Three-reflection set

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The three reflection rate or set of the three reflections is a mathematical theorem of geometry , which both the elementary geometry and the reflection geometry belongs. The sentence deals with the important question of the concatenation of reflections in the Euclidean plane .

Formulation of the sentence

The sentence can be represented as follows:

In the Euclidean plane the concatenation of three reflections is in turn one such when the three mirror axes lie in the tuft ; that is, when the three straight lines involved are parallel or have a common point of intersection .
If the mentioned condition is met, the mirror axis of the concatenation product belongs to the same tuft, i.e. it is parallel to the axes of the three given reflections or goes through their common point of intersection.

Conclusion from the three-reflection theorem

The three-reflection theorem entails the following representation set:

In the Euclidean plane, each congruence mapping is itself a reflection or can be represented as a chain of two or three reflections.
In short, one also says:
Every plane congruence mapping is a reflection or a double reflection or a triple reflection.
This means that the following four cases are possible for a plane congruence mapping:
It is a reflection.
It is a shift , i.e. a double reflection on parallel mirror axes.
It is a rotation , i.e. a double reflection on intersecting mirror axes.
It is a sliding mirror , i.e. the concatenation of a mirroring with a shift in the direction of the mirror axis.

Corresponding result in the room geometry

A corresponding sentence applies to congruence mapping in three-dimensional Euclidean space , the so-called representation sentence for spatial movements :

A spatial movement can always be represented as a concatenation of a maximum of four level reflections . If a spatial movement has a fixed point , three plane reflections are sufficient.

The correspondence to the plane three-reflection theorem is particularly valid here:

If the mirror planes of three plane reflections are parallel to each other or if they go through a common straight line, their concatenation is also a plane reflection.

In this context, the following interesting sentence about spatial rotations by Leonhard Euler from 1776 also falls :

If the two axes of rotation of two spatial rotations have a common point of intersection, the concatenation of the two is also a rotation and their axis of rotation in turn goes through this point of intersection.

Axiomatic context

As part of a structure of the levels of metric (absolute) geometry from the concept of reflection, the set of the three reflections is given the rank of an axiom. In this axiomatic approach, it plays a central role, as can be seen in particular from the studies by Johannes Hjelmslev , Gerhard Hessenberg , Arnold Schmidt and Friedrich Bachmann .

literature

  • Friedrich Bachmann: Structure of geometry from the concept of reflection (=  The basic teachings of the mathematical sciences . Volume 96 ). Springer Verlag, Berlin / Heidelberg / New York 1973, ISBN 3-540-06136-3 ( MR0346643 ).
  • Siegfried Krauter : Experience elementary geometry . A workbook for independent and active discovery. Spektrum Akademischer Verlag, Munich 2005, ISBN 3-8274-1644-2 .
  • H. Lenz : Fundamentals of elementary mathematics . 3rd, revised edition. Hanser Verlag, Munich ( inter alia ) 1976, ISBN 3-446-12160-9 ( MR0460009 ).
  • Erhard Quaisser: Movements in the plane and in space (=  mathematical student library . Volume 116 ). VEB Deutscher Verlag der Wissenschaften, 1983, ISSN  0076-5449 ( MR0739331 ).
  • Harald Scheid : Elements of Geometry (=  mathematical texts . Volume 3 ). BI Wissenschaftsverlag, Mannheim / Vienna / Zurich 1991, ISBN 3-411-14931-0 ( MR1168701 ).

References and comments

  1. ↑ Friedrich Bachmann: Structure of the geometry from the reflection concept. 1973, p. 5
  2. ^ Siegfried Krauter: Experience elementary geometry. 2005, p. 31
  3. ↑ Hanfried Lenz: Fundamentals of elementary mathematics. , 1976, pp. 204 ff, 208-209
  4. ↑ E. Quaisser: Movements in the plane and in space. , 1983, p. 54 ff.
  5. ↑ Harald Scheid: Elements of Geometry 1991, p. 115 ff.
  6. ^ Scheid, op. Cit., Pp. 117-118
  7. ↑ Quaisser, op. Cit., Pp. 60-61
  8. ↑ Quaisser, op.cit., P. 86
  9. ↑ Quaisser, op.cit., Pp. 89, 95
  10. ↑ Quaisser, op.cit., P. 90
  11. ↑ Bachmann, op. Cit., Foreword (X) and pp. 24, 33–34