# Doubling the cube with rational Meccano strips

In three monographs published in 2006, 2008 and 2014 Gerard 't Hooft considered "Meccano mathematics": how to construct specified distances and regular polygons by a rigid system of ideal Meccano strips, where the distance between adjacent holes on a strip is 1. He showed in the first monograph that all regular polygons can be constructed this way, but there is a stronger result:

Theorem (Maehara 1991). A positive real number can be constructed (braced) as the distance between two holes in a rigid Meccano linkage iff it is algebraic, even if all the strips are of length 1.

Based on this, here is a generalisation I have thought of:

1. The strips have no discrete holes on them and may be arbitrarily long
2. On the same strip all distances between hinges and measurement points (for the specified distance) must be rational numbers (but these numbers can be arbitrarily large, numerator or denominator alike)
3. The objective is to use as few strips as possible while keeping rigidity
4. Everything is in the Euclidean plane
5. No other tools (pencil, paper, compasses, sliders, etc.) are allowed; the strips may only be connected to each other through hinges

Here is an example valid construction for the golden ratio $$\frac{1+\sqrt5}2$$: The puzzle here is to brace $$\sqrt2$$ following the rules above, hence doubling the cube. Since the number is a root of $$x^3-2$$, it is algebraic and the theorem implies that a solution to the puzzle exists; answers should explain why $$\sqrt2$$ is exactly attained and why their system is rigid.

This puzzle is an offspring of some questions I've asked on the Maths Stack; cf. here, here, here and here.

There is a solution with 13 strips obtainable from 't Hooft's monographs, but I strongly suspect it can be done in fewer, perhaps as little as 5.

• Something like spoiler? Sep 21, 2021 at 22:27
• Maybe I misunderstand. Aren't the strips just like marked rulers? What specifically is not allowed in the linked construction? I don't see where a compass is required. This seems to be quite a specialist topic, so it would do no harm to explain for dummies. Sep 22, 2021 at 5:13
• Sorry, you completely lost me.What's the difference between drawing a triange and joining three strips of suitable lengths at their ends? Sep 22, 2021 at 5:22
• @loopywalt In the first case, you have implicitly used a piece of paper, which is not allowed. Sep 22, 2021 at 5:58
• @loopywalt Segments CD CG and CH probably have irrational lengths, which I guess isn't allowed by rule 2. Sep 24, 2021 at 1:08