A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.
In geometry and algebra, a real number is constructible if and only if, given a line segment of unit length, a line segment of length can be constructed with compass and straightedge in a finite number of steps. Equivalently, is constructible if and only if there is a closed-form expression for using only the integers 0 and 1 and the operations for addition, subtraction, multiplication, division, and square roots.
Straightedge and compass construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a pair of compasses.
In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle. The property extends to other related geometric objects.
Angle trisection is a classical problem of compass and straightedge constructions of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass.
Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square.
In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinitely many constructible polygons, but only 31 with an odd number of sides are known.
In mathematics, a quadratrix is a curve having ordinates which are a measure of the area of another curve. The two most famous curves of this class are those of Dinostratus and E. W. Tschirnhaus, which are both related to the circle.
In geometry, a trisectrix is a curve which can be used to trisect an arbitrary angle with ruler and compass and this curve as an additional tool. Such a method falls outside those allowed by compass and straightedge constructions, so they do not contradict the well known theorem which states that an arbitrary angle cannot be trisected with that type of construction. There is a variety of such curves and the methods used to construct an angle trisector differ according to the curve. Examples include:
Dinostratus was a Greek mathematician and geometer, and the brother of Menaechmus. He is known for using the quadratrix to solve the problem of squaring the circle.
Hippocrates of Chios was an ancient Greek mathematician, geometer, and astronomer.
The neusis is a geometric construction method that was used in antiquity by Greek mathematicians.
In geometry, a 257-gon is a polygon with 257 sides. The sum of the interior angles of any non-self-intersecting 257-gon is 45,900°.
In geometry, a 65537-gon is a polygon with 65,537 (216 + 1) sides. The sum of the interior angles of any non-self-intersecting 65537-gon is 11796300°.
Nicomedes was an ancient Greek mathematician.
A timeline of algebra and geometry
The quadratrix or trisectrix of Hippias is a curve, which is created by a uniform motion. It is one of the oldest examples for a kinematic curve, that is a curve created through motion. Its discovery is attributed to the Greek sophist Hippias of Elis, who used it around 420 BC in an attempt to solve the angle trisection problem. Later around 350 BC Dinostratus used it in an attempt to solve the problem of squaring the circle.
In geometry, an icositrigon or 23-gon is a 23-sided polygon. The icositrigon has the distinction of being the smallest regular polygon that is not neusis constructible.
The Ancient Tradition of Geometric Problems is a book on ancient Greek mathematics, focusing on three problems now known to be impossible if one uses only the straightedge and compass constructions favored by the Greek mathematicians: squaring the circle, doubling the cube, and trisecting the angle. It was written by Wilbur Knorr (1945–1997), a historian of mathematics, and published in 1986 by Birkhäuser. Dover Publications reprinted it in 1993.
Geometric Constructions is a mathematics textbook on constructible numbers, and more generally on using abstract algebra to model the sets of points that can be created through certain types of geometric construction, and using Galois theory to prove limits on the constructions that can be performed. It was written by George E. Martin, and published by Springer-Verlag in 1998 as volume 81 of their Undergraduate Texts in Mathematics book series.
