cube coordinates

how have the vertices of the cube and inbetween points been numbered?

what happens with the coordinates that are vertices of an equilateral triangle?

what coordinates could you have for a regular hexagon?

this diagram shows what seems to be a common convention for 3D coordinates:

the positive z axis coming out of the page rather than going into it, which is 'right-handed'

[the diagram with the coordinates above follows this convention - albeit in an unusual orientation]