The cube balloon, like the tetrahedron balloon, has the problem of having odd degree vertices. By having two balloons, four of the problem vertices can be dealt with. The other four will be dealt with by going through the interior point once for each balloon turning those vertices into degree 4 vertices. To symmetrically space these degree 4 vertices, I used the vertices so that none share an edge (the points form a tetrahedron).
3 4 1 5a 0b 2 6 7 8
6 5 8 4a 0b 7 3 2 1
This design thus use two balloons with eight segments each, which allows for large segments for a large cube. Notice that the two internal segments would theoretically be 0.8660 (root 3 over 2) the length of the larger sections but I think in practice it's more like 0.7 because of the bulging properties of balloons. With only 4 internal balloons this design is very sensitive to the ratio of the external and internal balloons. This allows for some interesting patterns that are topologically equivalent but that don't look like a cube.
An alternative design that would ensure a cube look would have all 8 vertices connected to the centre point:
0z 3 4 1 5a 0b 2 6 7 8 0z
0z 6 5 8 4a 0b 7 3 2 1 0z
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