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I asked this question over a year ago on Math.StackExchange but I didn't get an answer.

In his famous treatise On spirals, Archimedes used a spiral to square the circle and trisect an angle. There are known algebraic characterizations of the numbers constructed with compass and straightedge, or using origami (paper folding), see the nice survey [1]. What happens if a spiral is also used?

Is there a known algebraic characterization of the numbers constructed with compass, straightedge and spirals?

A real number $r$ is constructible if and only if, given a line segment of unit length, a line segment of length $|r|$ can be constructed with compass and straightedge in a finite number of steps. The set of constructible numbers is the smallest field extension of the rationals that includes the square roots of all of its positive numbers. Similarly, the field of origami constructible numbers, is the smallest subfield of $\Bbb C$ closed under square roots, cube roots and complex conjugation [1].

Suppose now that you are given the Archimedean spiral with polar equation $r=\theta$. Let us define an Archimedean number in the same way as a constructible number, but now allowing as basic steps intersections with the spiral and construction of the tangent at some point of the spiral. A more formal definition would be welcome, but I hope this is sufficient to convey the spirit of my question. I suppose that an algebraic characterization of the Archimedean numbers would involve extensions of the form $\Bbb Q[\cos \theta]$.

[1] Alperin, Roger C. A mathematical theory of origami constructions and numbers. New York J. Math. 6 (2000), 119--133

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    $\begingroup$ Given the Archimedean numbers will constitute some but not all transcendental numbers, I doubt there is a completely algebraic characterization. $\endgroup$
    – Wojowu
    Commented May 24, 2021 at 12:18
  • $\begingroup$ Could you describe more in detail a list of allowed constructions? Can you draw and intersect more spirals? Just translated copies or also rotated? And specular copies? $\endgroup$
    – Pietro Majer
    Commented Dec 29, 2023 at 0:25
  • $\begingroup$ I guess that in any case there is a family of equations, $F(x,c)=0$, each depending on finitely many real coefficients $c=(c_1,\dots,c_n)$. The set of Archimedean numbers should be the smallest additive subgroup S of numbers that contains 1 and all solutions x of every equation F(x,c)=0 with coefficients in S. $\endgroup$
    – Pietro Majer
    Commented Dec 29, 2023 at 0:51

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