Abstract. I'll
discuss the following problem: suppose we have several villages, and we
want to build roads with minimal total length such that one can go from any
village to any other village. How to do that?
a) If the number of villages is 4 and they are located at
the vertices of a square, then this problem was solved in ancient China (the
answer is not what you'd immediately think!)
b) If
the number of villages is 3, then this problem was solved by Fermat and
Torricelli in the mid 17th century, and the answer is very beautiful.
Problem
a) can be easily reduced to an optimization problem at the level of Calc 1.
Problem b) is harder.
In mid
19th century, a Swiss mathematician Jacob Steiner stepped in and offered an
amazing and very elementary solution to the general problem for any n.
I'll
describe the solutions of a) and b), and then of the general problem.
I
promise this talk will be a lot of fun. No background in anything is needed. If
you like paintings with villages (or
houses) and roads in it, then you'll like my talk!