# What is the geodesic beween two opposite corners of cube on its surface?

What is the shortest path between two opposing corners of cube (two corners that have no edge in common)?

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Visual solution:

Length $= \sqrt{2^2 + 1^2} =\sqrt 5.$

Here is a non-rigorous approach:

The geodesic must clearly consist of two straight lines segments on the two faces connected the opposing corners, and it should be pretty obvious (due to symmetry) that they should have the same “slope” w.r.t. the face that they are on.

This results in the first line segment going from the bottom-right corner of the first face (we just start here arbitrarily) to the middle of its top edge. This last point is then the middle of the lower edge of the second face through which the geodesic runs and it connects to its top-left corner.

If we’re dealing with a cube of unit side length, the total distance becomes
$$2\sqrt{1^2+\left(\frac{1}{2}\right)^2}=2\sqrt{\frac{5}{4}}=\sqrt{5}$$