**Dehn Planes**

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 Max Dehn (1900) and discussed by Hilbert (1902, p.127–130, or p. 42-43 in some later editions).

Read more about Dehn Planes: Dehn's Non-archimedean Field Ω(*t*), Dehn's Semi-Euclidean Geometry, Dehn's Non-Legendrian Geometry

### Other articles related to "dehn planes, dehn, plane":

**Dehn Planes**- 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 geometry over Ω(t) consists of the projective

**plane**over Ω(t), which can be identified with the affine

**plane**of points (xy1) together with the "line at infinity ... 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 ...

### Famous quotes containing the word planes:

“After the *planes* unloaded, we fell down

Buried together, unmarried men and women;”

—Robert Lowell (1917–1977)