The distant planet of Torus is basically the unit cube C = [0,1]^3 with corresponding points on opposite faces identified: (0,y,z) ~ (1,y,z), (x,0,z) ~ (x,1,z), (x,y,0) ~ (x,y,1). Or if you prefer, it's the quotient group of R^3 as an abelian group by the subgroup Z^3. ----- ----- It is desired to create a road network through Torus with two properties: I. It must be a connected network, and (When the torus is viewed as a cube:) II. There must be a circuit all around the torus from any point in the plane x = 0 to the corresponding point in x = 1, from any point in the plane y = 0 to the corresponding point in y = 1, and from any point in z = 0 to the corresponding point in z = 1. ----- For example, the 3D plus-sign P ("+") — consisting of the 3 segments connecting each face-midpoint of the cube to the corresponding face-midpoint on the opposite face — will do the job. The total length of the plus-sign is 3. However, it is not the shortest solution. Can you find a network satisfying I. and II., that also is shorter than P ??? How short can it be ??? —Dan