Steiner tree problem in the plane (2D) is explained on wiki that though there's no straight solution, the solution has some properties, namely
- points added to the graph (Steiner points) must have a degree of three, and the three edges incident to such a point must form three 120 degree angles (this is by instinct right as this is the Fermat point)
- the maximum number of Steiner points that a Steiner tree can have is N − 2, where N is the initial number of given points.
However, when things move to the space (3D), will the above conclusions change?
For example, in the case of Tetrahedron ABCD, it looks the solution X shall be a "3D Fermat point", that the lines of $AX$, $BX$, $CX$, $DX$ shall be symmetry.
However, when it is about a cube, people seem are looking for connected triangles, within each the (2D) Fermat points are used
What are the conclusions on 3D Steiner Tree problem?