Sphere Surface Dissection, Baseball, Tennis Ball, Math

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Looking at a baseball or a tennis ball, one wonders how to cut a spherical surface into two equal pieces. There are certainly many ways to do it, taking advantage of the spherical symmetries. Scott Elliot has some nice OpenScad code that performs such a dissection. I am looking at the project from a K-12 educational perspective. How can one dissect a spherical surface in Fusion 360 or similar 3D CAD environment?

Well, it turns out to be an appealing effort on Fusion 360, after a bit of mathematical tweaking. We can start with a hollow sphere (the result of subtracting two concentric spheres) and a cube of equal dimensions. After centering the hollow sphere and the cube, we can sketch a circle on a cube surface, whose diameter needs to be such that, if we were to make another circle of the same diameter on another orthogonal cube surface, the two circles are tangent to each other in space. A bit of algebra with the help of old Pythagoras leads to the conclusion: the circle diameter is the diameter of the outer sphere/ sqrt(2). This itself is an interesting math project for middle and secondary students.

Indeed, we need only half of that circle, plus two tangent lines for a “U” curve. Using the U curve, we could split the sphere into two pieces. These two pieces have the same outer surface, but are not exactly the same, because of the thickness. Take the half with an even thickness, construct a plane along the edge curve, sketch a 45-degree right triangle, and then perform a sweep cut along the whole path around the opening. Now, we have one half of the spherical surface (with some thickness). It takes two to make a hollow ball.
This should be an intermediate design project for 6-16 students with some guidance, as presented above. Have fun and try to come up another way to do it!

References:

A brief history of the baseball. https://www.smithsonianmag.com/arts-culture/a-brief-history-of-the-baseball-3685086/

Baseball. https://en.wikipedia.org/wiki/Baseball_(ball)

Elliot, Scott (2010). OpenScad code for spherical bisection, available at https://www.thingiverse.com/thing:3068/#files

Gardner Martin. (2001). The colossal book of mathematics. New York, NY: W. W. Norton & Co.