geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

Max Dehn Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Born to a Jewish family in Germany, Dehn's early life and career took place in Germany. ...

introduced 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 . A similar phenomenon occurs in

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...

, except that the sum of the angles of a triangle is less than . Dehn's examples use a non-Archimedean field, so that the

Archimedean axiom In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields. The property, typical ...

is violated. They were introduced by and discussed by .

Dehn's non-archimedean field Ω(''t'')

To construct his geometries, Dehn used a non-Archimedean ordered

Pythagorean field In algebra, a Pythagorean field is a field in which every sum of two squares is a square: equivalently it has Pythagoras number equal to 1. A Pythagorean extension of a field F is an extension obtained by adjoining an element \sqrt for some \lamb ...

Ω(''t''), a

Pythagorean closure In algebra, a Pythagorean field is a field in which every sum of two squares is a square: equivalently it has Pythagoras number equal to 1. A Pythagorean extension of a field F is an extension obtained by adjoining an element \sqrt for some \lamb ...

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

\omega \mapsto \sqrt

. 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

metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...

\, (x,y)\,  = \sqrt,

which takes values in Ω(''t''), gives a model of

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

. 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 . This is Dehn's semi-Euclidean geometry. It is discussed in .

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 . Riemann's

elliptic geometry Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines a ...

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 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 , but assumes Archimedes's axiom, and Dehn's example shows that Legendre's theorem need not hold if Archimedes' axiom is dropped.

References

* * * Planes (geometry) Non-Euclidean geometry {{SIA, mathematics