[btpqmyjz] navigating the diamond cubic crystal structure

in the diamond cubic crystal structure, each carbon atom connects to four others.  this provides a way to navigate 3-space with only a 4-way decision at each point: 4 buttons suffice, less than the 6 of going orthogonally.

2D analogue: the vertices of a hexagonal honeycomb.  each vertex has 3 neighbors, less than the 4 of the square honeycomb.

in both 2D and 3D, there are two different types of nodes (vertices) that alternate: one type with outgoing edges in one set of orientations, the other going the other way.  the 4 buttons (for 3D) will correspond to different directions depending on which type of node you are at.  perhaps repeatedly hitting the same button executes a 2-cycle, bouncing back and forth along an edge.

a dungeon whose rooms each have up to 4 exits (e.g., Zelda 1), but the topology is not a plane but 3-space.

the Voronoi cell is the triakis truncated tetrahedral honeycomb, corresponding to the equilateral triangular honeycomb in 2D, with every other triangle upside-down.

consider one tetrahedron in the tetrahedral-octahedral honeycomb.  go to another tetrahedron by traveling orthogonally through a face, through an octahedron, to the next tetrahedron.  this is possible because tetrahedra and octahedra alternate in the honeycomb, and an octahedron has opposite pairs of parallel faces.  I strongly suspect this graph among tetrahedron centers is the diamond cubic lattice.

are there analogues in higher dimensions?  first thought for 4D is the edges of the 24-cell honeycomb, 16 edges per vertex.  but the edges of a tesseract honeycomb is better, 8 edges per vertex.  it should be possible to do better.

any graph (starting from any lattice in any number of dimensions) can be made into a cubic graph (all node degrees 3 or less) by replacing nodes with too many edges by rings (traffic roundabouts).  3 buttons suffice.  roundabouts allow U-turns by going all the way around, so one does not need to provide a "go back" button.  2 buttons suffice: exit roundabout or keep going around.  we would still prefer roundabouts with fewer outgoing edges.