Which power means are constructible?

Question

The three classic Pythagorean means $A$, $G$, $H$ (arithmetic, geometric, and harmonic mean respectively) of positive real $a$ and $b$ have a cute geometric construction, as does the quadratic mean $Q$:

From this picture the ordering of these power means is directly visible.

What I'm curious about is whether there are other examples of such constructible power means (i.e. $M_q=\left(\frac{a^q+b^q}{2}\right)^{1/q}$). Obviously not all means will work; for instance, it will not be constructible if it can map two constructible numbers to an unconstructible number. The question is which ones are possible, and how they would be explicitly constructed.

Answer 1

For an algebraic number over $\mathbb Q$, in order to be constructable it's minimal polynomial should be of a degree $2^n$ , thus $M_{2^n}$

Answer 2

This is a construction for $M_4$:

$AC=a$ and $AK=b$.

Using a dummy unit length $(AB)$ you construct $GH=a^4$ and $NO=b^4$ (example for $a^2$ construction).

Let $BQ=a^4+b^4$, $R$ is the middle point for $BQ$ so $BR=(a^4+b^4)/2$.

Using the same dummy unit we construct the square root $BT$ of $BR$ and the square root $BW$ of $BU=BT$.

We now have your desired mean. The dummy unit is uninfluent, if you change the lenght of $AB$ the length of $BW$ does not change.