Wolfram|Alpha knows about lots of periodic tilings, e.g. hexgonal. Most of the information it displays is accessible though Mathematica's entities.

Entity["PeriodicTiling", "HexagonalTiling"]["Dataset"]

But the primitive cell is missing, even though W|A clearly knows it.

How can I get this information as computable data inside Mathematica? I can't find it among entities, and I can't seem to retrive it using WolframAlpha either. It gives me Missing["NoRawData"]. Try e.g. WolframAlpha["tiling primitive unit", {{"Result", 1}, "ComputableData"}, PodStates -> {"Result__More"}], which is generated by the W|A UI displayed in the notebook.