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 The Set - 3D representation The Set - flat representation

The Idea
This set consists of all all possible ways to place 0-6 connections on a cube that has a fixed oriëntation. Mirrored cubes above and below the red line are complementary. As in all puzzles based on the 'china principle', transcendental and starmap solutions are possible, but who cares. This one has as far as we know never been used other than to pack it, connections matching, into a 4x4x4 cube.
Both representations are precisely mapped onto each other, red being 'up/down'.

Showing a solution in 3D is not very convenient, but here's a solution in the 'flat representation'.

A China Cube Solution - flat representation

From bottomleft to topright there are 4 levels. Blue connections match on every level. Red connections match with the corresponding connection on the next level in that direction.
Flat solutions are the blueprints for actual solutions. In this case I've made one with a delibarate problem. On the ground level there are two connected cubes, disconnected from the rest - but at least they're on ground level (or rather, since they're in an edge, it's convenient to choose one of the two adjacent sides as ground level - of course a finished solution has six sides, any one of which may be considered ground level).
However, no orientation will keep the blank cube from 'hovering in the air', since it is inside the cube (on the third level).