Assuming I understood the puzzle, I found a network of length sqrt(8) based on four edges of the regular octahedron. -Veit
On Aug 3, 2017, at 9:19 PM, Dan Asimov <dasimov@earthlink.net> wrote:
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
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