![]() ![]() The final three chapters go beyond the straightedge and compass to other construction tools. These chapters also discuss the restriction of compasses to dividers, tools that can transfer line segments onto equal segments of other lines but cannot be used to find intersections of circles with other curves, or to rusty compasses, compasses that cannot change radius, and they use dividers to construct the Malfatti circles. The next four chapters study what happens when the use of the compass or straightedge is restricted: by the Mohr–Mascheroni theorem there is no loss in constructibility if one uses only a compass, but a straightedge without a compass has significantly less power, unless an auxiliary circle is provided (the Poncelet–Steiner theorem). They also include impossibility results for the classical Greek problems of straightedge and compass construction the impossibility of doubling the cube and trisecting the angle are proved algebraically, while the impossibility of squaring the circle and constructing some regular polygons is mentioned but not proved. The first two discuss straightedge and compass constructions, including many of the constructions from Euclid's Elements, and their algebraic model, the constructible numbers. ![]() Geometric Constructions has ten chapters. ![]() Martin, and published by Springer-Verlag in 1998 as volume 81 of their Undergraduate Texts in Mathematics book series.
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