University of Arizona DESIGNING A BASEBALL COVER
1860's: Patience, Trial, and Error
by Richard B. Thompson 1. HISTORY OF THE PROBLEM
Figure 1 Figure 2 Figure 3 Figure 4 We will refer to the region in Figure 1 as a flat . Two flats, ready for stitching, are shown in Figure 2 . The stitched pair of pieces, shown in Figure 3, will be called a preball . If the seam of the preball lies on a sphere of the same radius as the ball, then it will not be distorted when the leather is stretched to form a ball, as shown in Figure 4 . A preball with this property will be called acceptable , and its associated flat will also be called acceptable. There were other design parameters, that had to be met. Baseballs were expected to have the nominal circumference of 91/8", and two parts of the seam were to be located in such a way as to provide a good grip for the pitcher's fingers. Measurements of current balls indicate that this distance has an arc length of 13/16". It was also expected that flats were to be symmetrical about both their horizontal and vertical axes. Mr. Jackson's problem was to use experimentation to discover an acceptable flat, that met the requirements of dimension and symmetry. Our goal is to develop mathematical methods that solve the problem of finding an acceptable flat, without trial and error.
In the abstract, there are two ways to design a baseball cover. (i) We can draw a flat and then wrap two copies of this around the ball. (ii) We can draw the seam on the ball, and then unwrap the resulting regions to form flats. Mr. Jackson chose the first option, since it is simpler, and is the only practical plan for pen and ink experimentation. The main difficulty of drawing flats directly, is that it appears to be necessary to specify onefourth of the entire seam. This makes it very difficult to design an acceptable flat. The second plan removes this difficulty, since we design directly on the ball. As we will see, this requires us to specify only oneeighth of the total seam. Mathematical analysis and computational power eliminate trial and error and make, the apparently more complicated, second plan more attractive.
From this view, we can use the design parameters of circumference, C = 91/8", and minimum width between seams, = 1 3/16", to define the points (, ) and (, 0) that are at the ends of the upper part of the projected seam. We use the congruence of the two sections of the ball to determine the point (0, ) at which the projected seam crosses the yaxis.
Let
f: [, ] > R
be the function whose graph is the top half of the projected seam, and let p and q be the functions that constitute the right and left sides of
f
, respectively. Due to symmetry, a point on the right side of the seam has 3dimensional coordinates (
x, p(x)
, q(x)). Since the ball has fixed radius
R
, we see that
p
determines
q
and
f.
Our task is to define p: [0, ] > R such that p(0) = , and p() = . In this matter we have great freedom. As long as p is reasonably wellbehaved, and has a graph that stays in the first quadrant of the projected circle, there is no mathematical necessity for the selection of any particular function. Our choice must rest upon our mental picture of the usual baseball seam. Since construction of the ball is not uniquely determined, we can not say that we are either right or wrong with any particular function p that we select. The best measure of our success is to look at pictures, and see if we have something close to the usual appearance of a baseball. To do this, we will first parametrize the part of the seam that projects onto the graph of f.
At a point
x
, the seam crosses a slice of the sphere that has a radius
r
, with .
Knowing the second coordinate, f(x), we can determine the third coordinate and define parametric functions u, v, w: [, ] > R, for the seam
Having parametrized one fourth of the seam, we can splice together copies of our functions and describe the entire seam with x running from 0 to 8.
Our initial choice for
p
is dictated by two considerations. First, examination of an actual baseball indicates that the graph of
p
is to be relatively straight. Second, mathematicians like linearity! We define
p
to be the linear function,
connecting the two designated points. (This is the function that was used to generate all of the graphics that we have seen so far.) Rotation matrices can be used to display our results from several different angles. The Rawlings Company describes its major league baseballs as having, "108 stitches in all, with the first and last perfectly hidden." For cosmetic effect, we will show our seam with 104 visible stitches.
Referring to that figure, let be the length of the graph of
f
, over [, ]. Since the center line of the top and bottom flat lies directly along the graph of
f
, we will define a function
F: [0, ] > R whose graph is part of the edge of the upper portion of the flat. To start, we use arc length to define
L: [, ] > [0, ].
For convenience, denote the inverse function, , of L by M. This gives M: [0, ] > [, ] and allows us to define the part of the edge of the flat that projects toward the viewer from Figure 5 . Let F: [0, ] > R be given by F(a) = f(M(a)). The graph of F is shown in Figure 7 . Using symmetry, we can assemble four of these curves to form the boundaries of an acceptable flat. The resulting shape is shown in Figure 1 . We might want to go a step further than our original goal of drawing an acceptable flat. It is now quite easy to parametrize the image of our flat on the preball. We will give the mapping of the portion of the flat that is shown in Figure 7 .
Let D = {(a, b)  and
0b1} and define
,
,
: D > by the following:
(a, b) = M(a),
(a, b) = f(M(a)) and
(a, b) = b F(a) .
Since the flat is acceptable, the seam on the preball does not move under this expansion. The stretched preball is shown in Figure 4 . We have created a parametrization of the seam , in closed form. Our parametrization of the acceptable flat uses arc length and an inverse function. These are computable to any desired accuracy, but do not give F in closed form.
We start with a variable , and a function, P , whose graph connects (0, ,) and (, ). This is shown in Figure 8 .
Next, we define an arc length function
K: [0, ] > R , along the graph of
P, and let = K(), so that
. For convenience, denote the inverse function, ,
of K by H.
The most natural of the coordinates is v(t) = P(H(t)). The definition of u requires more careful thought. We want to preserve arc length, as the edge of the flat becomes the seam on the preball. When we differentiate both sides with respect to t , and square both sides of the resulting equation, we obtain . The derivatives of v and w can be computed and simplified. When these are substituted into the equation that we obtained from arc length, we have an implicit differential equation for u , with all other parts being known functions. Our initial condition is u(0) = . Since was left as a variable in the definition of K , we actually have a family of solutions u(t) that depends upon the choice of . Of these we pick the one that allows the image of the edge of the flat to end at (0, ). This means that we pick so that u() = 0. Once we have a parametrization of the seam (on the ball), we can use symmetry to extend u , v , and w to [0, 2], given a parametrization of onefourth of the seam. This part corresponds to the section of the seam that projects to the graphs of p and q , in Figure 5 .
Arc length of the seam, on the ball, is the same as arclength, on the edge of the flat. Hence, we can parametrize the edge of the flat that corresponds to the graph of
F
in
Figure 7
. This is done with
X, Y: [0, ] > R , where
It is time to stop and consider the practicality of our plan. We did not encounter a simple, gardenvariety of differential equation! It is not our purpose to give a theoretical discussion of the solvability of the equation. What we will show is that, if we start with a reasonable function P , then we can use implicit numerical solutions as part of Euler's method (750 steps), and obtain an approximate solution of the initial value problem.
We will illustrate this with a function
P
, that is part of a cosine curve, connecting (0, ) and (, ). This is the function whose graph is shown in
Figure 8
.
When the solutions were computed, we found that u() = 0, when = 1.950. Continued numerical work allows us to compute values for the parametric functions X and Y . The picture of our extended, acceptable flat is shown in Figure 9 .
In addition to the mathematics that we have used, we have also shown that we must think in different ways, if we are to take advantage of computation. We actually solved the original design problem in two ways. Of these, Mr Jackson's original plan, of designing in the plane, was the most difficult. A change of venue to the surface of the ball was impossible for Mr. Jackson, but it provided the most natural setting for our work. The following Mathcad PLUS 6.0 files demonstrate the above more fully.
Note: a revised version of this article appeared in The College Mathematics Journal, Vol. 29, No. 1 (January 1998); click here for abstract.
