I managed to extract the primitive cell data as follows:
Find out the exact name of the tiling, e.g. by finding the entity, and getting it's "Name"
property. For example,
name = "trihexagonal tiling";
Now if we type == trihexagonal tiling
, click the + in the Primitive unit subpod, and choose Computable data, we only get Missing[]
. I suspect this is an oversight and not an intentional limitation, but I'm unsure. All the other subpods are available as computable data. So instead we get it as "Subpod content".
This gives us WolframAlpha["trihexagonal tiling", {{"PrimitiveUnit", 1}, "Content"}]
, which returns a RawBoxes
expression that needs to be interpreted.
We can do this, and extract the actual polygons from the graphics, like this:
In[944]:= name = "trihexagonal tiling";
In[945]:= Cases[
ToExpression@
First@WolframAlpha[name, {{"PrimitiveUnit", 1}, "Content"}],
GeometricTransformation[polys_, ___] :> polys,
Infinity,
1
]
Out[945]= {{Polygon[{{1/2, Sqrt[3]/2}, {1, 0}, {1/
2, -(Sqrt[3]/2)}, {-(1/2), -(Sqrt[3]/2)}, {-1, 0}, {-(1/2), Sqrt[
3]/2}}], Polygon[{{1/2, Sqrt[3]/2}, {1, 0}, {3/2, Sqrt[3]/2}}],
Polygon[{{1/2, -(Sqrt[3]/2)}, {1, 0}, {3/2, -(Sqrt[3]/2)}}]}}