Finding center of tessellating hexagons

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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?

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1 Answer

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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.

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