Squares on Spheres
Name: James K.
A square on the surface of a sphere measuring 1
degree 1/360th. of a circle ) square.
How many of these squares cover the surface of that sphere?
There is a fundamental problem here. If you try to paste a postage stamp
on a soccer ball, it won't lay flat. That's why soccer balls have a
pentagonal surface. But assuming that the squares are small enough that the
above problem is negligible, here's how the problem goes:
The surface of a sphere is S=4*pi*r^2, where pi=3.14... and we take r=1
So the total surface area S=4*pi(radians^2). Now 360 (deg) = 2*pi(radians),
360^2 (deg^2) = (2*pi)^2 (radians) = 4*pi^2(radians^2).
So the total surface area S=4*pi (radians^2) * (360)^2 (degrees^2)/
4*pi^2 (radians^2). Canceling out 4*pi in the numerator and denominator
S = (360)^2/ pi (degree^2) = 41253 (degree^2) patches 1(degree)x1(degree).
This number is rounded to the nearest (degree^2) because no integer number
of tiny squares can evenly and completely cover a sphere.
I cannot tell exactly how many will fit due to the fact that there might be
a little bit of open space between some of the circles. A way to get a very
good approximation, and definitely a maximum, involves area. Choose a
radius. The side of a square will be 1/360th of the distance around a
circle. Calculate the area of such a square. Calculate the surface area of
a corresponding sphere: 4(pi)(radius)^2. See how many square areas will
fit into a sphere area.
Dr. Ken Mellendorf
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Update: June 2012