Do we extend the geometrically constructible numbers in a 3D space, where lines, circles, spheres and planes can be constructed?

Question

Do we extend the geometrically constructible numbers in a 3D space, where lines, circles, spheres and planes can be constructed?

In a 2D plane, we construct lines and circles only with compass and straightedge, and through this we give rise to constructible numbers hence: without arbitrary placement of points, any point that may be constructed onto the real number axis (a line defined by points representing 0 and 1) is a constructible real number. This set of numbers is contained in the algebraics but contains the rationals.

Now, just imagine this in 3D - suppose we have three or four points defining 0, 1, and perhaps an imaginary i or whatever significant point designators, etc. We have the ability to draw circles, lines, spheres and planes defined through any known points.

Im concerning myself with real numbers, specifically, and am curious what points can be plotted onto the real number line. Im curious if this 3D extension and new ruleset in any way extends the constructable numbers beyond what they already are.

Many tools have been used to extend the constructables. These are called neusis tools, but also non-constructiable curves drawn in the plane, etc. can accomplish much of the same. Im wondering if we may include higher dimensions among our neusis tools.

Answer 1

Yes. You get the points with constructible coordinates in space. See Book XIII Proposition 17 for the constructon of the regular dodecahedron.

Answer 2

You can extend the range of constructible length ratios in this way, and the Greeks actually explored such extensions -- but with cones instead of spheres. Such constructions were called solid constructions. Cones are superior to spheres because spheres can intersect any plane only in circles, whereas cones generate the full range of conic sections (circles, ellipses, parabolas, hyperbolas). The conic sections are then used to generate the constructions on paper.

The method solves all equations up to degree $4$ in integers or previously constructed quantities, thereby allowing the definition of angle trisections, cube root extractions and regular $n$-gons when the Euler totient of $n$ has prime factors of $2$ and $3$. This capability may even be generated from one properly chosen conic section, such as the parabola $y=x^2$, plus Euclidean construction.