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Visual Group Theory, Lecture 6.7: Ruler and compass constructions
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Visual Group Theory, Lecture 6.7: Ruler and compass constructions
Inspired by philosophers such as Plato and Aristotle, one of the chief purposes of ancient Greek mathematics was to find exact constructions for various lengths, using only the basic tools of a ruler and compass. However, the Greeks were unable to find constructions for three basic problems: (1) Squaring the circle, (2) Doubling the cube, and (3) Trisecting an angle. These problems remained unsolved for over 2000 years, until field theory was developed in the 19th century, which was able to establish their impossiblity. In this lecture, we see how the set of constructible numbers is a subfield of the complex numbers, and how if the complex number 'z' is constructable, then the degree [Q(z):Q] is a power of 2.
Inspired by philosophers such as Plato and Aristotle, one of the chief purposes of ancient Greek mathematics was to find exact constructions for various lengths, using only the basic tools of a ruler and compass. However, the Greeks were unable to find constructions for three basic problems: (1) Squaring the circle, (2) Doubling the cube, and (3) Trisecting an angle. These problems remained unsolved for over 2000 years, until field theory was developed in the 19th century, which was able to establish their impossiblity. In this lecture, we see how the set of constructible numbers is a subfield of the complex numbers, and how if the complex number 'z' is constructable, then the degree [Q(z):Q] is a power of 2.
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