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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 $Q$:

pythagorean means

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.

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    $\begingroup$ $n$-th roots are not constructible unless $n$ is a power of $2$. All integer powers are constructible so the constructible ones are $M_{2^k}$ $\endgroup$ – Darth Geek Aug 15 '14 at 14:44
  • $\begingroup$ @DarthGeek: Makes sense. Do you know how one would construct $M_{2^k}$ in the spirit of the above diagram? I imagine it gets tedious quickly, but a standard algorithm for such seems plausible. $\endgroup$ – Semiclassical Aug 15 '14 at 14:48
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    $\begingroup$ @DarthGeek I think you want $M_{\pm 2^{k}}$, because $M_{-1}=H$ above... $\endgroup$ – Thomas Andrews Aug 15 '14 at 14:48
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    $\begingroup$ @Semiclassical sure. I'll get to it! $\endgroup$ – Darth Geek Aug 19 '14 at 12:56
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    $\begingroup$ @DarthGeek: A reference unit should not be required, since all the means are scale-invariant: $M_q(c\cdot a,c\cdot b)=c\cdot M_q(a,b)$. Or, in other words, you can use whatever you want as the reference unit; the final answer will not be affected by your choice. $\endgroup$ – Matt Aug 20 '14 at 18:49
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For an algebraic number over $\mathbb Q$, in order to be constructable it's minimal polynomial should be of a degree $2^n$ , thus $M_{2^n}$

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This is a construction for $M_4$: construction

$AC=a$ 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.

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