Nov 18, 2016 · $\begingroup$ "With a marked ruler and neusis, you can trisect angles and double cubes (or in general extract cube roots), but the constructible points will still all have algebraic coordinates, so you can't square the circle that way." The proof of this statement was first published only 22 years ago!

Aug 24, 2018 · What's interesting is that the Tomahawk - which gives the same power as the marked ruler, e.g. in angle trisection - is fairly easy to use: So the above argument would not suffice to rule it out for reasons of "simplicity" of use: its exclusion looks more arbitrary than the exclusion of the marked ruler resp. the way to use it.

A marked ruler is known to allow the trisection of an arbitrary angle which is, in some precise sense, equivalent to solving an irreducible cubic (not all, of course). $\endgroup$ – user02138 Commented May 19, 2012 at 13:08

Jun 29, 2023 · The solution of this particular cubic equation using a marked ruler figures into the marked ruler/compasses construction of the regular hendecagon. Reference. Benjamin, Elliot & Snyder, Chip. (2014). "On the construction of the regular hendecagon by marked ruler and compass". Mathematical Proceedings of the Cambridge Philosophical Society.

Unmarked ruler - usual theory, only some extensions of degree $2^n$ can be done. Marked Ruler but **only allowed to use marks on lines **, only some extensions of degree $2^n3^m$ can be done. But, if one allows the use of the marks between circles, or between a …

If we allow a marked ruler or a non-rectractable compass (or we use origami...) we can construct numbers like $\sqrt[3]{2}$, but no trascendental numbers. However the quadratrix of Hippias allows us to construct $\pi$, albeit using a special instrument.

Feb 16, 2015 · Ancient Greeks were not able to trisect a general angle with compass and straightedge: now we know that it is impossible, since we would need to solve a cubic equation while only linear and quadratic

Apr 27, 2019 · $\begingroup$ To be more specific, the ruler is marked with only points that may be constructed with compass and unmarked straightedge, and points that may be constructed with ruler and compass. This may also be relevant: we only allow verging between lines, not lines and curves or curves and curves. $\endgroup$

Is there a formal definition for the class of non-cubic integers whose cube roots are constructible using a compass and a MARKED straightedge? If such a class exists, then $2$ would be a member of it. Are there other known members not derivable from $2$?

Jul 31, 2016 · Benjamin and Snyder proved the existence of a construction using a marked ruler and compasses in 2014 ...