Dehn plane
Encyclopedia
In geometry
, Dehn constructed two examples of planes, a semi-Euclidean geometry and a non-Legendrian geometry, that have infinitely many lines parallel to a given one that pass through a given point, but where the sum of the angles of a triangle is at least π. A similar phenomenon occurs in hyperbolic geometry
, except that the sum of the angles of a triangle is less than π. Dehn's examples use a non-Archimedean field, so that the Archimedean axiom is violated. They were introduced by and discussed by .
ordered Pythagorean field
Ω(t), a Pythagorean closure of the field of rational functions R(t), consisting of the smallest field of real-valued functions on the real line containing the real constants, the identity function t (taking any real number to itself) and closed under the operation ω → √(1+ω2). The field Ω(t) is ordered by putting x>y if the function x is larger than y for sufficiently large reals. An element x of Ω(t) is called finite if m<x<n for some integers m,n, and is called infinite otherwise.
which takes values in Ω(t), gives a model of Euclidean geometry
. The parallel postulate is true in this model, but if the deviation from the perpendicular is infinitesimal (meaning smaller than any positive rational number), the intersecting lines intersect at a point that is not in the finite part of the plane. Hence, if the model is restricted to the finite part of the plane (points (x,y) with x and y finite), a geometry is obtained in which the parallel postulate fails but the sum of the angles of a triangle is π. This is Dehn's semi-Euclidean geometry.
over Ω(t) consists of the projective plane over Ω(t), which can be identified with the affine plane of points (x:y:1) together with the "line at infinity", and has the property that the sum of the angles of any triangle is greater than π The non-Legendrian geometry consists of the points (x:y:1) of this affine subspace such that tx and ty are finite (where as above t is the element of Ω(t) represented by the identity function). Legendre's theorem
states that the sum of the angles of a triangle is at most π, but assumes Archimedes's axiom, and Dehn's example shows that Legendre's theorem need not hold if Archimedes' axiom is dropped.
Geometry
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 ....
, Dehn constructed two examples of planes, a semi-Euclidean geometry and a non-Legendrian geometry, that have infinitely many lines parallel to a given one that pass through a given point, but where the sum of the angles of a triangle is at least π. A similar phenomenon occurs in hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
, except that the sum of the angles of a triangle is less than π. Dehn's examples use a non-Archimedean field, so that the Archimedean axiom is violated. They were introduced by and discussed by .
Dehn's non-archimedean field Ω(t)
To construct his geometries, Dehn used a non-ArchimedeanArchimedean property
In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some ordered or normed groups, fields, and other algebraic structures. Roughly speaking, it is the property of having no infinitely large or...
ordered Pythagorean field
Pythagorean field
In algebra, a Pythagorean field is a field in which every sum of two squares is a square. A Pythagorean extension of a field F is an extension obtained by adjoining an element for some λ in F. So a Pythagorean field is one closed under taking Pythagorean extensions...
Ω(t), a Pythagorean closure of the field of rational functions R(t), consisting of the smallest field of real-valued functions on the real line containing the real constants, the identity function t (taking any real number to itself) and closed under the operation ω → √(1+ω2). The field Ω(t) is ordered by putting x>y if the function x is larger than y for sufficiently large reals. An element x of Ω(t) is called finite if m<x<n for some integers m,n, and is called infinite otherwise.
Dehn's semi-Euclidean geometry
The set of all pairs (x, y), where x and y are any (possibly infinite) elements of the field Ω(t), and with the usual metricMetric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
which takes values in Ω(t), gives a model of Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
. The parallel postulate is true in this model, but if the deviation from the perpendicular is infinitesimal (meaning smaller than any positive rational number), the intersecting lines intersect at a point that is not in the finite part of the plane. Hence, if the model is restricted to the finite part of the plane (points (x,y) with x and y finite), a geometry is obtained in which the parallel postulate fails but the sum of the angles of a triangle is π. This is Dehn's semi-Euclidean geometry.
Dehn's non-Legendrian geometry
In the same paper, Dehn also constructed an example of a non-Legendrian geometry where there are infinitely many lines through a point not meeting another line, but the sum of the angles in a triangle exceeds π. Riemann's elliptic geometryElliptic geometry
Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one...
over Ω(t) consists of the projective plane over Ω(t), which can be identified with the affine plane of points (x:y:1) together with the "line at infinity", and has the property that the sum of the angles of any triangle is greater than π The non-Legendrian geometry consists of the points (x:y:1) of this affine subspace such that tx and ty are finite (where as above t is the element of Ω(t) represented by the identity function). Legendre's theorem
Saccheri–Legendre theorem
In absolute geometry, the Saccheri–Legendre theorem asserts that the sum of the angles in a triangle is at most 180°. Absolute geometry is the geometry obtained from assuming all the axioms that lead to Euclidean geometry with the exception of the axiom that is equivalent to the parallel postulate...
states that the sum of the angles of a triangle is at most π, but assumes Archimedes's axiom, and Dehn's example shows that Legendre's theorem need not hold if Archimedes' axiom is dropped.