Wikipedia:Reference desk/Archives/Mathematics/2017 June 12

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June 12

Power of neusis construction

Based on the constructability of hendecagon by full neusis but not compass and straightedge with angle trisector, the power of full neusis is strictly greater. What precisely, then, is the set of neusis-constructible regular polygons? Perhaps more strongly, which field extensions are constructible by full neusis?--Jasper Deng (talk) 18:16, 12 June 2017 (UTC)

Neusis construction#Use of the neusis gives the start of the sequence and says it is sequence A122254 in the OEIS, recently (May 2014) augmented by the n = 11 case. OEIS calls this sequence "numbers with 3-smooth Euler's totient". Our article says the augmented sequence is iff. Loraof (talk) 19:26, 12 June 2017 (UTC)

"3-smooth Euler's totient" - this would be satisfying for regular polygons, but Euler's totient function evaluated at 11 is 10, which is not 3-smooth (as its largest prime factor is 5). The article implies there are no other exceptional cases (it seems that 11 is the only known exception to the rule), but since the result is recent, and I don't have access to the paper, I'm slightly uncomfortable accepting the articles iff.

It also still leaves open the possibility of neusis-constructible numbers which are not the cosine of any regular polygon's common angle, such as the cube root of 2.--Jasper Deng (talk) 19:58, 12 June 2017 (UTC)

Right, the linked section of our article says it can be used to double the cube−hence the cube root of 2 is neusis-constructible. Loraof (talk) 23:55, 12 June 2017 (UTC)

• Note that our article's augmented sequence is augmented not only by 11 but also by all multiples of 11. Loraof (talk) 00:12, 13 June 2017 (UTC)

• Technically, not all multiples of 11; according to the article, it has been proved that numbers with 3-smooth Euler's totients are all constructible by neusis, and numbers without 5-smooth Euler's totients are not, and it is only the 5-smooth case that is not yet fully characterised. In that case, the answer seems to be that Jasper Deng has asked an open problem: the article says "This is also a first result in a program to try to characterize all points constructible by marked ruler and compass, a problem whose solution seems to be a long way off."
• Note that the article's "iff" has long been known to be correct if you allow use of the marked ruler but not the compass (this was known to Pierpont). The fact that the two cases are not identical is quite surprising. Double sharp (talk) 04:03, 13 June 2017 (UTC)

@Double sharp: How can we have neusis without a compass, though? Do you mean paper folding? (Paper folding is known to be limited in such a way that 11 is excluded).--Jasper Deng (talk) 06:04, 13 June 2017 (UTC)

The hendecagon paper defines ruler-only neusis to allow the classical use of the straightedge or its use for verging, which they define as follows. "By verging through a point V between two geometric objects (for us lines and circles) we mean determining two points Q1 and Q2 obtained by drawing a line through the point V that intersects the two geometric objects at points Q1 on one object and Q2 on the other such that Q1 and Q2 are one unit apart. Notice that this can be done by a marked ruler once we are given a verging point and a pair of geometric objects. The points Qi are thus said to be constructible by verging." I suppose that when a circle needs to be constructed it suffices to determine it via three points: off the top of my head this may well be identical to paper folding, but I haven't really thought about it. Double sharp (talk) 06:09, 13 June 2017 (UTC)