

Equilateral triangles on a sphere
Name: julie a corder, pam swan, candice korb, and natalie bauman
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995
Question:
Our eighth grade geometry class has been assigned a sixweek project. We
have to cover the surface of a sphere with equilateral triangles of any
size. We are working in partners; one pair has a sphere with a diameter of
4 inches, the other 6. Our research has shown that since no straight lines
can be drawn on a sphere, and since spherical triangles all have angles with
measures with a sum greater than 180, getting plane triangles to lie on a
spherical surface is impossible. We have been told to just make the
triangles as small as possible, so that the error is not visible.
Replies:
Julie, Pam, Candice and Natalie  the solution to the problem is that you
can only "strictly speaking" do it correctly when the triangles are really
tiny  infinitesimally small in fact. By using small, but not infinitely
small, triangles, you are making an approximation. Did you actually measure
those angles for spherical triangles? Try comparing the sum of the angles
for a large spherical triangle with the sum for a smaller one. You should
find that the smaller one has a sum much closer to 180 degrees. Remember
that the lines for a spherical triangle must be "locally straight" so you
cannot just follow "lines of latitude" which are curved  the lines you draw
must be more like "lines of longitude."
Anyway, if you actually did try covering the sphere with little triangles,
congratulations! It has kind of like making a circle out of little
straightline segments, which is the sort of thing computers do all the time
of course.
asmith
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Update: June 2012

