Thanks to Steven Finch for forwarding this pointer to the online article. I re-read the article, and I think I discovered something about baseballs that seams not to have been noticed by the author Richard Thompson, and possible not by other authors he references. Thompson first rotates the baseball into a preferred position, wherein the projection of the seam onto the xy plane is a "C" -- the seam is symmetrical, so that the "C" is double-covered by the seam projection, except for the two ends of the "C". The "C" is, of course, symmetric about the x axis. The "C" is pictured inside the circle projected by the baseball itself. Thompson also notices that due to the symmetry required of the covers, the projected "C" extends equally far on either side of the y axis. Thompson then notices that on particular baseballs he examines, the arms of the "C" _to the right of the y axis_ are _straight lines_! Unfortunately, he doesn't plumb the consequences of this fact, but we will. The straightness of these lines imply that the seam can be embedded not just in the sphere of the baseball, but also in a (non-regular) tetrahedron, and in fact, the seam is the intersection of the sphere with this tetrahedron! This tetrahedron is semiregular, in that 4 edges have one length, and the remaining two (opposite) edges have another (shorter) length -- i.e., all faces are isosceles. The two opposite edges do not intersect the ball, while all four of the other edges intersect the ball at the vertices of a square in the yz plane. In other words, the extreme y values of the "C" indicate the points of intersection of the non-opposite edges with the ball, and also indicate the places where the seam starts following a different face of the tetrahedron. The limiting case of this embedding occurs when the tetrahedron is regular, in which case the ball also intersects the middle of the two opposite edges, and each of the two pieces of leather have zero thickness in their middle. By the way, what do you call the sphere that intersects the middles of each of the edges of a regular tetrahedron? FYI -- The left side of the "C" is an ellipse. There appears to be only one free parameter -- the angle of the arms of the "C" (or the ratio of the edges of the tetrahedron). I dont know if/when the "C" (or the seam) is smooth. At 02:38 PM 7/5/03 -0400, Steven Finch wrote:
Hello!
The particular web files you mentioned are still available online (zipped up, however) at:
http://www.mathcad.com/Library/LibraryContent/baseball.zip
I don't know why my former employer chose to make these interesting files less visible. Best wishes,
Steve Finch
From: Henry Baker <hbaker1@pipeline.com> Reply-To: math-fun <math-fun@mailman.xmission.com> To: "Bernie Cosell" <bernie@fantasyfarm.com> CC: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] NY State math exam flapment Date: Sat, 05 Jul 2003 09:08:50 -0700
The original article (Thompson, R.B. 1998. Designing a baseball cover. College Mathematics Journal 29(January):48) was pretty good. Unfortunately, it does not appear to be online anymore, since the widely referenced mathsoft site is gone. The author apparently works for the U. of Arizona, but he doesn't have it online either.
Perhaps someone on this list can get Dr. Thompson to post it on his web site?
At 11:01 AM 7/4/03 -0400, Bernie Cosell wrote:
On 4 Jul 2003 at 7:25, Henry Baker wrote:
I hope that you know enough about baseball to appreciate the elegance of the baseball's seam:
What property of the 'seam' problem for the baseball precludes just an equator for the seam as a possible solution? Is there some "minimization of necessary stretch" or other criterion that drives the more complicated shape?
/Bernie\