Does $^\circ\mathrm{C}^2$ have any physical consistency? I know that $\mathrm{K}^2$ is a valid unit even though I don't know in which context it can be used. But I feel strange about $^\circ\mathrm{C}^2$ because of the offset. My intuition about unit behaviour is that it is not a valid unit at all. I am looking for an scientific argument to answer if the unit $^\circ\mathrm{C}^2$ would be valid or not? Thanks.
Update
I am developing units
and quantity
classes for a project. I implemented units
as a vector of dimensions and a scale factor before a reference unit. I also implemented an offset in order to convert my $^\circ\mathrm{C}$ into $\mathrm{K}$. Now I am implementing multiplication and division. Then the question has risen.
No, I never met $^\circ\mathrm{C}^2$ before, and I believe it will never happen. I suspect it is because units are based on some kind of vector-space where the origin is a key concept which ill-defined units such as $^\circ\mathrm{C}$ break.
Update 2
Reading this post helped a bit.
I am not sure I understood what NowIGetToLearnWhatAHeadIs
have said. Anyway, distance have an origin when you measure it, because you make the origin of your rule to coincide with some arbitrary origin (a point of space that will endorse the responsibility of origin).
Measurements of area by the means of a rule will have consistency not mater if your rule is Imperial or International as long as you use units to track your computations.
But this will not be correct if you decide to start with the first digit graduation of your rule instead of the origin of your rule. Eg.: I always use $1\,\mathrm{cm}$ or $1\,\mathrm{in}$ to set my origin as I just read the rule as if I was using it the right way. Then, the area will be meaningless until you adjust for all your measurements of the origin shift you falsely introduced.
This is how I am now perceiving $^\circ\mathrm{C}$ units and I really doubt it is meaningful to square Celsius or Fahrenheit if they are taken as "absolute" measurement. If they are difference temperatures it looks right because you withdraw the offset.
Now asking the SE Physics community: Are my last new statements correct? Thank you.