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Trisections and Totally Real Origami
 American Math. Monthly
, 2005
"... The study of methods that accomplish trisections is vast and extends back in time approximately 2300 years. According to Knorr, [9], even Plato had a favorite method. My own favorite method of trisection from the Ancients is due to Archimedes. He performed a neusis between a circle and line. Basical ..."
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The study of methods that accomplish trisections is vast and extends back in time approximately 2300 years. According to Knorr, [9], even Plato had a favorite method. My own favorite method of trisection from the Ancients is due to Archimedes. He performed a neusis between a circle and line. Basically a marked ruler method allows the marking of points on constructed objects of unit distance apart using a ruler placed so that it passes through some known (constructed) point P. The standard marked ruler method allows only neusis between lines; a trisection method using a neusis between lines is due to Apollonius. Here is Archimedes ’ trisection method. Given an acute angle between rays r, s meeting at O, construct a circle of radius one at O; the ray r is extended to give a line which is a diameter of the circle; the circle meets the ray s at the point P. Now place a ruler through P with the unit distance CD lying on the circle at C and diameter at D, on the opposite ray to r. The angle \ODP is the desired trisection; you can easily check this using
Geometry and Number Theory on Clovers
"... (x 2 + y 2) 2 = x 2 − y 2 pictured in Figure 1, can be divided into n arcs of equal length by straightedge and compass if and only if n is a power of 2 times a product of distinct Fermat primes [1, p. 314]. By an earlier theorem of Gauss, these are exactly the values of n for which a regular ngon i ..."
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(x 2 + y 2) 2 = x 2 − y 2 pictured in Figure 1, can be divided into n arcs of equal length by straightedge and compass if and only if n is a power of 2 times a product of distinct Fermat primes [1, p. 314]. By an earlier theorem of Gauss, these are exactly the values of n for which a regular ngon is constructible by straightedge