Affinity Lattice

lets say there are 3 bits 000, the graph im interested in looks like this

            111 ------\

          /       \       \              

        011    101    |                 

       /    \      /  \    |            

       |     001     |   |

       |       |       |    |

       |     000     |    |

       \    /     \    /    |

         010    100    |

           \      /        /

             110 ------/

note that this is the graph where we go from one node to another if 1 bit changes. What i have drawn above is essentially a cube flattened out. The trouble is that i cannot draw a planar/non-overlapping graph/lattice for 4 dimensional codes.I am curious to find out if it is even possible at all to draw a hypercube on paper without overlapping edges.

Further, we cannot write this in a row-by-row table because the geometry of the table does not allow more than 2 neighbors. 

What i am essentially looking for are geometric objects into which these codes can be embedded, and i want to call those geometric objects as "Affinity Objects" perhaps Affinity lattices. 

Let me get back on this and find out the literature about this