Can the following sum be counted or expressed in terms of special functions?

ho.historical past overview – Are there any neusis-hard/neusis-complete issues? Answer

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ho.historical past overview – Are there any neusis-hard/neusis-complete issues?

I’ve currently been having fun with Richeson’s Tales of Impossibility (behold MAA evaluate), an accessible bespeak on the illustrious issues of Euclidean geometry together with angle trisection/dice doubling/heptagon development/and so forth. Richeson traces the historical past of those issues in antiquity by means of the center ages and the nineteenth century and past, with many tangents/chapters on instruments and strategies above and past these afforded by Euclid.

Richeson reveals how including strategies to the geometer’s toolkit usually, however not at all times, impacts the character of the issues that she will be able to decipher. For instance one of many first theorems of the Elements is {that a} ruler together with a collapsing perimeter is simply as highly effective as a ruler and a lockable perimeter; however empowering the geometer to employ a neusis or a carpenter’s sq. permits her to trisect an angle.

I’m too noticing a harsh affinity between the category of issues in aircraft geometry that one can decipher with a given clique of instruments and the category of issues in computational complexity one can decipher with a given clique of sources. Indeed mighty as one can create a Hasse-diagram primarily based on the Complexity Zoo I believe one may create an identical Venn-diagram for all of the strategies thought of by Richeson.

But can this affinity breathe taken additional? For instance mighty as computational complexity idea has given us the language of hardness and completeness for inescapable lessons of issues, wouldn’t it make sense to talk of a “neusis-complete” downside?

Is there an issue solvable or a round constructible with a perimeter, straightedge, and neusis such that another downside constructible with the above is in some sense reducible thereto? Similarly are there “origami-complete” constructions?

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