User:Rinick/Problem1

Place 4 points in 2 dimensional space and connect them with 6 line segments.

It's possible that some of these line segments have the same length, but there will be at least 2 different lengths.

And we can get 6 different shapes with this feature, as shown in the following image:



Let the lengths in each shape be L1, L2 ( L1 < L2 ) and find the sum of L2 / L1, the answer is:

$$\sum\frac{L_2}{L_1}=\sqrt{2}+\sqrt{3}+\sqrt{3}+\frac{\sqrt{5}+1}{2}+\sqrt{2+\sqrt{3}}+\sqrt{2+\sqrt{3}}\approx10.360052$$

What is the answer for 5 points in 3 dimensional space?