# Why cube roots or in general $(2n-1)$th roots are rarely seen in equations in physics?

I rarely saw any equation in physics which involved cube roots or odd roots.Even while solving problems I rarely saw any odd root or cube root.

So why nature prefers even powers of physical quantities?

• in studies of phase transitions you see many expressions of the form $x^a$ where $a$ is a real number, not just 1/2 or some rationals. – wcc Jun 13 at 5:48
• in whole physics I doubt you've see the whole of physics. Most non-trivial problems result in equations and expressions that are not even polynomials and cannot be solved by such simple methods. – StephenG Jun 13 at 5:56
• One of Kepler’s Laws has a cube root: $a \propto T^{2/3}$. – G. Smith Jun 13 at 6:21
• The time to cook a spherical turkey varies as the 2/3 power of its weight. – G. Smith Jun 13 at 6:48
• Don Page derived the following physics approximation for the height of a giraffe: $H \sim \alpha^{4/5}m_e^{2/5}m_p^{-4/5}g^{-3/5}$ in this amusing paper: arxiv.org/abs/0708.0573 – G. Smith Jun 13 at 6:59

## 2 Answers

I suspect this has to do with the fact that everybody knows the quadratic formula, while the general solution to higher degree polynomials is either impractical or non existent. These are rarer but they do occur: for instance in GR the horizons of the Schwarzschild-de Sitter metric are the solution of a third degree polynomial, while in fluid mechanics the radius of a circle of viscous fluid on a table grows like $$t^{1/8}$$

• there is always a bias in physics towards what is possible to solve, certain things can end up looking very important, while one of the reason they are omnipresent in literature is because they are more tractable than other cases – user2723984 Jun 13 at 7:12

As others pointed out, cube roots do occur in many physical problems. Still, the question remains, why are they so much less common than square roots?

One fundamental reason is the Pythagoras theorem: square roots pop up when calculating distances, regardless of the dimension. If you take a bit more general perspective, a great deal of examples also fit into the same category. For example, the $$\sqrt{n}$$ that pops up in the Central Limit theorem is the standard deviation of the sum of $$n$$ i. i. d. random variables, and the standard deviation is the (in fact, $$L^2$$) distance in the space of random variables. Similarly, in QM/QFT, you often get square roots because you work in $$L^2$$.

This MathOverflow question even goes as far as to ask whether there are any other natural reasons for square roots to pop up. It turns out there are, but they are not so trivial to pinpoint!

Now, what would be a reason for the third root to occur that would be as fundamental? For one thing, our world is $$3$$-dimensional, so that the volume scales as the third power of the linear dimension. This underlines the "turkey cooking time" example in the comments above.

Of course, there is a share of examples where you add some exponents, and occasionally get $$3$$ without any "deep" reason, just as the third simplest natural number. Then you can invert a problem and get a third root. In my view, Kepler's law falls in this cathegory: for a circular orbit of radius $$r$$, we have $$T=2\pi r/|v|$$, and the velocity $$v$$ needed to generate the centrifugal force to offset a $$\frac{1}{r^2}$$ gravity is $$1/\sqrt{r}$$. So, the ultimate reason for the $$\frac32$$ exponent is that $$\frac32=1+\frac12$$.

One "systematic" reason for the $$1/3$$ power to occur comes from saddle point approximations (a k. a Laplace/stationary phase methods). Assume that you are trying to compute the asymptotics of an integral of the form $$\int_\gamma g(z)e^{nf(z)}dz$$. over some contour $$\gamma$$ in the complex plane. The general recipe is to drag the contour so that it passes through a critical point $$z_0$$ of $$f$$, and then argue that only the part near $$z_0$$ contributes, where $$f$$ can be replaced by its Taylor polynomial. In the "generic" case, this Taylor series will be quadratic, so that the change of variables $$w=n(z-z_0)^2$$ which eliminates $$n$$ from the integral leads to $$n^{-\frac12}$$ factor. However, in the "second simplest" case, the first non-trivial term is cubic, leading instead to $$n^{-\frac13}$$ factor. To my understanding, this is the source of $$1/3$$ scaling and Tracey-Widom distributions in random matrices/interacting particle systems/ KPZ universality class.