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:
- The strips have no discrete holes on them and may be arbitrarily long
- 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)
- The objective is to use as few strips as possible while keeping rigidity
- Everything is in the Euclidean plane
- 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 $\sqrt[3]2$ 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 $\sqrt[3]2$ 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.
