# Neusis constructions

Is there some simple description of which complex numbers are "constructible" with straightedge and compass and neusis?

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Just as straightedge and compass constructions give the numbers in the closure of the rationals under square roots, neusis gives the closure of the rationals under square roots and cube roots.

For more details, also for an alternate characterization in terms of origami, see this paper by Roger Alperin.

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This would seem to be the same as the set of "conic-constructible" numbers, as per Videla, On points constructible from conics, Math Intelligencer 19 (1997) 53-57. –  Gerry Myerson Jul 16 '10 at 1:14
Gerry, yes it is. In fact, Alperin cites this paper. –  John Stillwell Jul 16 '10 at 1:46

The original question turns out to be a bit fuzzy and Stilwell's answer is potentially misleading. There are actually two flavors of "neusis constructions". In the weaker flavor, each of the two marks on the neusis straightedge must be constrained to lie along a line -- in particular, not along a circle. It is this weaker flavor of construction that is referred to in the Alperin paper; and this is the flavor of neusis construction that can extract both square roots and cube roots, but no more. So it is this weaker flavor that is equivalent to "conic constructability" (which Baragar calls "solid constructability").

In the stronger flavor of neusis construction, each mark may be constrained to lie along either a line or a circle. The Baragar article shows that the stronger flavor is strictly more powerful than the weaker flavor. In particular, the stronger flavor can construct a number whose extension field over the rationals has degree divisible by 5; so that number cannot be constructed by the weaker flavor.

Note that the famous Archimedes neusis trisection of the angle constrains one of the two marks to lie along a circle, so it is of the stronger flavor. But the results above imply that an angle can also be trisected using some neusis construction of the weaker flavor.

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