption that Williams with .406 in 1941 was the last of the breed. Numerous hypothetical reasons have been suggested as to why hitters are declining in their ability or why hitting has become progressively tougher. Explanations offered have included dumber hitters (by Williams), failure to hit to the opposite field (by Stan Musial), more night games, better relief pitching, larger fielders’ gloves, and exhaustion due to transcontinental travel.

Paleontologist Stephen Jay Gould has argued that hitting (as well as pitching and fielding) has been improving throughout baseball history, and the disappearance of .400 hitters results from a loss of variability in hitting proficiency (see S.J. Gould, Full House, Harmony Books, NY, 1996). Gould’s model, derived from studies of biological populations, assumes that a static major-league population is selected from an ever larger and more proficient player pool—bigger, stronger, quicker athletes with better coaching and training methods. It also assumes that batting averages are constrained by a biological limit to hitting proficiency. (This is similar to the biological limits on the size of elephants or the speed of cheetahs.) Gould refers to this as a “right wall” on the distribution of averages—hitters are limited as to how good they can become.

An increase in proficiency amongst the players who are selected to play in the majors decreases the variance of major-league batting averages. In other words, hitters begin to pile up against the biological barrier. Therefore, it becomes progressively harder for any hitter to separate himself from the rest of the hitters. Neither the median batting average nor the biological limit changes, but the position of the right wall relative to the batting average distribution changes. It moves toward the median.

Meanwhile, with the increasing number of highly proficient hitters coming into the majors, there is less roster room available for poorer hitters. The bell-shaped curve describing batting average distributions in earlier times becomes a mountain with steep slopes. The .400 hitter becomes rarer but so does the .150 hitter.

If this process continues long enough, most of the hitters in the majors will be clumped very close to the median (say, around .265). No one would stand out.  Perhaps a dozen or more hitters could tie for the league batting title at, say, .283. The majors will have approached parity.

Parity is a bad thing for professional sports. Pro teams need to attract paying customers, and the fans pay to see the stars. The rest of the players are there just to make it possible for the stars to shine. Parity is good for amateur sports, but it doesn’t produce the big money that major-league teams thrive on.

Before Gould died in 2002, I had several discussions with him about this model. We talked about the basic assumptions of his model and how those assumptions might be disrupted, necessitating the formulation of a new model. Gould had assumed that the balance in the offense-defense contention would be maintained, but he realized that future changes in rules or playing conditions could upset that balance and batting averages could soar as a result.

But a more interesting potential change would move the right wall farther toward the right for a subset of the hitters. Some biological advantage might be introduced that would give certain hitters an edge over the others. Gould and I discussed the various performance enhancing drugs that were known at that time (2000 – 2001), and we wondered whether that might be the disruptive factor that could invalidate his model.

The use of steroids, human growth hormones, etc., have yet to produce even one .400 hitter. This isn’t a proof of Gould’s model, of course. With each season we test it once again.

Will Joe Mauer be able to hit .400? Just as today’s hitters are better athletes than were the 1941 hitters, pitchers are bigger, stronger, and in better condition than the pitchers of seventy years ago. There aren’t as many “weak” pitchers in the majors now. And defensive players are better, too. It’s harder to get a hit these days. I wouldn’t guess whether Joe Mauer or any other current player might hit .400 some day, but it seems likely that it will be somewhat tougher than it was for Ted Williams.

DL: How large is the sweet spot of the bat?

DB: First, let’s define the “sweetness” of a point on the bat’s face (hitting surface) as the probability of the batter getting a base hit given the ball makes contact at that point. Then, the face of the bat can be considered a field of points, each with a specific sweetness value. The highest sweetness values will fall in the middle of the field, with sweetness diminishing in gradients outward to the edges of the face.

To simplify this, let’s look at how sweetness is distributed around the bat’s transverse (vertical) curve (fig.3). The sweetest point (where the probability of getting a base hit is highest) on that vertical arc will be at or near the middle of the bat’s face. From that point, sweetness decreases in two gradients around the arc—one lies over the upper shoulder of the bat, the other under the lower shoulder. If the batter swings too low, the contact point on the bat will be offset above the bat’s sweetest point and the ball will be launched upward. If the swing is too high, the offset will be below the sweetest point, and the ball will be directed toward the turf.

Curt Simmons used to say that the pitcher’s job is to “take the sting off the ball.” By that he meant that the pitcher is trying to trick the batter into hitting the ball with the upper or lower shoulder of the bat. The pitcher is hoping for a fairly large offset, either above or below the sweetest point. A swing that’s too low results in a pop-up or a fly out; a swing too high results in a ground ball with the ball/earth impact absorbing much of the batted ball’s energy. In either case, the pitcher figures he’s done his job (so long as the fly is playable, of course). On the other hand, if the offset is very small, resulting in a line drive, the sting hasn’t been taken off the ball, and the batter is likely to get a base hit.

The vertical sweetness gradients are only partially determined by properties of the bat—constants such as the bat’s mass, its radius at the contact point, and the coefficient of restitution (bounciness) of the wood. Variables that shape the vertical sweetness gradients include the velocity vectors of bat and ball, the angle of the swing, the angle of the pitch, and the spin velocity of the ball.

The longitudinal (horizontal) sweetness gradients are more difficult to describe. They are influenced by several bat constants such as the distribution of the bat’s mass, vibrational properties, and energy transfer/loss properties. A number of variables affect horizontal sweetness, including the bat speed and the horizontal angle of the bat when contact is made. Of course, both the vertical and horizontal gradients can be modified by a batter who is also a good carpenter.

We can say that the “sweet spot” is actually a region within the bat’s sweetness gradient field. We could call this region a “sweet patch.” Determining the size of the “sweet patch” requires specification of a threshold probability of success. For each point of the bat’s face, this threshold would be used for deciding whether that point should be included in the sweet patch.  We could then describe the collection of points satisfying the specified minimum sweetness. Because the threshold is arbitrary, many such sweet patches can be defined. They will shift, too, as the variables (angle and velocity of swing, etc.) change. As a rough approximation, though, we can say that the length of the sweet patch is around three to four inches and the width is about half an inch, more or less. Of course, it all depends upon who is swinging the bat. If Hank Aaron (who had the fastest bat I’ve ever seen) is hitting, the sweet patch is very large, whereas if Dave Baldwin is at bat, it disappears altogether.

Fig. 3 Common outcomes for some particular launch angles and bat-ball offsets. Copyright © 2004, A.T. Bahill, from http://www/sie.arizona.edu/sysengr/slides/ (used with permission).

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