The author here writes:
Little known fact, the system of measuring length in the US is Turing complete
My question is: Is the system of measuring length in the US Turing complete?
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The post you're referring to is a joke: the US (and similar British Imperial) measurement systems are not Turing complete and the claim is an example of hyperbole.
A key feature that's necessary for a system to be Turing complete is that the system must include computations that do not terminate ("infinite loops"). Although it's not clear exactly what computations can be modeled by unit conversions (or what that even means!), any unit conversion within the diagram takes a finite number of steps.