Can you mix units of $t$ and $t^2$ when constructing a shape?

Question

Let's say I have a length (possibly a radius), we'll call it $y$ and is of unit $t$. My data suggests to me that the distance around this shape (possibly a circumference), is $y^2$, which means the units of this space is $t^2$. Is it possible that the data suggests an underlying geometry? Can you create a shape and measure the curvature of something composed of $t$ and $t^2$?

Answer 1

Mixing $t$ and $t^2$ directly has problems since the units don't match. In a comment you indicated a correction factor $\phi$ in front of the quadratic term $t^2$, and with that you can certainly express things which correlate linear grow in one direction with quadratic grow in another direction.

From the comments I learned that you essentially want space ($x,y,z$) to grow quadratically with time ($t$). So you want something like

$$\sqrt{x^2+y^2+z^2}=\phi t^2$$

or expressed as a polynomial

$$x^2+y^2+z^2-\phi^2t^4=0$$

This is an algebraic variety, and as the highest degree is $4$, it's a quartic. It has some specific properties with regards to the symmetry and degrees of the spatial dimensions, so there may well be a more specific name for this, but I don't know it.

You can have Wolfram Alpha plot $x^2+y^2-z^4=0$ for an idea of what the 3d counterpart of this 4d surface looks like. If you click on the link “Open code” you get a version where you can rotate the plot by mouse, and adjust the portion of space to view.

Regarding your intended physical interpretation, note that the existence of a mathematical description does not explain its derivation from the laws of physics. So if you state that the universe for some reason follows a structure such as this, then I expect the typical reaction to be one of “fine, but why?” That question would be better suited for the physics stack exchange, though.