Creating a Straightedge, Compass and "New Tool" problem.

Question

I've been trying to create a problem that involves similar concepts as straightedge and compass construction.

The idea is to add a new tool that allows for a different type of graph to be drawn besides circles and lines, and then to ask whether previously impossible constructions such as squaring the circle or trisecting the angle have become possible (using the new tool).

Sadly, I haven't been able to find another kind of function (ellipses excluded) such that when one of its variables $x$ or $y$ is isolated and plugged in into another function of the same kind a polynomial is formed.

To be more specific, what I'm looking for is a function $f$ of two variables such that if

$$y-k=f(x-h, r)$$ and $$y-p=f(x-q, s)$$

then $$f(x-h, r)+k-f(x-q, s)-q = P(x) = 0$$

for some polynomial $P(x)$ with coefficients in $ℚ(k, h, p, q)$

The function doesn't have to form polynomials when the same process is followed but its variables are instead plugged in the equation of a line or a circle.

In such case one could simply not allow the use of compass and straightedge to draw circles and lines respectively, yet still ponder if some constructions have become possible.

I would really appreciate any help/thoughts

One standard thing is to introduce Paper Folding. If memory serves, this is equivalent to adding an angle trisector. — lulu, Jun 07 '18 at 19:27
Are you sure there is no typo? There is no $p$ in the last equation. — user58697, Jun 07 '18 at 22:03
The classical quadratrix of Hippias is a "kinematic curve" that one can use the trisect angles and square circles. — Blue, Jul 05 '18 at 11:42
Paper folding is not exactly the same as adding an angle trisector. The former solves equations like $x^3+x-1=0$ that the trisector can't do because the "trisected angle" would be complex. — Oscar Lanzi, Jul 05 '18 at 11:42

Rosie F · Answer 1 · 2018-07-05T11:28:10.147

One possible way to make some "previously impossible constructions ... possible" is not to introduce a new tool but allow a new way of using the existing tools of straight edge and compass: neusis. A length $a$ is determined somehow (e.g. the distance between two points constructed previously), and two lines (straight lines or circles) $m, n$ are drawn. Then a straight line $l$ is found which goes through some known point $P$, and where the points where $l$ intersects $m$ and $n$ are distance $a$ apart. Testing this latter condition could be done by making two marks on the straight-edge at distance $a$ apart, or setting the compasses to radius $a$ and placing them against the straight edge. Neusis is also described here.

Neusis makes it possible to solve cubics and thus

duplicate the cube
trisect any angle
construct a regular $p$-gon where $p$ is a Pierpont prime, i.e. a prime where no prime $q>3\mid p-1$; also where $p=11$, according to Benjamin and Snyder[1]

[1] Benjamin, Elliot and Snyder, C.. On the Construction of the Regular Hendecagon by Marked Ruler and Compass. Math. Proc. Camb. Phil. Soc., vol. 156 (2014), 409-424.

"Neusis makes it possible to solve cubics", also quartics and at least some quintics and sextics, the "some quintics and sextics" is where the Benjamin-Snyder theory for the 11-gon comes from. — Oscar Lanzi, Jul 05 '18 at 13:08
Adding neusis is equivalent to adding a tool that draws conchoids. — Szczepan Hołyszewski, May 29 '22 at 03:25

Creating a Straightedge, Compass and "New Tool" problem.

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