Given a hexagon such that opposite angles and radii are equal, how can I find the center point of any number of other hexagons (of the same dimensions) that form a tessellation?
In this (very ugly) drawing, we have hexagon with center P. Opposing angles are equal.
APC = EPD
BPC = FPE
CPD = FPA
And also, the opposing radii are equal.
AP = PD
BP = PE
CP = PF
How can I find the center points of the other (?) hexagons, and more generally, an arbitrary number of hexagons tessellating out from this initial shape?
The centers of all hexagons in your tesselation are $n\vec t_1+m\vec t_2$, where $n$ and $m$ are arbitrary integers, and $\vec t_1$ and $\vec t_2$ are two translation vectors. You may select those in a multitude of ways. From what we have here, $\vec t_1=\vec{PB}+\vec{PC}$ and $\vec t_2=\vec{PC}+\vec{PD}$ looks like a reasonable choice.