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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
www.elsevier.com/locate/jal Axiomatizing geometric constructions
, 2007
"... In this survey paper, we present several results linking quantifierfree axiomatizations of various Euclidean and hyperbolic geometries in languages without relation symbols to geometric constructibility theorems. Several fragments of Euclidean and hyperbolic geometries turn out to be naturally occu ..."
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In this survey paper, we present several results linking quantifierfree axiomatizations of various Euclidean and hyperbolic geometries in languages without relation symbols to geometric constructibility theorems. Several fragments of Euclidean and hyperbolic geometries turn out to be naturally occurring only when we ask for the universal theory of the standard plane (Euclidean or hyperbolic), that can be expressed in a certain language containing only operation symbols standing for certain geometric constructions.
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"... trisections is vast and extends back in time approximately 2300 years. My own favorite method of trisection from the Ancients is due to Archimedes, who performed a “neusis ” between a circle and line. Basically a neusis (or use of a marked ruler) allows the marking of points on constructed objects ..."
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trisections is vast and extends back in time approximately 2300 years. My own favorite method of trisection from the Ancients is due to Archimedes, who performed a “neusis ” between a circle and line. Basically a neusis (or use of a marked ruler) 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. Here is Archimedes ’ trisection method (see Figure 1): Given an acute angle between rays r and s meeting at the point O, construct a circle K of radius one at O, and then extend r to produce a line that includes a diameter of K. The circle K meets the ray s at a point P. Now place a ruler through P with the unit distance CD lying with C on K and D on the ray opposite to r. That the angle ODP is the desired trisection is easy to check using the isosceles triangles DCO and COP and the exterior angle of the triangle PDO. As one sees when trying this for oneself, there is a bit of “fiddling” required to make everything line up as desired; that fiddling is also essential when one does origami.
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