Hjelmslev's theorem
Encyclopedia
In geometry
, Hjelmslev's theorem, named after Johannes Hjelmslev
, is the statement that if points P, Q, R... on a line are isometrically
mapped to points P´, Q´, R´... of another line in the same plane, then the midpoints of the segments PP`, QQ´, RR´... also lie on a line.
The proof is easy if one assumes the classification of plane isometries
. If the given isometry is odd, in which case it is necessarily either a reflection in a line or a glide-reflection (the product of three reflections in a line and two perpendiculars to it), then the statement is true of any points in the plane whatsoever: the midpoint of PP´ lies upon the axis of the (glide-)reflection for any P. If the isometry is even, compose it with reflection in line PQR to obtain an odd isometry with the same effect on P, Q, R... and apply the previous remark.
The importance of the theorem lies in the fact that it has a different proof that does not presuppose the parallel postulate
and is therefore valid in non-Euclidean geometry
as well. By its help, the mapping that maps every point P of the plane to the midpoint of the segment P´P´´, where P´and P´´ are the images of P under a rotation (in either sense) by a given acute angle about a given center, is seen to be a collineation mapping the whole hyperbolic plane
in a 1-1 way onto the inside of a disk, thus providing a good intuitive notion of the linear structure of the hyperbolic plane. In fact, this is called the Hjelmslev transformation
.
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 ....
, Hjelmslev's theorem, named after Johannes Hjelmslev
Johannes Hjelmslev
Johannes Trolle Hjelmslev was a mathematician from Copenhagen, Denmark. Hjelmslev worked in geometry and history of geometry...
, is the statement that if points P, Q, R... on a line are isometrically
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
mapped to points P´, Q´, R´... of another line in the same plane, then the midpoints of the segments PP`, QQ´, RR´... also lie on a line.
The proof is easy if one assumes the classification of plane isometries
Euclidean plane isometry
In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length...
. If the given isometry is odd, in which case it is necessarily either a reflection in a line or a glide-reflection (the product of three reflections in a line and two perpendiculars to it), then the statement is true of any points in the plane whatsoever: the midpoint of PP´ lies upon the axis of the (glide-)reflection for any P. If the isometry is even, compose it with reflection in line PQR to obtain an odd isometry with the same effect on P, Q, R... and apply the previous remark.
The importance of the theorem lies in the fact that it has a different proof that does not presuppose the parallel postulate
Parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry...
and is therefore valid in non-Euclidean geometry
Non-Euclidean geometry
Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
as well. By its help, the mapping that maps every point P of the plane to the midpoint of the segment P´P´´, where P´and P´´ are the images of P under a rotation (in either sense) by a given acute angle about a given center, is seen to be a collineation mapping the whole hyperbolic plane
Hyperbolic space
In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...
in a 1-1 way onto the inside of a disk, thus providing a good intuitive notion of the linear structure of the hyperbolic plane. In fact, this is called the Hjelmslev transformation
Hjelmslev transformation
In mathematics, the Hjelmslev transformation is an effective method for mapping an entire hyperbolic plane into a circle with a finite radius. The transformation was invented by Danish mathematician Johannes Hjelmslev...
.
External links
- Hjelmslev's Theorem by Jay Warendorff, the Wolfram Demonstrations ProjectWolfram Demonstrations ProjectThe Wolfram Demonstrations Project is hosted by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience. It consists of an organized, open-source collection of small interactive programs called Demonstrations, which are meant to visually and...
. - Hjelmslev's Theorem from cut-the-knotCut-the-knotCut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...