Angle trisection

Angle trisection is the problem of dividing an arbitrary angle into three equal angles using specified geometric tools. With an unmarked straightedge and compass alone, it is impossible in general.

Straightedge and compass construction: A geometric construction using only an unmarked straightedge and compass. These are the classical tools for which arbitrary angle trisection is impossible.
Euclidean geometry: The geometry of points, lines, circles, and planes governed by Euclid's parallel postulate. The classical trisection problem is posed within Euclidean geometric construction.
Trisection of a 90-degree angle: The construction of three 30-degree angles from a right angle. This familiar special case is constructible despite the general impossibility.
Origami mathematics: The study of mathematical constructions and proofs using paper folding. Certain folds solve angle trisections that straightedge and compass cannot.
Field extension: An inclusion of fields that makes one field a vector space over another. Constructible lengths arise through restricted sequences of quadratic field extensions.
Euclid: An ancient Greek mathematician traditionally associated with the textbook Elements. The construction rules in Euclid's Elements established the classical straightedge-and-compass framework.
Trisecting 60 degrees: The problem of constructing a 20-degree angle from a 60-degree angle. Its required angle is not straightedge-and-compass constructible, illustrating the obstruction.
Neusis construction: A geometric construction that uses a ruler with two prescribed points aligned to fixed targets. This stronger construction technique can trisect any angle.
Constructible number: A real number obtainable from rational numbers by arithmetic operations and square roots. Trisecting an angle requires constructing a particular cosine that is not always constructible.
Greek mathematics: Mathematical traditions developed in ancient Greek-speaking societies. Angle trisection belongs to the famous construction problems of ancient Greek geometry.

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