Unsolved problem in mathematics:
Can a Robbins pentagon have irrational diagonals?
In geometry, a Robbins pentagon is a cyclic pentagon whose side lengths and area are all rational numbers.
Can a Robbins pentagon have irrational diagonals?
In geometry, a Robbins pentagon is a cyclic pentagon whose side lengths and area are all rational numbers.
Robbins pentagons were named by Buchholz & MacDougall (2008) after David P. Robbins, who had previously given a formula for the area of a cyclic pentagon as a function of its edge lengths. Buchholz and MacDougall chose this name by analogy with the naming of Heron triangles after Hero of Alexandria, the discoverer of Heron's formula for the area of a triangle as a function of its edge lengths.
Every Robbins pentagon may be scaled so that its sides and area are integers. More strongly, Buchholz and MacDougall showed that if the side lengths are all integers and the area is rational, then the area is necessarily also an integer, and the perimeter is necessarily an even number.
Buchholz and MacDougall also showed that, in every Robbins pentagon, either all five of the internal diagonals are rational numbers or none of them are. If the five diagonals are rational (the case called a Brahmagupta pentagon by Sastry (2005)), then the radius of its circumscribed circle must also be rational, and the pentagon may be partitioned into three Heron triangles by cutting it along any two non-crossing diagonals, or into five Heron triangles by cutting it along the five radii from the circle center to its vertices.
Buchholz and MacDougall performed computational searches for Robbins pentagons with irrational diagonals but were unable to find any. On the basis of this negative result they suggested that Robbins pentagons with irrational diagonals may not exist.
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or plane area refers to the area of a shape or planar lamina, while surface area refers to the area of an open surface or the boundary of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analogue of the length of a curve or the volume of a solid.
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers (arithmetic).
In geometry, a polygon is a plane figure that is described by a finite number of straight line segments connected to form a closed polygonal chain. The bounded plane region, the bounding circuit, or the two together, may be called a polygon.
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal ; or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as ABCD.
In geometry, Heron's formula, named after Hero of Alexandria, gives the area of a triangle when the lengths of all three sides are known. Unlike other triangle area formulae, there is no need to calculate angles or other distances in the triangle first.
In Euclidean geometry, Brahmagupta's formula is used to find the area of any cyclic quadrilateral given the lengths of the sides; its generalized version can be used with non-cyclic quadrilateral.
In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.
In geometry, an equilateral polygon is a polygon which has all sides of the same length. Except in the triangle case, it doesn’t need to be equiangular, but if it does then it is a regular polygon. If the number of sides is at least five, an equilateral polygon doesn’t need to be a convex polygon: it could be concave or even self-intersecting.
Brahmagupta was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the Brāhmasphuṭasiddhānta, a theoretical treatise, and the Khaṇḍakhādyaka, a more practical text.
In geometry, an isosceles triangle is a triangle that has two sides of equal length. Sometimes it is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids.
A Heronian tetrahedron is a tetrahedron whose edge lengths, face areas and volume are all integers. The faces must therefore all be Heronian triangles. Every Heronian tetrahedron can be arranged in Euclidean space so that its vertex coordinates are also integers.
In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ABCD would be denoted ABCD.
In geometry, a Heronian triangle is a triangle that has side lengths and area that are all integers. Heronian triangles are named after Hero of Alexandria. The term is sometimes applied more widely to triangles whose sides and area are all rational numbers, since one can rescale the sides by a common multiple to obtain a triangle whose sides and area are integers.
In geometry, Brahmagupta's theorem states that if a cyclic quadrilateral is orthodiagonal, then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side. It is named after the Indian mathematician Brahmagupta (598-668).
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
David Peter Robbins was an American mathematician. He is most famous for introducing alternating sign matrices. He is also known for his work on generalizations of Heron's formula on the area of polygons, due to which Robbins pentagons were named after him.
In mathematics, an associahedronKn is an (n – 2)-dimensional convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a string of n letters, and the edges correspond to single application of the associativity rule. Equivalently, the vertices of an associahedron correspond to the triangulations of a regular polygon with n + 1 sides and the edges correspond to edge flips in which a single diagonal is removed from a triangulation and replaced by a different diagonal. Associahedra are also called Stasheff polytopes after the work of Jim Stasheff, who rediscovered them in the early 1960s after earlier work on them by Dov Tamari.
In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.